Measuring the Effects of a Shock to Monetary Policy - Humboldt ...

Measuring the Effects of a Shock to Monetary Policy: 

A Factor-Augmented Vector Autoregression (FAVAR) 

Approach with Agnostic Identification 

Diplomarbeit 

zur Erlangung des Grades 

eines Diplom-Volkswirtes 

an der Wirtschaftswissenschaftlichen Fakultät 

der Humboldt-Universität zu Berlin 

vorgelegt von 

Pooyan Amir Ahmadi 

(Matrikel Nr.: 174701) 

Prüfer : Prof. Harald Uhlig Ph.D. 

Berlin, 26. August 2005

Abstract 

In this thesis I try to measure the dynamic effects of a shock to monetary policy in a 

Bayesian FAVAR framework. The innovation is to combine the Bayesian FAVAR with 

the agnostic identification introduced by Uhlig [2005] which has not been done yet. This 

identification scheme provides reasonable results and furthermore the possibility to im- 

pose a broader set of sign restriction on variables, proposed by Uhlig that are consistent 

with the conventional wisdom. Due to the greater information set it is possible to set 

the sign restrictions on several prices, monetary aggregates and short term interest rates 

considered in the dataset. In this vein one can narrow down the space of reasonable 

impulse responses in order to disentangle precisely the quantitative effects induced by 

contractionary monetary policy. Although the agnostic identification is a ”weaker” one 

with respect to the structure and restrictions imposed, this identification scheme com- 

bined with Markov chain Monte Carlo simulation methods delivers results that appear to 

be reasonable for a broad set of variables and with a higher accuracy than the alternative 

results provided by Bernanke, Boivin and Eliasz [2005]. Combining the two methodologies 

hold the enticing promise to measure the effects of a shock to monetary policy very pre- 

cisly when applying it to large panels of data. From the results one can conclude that the 

identification scheme is crucial for a succesful identification especially when the dataset 

considered is large. However with increasingly restrictions the results are delivered in- 

creasingly infrequent. Additionally I provide a Matlab code for the estimation procedure. 

Acknowledgements 

I would like to thank Harald Uhlig for excellent guidance and also Albrecht Ritschl and 

Bartosz Maćkowiak for supportive discussions. The material provided by Piotr Eliasz is 

thankfully acknowledged. I cordially thank Samad Sarferaz for proofreading this thesis 

and for helpful discussions. But of most I am indebted to Alborz Radmanesch to who I 

am sincerely greatful for invaluable support.

2 Bayesian FAVARs with Agnostic Identification 

Contents 

1 Introduction 4 

2 Literature 7 

3 Dynamic Factor Models 11 

4 The Econometric Framework 16 

4.1 FAVARs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 

4.2 FAVAR Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 

4.3 Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 

4.3.1 Generalized Dynamic Factor Model . . . . . . . . . . . . . . . . . . 21 

4.3.2 Two-Step Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 22 

4.3.3 Likelihood-Based Estimation . . . . . . . . . . . . . . . . . . . . . . 23 

4.3.4 Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . 23 

4.3.5 The Gibbs Sampler . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 

5 The Econometric Model 25 

5.1 The Bayesian Approach versus the Frequentists Approach . . . . . . . . . 25 

5.2 State-Space Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 26 

5.3 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 

6 Structural FAVARs 33 

6.1 Identification of Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 

6.2 Identification Schemes in SVARs . . . . . . . . . . . . . . . . . . . . . . . . 34 

6.3 Identification in DFMs and FAVARs . . . . . . . . . . . . . . . . . . . . . 34 

7 Empirical Results 38 

8 Discussion 49 

9 Summary and Concluding Remarks 51

Bayesian FAVARs with Agnostic Identification 3 

10 Matlab Implementation 51 

References 63 

Appendix A: Data 69 

Appendix B: Figures 76 

Appendix C: Matlab Code 88


1 Introduction 

What are the dynamic effects of a shock to monetary policy on business cycle fluctua- 

tions? Are there any real effects or are the real aggregates independent from the monetary 

sector? Whether there is a link between the monetary and the real aggregates over the 

business cycle, has been a question economist deal with since the early work by Friedman 

and Schwartz (1963). In their influential book they arrive at an affirmative conclusion and 

hence postulate that there is a link between monetary policy and real economic activity. 

There have been many approaches that dealed with this question in an advanced manner. 

Most of the research concerned with this question faced the limitation that they were 

only able to consider a limited information set that cannot capture the ”data-rich envi- 

ronment” 1 in which central bankers decision are assumed to take place. This limitation is 

entailed by the econometric frameworks mostly applied. Most of the methodologies and 

models applied provide the conclusion that the effects of monetary policy are not of great 

importance for the real sector. 

The recent literature advances applying dynamic factor models to deal with large pan- 

els of data in applied macroeconomics. These models extract from large datasets a few 

factors that capture the main driving forces of the economy. In the strict sense the dynam- 

ics of the data is then explained by a common component and an idiosyncratic component, 

which is series specific. Bernanke, Boivin and Eliasz [2005] introduced a combination of 

the recent advances of dynamic factor models with the standard VAR analysis, in a unify- 

ing framework namely the factor-augmented vector autoregression (henceforth FAVAR). 

In their paper, Bernanke, Boivin and Eliasz [2005] also aim to disentangle the dynamic 

propagation of a shock to monetary policy throughout the economy. The identification 

scheme they apply is a standard recursive one implemented in the FAVAR framework, and 

the results they provide are based on a nonparametric two-step estimation using principal 

component analysis. The results they achieve from a joint likelihood approach via Gibbs 

1 see Bernanke and Boivin [2003]


sampling seems to be unfavorable. 

The aim of this paper is threefold. First I critically examine the result by BBE and 

provide an answer why they arrive at inferior results and how one should apply this pro- 

cedure to receive more reasonable results. I like to combine the Bayesian FAVARs with 

the agnostic identification by Uhlig (2005). To my best knowledge this has not been done 

yet in the current literature on empirical macroeconomics. Hence I tackle the question 

raised above in a complete consistent Bayesian framework. Not only the factors and the 

parameters of interest are estimated with Bayesian methods, more importantly I apply 

the agnostic identification in order to identify the specific effects induced by contrac- 

tionary monetary policy. Here I try to be as precise as possible and as close as possible 

to the conventional wisdom. This is accomplished by setting different ”block criteria” 

that successively become stricter, while imposing more restrictions. This approach holds 

an enticing promise in that one can narrow down the space of economically reasonable 

dynamic reactions as accurate as the data allows. We do not have to set restrictions on 

only one price variable, one of the monetary aggregates and one short term interest rate 

as it has been applied so far and is common practice when applying sign-restrictions in 

the VAR framework. Having available such a large datasets, provides the possibility to 

be more strict in imposing e.g. not only the CPI to react non-positively after a monetary 

policy shock but also other price variables that are incorporated in the estimation proce- 

dure. This serves the possibility to identify the structurally, reaction of the economy in 

an more exact manner than other identification approaches common in the literature on 

empirical macroeconomics. 

The second contribution and also the most time consuming one was to provide a Mat- 

lab code that does the estimation and identification. Here the major challenge has been 

to provide a code that is as efficient as possible, especially with respect to the calculating 

time and the memory required in a way that also students can run the program on PCs of 

”common” capacity without waiting a week for the results. The Gibbs sampling procedure


itself is a very cumbersome computer intensive and memory intensive estimation method 

that applied to such large datasets might take very long to produce results. The challenge 

is to combine the Gibbs sampling with the agnostic identification procedure which itself 

is time consuming. How the code accomplishes this challenge is explained on the section 

about the Matlab implementation. Third we present results that seem reasonable and 

consistent with the conventional wisdom in like the ones by BBE. This indicates that 

the key to the question above is the identification scheme. More importantly the results 

confirm the lead of Sims to avoid unreasonable identifying assumption. Hence opposed to 

the conclusion of BBE our results confirm that not only information is very important but 

one receives reasonable estimation results in combination with economically sound identi- 

fication schemes such as the agnostic identification using a broader set of sign-restrictions. 

The main results are that the Bayesian estimation of FAVARs combined with the ag- 

nostic identification of Uhlig [2005] deliver results according to the conventional wisdom 

where no prize puzzle arises. It seems to be the crucial issue why Bernanke, Boivin and 

Eliasz [2005] arrive at inferior results compared to the two-step principal component esti- 

mation. They apply a standard recursive identification scheme. Furthermore the greater 

information set allows the researcher to be more strict w.r.t. to the sign restrictions im- 

posed, in order to disentangle the dynamic effects of a shock to monetary policy more 

exactly. 

However one should be cautious with the conclusion, due to the fact that one should 

collect results for longer Gibbs iterations in order to assure the results. One should also 

try out several number of factors and also report their contribution via variance decom- 

position. 

The next section gives an overview of the relevant literature followed by an introdution 

into dynamic factor models in section 3. Section 4 starts to explain the FAVAR framework 

with its different identification and various estimation approaches. Furthermore a short


introduction for Markov chain Monte Carlo simulation methods and the Gibbs sampler 

is provided. Section five elaborates on the Bayesian estimation procedure for the FAVAR 

methodology and the inference on the factors and parameters to be estimated. The 

Identification schemes for identifying the effecst of a policy shock are presented in section 

6. The empirical results and the discussuion are provided in section 7 and 8 respectively. 

Section 9 concludes and in the last section exlpaines the attached Matlab code with the 

help of sequence diagrams. 

2 Literature 

The question what the effects of a monetary policy contration are, still is a very important 

one to economists. Regarding the qualitative effects there has come up a broad consensus 

among economists, but the quantitative measure is still subject to controverse discus- 

sions. Since Friedman and Schwartz [1963] examined the question raised above, there 

have been several empirical and theoretical approaches trying to tackle this matter, in a 

more advanced manner, focusing on the accurate quantitative measure of the effects of a 

shock to monetary policy. Regarding empirical studies dedicated to the question raised 

above, Sims [1992], Sims and Zha [1996], Leeper, Sims and Zha [1996], Sims [1998], Temin 

[1998], Christiano Eichenbaum and Evans [1999], Canova and De Nicolo [2002] and Uhlig 

[2005] provide advanced approaches. These studies in a broad sense all deal with the 

examination of the link between monetary policy and the real sector. They mostly differ 

in the methodology applied to unriddle the mere effect of a shock induced by monetary 

policy. Theoretical approaches are provided amongst others by Ireland [2001], Godfreind 

and King [1997], Clarida, Gali and Gertler [2000] and Gali [2002]. 

Over the years, there has come up a broad consensus about the monetary transmis- 

sion mechanism. Most of the literature agrees that monetary policy does not play an 

important role for business cycle fluctuations or more precisely does not affect output un-


ambiguously 2 3 . In the current literature there are also approaches to consider an index 

extracted with dynamic factor models, representing the relevant dynamics out of many 

macroeconomic variables associated with ”economic activity”, such as Korenok and Rad- 

chenko [2004]. The consensus seems to be consistent with the ”monetary neutrality” that 

monetary policy shocks only affect the nominal but not the real sector in the long-run. 

Contrariwise Canova and De Nicolo [2002] show that monetary policy had a significant 

effect in reducing the output in the G7 industry countries, hence they implicitly disagree 

with the ambiguous effects of monetary policy shocks mentioned by the papers above. The 

contentious issue raising the overall dissentation is rather the quantitative magnitude of 

the reaction 4 instead of its mere direction. There are methods required to address such 

issues quantitatively, which is important for central bankers who have to decide how to 

react to the state of the economy. In order to accomplish a sound decision, the monetary 

authorities have to be certain about the precise reaction of the economy. For example 

in a recession, it might be required to know about the accurate effects, in order to avoid 

a policy decision that might rather be a drag on the recovery. The broader the relevant 

information set the more accurate is analysis the for central bankers. 

The ”workhorse” in applied empirical macroeconomics for accomplishing the question 

at hand is the structural vector autoregression (Hence forth SVAR). The problem is that 

the VAR framework with its limitations such as the ”curse of dimensionality” 5 cannot 

mirror the reality of central bankers decision making, particularly w.r.t. the amount of 

information exploited which apparently is relevant for the economy. This might be very 

important for researchers because there is little hope that economists can evaluate al- 

ternative theories of the monetary policy transmission mechanism, or obtain quantitative 

2 See Uhlig [2005] 

3 Some researchers measure the effects on output, some rather on industrial output as a representative 

”proxy” or ”index” for economic activity (see Sims). 

4 In my thesis I only consider the effects of contractionary monetary policy shocks. 

5 This was raised by Sims (1980), where he states that with increasing number of variables included in 

the VAR, the parameters to be estimated increase quadratically and soon become intractable.


estimates of the impact of monetary policy shocks and changes in policy on various sectors 

of the economy, if there exists no reasonable objective means of determining the direction 

and size of changes in policy stances 6 . 

Monetary policy decisions are implicitly assumed to take place in a ”data-rich envi- 

ronment” 7 , reflecting that the central bankers analyze a vast amount of economic time 

series 8 prior to their policy decisions taken. The fact that central bankers bear this huge 

cost to analyze such a plethora of data and assuming that they do not have a great 

irrational interest in wasting their time with dispensable data analysis, should provide 

the indication that a broad set of ”information” is of great relevance for the decisions 

to be taken. Hence much more variables would have to be considered for example if a 

researcher would be interested to model the monetary transmission mechanism, or the 

Fed’s reaction function. The limitation does not come from the modeling researchers ig- 

noring the importance of the large data set, but from the methodologies mainly applied. 

The advantage of the SVAR framework is that it is small and therefore more tractable 

and easier to compute. Bernanke and Boivin (2003) were the first who tried to tackle the 

problem of ”information dimensions” in the context of monetary policy in a framework 

that combines the advances of dynamic factor models with standard VAR analysis. They 

”explore the feasibility to incorporate a richer information set into the analysis of positive 

and normative Fed policy making” 9 . They achieve this through a factor model approach 

based on the work of Stock and Watson (1999,2002) and Watson (2000). Dynamic factor 

models that were introduced to economics by the seminal papers of Geweke (1977) and 

Sargent and Sims (1977), put the classical factor model, that only regards cross-sectional 

data, in a dynamic setting. These models have become increasingly popular since the work 

by Stock and Watson (1989,1999,2003) that outperformed the forecasting accuracy of the 

6 See Bernanke and Mihov (1998a) 

7 see Bernanke and Boivin (2003) 

8 Bernanke and Boivin (2003), state that central bankers literally monitor several hundreds or even up 

to thousand time series. 

9 See Bernanke, Boivin [2003]


standard autoregression approach, but only for the real variables except employment. A 

lot of good research has revealed advances to these models, in particular w.r.t. the esti- 

mation procedure that allows for large datasets, in length and dimension, so that it can 

be applied to the economic question at hand. Hence one can overcome the dimensionality 

problem faced w.r.t. the large datasets to be included. The advantage of factor models 

is that the main driving forces in a large set of cross-sectional data, which in our case is 

required to consider, can be represented by a much smaller number of ”factors” extracted 

from the dataset. For our analysis I decide to apply the so-called FAVAR 10 methodology 

which was introduced in Bernanke and Boivin (2003), advanced in Bernanke, Boivin and 

Eliasz (2005). Stock and Watson (2005) added some variations to it, and provide a broad 

survey on its different approaches. These models exploit the advances of dynamic factor 

models and combine them with the VAR methodology. And the crucial innovation is to 

combine the Bayesian FAVAR with an Agnostic identification. 

The question of identification is an issue that has been covered a huge body of lit- 

erature and applied to SVAR. The most prevalent schemes are the Recursive Cholesky 

identification, the Long-run Identification, the combination of the two (Zero restrictions; 

Leeper, Sims and Zha [1996]) and the Agnostic Identification introduced by Uhlig [2005]. 

In his paper Uhlig also seeks to measure the effects of a shock to monetary policy, strictly 

speaking to a ”contractionary” monetary policy shock, in particular he focuses on the 

effects on Output and finds out that there is no clear effect on output, and that the 

neutrality of monetary policy shocks are not inconsistent with the data. But what is so 

crucial about the paper by Uhlig is the new more sophisticated identification scheme he 

introduces, namely the ”agnostic identification” 11 which imposes for a certain period of 

time sign-restrictions on the impulse responses, that are consistent with the conventional 

wisdom. 

10 This terminology comes from Bernanke, Boivin and Eliaz [2005] 

11 The term agnostic refers to the missing restriction on the variable to be analysed.


One of the main contributions of this thesis is to disentangle the dynamic propagation 

of a monetary policy shock on the macroeconomic variables is approached in a fully 

Bayesian perspective. First the model is estimated with a likelihood based approach via 

Gibbs sampling, and second I try to measure the dynamic effects through an agnostic 

identification scheme, using the sign-restriction approach advanced by Uhlig (2005) who 

imposes for a certain period of time sign restrictions on the impulse responses of prices, 

nonborrowed reserves and the federal funds rate in response to contractionary monetary 

policy shock 12 13 . Furthermore I provide an easily extendable Matlab code that does the 

model estimation and identification. 

3 Dynamic Factor Models 

The key idea behind the factor models is, to represent the movements in a large set of 

cross-sectional data by only a limited number of common shocks, which apparently are 

sufficient to represent the crucial dynamics, and an idiosyncratic component that reflects 

the variable specific part. The common shocks are referred to the common component, 

which consists of the factors and the respective factor loadings. 

In the recent few years there has been a surge in the research on dynamic factor models 

(henceforth DFM) where advances and several extensions have been introduced. It has 

become an important tool in empirical macroeconomics since it provides a possibility to 

break down the dimensionality of the large amounts of economic time series, which have 

become available, into a few series of factors. Large datasets are important in various 

fields in economics, such as in the study of the effects of trade (see Justiniano [2004]) and 

cross-country business cycle synchronization (Kose, Otrok and Whiteman [2003a, 2003b] 

12 To my best knowledge this combination has not been done by anyone. Neither in the DFM method- 

ology nor in the FAVAR methodology. 

13 As previously stated I follow the approach to apply the FAVAR methodology, but the application of 

the sign-restriction approach can be applied to DFMs in an equivalent and straight forward manner.


among others). In the next subsections I am first going to explain DFMs on which the 

FAVAR methodology grounds. Then I explain the FAVAR and the required identification 

assumptions in order to identify the factors and factor loading separately. The last sub- 

section elaborates on the estimation procedures and in particular on the likelihood-based 

joint estimation. 

The classical (static) factor model applied only to cross-sectional data, was first set in 

a dynamic setting and introduced to economics by the seminal work of Sargent and Sims 

[1977] and Geweke [1977]. Sargent and Sims [1977] applied DFM to a low dimensional set 

of macroeconomic variables in order to explain the mutual comovements. The dynamic 

factors parsimoniously summarize the dynamics and information of a large panel of data. 

Due to the small dimensionality of the data the estimation could be accomplished by 

maximum likelihood estimation methods (MLE). The MLE will soon arrive at its limits 

with increasing number of time series. Quah and Sargent [1993] apply the EM algorithm 

for a set of 60 variables which is the biggest application considered in a MLE framework. 

Due to the complicated nature of the shape of the likelihood in a high dimensional case 

it becomes soon infeasible to apply ML methods. Stock and Watson [1998,2001] show 

that factor 14 extracted from eight 15 macroeconomic variables, improve the forecasting ac- 

curacy of inflation. Their results outperform the standard AR approach but only in a 

supportive manner for the real variables except for employment. Since then DFMs have 

gained an increasing attention in the academic world of empirical macroeconomics and 

seem to become an important alternative to the VAR, which is still the ”workhorse” in 

empirical macroeconomics and most widely applied. 

Important extensions and advances have been achieved especially w.r.t. the estimation 

14 The authors sometimes refer to the factors as diffusion indexes that summarize the inherent risk of 

the variables considered 

15 In their first paper the consider four variables and extend there data and variables in Stock and 

Watson [2001]


procedures. Stock and Watson [1989,1991,1991,2001,2003,2005], study a static version of 

the DFM estimated via principal component analysis. In the proceeding papers they 

consider a two-step estimation procedure. Forni,Hallin,Lippi and Reichlin (2001), provide 

a dynamic version of the PCA which is known as the generalized dynamic factor model 

(henceforth GDFM). Some refer to it also as the dynamic principal component analysis 

(DPCA) where the model is considered in the frequency domain. Kim and Nelson (1998), 

Otrok and Whiteman (1998) tackle the model estimation from a Bayesian perspective 

via Markov chain Monte Carlo (MCMC) simulation methods, in particular applying the 

Gibbs sampler. This will be the approach I apply in my thesis in order to extract the 

factors and do inference on the models parameters. One of the most recent advances 

have been the so-called factor augmented vector autoregression (FAVAR) which has been 

introduced by Bernanke and Boivin [2003], and advanced in Bernanke, Boivin and Eliasz 

[2005], a framework in which the advantages of DFMs are combined with the analysis of 

SVARs. The various model specifications and estimation procedures of large data sets on 

which the FAVAR builds are briefly explained in the following subsection where I provide 

an overview of the most important and influencing ones and briefly explain the different 

approaches. 

Dynamic factor models can be considered either in the frequency domain representa- 

tion or in the state-space representation depending on the estimation approach desired. 

The model cast in the frequency domain representation are introduced and explained 

in several papers by Forni,Hallin,Lippi and Reichlin. They use an approximate DFM, 

and compute the eigenvector-eigenvalue decomposition of the spectral density matrix fre- 

quency by frequency and inverse-Fourier transform the eigenvectors to create polynomials 

in the lag operator which when applied to the observables, yields estimates of the dynamic 

principal components (DPCA). 

The latter is a generalization that captures all time series models, such as the au- 

toregressive integrated moving average model (ARIMA), and consists of one observation


equation and one state equation 16 which is part of the observation equation and itself 

driven by a stochastic process: 

Xit = λift + eit (1) 

ft = φ(L)ft−1 + vit (2) 

where the subscript i = 1, ..., N stands for the observable variables and t = 1, ..., T 

denotes time. Equation (1) characterizes the stationary data or the economic variables 

(time series) that is driven by the unobservable factors ft which itself is driven by a 

stochastic process. λi denotes the factor loading . Strictly speaking Xit is driven by a 

distributed lag of a small number of factors (K


limN→∞N −1 

N N 

|E(eitejt)| < ∞ 

i=1 j=1 

The decision which approach to choose will depend on the estimation procedure de- 

sired to apply. The frequentists DPA approach allows for an approximate DFM, the 

Bayesian requires an exact DFM specification. From my perspective, the question which 

approach rather to pursue has not yet been sufficiently and conclusively enough answered 

in the literature. BBE provide results for both specifications and estimation approaches. 

Based on their results, which favors the classical nonparametric two-step estimation, they 

conclude implicitly that the approximate specification might be the better one. Here 

one should be very cautious, because the results BBE compare, might not necessarily be 

representative enough to draw a conclusion w.r.t. to the model choice. This will become 

clear when comparing the likelihood based results with the two alternative identification 

schemes. As the results differ qualitatively the conclusion based on the results becomes 

invalid. In the section on the empirical results I will show, that with reasonable identify- 

ing assumptions such as the ”agnostic identification” one can get very reasonable results 

in an Bayesian framework. This does not give an obvious hint what the correct approach 

should be, but at least one can conclude that the Bayesian approach does not suffer from 

the structure it imposes on the idiosyncratic component. Therefore the structure imposed 

must not be an unreasonable restriction on the model. It remains to further research on 

this specific issue to prove conclusively. More precise assumptions of DFMs can be found 

in the section on identification and normalization.


4 The Econometric Framework 

4.1 FAVARs 

As already stated, the idea behind the FAVARs is to combine the standard structural 

VAR analysis with the recent developed and advanced features of dynamic factor models 

estimating a joint VAR that contains factors extracted from large panel of informational 

data and in addition perfectly observable time series that have pervasive effects on the 

economy such as the short-term interest rate set by the central. Therefore BBE labeled 

the model in a straight forward manner ”factor-augmented VAR”. This approach is well 

suited for structural analysis such as impulse response analysis and variance decomposi- 

tion (in particular for the problem at hand). For the estimation procedure the model has 

to be cast into a state-space representation. For the rest of the thesis I will mostly follow 

the approach and notation of BBE otherwise explicitly stated. 

The model consists of the two equations (1) and (2) introduced in the previous section. 

The FAVAR equation (2) already has the form to build the state equation to which one 

also refers often to as the transition equation. Equation (2) represents the joint dynamics 

of factors and the observable pervasive variables (Ft, Yt). 

⎡ 

⎢ 

⎣ Ft 

Yt 

⎤ 

⎡ 

⎥ ⎢ 

⎦ = Φ(L) ⎣ Ft−1 

Yt−1 

⎤ 

⎥ 

⎦ + vt 

(3) 

vt ∼ N(0, Q) (4) 

Here the variable Yt denotes the [M × 1] vector of observable economic variables 

that having pervasive effects throughout the economy. The index t = 1, ..., T represents 

the time the term Φ(L) represents a conformable lag polynomial of order d. In our 

specification that follows BBE Yt is assumed to represent the policy instrument,e.g. the


federal funds rate in the US case. But it can also represent economic concepts such as 

”economic activity” an so forth. In fact one can let Yt represent a complete VAR, with 

the specification desired or that is standard in the literature instead of one single variable. 

If the observation equation (1) would only consist of Yt we would be in the well known 

standard VAR framework the factor augmentation would be missing. Hence one could 

apply standard SVAR analysis or other multivariate time series estimation using only 

data for Yt. 

The cross-sectional dynamics of the data are represented by the parsimoniously ex- 

tracted factors. Through the factors one can deduce the dynamic effects of a shock to 

monetary policy. That are the relevant comovements not captured by Yt. The dynamics 

of the whole economy and its reaction induced by an unexpected shock are captured by 

the parsimoniously extracted factors. 

The number of the factors, K should be small 17 compared to the number of time 

series considered. One can think of the unobserved factors as diffuse concepts such as 

”economic activity” or ”credit conditions” which are rather represented by a range of eco- 

nomic variables than by only one or a few. This hint given by BBE might be interesting to 

pursue and further explored in future research when it comes to question of structurally 

identifying the factors. One approach to give the factors a structural interpretation as has 

been done by Belviso and Milani (2005), is to explicitly model the factor loadings only 

to load on specified factor that is extracted from a subset of the sorted data associated 

with an economic concept. Belviso and Milani [2005], follow Stock and Watson [2005], 

regarding the number of factors required and relevant to represent the driving force of the 

economy, and try to structurally identify seven Factors which are supposed to represent 

17 To have a reference for the term ”small”, Gianone, Reichlin ans Sala [2004] assume that the driving 

forces of the US economy can be represented by two factors opposed to Stock and Watson [2005] who 

argue the fundamental driving forces of the US economy should be represented by seven factors. These 

factors were extracted out a large panel of 173 and 132 variables respectively


the main dynamics of the US economy. 

Regarding the term Φ(L) BBE note that one may set a priori restrictions as it is well 

known and done in the structural VAR literature, that would reduce the number of param- 

eters to be estimated. The vector of error terms vt has mean 0 and variance-covariance 

Q. [ The above equation represents a VAR w.r.t. the joint dynamics of (Ft, Yt) to which 

BBE refer to as a factor-augmented vector autoregression henceforth FAVAR.] Note that 

if the block of the lag polynomial that relate Yt to Ft−1 is 0 the system reduces to 

the standard VAR as there would assume that Yt and Ft are independent to each other 

and hence not relevant to explain its dynamics. In this way one can measure the direct 

contribution of incorporating more economic information into the observation equation 

via Ft. 

The results for the contribution of adding further factors are represented in section 

empirical results. If the true data generating process is a FAVAR, the standard VAR sys- 

tem in Yt will lead to biased estimates (especially w.r.t. IR-coefficients). This is straight 

forward as this implies that relevant information is not included. In order to be able to 

estimate the FAVAR one needs the unobservable factors Ft which will be extracted from 

the set of ”background” or ”informational” time series denoted by the [N × 1] vector 

Xt. The number of N might be very large, even larger than the number of observations 

or time periods T . Furthermore it is assumed to be much greater than the number of 

Factors and pervasive observables (K + M


the statement that more data necessarily means more information and hence better for 

the analysis seems not to be the end of the story. 

The dynamics of the ”informational” variables is assumed to be like the following: 

X ′ t = Λ f F ′ 

t + Λ y Y ′ 

t + e ′ t 

(5) 

et ∼ N(0, R) (6) 

Here Λ f denotes the matrix of factor loadings with dimension [N × K] and Λ y is 

[N × M]. The error term is et with mean 0 and covariance R. Note that et and vt are 

independent and that R is diagonal which means that the error terms of the observable 

variables are mutually uncorrelated. At this point one has to make a clear stand which 

assumption one follows when it comes to the issue of error correlation. One can think 

of the error terms to be weakly correlated or completely uncorrelated. We had this 

previously in the discussion of exact or approximate dynamic factor models. The standard 

assumptions in the literature with respect to dynamic factor models has been introduced 

in the previous section. As we follow the Bayesian likelihood-based approach we decide to 

set the assumption of uncorrelated error terms. Hence we model an exact dynamic factor 

model in the vein of Sargent and Sims [1977]. The distinction between the observation 

equations of the DFMs we have seen so far and (1) is that there, the dynamics of the data 

are supposed to be driven by Ft and Yt which in fact can be correlated. Here Xt only 

depends on the current and not lagged values of Ft. BBE state that this implication is 

not restrictive in practice as the factors can be interpreted as including arbitrary lags of 

the fundamental factors. 

4.2 FAVAR Identification 

Identifying restrictions have to be set, in order to distinguish the idiosyncratic from the 

common component. Additionally one can set further identifying assumptions in order to


identify the factors and the loadings, separately, and furthermore distinguish the single 

factors. In this thesis I follow the standard identification restrictions 18 either on the 

coefficient matrix Λ or on the factors Ft employed by BBE in order to identify the 

factors and the factor loadings uniquely which looks like the following : 

Λ ′ fΛ ′ f 

N = I or F ′ F ′ 

T 

The crucial assumption is that Yt (in our baseline model the policy instrument FFR) 

does not react to the X’s contemporaneously. The channels are restricted if the upper 

K × K block of Λ f is set to an identity matrix and the upper K × M block is set 

to a zero matrix. This restricts the impact of Yt on only those K variables that react 

contemporaneously and therefore such variables should be chosen for the respective block 

that do not react contemporaneously. Since factors are estimated up to a rotation, the 

choice of the K × K that is set to an identity matrix should not affect the space spanned 

by the estimated factors 19 . 

For some proposes it is useful to separately identify the common shocks and the fac- 

tor loadings. But as only the product of the two is known, a rotation has to be chosen 

when one is interested in identifying the factors and the loadings separately. In my case 

I am interested in the separate identification and in the following section I describe the 

= I 

standard approach chosen by BBE that I also decided to choose. 

Digression on Factor identification In order to identify the factors against rotation 

BBE impose the factor restriction F ′ F ′ 

T = I, obtaining ˆ F = √ T ˆ Z, where Z are the first 

K largest eigenvectors sorted in descending order. In the joint estimation case the specified 

identification against rotation requires that the Factors are identified in the following form: 

18 The factor identification should not be confused with the identification of the structural shocks of 

e.g. monetary policy. 

19 see BBE [2005]


F ∗ 

t = AFt − BYt 

Here is a nonsingular K × K] matrix and is of dimension K × M]. Restrictions 

are only imposed on the observation equation. Here BBE substitute F ∗ into (1) due to 

the fact that restrictions should not be imposed on the VAR dynamics and hence arrive 

at 

Xt = Λ f A −1 F ∗ 

t + 

Λ y + Λ f A −1 B 

Yt + et 

Now for the factors and and their loadings to be identified uniquely it is required that 

Λ f A −1 = Λ f and Λ y + Λ f A −1 B = Λ y . 

When it comes to the identification of the VAR part or the FAVAR equation (2), one is 

concerned with the identification of the innovation, strictly speaking the innovation to Yt 

which is the shock to monetary policy. As this is the main issue of this paper we will only 

consider this case, but the agnostic identification by Uhlig [2005] using sign-restrictions 

is extendable to any other shock desired. The identification schemes are elaborately 

explained in section following section. 

4.3 Estimation Procedure 

There are two estimation procedures considered by BBE, the nonparametric two-step 

principal component estimation and the parametric Bayesian approach to which they refer 

to as the one-step estimation or the likelihood-based estimation (Eliasz [2005]). As the 

aim of this thesis is to tackle the question raised by the title from a Bayesian perspective, 

we will mention the two-step estimation procedure only very briefly and elaborate on the 

likelihood-based estimation procedure. Here I mostly rely on BBE [2005] and Eliasz[2005]. 

4.3.1 Generalized Dynamic Factor Model 

Factor models can be decomposed into a common component and an idiosyncratic com- 

ponent. The authors Forni, Hallin, Lippi and Reichlin (2001) tried to combine the ap- 

proximate DFM of Chamberlain (1983) and Chamberlain and Rothschild (1984), which


is static 20 and allows for some cross-correlation of the idiosyncratic components, and the 

dynamic version of Geweke(1977) and Sargent and Sims (1977) which assumes orthog- 

onalized idiosyncratic components. Their concept is mostly known as the generalized 

dynamic factor model and in the current literature also known as the dynamic principal 

component analysis (DPCA). Here they do a principal component analysis of the first 

Q eigenvalues and eigenvectors which are calculated from the variance-covariance matrix 

(VCV) of the data set. The GDFM or the DPCA can be summarized as a concept that 

has a similar representation as the static factor model but with a dynamic setting with 

respect to the factor loadings. 

4.3.2 Two-Step Estimation 

This approach is analogous to the estimation procedure used in Stock and Watson [2002], 

where they used DFMs to forecast inflation. In order to uncover the space spanned by the 

common components Ct = (F ′ 

t, Y ′ 

t ) ′ as a first step the first [K+M] principal components 

(henceforth PC) of Xt are estimated. Here the attentive reader should note that this 

first step does not exploit the fact that Yt is observed 21 Furthermore the number of the 

informative variables N has to be large and the number of the PCs have to be at least as 

large as the true number of the factors for the PCs to recover the space spanned Ft and 

Yt consistently. As a further disadvantage one should state that the two-step approach 

implies the presence of ”generated regressors” in the second step 22 . Ft is obtained as 

the part of the space covered by Ct that is not covered by Yt. The advantage of this 

approach is that it is easy to implement and computationally simple as opposed to the 

computer intensive burdensome Gibbs-Sampling approach. Stock and Watson state that 

it also imposes few distributional assumptions and allows for some degree correlation in 

20 In this context, the static version of the factor model means that the common shocks only affect the 

series contemporaneously. 

21 For our baseline model this means that the fed’s policy action is not taken into account contempo- 

raneously. 

22 For more details please refer to BBE [2005].


the idiosyncratic error term et. 

4.3.3 Likelihood-Based Estimation 

The alternative to the approach discussed above is to use the joint estimation by likelihood- 

based Gibbs-Sampling techniques. The problem of such models with high dimensions is 

that it is very difficult get the joint marginal distribution by integration. As BBE [2005] 

state, the irregular nature of the likelihood function makes maximum likelihood estima- 

tion (henceforth MLE) infeasible in practice. In order to understand better why the Gibbs 

sampling is considered as a useful tool to estimate large DFMs and FAVARs, I will briefly 

discuss the technical background of the estimation procedure. The idea of this section is 

to give the interested reader an introduction to the Markov Chain Monte Carlo methods 

(MCMC in the following) afterward elaborate on the Gibbs sampler and on the multi 

move version of the Gibbs sampler in order to convey its usefulness for DFMs. 

4.3.4 Markov Chain Monte Carlo 

The estimation of the parameter space (of LDFMs) is about integrating statistics, espe- 

cially in the Baye’s statistic or in Bayesian approaches obtaining the posterior distribution, 

which contains all relevant information on the unknown parameters given the observed 

data, requires the integration of high-dimensional functions. The statistical inference can 

be deduced from posterior distributions. The integration problem is one of the crucial 

ones and can be computationally very cumbersome. One can do the integration (evaluate 

the integrals) through approximation via numerical methods 23 but when the parameter 

space is multidimensional even numerical methods may fail. One can make use of stochas- 

tic algorithms such as the Monte Carlo Integration techniques. The MCMC methods use 

computer simulations of Markov chains in the parameter chain. For random variables of 

higher dimensions 24 , one has to solve multiple integrals 25 . The problem of such models 

23 Such as the Simpson or Trapez method 

24 as it is the case in FAVARs and (L)DFMs 

25 In such cases the integration via numerical methods is very hard if solvable at all.


with high dimensions is that it is very difficult get the joint marginal distribution by 

integration. As BBE (2005) state, the irregular nature of the likelihood function makes 

integration infeasible in practice. Bayesian analysis usually requires integration to get 

the marginal posterior distribution of the individual parameters from a joint posterior 

distribution of all unknown parameters of the model in which statistical practioneers are 

interested in. As already stated these integrals in a high dimensional case are very hard 

to solve. The joint posterior density itself is very difficult to derive, from which marginals 

are to be derived. The MCMC methods such as the Gibbs sampler serve the solution 

to such high dimensional problems, in that they allow to implement posterior simulation 

that allow the point wise evaluation of prior distribution and likelihood function. 

Markov chain refers to the sequence of random variables (Z0, ..., Zn) that are generated 

by a Markov process. The MCMC methods attempt to simulate direct draws from some 

complex distribution of interest. Here the Gibbs sampling methodology offers an easy way 

to solve the problem, given that conditional posterior distributions are readily available. 

It may be employed to obtain easily marginals of the parameters without integration and 

without having to know the joint density. 

4.3.5 The Gibbs Sampler 

The Gibbs sampling methodology offers an easy way to to solve the dimensionality prob- 

lem given that conditional posterior distribution are readily available. In order to exem- 

plify the Gibbs sampler, think of θ as the parameter space. Furthermore assume that 

p(θ | YT ) represents the joint probability density function where YT denotes the data. 

Following a cyclical iterative pattern, the Gibbs sampler generates the joint distribution 

of p(θ | YT ) which can be also referred to as the target distribution that one tries to 

approximate empirically via a Markov chain. The Gibbs sampler begins with a parti- 

tioning or blocking of the parameters in d subvectors θ ′ = (θ1, . . . , θd). In practice the 

blocking is chosen so that it is feasible to draw from each of the conditional pdf’s so


that p(θk | YT , θ.=k) where θ j 

.=k 

= (θj 1, . . . , θ j 

k−1 

, θj−1 k+1 

, . . . , θj−1 

d , ). The blocking can arise 

naturally, if the prior distribution θk are independent and each conditionally conjugate. 

Given an arbitrary set of starting values θ ′0 = (θ 0 1, . . . , θ 0 d) set the iteration index j to zero 

and repeat the following cycle J times. 

j = 1 

draw θ (j) 

(1) from p(θ1 | θ j−1 

2 , . . . , θ j−1 

k , YT ) 

draw θ (j) 

(2) from p(θ2 | θ j 

1, θ j−1 

3 , . . . , θ j−1 

k , YT ) 

. 

draw θ (j) 

(k) from p(θk | θ j 

1, θ j 

2, . . . , θ j−1 

k−1 , YT ) 

j = j + 1 

After each cycle j is increased by one. Thus each subvector is updated conditional on 

the most recent value of θ for all other components. The Gibbs sampler produces a series 

of j = 1, . . . , B, . . . , B + M conditioning drawings by cycling through the conditional 

posteriors. In order to avoid the effect of the starting values on the desired joint density 

and to ensure convergence the first B draws should be discarded. Hence the last M cycles 

are considered as the approximate empirical simulated sample from p(θ | YT ). 

5 The Econometric Model 

In this part I will specify the model more precisely and show the steps required to prepare 

the model for the estimation procedure. 

5.1 The Bayesian Approach versus the Frequentists Approach 

Why do I favor the Bayesian approach rather than the classical approach in dynamic factor 

models? The Bayesian exploits the available information in a more efficient manner and 

does not ignore in case of having a priori information about the parameters of interest. 

In the classical approach inference about the unobserved state vector is based on the


estimated values of the hyperparameters 26 of the model, which are obtained with the 

maximum likelihood method. One has to treat them as they were the true values of 

the models nonrandom hyperparameters. This is a disadvantage towards the Bayesian 

approach where in the vein of Bayesian data analysis either the models hyperparameters 

and the unobserved state vector are treated as random variables. The classical approach 

does not exploit the fact that Yt is observed and furthermore does not exploit the structure 

of the state equation in the estimation of the factors. 

5.2 State-Space Representation 

This part elaborates on the specific model and the required transformations in order to 

estimate the model via Gibbs sampling. For more details the reader is referred to Eliasz 

[2005] and Kim and Nelson [1999] for a very good and elaborate survey. In order to prepare 

(1) and (2) for the estimation the model has to be cast into the following state-space form: 

⎡ 

⎢ 

⎣ Xt 

Yt 

⎤ 

⎥ 

⎦ = 

⎡ 

⎢ 

⎣ Ft 

Yt 

⎤ 

⎡ 

⎢ 

⎣ Λf Λy 0 I 

⎤ ⎡ 

⎥ ⎢ 

⎦ ⎣ Ft 

Yt 

⎡ 

⎥ 

⎢ 

⎦ = Φ(L) ⎣ Ft−1 

Yt−1 

⎤ 

⎥ 

⎦ + 

⎤ 

⎡ 

⎢ 

⎣ et 

0 

⎥ 

⎦ + vt 

⎤ 

⎥ 

⎦ , (7) 

The respective variables are the same as explained in the preceding sections. The 

loadings Λ f and Λ y are restricted and identified against rotational indeterminacies as 

it has been implemented by BBE and described in the previous section. According 

to BBE the inclusion of the policy instrument Yt in both equations will not change 

the model, it merely should serve notational and computational simplification. The 

Bayesian econometricians treats the parameters of the model, interested to do inference 

on, as random variables. We are interested in doing inference on the parameter space 

θ = 

Λf , Λy , R, vec(Φ), Q 

. Note that vec(Φ) is the vectorized finite order conformable 

lag polynomial, i.e. Φ is columnwise stacked to have a vectorized form 27 . To apply the 

26 Hyperparameters are the elements of the parameter space to be estimated 

27 for more details about the vec operator please refer to Lütkepohl [1993] 

(8)


multi move version of the Gibbs sampler one has to prepare the model further which is 

done step by step in the following. The multi move Gibbs Sampling, alternately samples 

the parameters θ and the factors Ft given the data. We use the multi move version of the 

Gibbs sampler because this approach allows us as, a first step to estimate the unobserved 

common components, namely the factors via the Kalman filtering technique conditional 

on the given hyperparameters and as a second step calculate the hyperparameters of the 

model given the factors via the Gibbs sampler in the respective blocking 28 . 

For the state space representation we define X ′ 

t = (X ′ t, Y ′ 

t ) , e ′ t = (e ′ t, 0) ′ and F ′ 

t = 

(F ′ 

t, Y ′ 

t ). For the case that Φ(L) is of order one, the model can be rewritten as: 

with 

Λ = 

⎡ 

⎢ 

⎣ Λf Λy 0 I 

Xt = ΛFt + et 

Ft = Φ(L)Ft−1 + vt 

⎤ 

⎥ 

⎦ , R = 

⎡ 

⎢ 

⎣ 

R 0 

0 0 

But in most applications one can expect the order to be d > 1, so is the case in 

the dataset I analyze. The dataset I analyze is in monthly frequency therefore I chose 

a lag order of 12 for Φ(L). The FAVAR equation has to be transformed into a first- 

order Markov process, in order to be able to draw the factors via Bayesian Kalman 

filtering. For that we define Φ(L) = Φ1L+Φ2L 2 +...+ΦdL d , ¯ Ft = (F ′ 

t, F ′ 

t−1, ..., F ′ 

t−1−d) ′ 

and ¯vt = (vt, 0, ..., 0) ′ . The lag polynomial of the FAVAR equation in the first-order 

representation changes to: 

28 Please note that we always also condition on the data due to notational convenience it is left out but 

is implicitly assumed and not further explicitly written 

⎤ 

⎥ 

⎦ . 

(9) 

(10)


⎡ 

⎢ Φ1 Φ2 . . . Φd−1 Φd ⎥ 

⎢ 

⎥ 

⎢ 

⎥ 

⎢ IK+M 0 . . . 0 0 ⎥ 

⎢ 

⎥ 

⎢ 

⎥ 

¯Φ = ⎢ 0 IK+M . . . 0 0 ⎥ 

⎢ 

⎥ 

⎢ 

⎥ 

⎢ . . . . . . . . . . . . . . . ⎥ 

⎣ 

⎦ 

0 0 . . . IK+M 0 

Now we have to transform the VCV of the FAVAR disturbances with 0’s in a straight- 

forward way to adjust the dimensions of the state equation which results in the following 

matrix: 

⎡ 

⎤ 

⎢ Q 

⎢ 0 

¯Q = ⎢ . . . 

⎣ 

0 

0 

. . . 

. . . 

. . . 

. . . 

0 ⎥ 

0 ⎥ 

. . . ⎥ 

⎦ 

0 0 . . . 0 

where the 0 ′ s and ¯ Q have dimension [(K + M) × (K + M)] , and [d(K + M) × 

d(K + M)] respectively. 

This results in the final observation equation of the following form: 

And finally the last extensions 

Xt = ¯ Λ ¯ Ft + et 

⎤ 

(11) 

(12) 

(13) 

¯Λ = [Λ 0 . . . 0] (14) 

The final state-space representation prepared to fit the estimation procedure are: 

¯Ft = ¯ Φ ¯ Ft−1 + ¯vt 

(15)


Xt = ¯ Λ ¯ Ft + et 

According to the Bayesian approach the parameter space with the respective hyper- 

parameters 29 and the factors {Ft} T 

t=1 

histories of X and F from period 1 through T are defined by 

5.3 Inference 

(16) 

are treated as random variables. The respective 

˜XT = (X1, X2, . . . , XT ) 

˜FT = (F1, F2, . . . , FT ) 

This part is very close to BBE [2005] and Eliasz [2005]. For completenes the single steps 

are presented at this stage. The task as it was described in the section about Gibbs 

sampling, is to derive the posterior densities. The aim is to empirically approximate the 

marginal posterior densities of 

p( ˜ FT ) = p( ˜ FT , θ)dθ and p(θ) = p( ˜ FT , θ)d ˜ FT where p( ˜ FT , θ) 

is the joint posterior density and the integrals are taken with respect to the supports of 

θ and ˜ FT respectively. The procedure applied to obtain the empirical approximation 

of the posterior distribution is the previously explained multi move version of the Gibbs 

sampling technique by Carter and Kohn [1994]. BBE also apply this estimation procedure 

that is surveyed by Kim and Nelson [1999]. 

Choosing the Starting Values θ 0 

In general one can start the iteration cycle with any arbitrary randomly drawn set of 

parameters, as the joint and marginal empirical distributions of the generated parame- 

ters will converge at an exponential rate to its joint and marginal target distributions as 

S → ∞. This has been shown by Geman and Geman [1984]. Following the advice of 

Eliasz [2005] one should judiciously select the starting values in the framework of large 

29 The hyperparameters refer to the elements of the parameter space


dimensional models, due to the fact that in case of large cross-sections, highly dimen- 

sional likelihoods make irregularities more likely. This can reduce the number of draws 

relevant for convergence and hence saves time, which in a computer-intensive statistical 

framework is of great relevance. I follow the suggestions of Eliasz [2005] and apply the 

first step estimates of PCA to select the starting values. A detailed description how to 

obtain the starting values via the first step PCA can be found his paper. Since Gelman 

and Rubin [1992] have shown that a single chain of the Gibbs sampler might give a ”false 

sense of security ”, it has become common practice to try out different starting values, at 

best from a randomly (over)dispersed set of parameters and then check the convergence 

verifying that they lead to similar empirical distributions. The Inference part is very close 

to BBE [2005] and Eliasz [2005]. This part can also be found in a slightly more elaborate 

verion in their papers. But for completenes it is stated here. 

Conditional density of the factors 

In this subsection we want to draw from 

pF ( ˜ FT | ˜ XT , θ) 

assuming that the hyperparameters of the parameter space θ are given, hence I describe 

Bayesian Inference on the dynamic evolution of the factors Ft conditional on Xt for 

t = 1, . . . , T and conditional on θ. The transformations that are required to draw the 

factors have been done in the previous section. The conditional distribution, from which 

the state vector is generated, can be expressed as the product of conditional distributions 

by exploiting the Markov property of state space models in the following way 

pF ( ˜ FT | ˜ XT , θ) = pF (FT | ˜ XT , θ)pF (FT −1 | FT , ˜ XT , θ), . . . , pF (F1 | F2, ˜ XT , θ) 

pF ( ˜ FT | ˜ XT , θ) = pF (FT | ˜ XT , θ) T −1 

t=1 pF (Ft | Ft+1, ˜ XT , θ) 

At this point it is important to note that the conditioning is on the first [K + M] rows of 

FT only, since otherwise for the case of d > 1 the VCV of the density would be singular.


This is an important hint, which is very relevant for the implementation into Matlab, and 

was found in Eliasz [2005] but was not explicitly stated in BBE [2005]. The state space 

model is linear and Gaussian, hence we have: 

FT | ˜ 

XT , θ ∼ N(FT |T , PT |T ) 

Ft | Ft+1 ˜ 

XT , θ ∼ N(Ft|t,Ft+1, Pt|t,Ft+1) 

where the first holds for the Kalman filter for t = 1, . . . , T and the second holds for the 

Kalman smoother for t = T − 1, T − 2, . . . , 1. The derivation of the Kalman filter and 

smoother can be found in an elaborate manner in Eliasz [2005], therefore I do not repeat 

it here at this place. 

Inference on the parameters θ 

Drawing from the conditional 30 distribution p(θ | ˜ XT , ˜ FT ) 

This part refers to to observation equation of the state space model which conditional on 

the estimated factors and the data given specifies the distribution of Λ and R. Here we 

can apply equation by equation OLS in order to obtain ˆ Λ and ê. This is feasible due to 

the fact that the errors are uncorrelated. According to the specification by BBE we also 

assume a proper (conjugate) but diffuse Inverse-Gamma(3,0.001) prior for Rii. Note that 

R is assumed to be diagonal. The posterior then has the following form: 

. 

Rii | XT , FT ∼ iG( ¯ Rii, T + 0.001) 

where ¯ Rii = 3 + ê ′ iêi + ˆ Λ ′ i[M −1 

0 + ( F ′ T (i) F (i) 

T ) −1 ] −1Λi ˆ and M −1 

0 

denoting the variance 

parameter in the prior on the coefficients of the i-th equation of Λi. The normalization 

discussed in section (4) in order to identify the factors and the loadings separately re- 

quires to set M0 = I. Conditional on the drawn value of Rii the prior on the factor 

loadings of the i-th equation is Λ prior 

i N(0, RiiM −1 

0 ). The regressors of the i-th equation 

are represented by ˜ F (i) 

T . The values of Λi are drawn from the posterior N( ¯ Λi, Rii ¯ M −1 

i ) 

30 The following part is very close to BBE [2005]


where ¯ Λi = ¯ M −1 

i ( F ′ T (i) F i T ) ˆ Λi and ¯ M −1 

i ( F ′ T (i) F i T ). 

The next Gibbs block requires to draw vec(Φ) and Q conditional on the most cur- 

rent draws of the factors, the R ′ iis and Λ ′ is and the data. As the FAVAR equation has a 

standard VAR form one can likewise estimate vec( ˆ Φ) and ˆ Q via equation by equation OLS. 

BBE impose a diffuse conjugate Normal-Wishart prior : 

Posterior Q is drawn from: 

vec(Φ) | Q ∼ N(0, Q ⊗ Ω0), Q ∼ iW (Q0, K + M + 2) 

iW ( ¯ Q, T + K + M + 2); 

In order to assume a prior in the vein of the Minnesota prior that expresses more 

distant lags to have less impact, hence be more likely zero, they follow Kadiyala and 

Karlsson [1997] 31 . First we draw Q from the Inverse-Wishart, iW ( ¯ Q, T + K + M + 2), 

where ¯ Q = Q0 + ˆ V ′ ˆ V + ˆ Φ ′ [Ω0 + ( ˜ F ′ T −1 ˜ FT −1) −1 ] −1 ˆ Φ and ˆ V the matrix of OLS residuals. 

The conditional on the drawn Q we draw vec(Φ) from the conditional normal according 

to 

vec(Φ) ∼ N(vec( ¯ Φ), Q ⊗ ¯ Ω) 

Here ¯ Φ = ¯ Ω( ˜ F ′ T −1 ˜ FT −1) ˆ Φ and ¯ Ω = (Ω −1 

0 + ( ˜ F ′ T −1 ˜ FT −1)) −1 . It is straight forward that 

we truncate the draws to only acceptable values for Φ less than one in absolute values in 

order to ensure stationarity. This block on Kalman filter and smoother and the block on 

drawing the parameter space are iterated until convergence is achieved. 

31 for a detailed description please refer to BBE [2005]


6 Structural FAVARs 

6.1 Identification of Shocks 

The issue of identifying structural shocks from the reduced form VAR innovations, and 

in particular identifying a shock to monetary policy has been dealt with in a huge body 

of literature. There have been introduced a lot of variations on how to achieve identifica- 

tion. The most prominent ones are explained below. As there are various approaches to 

deal with the same question it seems clear that there is also a controversial debate about 

which scheme to choose in order to reveal the true propagation mechanism attributable 

to a monetary policy shock. After considering the different approaches available, it seems 

to for me to be advisable to head the challenge of identification through applying the 

agnostic identification using sign restrictions 32 . Especially from the perspective of an 

economist it seems to me plausible to have an identification scheme that incorporates 

economic theory and through imposing the impulse responses to satisfy the conventional 

wisdom. Although this is a weaker identification scheme 33 . In this section the well known 

identification schemes are presented, afterwards I show briefly how they were extended to 

be applicable to large scale DFMs and to the FAVAR framework. And finally in the last 

part of this section I elaborate on the extension of the sign restriction of Uhlig (2005) to 

the FAVAR framework that incorporates the Gibbs sampling. 

In the common VAR framework one is required to deduce the structural shocks for 

the VAR innovations. In the DFM and FAVAR framework the task is actually the same 

with the main distinction that the structural shocks are not required to be deduced from 

the reduced form VAR innovation, but from the FAVAR innovation, including the factors 

that drive the dynamics of the informational variables or the observed data. 

32 Agnostic because no restriction on output. 

33 weaker in a sense that only restriction is on the mentioned variables according to the conventional 

wisdom. The aim is to restrict as little as necessary a priori.


6.2 Identification Schemes in SVARs 

In the well know SVAR framework, which is the mostly known and applied framework for 

the identification of the monetary policy shock. The widely applied ones are the recursive 

Cholesky identification that was advanced CEE [1999], the long-run identification that 

goes back to Blanchard and Quah [1989], and the combination of the previous two that 

sets zero restrictions on the coefficient matrix introduced by Leeper, Sims and Zha [1996]. 

Very good surveys about the Identification in SVARs can be found in CEE [1999] and 

Leeper, Sims and Zha [1996]. They document the progress that has been done over the 

time what versions there are around and what the state-of-the-art is. 

6.3 Identification in DFMs and FAVARs 

Equivalently to the SVAR case, the structural shocks in DFMs and FAVARs have to be 

derived from the reduced form innovation, with the distinction that here one refers not 

to the VAR but to the innovation in the FAVAR strictly speaking on the Yt or in case 

that Yt consists of more than one variable, the one that the researcher is interested in. 

There are already some identification schemes applied to the framework of DFMs and 

FAVARs around which shall be discussed here very briefly. There is a very good survey 

by Stock and Watson (2005), which introduces broadly the different approaches and how 

to set restrictions to identify factors and factor loadings. Furthermore they elaborate 

on the different identification schemes to figure out the structural shocks in DFMs and 

FAVARs. As they mostly deal with the nonparametric and the frequentists approach, the 

reader is kindly referred to this reference for further details. My thesis concentrates on 

the Bayesian approach combined also with a Bayesian identification, namely the agnostic 

identification. Therefore in the following the other approaches are mentioned briefly with 

the relevant references, and for the rest I focus on the model and methodology that I 

chose to analyze described above. 

The identification schemes that are around are surveyed by Stock and Watson (2005).


There is the BBE FAVAR identification scheme, also applied in this thesis, and a slightly 

modified version that is applied by Stock and Watson (2005). Furthermore the approach 

of Favero and Marcellino (2005) and Favero, Marcello and Neglia (2004) is introduced in 

the survey by Stock and Watson (2005). 

Here as in the VAR case the factor’s structural shocks are assumed to be linearly 

related to the reduced form factor innovations. 

vt = Qut 

where Q is a an (orthonormal) invertible [q × q] matrix. For identifying the trans- 

formation matrix Q there are two ways. The one is the full system identification by 

Blanchard and Watson (1986) who strive to identify all elements of Q. The other ap- 

proach is the single-equation identification where only one row of Q is required in order 

to identify the one respective shock. The latter one is the relevant one for us as, we are 

interested only in the identification of the shock attributable to monetary policy. There- 

fore we interested in a single row qs of the orthonormal matrix Q . 

Uhlig’s Sign Restriction 

As already introduced the sign restriction approach in its version advanced by Uhlig 

(2005) is the most reasonable approach in my view. Conventional wisdom says that after 

a monetary policy contraction the federal funds rate should increase, the prices should fall, 

and finally real output should fall. In other identification schemes that do not accomplish 

this wisdom, the researchers tend call this empirical observation a ”puzzle”. There are 

even some researchers that try to build a model producing such puzzles out of a model as 

has been done by CEE(2005). This seems unreasonable to me, because here one has to be 

very certain about the chosen identification scheme, and neglect any possible estimation 

mistakes not accomplished by the identification scheme chosen. Sims gives the advice 

to avoid unreasonable identification schemes. The approach by Uhlig seems reasonable,


especially regarding the application FAVARs of DFMs. Because in these frameworks one 

incorporates by far more relevant information than in the VAR methodology, and therefore 

one can set more restrictions in that sense that for instance not only CPI should fall after 

a monetary policy contraction but all the prices considered. Of course this becomes 

computationally by far more demanding than the other identification schemes and also 

than in the VAR case, because here we have now by far more information and respectively 

more ”reasonable” restrictions to set. It is straightforward that this method generates 

very few relevant candidate impulse responses due to the fact that the set of acceptable 

impulse responses is reduced due to the increased number of restrictions. It might and in 

practice is a difficult task to find an impulse vector that satisfies the ”stricter” restriction. 

This should not be considered as a disadvantage, it simply reflects that the economy is 

multi-causal where many things happen simultaneously and dynamically interact. From 

this fact one can deduce that the task to disentangle the effects of one single shock is very 

difficult, in particular has to satisfy a lot of ”economic conventional wisdom” in order to 

be identified as a mere effect of a single cause out of many. I would not go so far and 

state that with this method the shock is perfectly identified, but to me it seems that this 

approach is one of the more reasonable ones that are available especially with respect to 

the large set of information available, and furthermore provides at least the possibility to 

figure out very precise respones after a shock induced by the monetary authority. 

The task is first to identify the structural shock wt according the FAVAR innovation vt. 

It is actually the same concept as in the VAR case, except that the innovations one has 

are factor and not VAR variable innovations. The relation between the reduced form 

dynamic factor innovation vt and the structural factor innovation wt is given by 

wt = Avt 

The matrix A is an orthogonal invertible matrix of order [(K + M) × (K + M)]. We 

are only interested in identifying one single shock therefore it is sufficient to identify one 

single row as where s refers to the respective shock. This single-equation identification 

is the more common approach that most of the recent literature pursue. The alternative


would be to identify one row but the whole matrix A, which means to identify the full 

system. This approach goes back to Blanchard and Watson [1986]. 

The structural FAVAR can be arrived at when we premultiply the reduced form version 

with the rotation matrix A, which results in: 

AFt = AΦA −1 AFt−1 + Avt 

F ∗ 

t = Φ ∗ F ∗ 

t−1 + wt 

The crucial step is to represent the one-step ahead prediction error vt as a linear com- 

bination of orthogonalized structural shocks 34 . The fundamental innovations are mutually 

independent and normalized to have variance 1. Hence E[wtw ′ t] = I. The restriction on 

A emerges from its covariance structure of the factor reduced form factor innovation which 

results in: 

Σv = E[vtv ′ t] = AE[wtw ′ t]A ′ = AA ′ 

The reader can find an in-depth derivation and explanation of the sign restriction in 

Uhlig [2005]. Therefore we state very briefly the technical derivation very briefly and 

rather state its implementation for FAVARs. The steps are the following: 

First do a Cholesky decomposition of the variance-covariance matrix of the factor 

innovations ÃÃ′ = Σv, where 

Ã is the lower triangular Cholesky factor. Then a is an 

impulse vector if there exists an [K + M]-dimensional vector α of unit length so that 

a = Ãα 

Given the impulse vector a we can calculate the impulse responses of the factors to an 

innovation in for example the Federal funds rate. We collect the responses of the factors 

in order to estimate the impulse responses of the variables of interest. For exemplification 

34 See Uhlig [2005]


let us define the case of two factors with Ft = (F 1 

t , F 2 

t ) ′ the impulse responses of prices, 

that evolve for the horizon s accordingly: 

F0 = a1; F1 = ΦF0; . . . ; Fs = ΦFs−1 

Here Φ is the lag polynomial of the factor equation. As a final step we calculate the 

impulse responses of the informational variables given the factor responses. The case of 

a price variable would look look the following way: 

P0 = ΛP 1F 1 0 + ΛP 2F 2 0 

P1 = ΛP 1F 1 1 + ΛP 2F 2 1 

. . . 

Ps = ΛP 1F 1 s + ΛP 2F 2 s 

Our final task is to find that respective as of length unity that satisfies the sign 

restrictions explained above for the horizon previously specified. Only those impulse 

responses that satisfy the restrictions for the given horizon are stored otherwise discarded. 

For a detailed describtion of the methodology please refer to Uhlig [2005]. 

7 Empirical Results 

The dataset is an updated version from Stock and Watson [1998,1999], which consists of 

a balanced panel of 120 variables that are tabulated in Appendix A. The federal funds 

rate is interpreted as the monetary policy instrument and considered as the only variable 

that has pervasive effects on the economy. Alternative specifications are provided by BBE 

with respect to Yt. They further state that the ferdaral funds rate should not suffer from 

measurement error issues, which is straight-forward and therefore can be considered as 

having pervasive effects, and imply no idiosyncratic component. The monetary policy 

shock is standardized to correspond to a 25-basis-point innovation in the federal funds 

rate and the responses presented are reported in standard deviation units.


Before presenting the main results such as the plots for the impulse responses I provide 

the plots that show whether the chains of the single factors of the Gibbs iteration converge 

or not. There are several convergence criteria that can be applied to check the convergence 

of the algorithm for different starting values. To assure convergence of the Gibbs algorithm 

I also imposed on the one hand, the proper priors BBE imposed. They are reported in 

section (5.3) on Inference. The task of convergence diagnostics is an important one, but 

its formal implementation would have gone beyond the scope of this thesis. I decided to 

choose a less formal method where the first half of the median of the Gibbs sampling draws, 

of a single factor, are plotted against the second half after having discarded sufficient initial 

draws of Gibbs sampler in order to avoid the influence of the initial conditions. If the 

second half does not deviate too much from the first half, one might conclude as a first 

check that this single chain has converged. It is straight-forward that the convergence 

of the Gibbs chains should be checked for different starting values in order to assure the 

convergence of the respective Gibbs iteration with the respective specifications such as 

the number of factors, the number of draws, the number of initial draws to be discarded 

and so forth. Convergence has been tested for different starting values. The Figure 1-3 

provide provide the results for the single chains and the single factors.


Figure 1: Here the convergence is checked for the 2 factors specification. It is obvious 

that each of the factors have converged as the second half of the median of the factors 

generated does not deviate from the first half. This can be considered as an indication 

that the empirical distribution has approximated the marginal distribution of the factors 

sufficiently accurate.


Figure 2: Here the convergence is checked for the 5 factors specification. Each of the 

factors have converged as the second half of the median of the factors generated does 

not deviate from the first half. This can be considered as an indication that the empirical 

distribution has approximated the marginal distribution of the factors sufficiently accurate.


Figure 3: Here the convergence is checked for the 7 factors specification. Each of the 

factors have converged as the second half of the median of the factors generated does not 

deviate from the first half. This can be considered as an indication that the empirical dis- 

tribution has approximated the marginal distribution of the factors sufficiently accurate. 

On the whole I present results for factor specification of two, five and seven factors, 

in order to check on the one hand the impact of the number of the chosen factors for 

the reaction of the economy, visualized with the impulse response analysis. On the other 

hand I give an comment on the discussion how many factors are relevant and sufficient 

to capture the dynamics of the US economy. Gianone, Reichlin, and Sala [2004] find evi- 

dence that the US comovements and the dynamics of the US economy can be described 

by two factors whereas Stock and Watson [2005] provide evidence that more, that is to


say seven factors are required. My results are not supposed to give an conclusive answer 

to this discussion, rather they are supposed to give empirical evidence w.r.t. the impulse 

response analysis in favor of the one that provides more reasonable results. The specifi- 

cation used for the lags is 12, because the frequency of the data considered is monthly, 

however BBE report that even seven lags lead to similar results. 

Increasing the number of the factors had an impact in so far that the responses tended 

to be smoother and with a lower amplitude respectively. The qualitative results are fairly 

the same. Altogether the specification with seven factors produced the most reasonable 

results especially with respect to the prices. With this specification the reaction of the 

prices, in particular the commodity price index, had the most reasonable reaction. There 

is no prize puzzle prevalent. The restriction on the commodity price index appeared to 

be the one that had the least number of accepted draws with respect to the sign restriction. 

The most interesting results to me seem, however the importance of the identification 

scheme, as this approach combined with Gibbs sampling provides more accurate results 

than with the standard one applied by BBE. The critique by BBE that the parametric 

approach with a joint likelihood-based estimation might impose too much structure from 

which it suffers and therefore generates the inferior results compared to the nonpara- 

metric approach, cannot be approved. When applying the sign restriction approach in 

order to figure out the dynamic effects of a shock to monetary policy the results seem 

more reasonable and consistent with the conventional wisdom. In particular it is inter- 

esting that is delivers fairly tight error bands/confidence bands. The responses with the 

highest uncertainty are the output variables which is consistent with Uhlig [2005] and 

others who state that output does not have an unambiguous effect. It is fairly small.


Figure 4: Here the impulse responses for 20 selected variables are presented for seven fac- 

tors and the Block criteria 2 which has the following restrictions: The block criteria 1 sets 

the restriction consumer price index on nonborrowed reserves, M3 and the federal funds 

rate. The block criteria 2 sets the restrictions on consumer price index, nonborrowed 

reserves, M3, monetary base, and the federal funds rate. 

In order to get a better impression about the uncertainty of the impulse responses 

reported I provide mesh plots in figure 5-7. This is a possibility to visualize how certain 

one can be with median impulse responses. For that I collected all the accepted draws 

and sorted them in ascending order. Now one can directly see the uncertainty, not only 

for one specified percentile for error bands. The wider and flater the area in the center of 

the mesh the more certainty can be associated with mean response as most of the impulse 

resposes react alike akin to the same amplitude.


As the sign-restrictions approach delivers results with comparably tighter error bands 35 , 

than the BBE identification, one can conclude once again that the sign restriction is cru- 

cial for the structual anlysis. My results support the approach by Stock and Watson 

[2005] in so far that the results with seven factors provide more reasonable results with 

a relative higher degree of certainty compared to the Gibbs sampling results by BBE 

[2005]. The impulse responses for selected variables are reported in figure (4). Industrial 

Production declines after the shock for 1.5 years and then converges to the zero line but 

stayes under it in the version with seven factors. However the impulse responses on out- 

put stay ambigously regarding other output variables considered. Considering the plots 

in the appendix where impulse responses for further output variables are provided. Some 

of the variables react positively whereas others react negatively. 

Unfortunately the Matlab code appeared to be more challenging with respect to an 

efficient implementation, therefore it was accomplished fairly late so that the most fairly 

restricted Block criteria could not be provided where I impose all the relevant prices, 

all the money aggregates and the short term interest rates to satisfy the respective sign 

restrictions. This can be done in future work and can be approached by any user of 

the Matlab code attached. The enticing promise is that with some patience one can 

disentangle quite exactly the dynamic effects of a shock to monetary policy that is merely 

due it. This seems to me an important task and advantage of the combination of Bayesian 

FAVARs that are identified with an agnostic identification using the sign restriction. The 

fact that one can narrow down the space of the reactions might hold the enticing promise 

of providing an exact answer also with respect to the quantitative measure. 

35 The error bands report the 16% lower and 84% upper bound of the responses


Mesh of Impulse Responses of selected variables sorted for seven factors in ascending or- 

der. Please note that the number of the accepted draws are ten times higher than shown 

on the mesh. The picture looks the same with all draws included but takes much more 

time and memory for Matlab to load.


Mesh of Impulse Responses of selected variables, for five factors sorted in ascending or- 

der. Here one might conclude from the meshs how uncertain the responses are as they 

are given for each of the accepted draws. The wider the area in the center part the less 

volatile the impulse responses are over the draws. Such pictures serve the possibility to 

get an impression how certain the responses are with respect to the error bands.


Mesh of Impulse Responses of selected variables, for two factors sorted in ascending order. 

It is also worth to have look at the money aggregates that show a negative reaction for the 

whole 48 periods. Altogether the results improved where more factors were considered, 

in so far that the responses have more supportive error bands. Furthermore the results 

improved when I increased the number of variables on which sign-restrictions were im- 

posed. But one should note that it is computationally cumbersome to receive the results 

with more restrictions. The more variables are imposed with sign restrictions the less can 

be accepted that satisfy them. I conclude that this mirros the difficulty to identify the 

shock in a quantitativly precise manner. Therefore the critique on the Gibbs sampling 

approach by BBE [2005] seem not to be valid. The structure imposed is not appears not 

to be the restriction when combined with the alternative identification scheme.


Regarding the impulse responses one can see that most of them deliver tighter error 

bands than the results by BBE, which favors the alternative identification. In partic- 

ular regarding the commodity price index and the capacity utilization rate the results 

with Gibbs sampling combined with the agnostic identification seem not only more rea- 

sonable compared to the Gibbs sampling approach by BBE but also compared to their 

results estimated with the two-step PCA. There e.g. the capacity utilization rate and and 

the commodity price index increase increases directly after a shock. One of the striking 

results is the certainty with which the reactions of output (IP), CPI react and in par- 

ticular the moneytary aggregates react. Alltogether one can conclude that output still 

has an ambigous effect in that sense that all output variables considered in the dataset 

react identically and in the same direction. There are some that increase, but only very 

slightly. Therefore one can conclude that the results are not necessarily inconsistent with 

the monetary neutrality. 

All the responses have been calculated for the different horizons for sign restriction 

imposed. The results approved the ones by Uhlig [2005] in so far that the longer the 

restriction horizon the stronger is the reaction. The results reported have a restriction 

horizon of six month. As the results approve Uhlig and serve no new insights I decided, 

due to space limitations not provide the results in this thesis. 

There are more impulse responses provided in the Appendix(B) for different block criteria 

and number of factors specified. There results reporte Impulse responses for 42 variables 

out of the panel of 120. They are lables with the respective mnemonics reported in the 

data tabel in appendix(B). 

8 Discussion 

This section will provide a brief discussion of the results presented and a critical assess- 

ment. Furthermore some suggestions for future research are given. The impulse responses


presented serve the indication that, for the researcher interested in measuring the effects 

of a shock to monetary policy, it is crucial to apply identifying restrictions that are 

consistent with the conventional wisdom, such as the agnostic identification using sign 

restrictions. When comparing the results from the Gibbs sampling approach and com- 

pare them with the ones provided by BBE it is quite evident that the results seem more 

reasonable especially w.r.t. the quantitative measure and the certainty with with the 

results are reported. They do not show the great uncertainty as the results generated eith 

standard identification. The results are even more accurate some variables compared to 

the principal componant approach. Such variables are e.g. the commodity price index 

and the capacity utilization rate. However one should be cautious and still try out draws 

of at least 10000 in order to be conclusively certain with respect to the accuracy of the 

results. Although I have tried out many versions and several runs producing very similar 

results I think the results will still have to be confirmed with an iteration of say 10000 

draws to be completely sure. This was not feasible due to severe time constraints and the 

lack of time an appropriate (fast) computer that has no memory problems. It is advisable 

to use an unix based system. 

The most interesting suggestion to me seem to be even more strict with the restriction 

in that sense that one should set the restrictions for many prices, money aggregates the 

some short term interest rates. This could be accomplished only partly as the acceptance 

of ”reasonable” impulse responses decreases sharply. Hence one should be patient whiöe 

waiting for the results. Further extensions w.r.t. the model could be to model time vary- 

ing factor loadings and stochastic volatility e.g.in order to analyze the change of monetary 

policy in a ”data-rich environment” over time 36 As a next step one could also start to 

identify further shocks and measure the respective effects like the one to fiscal policy in 

a FAVAR framework, as it has been done by Mountford and Uhlig [2004] in the VAR 

framework. 

36 See Cogley and Sargent [2003].


9 Summary and Concluding Remarks 

In this thesis I combined the likelihood-based estimation of the FAVAR framework with 

the agnostic identification scheme to estimate the effects of a shock to monetary policy 

imposing sign restriction on the impulses responses on prices, nonborrowed reserves and 

the federal funds rate. Furthermore some combinations of restrictions on more than one 

monetary aggregate and price was tried out. I stay agnostic w.r.t. the output variables. 

The results seem produce results that appear to be more reasonable and more accurate 

than with standard identification schemes as provided by BBE. The accuracy increases 

with the increasing restrictions, however the number of accepted responses according to 

the agnostic identification decreases sharply. 

I suggest to be more strict with the restriction to set in so far that one imposes not only 

single variables but several variables such as prices to react according to the conventional 

wisdom. This should serve the possibility to narrow down the space of reasonable reactions 

that are merely according to a shock to monetary policy. Price and monetary aggregates 

show reasonable responses after a monetary policy shock. 

10 Matlab Implementation 

This part is supposed to explain the attached Matlab code. The code uses some codes 

written by Christopher Sims. I furthermore used some codes written by Piotr Eliasz and 

Jean Boivin. The part on the Gibbs sampling and Kalman filtering, in parts draws on the 

code written by Piotr Eliasz. Also some codes written by Bartosz Maćkowiak provided 

for the course ”Empirical Macroeconomics” have been also a great help. All the codes 

used by others are in an seperated folder in the attached cd-rom.


BAYESIAN_FAVAR.m 

The main Script. After setting the global GLOG_MODE (see description of function GLOG) 

the functions DO_INPUT, DO_CALCULATION and DO_RESULTS are called. The output of each 

function is given to the following functions as their input parameter. DO_IMPORTgenerates 

a structure called input (see description of the used data structure in the attached cd- 

rom), which contains information about user entries to choose a set of presets and spec- 

ification data. This structure is passed to the DO_CALCULATION function as its input. In 

DO_CALCULATION the structure results is created, which contains the results of the cal- 

culation process. DO_RESULTS uses this two data structures to present the results to the 

user. 

DO_INPUT.m 

This function returns the input data structure to the main function. To separate dif- 

ferent sources and groups of input information, the DO_INPUT function is separated into 

subfunctions, where each returns one part of the input structure.


DO_INPUT_VERSION.m


This function allows the user to load a set of parameters to replicate the results of the 

thesis or enter his own settings. All data is stored into the input.version structure. 

DO_INPUT_DATA.m 

Writes the Datasource into input.data. 

DO_INPUT_SPECIFICATIONS.m 

The aim of this function is to set all specification parameters for the calculation process 

including the Gibbs Sampler and Impulse Response Analysis. All parameters are stored 

in the input.specification structure. 

DO_INPUT_GENERATEXDATA.m 

This function returns the matrix input.xdata which is input.data excluding the data 

column of the perfectly observable that has pervasive effects on the economy. 

DO_INPUT_STARTINGVALUES.m 

Returns starting values for all variables included in the input.startingvalues structure. 

These are F, lam_f, lam_y, R, Phi_lags and Q. 

DO_CALCULATION.m 

In this function all calculation processes are started and their outputs are stored. These 

processes are initialized by calling the functions DO_CALCULATION_SETMODEL, DO_CALCULATION_CREATEST 

DO_CALCULATION_GIBBS_SAMPLING and DO_CALCULATION_IR where the last two are the 

main calculation processes. To save memory the data structure calculation is declared 

as global in all calculation functions. In this way all functions can refer to it as an input 

and output parameter without moving this big sized structure.


DO_CALCULATION_SETMODEL.m 

Initializes calculation.stateSpaceStructure. 

DO_CALCULATION_CREATESTRUCTURE.m 

Initializes 

calculation.Phi_bar_collect


calculation.QQ_bar_collect 

calculation.F_bar_collect 

calculation.Lam_collect. 

DO_CALCULATION_GIBBS_SAMPLING.m 

This function does the Gibbs Sampling by calling the functions 

DO_CALCULATION_GIBBS_SAMPLING_BK_FILTER, 

DO_CALCULATION_GIBBS_SAMPLING_BK_SMOOTHER, 

DO_CALCULATION_GIBBS_SAMPLING_PARAM_PREC_OBS and 

DO_CALCULATION_GIBBS_SAMPLING_PARAM_PREC_FAC 

for each Gibbs iteration. After each Iteration the results are stored in the global calculation 

data structure after ignoring the first input.version.burn_in draws.


DO_CALCULATION_GIBBS_SAMPLING_BK_FILTER.m 

Bayesian Kalman Filter


DO_CALCULATION_GIBBS_SAMPLING_BK_SMOOTHER.m 

Bayesian Kalman Smoother 

DO_CALCULATION_GIBBS_SAMPLING_PARAM_PREC_OBS.m 

Inference on Observation Equation 

DO_CALCULATION_GIBBS_SAMPLING_PARAM_PREC_FAC.m 

Inference on State Equation 

DO_CALCULATION_IRA.m 

This function starts the DO_CALCULATION_IRA_UHLIG or the DO_CALCULATION_IRA_BBE 

function depending on the value of input.version.ira_mode which contains information 

about the selected Impulse Response Mode to run. 

DO_CALCULATION_IRA_UHLIG.m 

Impulse Response Analysis with Uhlig (2005) Sign Restrictions. Returns finalresponse 

which is a vector with the length of the total number of block criteria. Responses are 

checked to satisfy each block criteria, which are set in input.specification.IRA.BC. 

Accepted Responses are added to finalresponses().response where is 

the block criteria satisfied by the responses.


To keep the memory usage of the finalResponses().response in efficient limits, 

I first initialize it with the initial size of 3% of the the [draws × α].


If the candidate satisfies the block of sign restrictions in the current block criteria, it is 

added to finalResponses().response. If the position the candidate is added to 

is the last element of finalResponses().response the size of .response matrix 

is increased by fr_add_length which has a default value of 1% of the the [draws × α].


Finally the size of finalResponses().response is reduced to set free the unused 

occupied memory. Also the size of finalResponses().response represents the 

number of accepted candidates.


DO_CALCULATION_IRA_UHLIG_CHECK_SIGNRESTRICTION.m 

This function checks if a given response satisfies the block criteria . Returns 1 if 

response is accepted, otherwise 0. 

GLOG.m 

This Function is used to log an output string depending on its log level glog_type. The 

global variable GLOG_MODE signifies the global minimum level for outputs and is set di- 

rectly in BAYESIAN_FAVAR. If the glog_type of an output string is less than GLOG_MODE 

the output is ignored. Otherwise GLOG displays the output string.


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in Practice”; Chapman and Hall, London


Appendix A: Data Appendix 1: Data Description 

The data is the one that Bernanke, Boivin and Eliasz [2005] use in their paper. Format 

is as in Stock and Watson’s papers: series number; series mnemonic; data span; trans- 

formation code and series description as appears in the database. The transformation 

codes are: 1 – no transformation; 2 – first difference; 4 – logarithm; 5 – first difference 

of logarithm. An asterisk, ‘*’, next to the mnemonic denotes a variable assumed to be 

“slow-moving” in the estimation. 

Real output and income 

1. IPP* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: PRODUCTS, TOTAL (1992=100,SA) 

2. IPF* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: FINAL PRODUCTS (1992=100,SA) 

3. IPC* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: CONSUMER GOODS (1992=100,SA) 

4. IPCD* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: DURABLE CONS. GOODS (1992=100,SA) 

5. IPCN* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: NONDURABLE CONS. GOODS (1992=100,SA) 

6. IPE* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: BUSINESS EQUIPMENT (1992=100,SA) 

7. IPI* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: INTERMEDIATE PRODUCTS (1992=100,SA) 

8. IPM* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: MATERIALS (1992=100,SA) 

9. IPMD* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: DURABLE GOODS MATERIALS (1992=100,SA) 

10. IPMND* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: NONDUR. GOODS MATERIALS (1992=100,SA) 

11. IPMFG* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: MANUFACTURING (1992=100,SA) 

12. IPD* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: DURABLE MANUFACTURING (1992=100,SA) 

13. IPN* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: NONDUR. MANUFACTURING (1992=100,SA) 

14. IPMIN* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: MINING (1992=100,SA) 

15. IPUT* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: UTILITIES (1992-=100,SA) 

16. IP* 1959:01-2001:08 5 INDUSTRIAL PRODUCTION: TOTAL INDEX (1992=100,SA) 

17. IPXMCA* 1959:01-2001:08 1 CAPACITY UTIL RATE: MANUFAC.,TOTAL( 

18. PMI* 1959:01-2001:08 1 PURCHASING MANAGERS’ INDEX (SA)


19. PMP* 1959:01-2001:08 1 NAPM PRODUCTION INDEX (PERCENT) 

20. GMPYQ* 1959:01-2001:08 5 PERSONAL INCOME (CHAINED) (SERIES #52) (BIL 92$,SAAR) 

21. GMYXPQ* 1959:01-2001:08 5 PERSONAL INC. LESS TRANS. PAYMENTS (CHAINED) (#51) (BIL 92$,SAAR) 

Employment and hours 

22. LHEL* 1959:01-2001:08 5 INDEX OF HELP-WANTED ADVERTISING IN NEWSPAPERS (1967=100;SA) 

23. LHELX* 1959:01-2001:08 4 EMPLOYMENT: RATIO; HELP-WANTED ADS:NO. UNEMPLOYED CLF 

24. LHEM* 1959:01-2001:08 5 CIVILIAN LABOR FORCE: EMPLOYED, TOTAL (THOUS.,SA) 

25. LHNAG* 1959:01-2001:08 5 CIVILIAN LABOR FORCE: EMPLOYED, NONAG.INDUSTRIES (THOUS.,SA) 

26. LHUR* 1959:01-2001:08 1 UNEMPLOYMENT RATE: ALL WORKERS, 16 YEARS & OVER ( 

27. LHU680* 1959:01-2001:08 1 UNEMPLOY.BY DURATION: AVERAGE(MEAN)DURATION IN WEEKS (SA) 

28. LHU5* 1959:01-2001:08 1 UNEMPLOY.BY DURATION: PERS UNEMPL.LESS THAN 5 WKS (THOUS.,SA) 

29. LHU14* 1959:01-2001:08 1 UNEMPLOY.BY DURATION: PERS UNEMPL.5 TO 14 WKS (THOUS.,SA) 

30. LHU15* 1959:01-2001:08 1 UNEMPLOY.BY DURATION: PERS UNEMPL.15 WKS + (THOUS.,SA) 

31. LHU26* 1959:01-2001:08 1 UNEMPLOY.BY DURATION: PERS UNEMPL.15 TO 26 WKS (THOUS.,SA) 

32. LPNAG* 1959:01-2001:08 5 EMPLOYEES ON NONAG. PAYROLLS: TOTAL (THOUS.,SA) 

33. LP* 1959:01-2001:08 5 EMPLOYEES ON NONAG. PAYROLLS: TOTAL, PRIVATE (THOUS,SA) 

34. LPGD* 1959:01-2001:08 5 EMPLOYEES ON NONAG. PAYROLLS: GOODS-PRODUCING (THOUS.,SA) 

35. LPMI* 1959:01-2001:08 5 EMPLOYEES ON NONAG. PAYROLLS: MINING (THOUS.,SA) 

36. LPCC* 1959:01-2001:08 5 EMPLOYEES ON NONAG. PAYROLLS: CONTRACT CONSTRUC. (THOUS.,SA) 

37. LPEM* 1959:01-2001:08 5 EMPLOYEES ON NONAG. PAYROLLS: MANUFACTURING (THOUS.,SA) 

38. LPED* 1959:01-2001:08 5 EMPLOYEES ON NONAG. PAYROLLS: DURABLE GOODS (THOUS.,SA) 

39. LPEN* 1959:01-2001:08 5 EMPLOYEES ON NONAG. PAYROLLS: NONDURABLE GOODS (THOUS.,SA) 

40. LPSP* 1959:01-2001:08 5 EMPLOYEES ON NONAG. PAYROLLS: SERVICE-PRODUCING (THOUS.,SA) 

41. LPTU* 1959:01-2001:08 5 EMPLOYEES ON NONAG. PAYROLLS: TRANS. & PUBLIC UTIL. (THOUS.,SA) 

42. LPT* 1959:01-2001:08 5 EMPLOYEES ON NONAG. PAYROLLS: WHOLESALE & RETAIL (THOUS.,SA)


43. LPFR* 1959:01-2001:08 5 EMPLOYEES ON NONAG. PAYROLLS: FINANCE,INS.&REAL EST (THOUS.,SA 

44. LPS* 1959:01-2001:08 5 EMPLOYEES ON NONAG. PAYROLLS: SERVICES (THOUS.,SA) 

45. LPGOV* 1959:01-2001:08 5 EMPLOYEES ON NONAG. PAYROLLS: GOVERNMENT (THOUS.,SA) 

46. LPHRM* 1959:01-2001:08 1 AVG. WEEKLY HRS. OF PRODUCTION WKRS.: MANUFACTURING (SA) 

47. LPMOSA* 1959:01-2001:08 1 AVG. WEEKLY HRS. OF PROD. WKRS.: MFG.,OVERTIME HRS. (SA) 

48. PMEMP* 1959:01-2001:08 1 NAPM EMPLOYMENT INDEX (PERCENT) 

Consumption 

49. GMCQ* 1959:01-2001:08 5 PERSONAL CONSUMPTION EXPEND (CHAINED) - TOTAL (BIL 92$,SAAR) 

50. GMCDQ* 1959:01-2001:08 5 PERSONAL CONSUMPTION EXPEND (CHAINED) – TOT. DUR. (BIL 96$,SAAR) 

51. GMCNQ* 1959:01-2001:08 5 PERSONAL CONSUMPTION EXPEND (CHAINED) – NONDUR. (BIL 92$,SAAR) 

52. GMCSQ* 1959:01-2001:08 5 PERSONAL CONSUMPTION EXPEND (CHAINED) - SERVICES (BIL 92$,SAAR) 

53. GMCANQ* 1959:01-2001:08 5 PERSONAL CONS EXPEND (CHAINED) - NEW CARS (BIL 96$,SAAR) Housing starts 

and sales 

54. HSFR 1959:01-2001:08 4 HOUSING STARTS:NONFARM(1947-58);TOT.(1959-)(THOUS.,SA 

55. HSNE 1959:01-2001:08 4 HOUSING STARTS:NORTHEAST (THOUS.U.)S.A. 

56. HSMW 1959:01-2001:08 4 HOUSING STARTS:MIDWEST(THOUS.U.)S.A. 

57. HSSOU 1959:01-2001:08 4 HOUSING STARTS:SOUTH (THOUS.U.)S.A. 

58. HSWST 1959:01-2001:08 4 HOUSING STARTS:WEST (THOUS.U.)S.A. 

59. HSBR 1959:01-2001:08 4 HOUSING AUTHORIZED: TOTAL NEW PRIV HOUSING (THOUS.,SAAR) 

60. HMOB 1959:01-2001:08 4 MOBILE HOMES: MANUFACTURERS’ SHIPMENTS (THOUS.OF UNITS,SAAR) Real in- 

ventories, orders and unfilled orders 

61. PMNV 1959:01-2001:08 1 NAPM INVENTORIES INDEX (PERCENT) 

62. PMNO 1959:01-2001:08 1 NAPM NEW ORDERS INDEX (PERCENT) 

63. PMDEL 1959:01-2001:08 1 NAPM VENDOR DELIVERIES INDEX (PERCENT) 

64. MOCMQ 1959:01-2001:08 5 NEW ORDERS (NET) - CONSUMER GOODS & MATERIALS, 1992 $ (BCI) 

65. MSONDQ 1959:01-2001:08 5 NEW ORDERS, NONDEFENSE CAPITAL GOODS, IN 1992 DOLLARS (BCI) Stock prices


66. FSNCOM 1959:01-2001:08 5 NYSE COMMON STOCK PRICE INDEX: COMPOSITE (12/31/65=50) 

67. FSPCOM 1959:01-2001:08 5 S&P’S COMMON STOCK PRICE INDEX: COMPOSITE (1941-43=10) 

68. FSPIN 1959:01-2001:08 5 S&P’S COMMON STOCK PRICE INDEX: INDUSTRIALS (1941-43=10) 

69. FSPCAP 1959:01-2001:08 5 S&P’S COMMON STOCK PRICE INDEX: CAPITAL GOODS (1941-43=10) 

70. FSPUT 1959:01-2001:08 5 S&P’S COMMON STOCK PRICE INDEX: UTILITIES (1941-43=10) 

71. FSDXP 1959:01-2001:08 1 S&P’S COMPOSITE COMMON STOCK: DIVIDEND YIELD ( 

72. FSPXE 1959:01-2001:08 1 S&P’S COMPOSITE COMMON STOCK: PRICE-EARNINGS RATIO ( 

73. EXRSW 1959:01-2001:08 5 FOREIGN EXCHANGE RATE: SWITZERLAND (SWISS FRANC PER U.S.$) 

74. EXRJAN 1959:01-2001:08 5 FOREIGN EXCHANGE RATE: JAPAN (YEN PER U.S.$) 

75. EXRUK 1959:01-2001:08 5 FOREIGN EXCHANGE RATE: UNITED KINGDOM (CENTS PER POUND) 

76. EXRCAN 1959:01-2001:08 5 FOREIGN EXCHANGE RATE: CANADA (CANADIAN $ PER U.S.$) Interest rates 

77. FYFF 1959:01-2001:08 1 INTEREST RATE: FEDERAL FUNDS (EFFECTIVE) ( 

78. FYGM3 1959:01-2001:08 1 INTEREST RATE: U.S.TREASURY BILLS,SEC MKT,3-MO.( 

79. FYGM6 1959:01-2001:08 1 INTEREST RATE: U.S.TREASURY BILLS,SEC MKT,6-MO.( 

80. FYGT1 1959:01-2001:08 1 INTEREST RATE: U.S.TREASURY CONST MATUR. ,1-YR.( 

81. FYGT5 1959:01-2001:08 1 INTEREST RATE: U.S.TREASURY CONST MATUR., 5-YR.( 

82. FYGT10 1959:01-2001:08 1 INTEREST RATE: U.S.TREASURY CONST MATUR.,10-YR.( 

83. FYAAAC 1959:01-2001:08 1 BOND YIELD: MOODY’S AAA CORPORATE ( 

84. FYBAAC 1959:01-2001:08 1 BOND YIELD: MOODY’S BAA CORPORATE ( 

85. SFYGM3 1959:01-2001:08 1 Spread FYGM3 - FYFF 

86. SFYGM6 1959:01-2001:08 1 Spread FYGM6 - FYFF 

87. SFYGT1 1959:01-2001:08 1 Spread FYGT1 - FYFF 



90. SFYAAAC 1959:01-2001:08 1 Spread FYAAAC - FYFF 

91. SFYBAAC 1959:01-2001:08 1 Spread FYBAAC - FYFF 

Money and credit quantity aggregates


92. FM1 1959:01-2001:08 5 MONEY STOCK: M1 (BIL$,SA) 

93. FM2 1959:01-2001:08 5 MONEY STOCK: M2 (BIL$, SA) 

94. FM3 1959:01-2001:08 5 MONEY STOCK: M3 (BIL$,SA) 

95. FM2DQ 1959:01-2001:08 5 MONEY SUPPLY - M2 IN 1992 DOLLARS (BCI) 

96. FMFBA 1959:01-2001:08 5 MONETARY BASE, ADJ FOR RESERVE REQUIREMENT CHANGES(MIL$,SA) 

97. FMRRA 1959:01-2001:08 5 DEPOSITORY INST RESERVES:TOTAL,ADJ FOR RES. REQ CHGS(MIL$,SA) 

98. FMRNBA 1959:01-2001:08 5 DEPOSITORY INST RESERVES:NONBOR. ,ADJ RES REQ CHGS(MIL$,SA) 

99. FCLNQ 1959:01-2001:08 5 COMMERCIAL & INDUST. LOANS OUSTANDING IN 1992 DOLLARS (BCI) 

100. FCLBMC 1959:01-2001:08 1 WKLY RP LG COM. BANKS: NET CHANGE COM & IND. LOANS(BIL$,SAAR) 

101. CCINRV 1959:01-2001:08 5 CONSUMER CREDIT OUTSTANDING NONREVOLVING G19 

Price indexes 

102. PMCP 1959:01-2001:08 1 NAPM COMMODITY PRICES INDEX (PERCENT) 

103. PWFSA* 1959:01-2001:08 5 PRODUCER PRICE INDEX: FINISHED GOODS (82=100,SA) 

104. PWFCSA* 1959:01-2001:08 5 PRODUCER PRICE INDEX:FINISHED CONSUMER GOODS (82=100,SA) 

105. PWIMSA* 1959:01-2001:08 5 PRODUCER PRICE INDEX:INTERMED MAT.SUP & COMPONENTS(82=100,SA) 

106. PWCMSA* 1959:01-2001:08 5 PRODUCER PRICE INDEX:CRUDE MATERIALS (82=100,SA) 

107. PSM99Q* 1959:01-2001:08 5 INDEX OF SENSITIVE MATERIALS PRICES (1990=100)(BCI-99A) 

108. PUNEW* 1959:01-2001:08 5 CPI-U: ALL ITEMS (82-84=100,SA) 

109. PU83* 1959:01-2001:08 5 CPI-U: APPAREL & UPKEEP (82-84=100,SA) 

110. PU84* 1959:01-2001:08 5 CPI-U: TRANSPORTATION (82-84=100,SA) 

111. PU85* 1959:01-2001:08 5 CPI-U: MEDICAL CARE (82-84=100,SA) 

112. PUC* 1959:01-2001:08 5 CPI-U: COMMODITIES (82-84=100,SA) 

113. PUCD* 1959:01-2001:08 5 CPI-U: DURABLES (82-84=100,SA) 

114. PUS* 1959:01-2001:08 5 CPI-U: SERVICES (82-84=100,SA) 

115. PUXF* 1959:01-2001:08 5 CPI-U: ALL ITEMS LESS FOOD (82-84=100,SA)


116. PUXHS* 1959:01-2001:08 5 CPI-U: ALL ITEMS LESS SHELTER (82-84=100,SA) 

117. PUXM* 1959:01-2001:08 5 CPI-U: ALL ITEMS LESS MIDICAL CARE (82-84=100,SA) 

Average hourly earnings 

118. LEHCC* 1959:01-2001:08 5 AVG HR EARNINGS OF CONSTR WKRS: CONSTRUCTION ($,SA) 

119. LEHM* 1959:01-2001:08 5 AVG HR EARNINGS OF PROD WKRS: MANUFACTURING ($,SA) 

Miscellaneous 

120. HHSNTN 1959:01-2001:08 1 U. OF MICH. INDEX OF CONSUMER EXPECTATIONS(BCD-83) 

Appendix B: Figures The figure that are presented in the following are either labeled 

with variable names used in the main text or with the mnemonic that are reported in the 

data table. The block criteria 1 sets the restriction consumer price index on nonborrowed 

reserves, M3 and the federal funds rate. The block criteria 2 sets the restrictions on 

consumer price index, nonborrowed reserves, M3, monetary base, and the federal funds 

rate.


Impulse Responses for 20 Variables with 5 factors and the Block Criteria 1






Impulse Responses of the first 20 variables of the selected Variables out of 43 with 7 fac- 

tors and the Block Criteria 1


Impulse Responses of the 21st to 40th variables of the selected Variables out of 43 with 7 

factors and the Block Criteria 1


Impulse Responses of the last 3 variables of the selected Variables out of 43 with 7 factors 

and the Block Criteria 1


Impulse Responses of the 21st to the 40th variables of the selected Variables out of 43 

with 5 factors and the Block Criteria 1


Impulse Responses of the last 3 variables of the selected Variables out of 43 with 5 factors 

and the Block Criteria 1


Impulse Responses for the Variables 41-52 with 5 factors and the Block Criteria 1






Impulse Responses for 2 factors and the Block Criteria 1 

Appendix C: Matlab Code 

%%%%%%**********************************************************%%%%% 

%%%%%% Bayesian FAVAR Code August 26th %%%%% 

%%%%%%**********************************************************%%%%% 

%%%%%% The main Script. After setting the global GLOG_MODE 

%%%%%% (see description of function GLOG) the functions 

%%%%%% DO_INPUT, DO_CALCULATION AND DO_RESULTS are called. 

%%%%%% The output of each function is given to the following 

%%%%%% functions as their input parameter. DO_IMPORT generates 

%%%%%% a structure called "input" (see description of the 

%%%%%% datastructure), which contains information about user 

%%%%%% entries to choose a set of presets and specification data. 

%%%%%% This structure is passed to the DO_CALCULATION function 

%%%%%% as its input. In DO_CALCULATION the structure "results" 

%%%%%% is created, which contains the results of the calculation 

%%%%%% process. DO_RESULTS uses this two data structures to 

%%%%%% present the results to the user. 

%profile on -detail builtin 

clear all;


clc; 

% Declarations 

% Declare global 

% 1 : Internal Var Monitoring 

% 2 : Information 

% 3 : Warnings 

% 4 : Errors 

global GLOG_MODE; 

GLOG_MODE = 2; 

GLOG (’Begin of Bayesian FAVAR Estimation’,2); 

% Main 

[input] = DO_INPUT; % see Sequence Diagram - Block A 

[results] = DO_CALCULATION (input); % see Sequence Diagram - Block B 

DO_RESULTS (input,results); % see Sequence Diagram - Block C 

GLOG (’End of Bayesian FAVAR Estimation’,2); 

%profile viewer 

% 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO _INPUT_VERSION %%%%% 

%%%%%% see Sequence Diagram Block A.1 %%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% version.versionId %%%%% 

%%%%%% version.nGibbsit %%%%% 

%%%%%% version.burn_in %%%%% 

%%%%%% version.ira_mode %%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% This function allows the user to load a set of 

%%%%%% parameters to replicate the results of the thesis 

%%%%%% or enter his own settings. All data is stored into 

%%%%%% the input.version structure. 

function [version] = DO_INPUT_VERSION () 

%function [version] = DO_INPUT_VERSION () 

disp(’%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ’) 

disp(’%% You are asked to type in the number of iteratrions or to choose a %% ’) 

disp(’%% one of the following specifications for replicating the results in %% ’) 

disp(’%% my thesis which are the following: %% ’) 

disp(’%% %% ’) 

disp(’%% Version 1: Type in 1 %% ’) 




disp(’%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ’) 

disp(’ ’) 

%disp(’%% Hit any key when ready... %% ’) 

%disp(’ ’) 

%pause;


%% --> EXTEND FOR MORE OPTIONS HERE ! 

version.versionId = input(’%% Please choose one of the above specification = ’); 

disp(’ ’) 

switch version.versionId % switch to choose number of iteration 

case 1 % Version 1 

disp(’%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%’) 

disp(’%% YOU HAVE CHOSEN VERSION 1 %%’) 

disp(’%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%’) 

disp(’ ’) 

version.nGibbsit=10000; 

version.burn_in=4000; 

version.ira_mode = 1; %ira_mode: 1 -> Uhlig (2005), 2 -> BBE (2005) 


disp(’%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%’) 


disp(’%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%’) 

disp(’ ’) 





disp(’%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%’) 


disp(’%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%’) 

disp(’ ’) 




case 4 % Interactively to be chosen by user 

version.nGibbsit=input(’Please type in the number of iterations = ’);version.nGibbsit 

version.burn_in=input(’Please type in the number of iterations to be discarded = ’) 


%version.ira_mode=input(’Please chosse mode of IRA (1: Uhlig (2005) or 2: BBE (2005)) ’) 

otherwise 

disp(’Please check whether you have chosen ’) 

disp(’a correct version or a correct ’) 

disp(’(natural) number. Please try again ’) 

end % end of switch for Iteration number 

GLOG (version.nGibbsit,1); 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO _INPUT %%%%% 

%%%%%% see Sequence Diagram Block A %%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% This function returns the input data structure to the 

%%%%%% main function. To seperate different sources and goups 

%%%%%% of input information, the DO_INPUT-function is seperated 

%%%%%% into five subfunctions, which each return one part of 

%%%%%% the input structure. 

function [input] = DO_INPUT () 

%function [input] = DO_INPUT () 

[input.version] = DO_INPUT_VERSION; 

% see Sequence Diagram - Block A.1 

[input.data] = DO_INPUT_DATA;



[input.specification] = DO_INPUT_SPECIFICATIONS (input); 


[input.xdata] = DO_INPUT_GENERATEXDATA (input); 


[input.startingvalues] = DO_INPUT_STARTINGVALUES (input); 


%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO _INPUT_DATA %%%%% 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% Loads the Datasource into input.data. 

function [data] = DO_INPUT_DATA () 

%function [data] = DO_INPUT_DATA () 

%*************************% 

% Load data directly % 

%*************************% 

load Datasource.txt; 

data = Datasource; 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO _INPUT_GENERATEXDATA %%%%% 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% This Function returns the matrix input.xdata which 

%%%%%% is input.data excluding the datacolumn of the perfectly 

%%%%%% observable thathas pervasive effects on the economy. 

%%%%%% 

%%%%%% xdata is data - col (varY) 

%%%%%% 

function [xdata] = DO_INPUT_GENERATEXDATA (input) 

%function [xdata] = DO_INPUT_GENERATEXDATA () 

xdata = input.data; 

xdata(:,input.specification.varY ) = []; 

xdata = xdata - repmat(mean(xdata),input.specification.dim.T,1); 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO _INPUT_SPECIFICATION %%%%% 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% The aim of this function is to set all specification 

%%%%%% parameters for the calculation prozess including the 

%%%%%% Gibbs Sampler and Impulse Response Analysis. 

%%%%%% All parameters are stored into the input.specification 

%%%%%% structure. 

%%%%%% 

%%%%%% specification 

%%%%%% |---------- time 

%%%%%% |---------- varY 

%%%%%% |---------- y


%%%%%% |---------- dim 

%%%%%% | |--- T 

%%%%%% | |--- M 

%%%%%% | |--- N 

%%%%%% | 

%%%%%% |---------- model 

%%%%%% | |--- draws 

%%%%%% | |--- K 

%%%%%% | |--- d 

%%%%%% | 

%%%%%% |-----------IRA 

%%%%%% | |--- nsteps 

%%%%%% | |--- tsteps 

%%%%%% | |--- zeroline 

%%%%%% | |--- alpha_draws 

%%%%%% | |--- var_index_sr 

%%%%%% | |--- sr_horizon 

%%%%%% | | 

%%%%%% | |--- BC (indexed) 

%%%%%% | |--- priceIndex 

%%%%%% | |--- moneyIndex 

%%%%%% | |--- interestIndex 

%%%%%% | 

%%%%%% |---------- VARNAMES_BBE 

%%%%%% |---------- ALL_VARNAMES 

function [specification] = DO_INPUT_SPECIFICATIONS (input) 

%function [specification] = DO_INPUT_SPECIFICATIONS () 

specification.time = 1959.1667:1/12:2001.6667; % 

specification.varY = [77]; % variable to be chosen for Y(t)(most of the cases FFR) 

y = input.data(:,specification.varY); % observable (VAR) variables 

[T,M] = size(y); 

dim.T = T; 

dim.M = M; 

[N] = size(input.data,2) - size(specification.varY,2) 

dim.N = N; 

y = y - repmat(mean(y),T,1); 

specification.y = y; 

% Dim short cuts 

specification.dim = dim; 

model.draws = input.version.nGibbsit - input.version.burn_in; 

% final number of iterantions that counts 

model.K = 7; % number of factors ; [GRS : 2 ; Stock&Watson : 7] 

model.d = 12; % finite order of conformable lag polynomial 

specification.model = model;


%**********************************% 

% IMPULSE RESPONSE SPECIFICATION % 

%**********************************% 

IRA.nsteps = 48; 

IRA.tstep = 1:IRA.nsteps; 

IRA.zeroline = zeros(IRA.nsteps,1); 

IRA.alpha_draws = 300; 

IRA.var_index_sr = [ 77 1;16 5;108 5;78 1;81 1;96 5;93 5;74 5;102 1;17 1;49 5; ... 

50 5;51 5;26 1;48 1;118 5;54 4;62 1;71 1;120 1]; 

%IRA.var_index_sr = [ 16 5;17 1;26 1;48 1;49 5;50 5;51 5;54 4;... 

62 1; 71 1;74 5;77 1;78 1;79 1;81 1;92 5;93 5; ... 

% 94 5;95 5;96 5;97 5;98 5;99 5;100 1;101 5;... 

102 1;103 5;104 5;105 5;106 5;107 5;108 5;109 5;110 5; ... 

% 111 5;112 5;113 5;114 5;115 5;116 5;117 5;118 5;120 1]; 

%%% Extended variables to be considered 

%IRA.var_index_sr = [1 5;2 5;3 5;4 5;5 5;11 5;16 5;17 1;26 1;... 

48 1;49 5;50 5;51 5;54 4;62 1; 71 1;73 5;74 5;... 

75 5;76 5; ... 

% 77 1;78 1;79 1;81 1;92 5;93 5;94 5;95 5;96 5;... 

97 5;98 5;99 5;100 1;101 5;102 1;103 5;104 5;... 

105 5;106 5;107 5; ... 

% 108 5;109 5;110 5;111 5;112 5;113 5;114 5;115 5;116 5;... 

117 5;118 5;120 1]; 

IRA.sr_horizon = 6; 

%%%%% set block criteria 

%Block Criteria 1 

%BC(1).priceIndex = [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41]; 

%BC(1).moneyIndex = [16,17,18,19,20,21,22,23,24,25]; 

%BC(1).interestIndex = [12,13,14]; 

%Block Criteria 2 

%BC(1).priceIndex = [26]; 

%BC(1).moneyIndex = [16]; 


%Block Criteria TEST SR 



%BC(1).interestIndex = [12]; 

%Block Criteria SR - 1 




%Block Criteria SR - 2 

%BC(2).priceIndex = [26,28]; 

%BC(2).moneyIndex = [16,17]; 


%Block Criteria SR - 3


%BC(3).priceIndex = [26,28,29,39]; 

%BC(3).moneyIndex = [16,17,18,19,20]; 


%Block Criteria SR - 4 - Relaxed Max 

%BC(4).priceIndex = [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41]; 

%BC(4).moneyIndex = [16,17,18,19,20,21,22,23,24,25]; 


%Block Criteria SR - 5 - Max 

%BC(5).priceIndex = [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41]; 

%BC(5).moneyIndex = [16,17,18,19,20,21,22,23,24,25]; 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%Block Criteria BBE GOOD1 

BC(1).priceIndex = [3]; 

BC(1).moneyIndex = [6]; 

BC(1).interestIndex = [1]; 


BC(2).priceIndex = [3]; 

BC(2).moneyIndex = [6,7]; 



BC(3).priceIndex = [3,9]; 

BC(3).moneyIndex = [6,7]; 


%%%%%%%%%%%%%%%%%Extended variables to be considered %%%%%%%%%%%%%%%%%%% 


%BC(1).moneyIndex = [31]; %M3/ NBR 


%BC(2).priceIndex = [35,41]; 



%************************************* 

IRA.BC = BC; 

specification.IRA = IRA; 

specification.VARNAMES_BBE = {’FFR’,’IP’,’CPI’,’3m TREASURY BILLS’,’5y TREASURY BONDS’,’MONETARY BASE’,’M2’,... 

’EXCHANGE RATE YEN’,’COMMODITY PRICE INDEX’,’CAPACITY UTIL RATE’,... 

’PERSONAL CONSUMPTION’,’DURABLE CONS’,’NONDURABLE CONS’,’UNEMPLOYMENT’,’EMPLOYMENT’,’AVG HOURLY EARNINGS’,... 

’HOUSING STARTS’,’NEW ORDERS’,’DIVIDENDS’,’CONSUMER EXPECTATIONS’}; 

specification.ALL_VARNAMES = {’IPP’,’IPF’,’IPC’,’IPCD’,’IPCN’,’IPE’,’IPI’,’IPM’,’IPMD’,’IPMND’,’IPMFG’,’IPD’,’IPN’,’IPMIN’, ... 

’IPUT’,’IP’,’IPXMCA’,’PMI’,’PMP’,’GMPYQ’,’GMYXPQ’,’LHEL’,’LHELX’,’LHEM’,’LHNAG’,’LHUR’,’LHU680’, ... 

’LHU5’,’LHU14’,’LHU15’,’LHU26’,’LPNAG’,’LP’,’LPGD’,’LPMI’,’LPCC’,’LPEM’,’LPED’,’LPEN’,’LPSP’, ... 

’LPTU’,’LPT’,’LPFR’,’LPS’,’LPGOV’,’LPHRM’,’LPMOSA’,’PMEMP’,’GMCQ’,’GMCDQ’,’GMCNQ’,’GMCSQ’,’GMCANQ’, ... 

’HSFR’,’HSNE’,’HSMW’,’HSSOU’,’HSWST’,’HSBR’,’HMOB’,’PMNV’,’PMNO’,’PMDEL’,’MOCMQ’,’MSONDQ’, ... 

’FSNCOM’,’FSPCOM’,’FSPIN’,’FSPCAP’,’FSPUT’,’FSDXP’,’FSPXE’,’EXRSW’,’EXRJAN’,’EXRUK’,’EXRCAN’, ... 

’FYFF’,’FYGM3’,’FYGM6’,’FYGT1’,’FYGT5’,’FYGT10’,’FYAAAC’,’FYBAAC’,’SFYGM3’,’SFYGM6’,’SFYGT1’, ...


’SFYGT5’,’SFYGT10’,’SFYAAAC’,’SFYBAAC’,’FM1’,’FM2’,’FM3’,’FM2DQ’,’FMFBA’,’FMRRA’,’FMRNBA’,’FCLNQ’, ... 

’FCLBMC’,’CCINRV’,’PMCP’,’PWSFA’,’PWFCSA’,’PWIMSA’,’PWCMSA’,’PSM99Q’,’PUNEW’,’PU83’,’PU84’,’PU85’, ... 

’PUC’,’PUCD’,’PUS’,’PUXF’,’PUXHS’,’PUXM’,’LEHCC’,’LEHM’,’HHSNTN’}; 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO _INPUT_STARTINGVALUES %%%%% 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% Returns starting values for all variables included in 

%%%%%% the input.startingvalues structure. These are F, lam_f, 

%%%%%% lam_y, R, Phi_lags and Q. 

%%%%%% 

%%%%%% startingvalues 

%%%%%% |---------- F 

%%%%%% |---------- lam_f 

%%%%%% |---------- lam_y 

%%%%%% |---------- R 

%%%%%% |---------- Phi_lags 

%%%%%% |---------- Q 

function [startingvalues] = DO_INPUT_STARTINGVALUES (input) 

%function [startingvalues] = DO_INPUT_STARTINGVALUES (input) 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’% % 

%’’’ function Get_Starting_Values = [Input_Structure] ’’’% % 

%’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’% % 

% % 

%switch mode ==1 % 

% case Get_Starting_Values(Generated) % 

% Statement 1 % 

% case Get_Starting_Values(Dispersed distribution) % 


% case Get_Starting_Values(zero_values) % 


% otherwise % 


% break % 

%end; %switch % 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

X_st = input.xdata ./ repmat(std(input.xdata,1),input.specification.dim.T,1); 

Y_st = input.specification.y ./ repmat(std(input.specification.y,1),input.specification.dim.T,1); 

% first step - extract PC from X 

[F,lam_f] = extract(X_st,input.specification.model.K); 

% regress X on F0 and Y, obtain loadings 

Lfy = olssvd(X_st(:,input.specification.model.K+1:input.specification.dim.N),[F Y_st])’; 

% upper KxM block of Ly set to zero 

lam_f=[lam_f(1:input.specification.model.K,:);Lfy(:,1:input.specification.model.K)]; 

lam_y=[zeros(input.specification.model.K,input.specification.dim.M);... 

Lfy(:,input.specification.model.K+1:input.specification.model.K+input.specification.dim.M)]; 

% transform factors and loadings for LE normalization 

[ql,rl]=qr(lam_f’); 

lam_f=rl; % do not transpose yet, is upper triangular


F=F*ql; 

% need identity in the first K columns of Lf, call them A for now 

A=lam_f(:,1:input.specification.model.K); 

lam_f=[eye(input.specification.model.K),inv(A)*lam_f(:,... 

(input.specification.model.K+1):input.specification.dim.N)]’; 

F=F*A; 

% obtain R: 

e=X_st-Y_st*lam_y’-F*lam_f’; 

R=e’*e ./ input.specification.dim.T; 

R=diag(diag(R)); 

% run a VAR in [F,Y], obtain initial B and Q 

[Phi_lags,Bc,v,Q,invFYFY]=estvar([F,input.specification.y],input.specification.model.d,[]); 

%----------------------------------------------------------------------------------------% 

startingvalues.F = F; 

startingvalues.lam_f = lam_f; 

startingvalues.lam_y = lam_y; 

startingvalues.R = R; 

startingvalues.Phi_lags = Phi_lags; 

startingvalues.Q = Q; 

%----------------------------------------------------------------------------------------% 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO_CALCULATION_SETMODEL %%%%% 

%%%%%% see Sequence Diagram Block B.1 %%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% Initializes calculation.stateSpaceStructure 

%%%%%% 

%%%%%% stateSpaceStructure.XX = XX; 

%%%%%% stateSpaceStructure.Lam = Lam; 

%%%%%% stateSpaceStructure.Xsi_in = Xsi_in; 

%%%%%% stateSpaceStructure.P_in = P_in; 

%%%%%% stateSpaceStructure.Phi_bar = Phi_bar; 

%%%%%% stateSpaceStructure.QQ_bar = QQ_bar; 

%%%%%% stateSpaceStructure.RR = RR; 

%%%%%% stateSpaceStructure.F_bar = F_bar; 

function DO_CALCULATION_SETMODEL (input) 

%function DO_CALCULATION_SETMODEL (input) 

global calculation; 

specM = input.specification.dim.M; 

specK = input.specification.model.K; 

specd = input.specification.model.d; 

specT = input.specification.dim.T; 

XX = [input.xdata, input.specification.y]; 

FF = [input.startingvalues.F, input.specification.y]; 

% initialize Factors its covarianvce matrix for Bayesian Kalman Filter & Smoother 

F_bar = [FF zeros(specT,((specd-1)*(specK+specM)))]; 

Xsi_in = zeros((specK+specM)*specd,1); 

P_in = eye((specK+specM)*specd); % for Kalman Filter&Smoother 

Lam = [input.startingvalues.lam_f input.startingvalues.lam_y; zeros(specM,specK) eye(specM)]; 

Lam_bar = [Lam zeros((input.specification.dim.N+specM),((specd-1)*(specK+specM)))];


RR=diag([diag(input.startingvalues.R);zeros(specM,1)]); %(N+M)x(N+M) 

Phi_lags = cat(2,input.startingvalues.Phi_lags(:,:)); 

Phi_bar = [Phi_lags ; eye((specd-1)*(specK+specM)) zeros((specd-1)*(specK+specM),(specK+specM))]; 

v = zeros(specT,(specK+specM)); 

v_bar = [v zeros(specT,((specd-1)*(specK+specM)))]; 

QQ_bar = [input.startingvalues.Q zeros((specK+specM),(specd-1)*(specK+specM)); zeros((specd-1)*(specK+specM),(specd*(specK+specM)))]; 

%----------------------------------------------------------------------------------------% 

stateSpaceStructure.XX = XX; 

stateSpaceStructure.Lam = Lam; 

stateSpaceStructure.Xsi_in = Xsi_in; 

stateSpaceStructure.P_in = P_in; 

stateSpaceStructure.Phi_bar = Phi_bar; 

stateSpaceStructure.QQ_bar = QQ_bar; 

stateSpaceStructure.RR = RR; 

stateSpaceStructure.F_bar = F_bar; 

stateSpaceStructure.Lam_bar = Lam_bar; 

%----------------------------------------------------------------------------------------% 

calculation.stateSpaceStructure = stateSpaceStructure; 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO_CALCULATION %%%%% 

%%%%%% see Sequence Diagram Block B %%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% In this function all calculationprocesses are started 

%%%%%% and their output stored. These processes are started 

%%%%%% by calling the functions DO_CALCULATION_SETMODEL, 

%%%%%% DO_CALCULATION_CREATESTRUCTURE, 

%%%%%% DO_CALCULATION_GIBBS_SAMPLING and DO_CALCULATION_IR 

%%%%%% where the last two are the main calculation processes. 

%%%%%% To save memory the datastructure calculation is declared 

%%%%%% as global in all functions. In this way all functions 

%%%%%% can refer to it as an input and out parameter without 

%%%%%% moving this big sized structure. 

function [results] = DO_CALCULATION (input) 

%function [results] = DO_CALCULATION (input) 

% declare calculation As global structure 


%the following DO_CALCULATION_x write their results directly into the 

%global calculation structure 

DO_CALCULATION_SETMODEL (input); % see Sequence Diagram - Block B.1 

DO_CALCULATION_CREATESTRUCTURE (input); % see Sequence Diagram - Block B.2 

DO_CALCULATION_GIBBS_SAMPLING (input); % see Sequence Diagram - Block B.3 

[results.ira] = DO_CALCULATION_IRA (input); % see Sequence Diagram - Block B.4 

% clear calculation 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO_CALCULATION_CREATESTRUCTURE %%%%% 

%%%%%% see Sequence Diagram Block B.2 %%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% Initializes calculation.Phi_bar_collect 

%%%%%% calculation.QQ_bar_collect 

%%%%%% calculation.F_bar_collect 

%%%%%% calculation.Lam_collect 

%%%%%% 

%%%%%% calculation.Phi_bar_collect = zeros(input.specification.nGibbsit,specK+specM,specK+specM,specd); 

%%%%%% calculation.QQ_bar_collect = zeros(input.specification.nGibbsit,specK+specM,specK+specM); 

%%%%%% calculation.F_bar_collect = zeros(input.specification.nGibbsit,specT,specK); 

function DO_CALCULATION_CREATESTRUCTURE (input) 

%function DO_CALCULATION_CREATESTRUCTURE (input) 




specd = input.specification.model.d; 

specT = input.specification.dim.T; 

specN = input.specification.dim.N; 

specDraws = input.specification.model.draws; 

calculation.Phi_bar_collect = zeros(specDraws,specK+specM,specK+specM,specd); 

calculation.QQ_bar_collect = zeros(specDraws,specK+specM,specK+specM); 

calculation.F_bar_collect = zeros(specDraws,specT,specK); 

calculation.Lam_collect = zeros(specDraws,specN+specM,specK+specM); 

for i=1:specM 

end 

calculation.Lam_collect(:,input.specification.varY(i),specK+i)=1; 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO_CALCULATION_GIBBS_SAMPLING %%%%% 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% This function does the Gibbs Sampling by calling the 

%%%%%% functions DO_CALCULATION_GIBBS_SAMPLING_BK_FILTER, 

%%%%%% DO_CALCULATION_GIBBS_SAMPLING_BK_SMOOTHER, 

%%%%%% DO_CALCULATION_GIBBS_SAMPLING_PARAM_PREC_OBS 

%%%%%% and DO_CALCULATION_GIBBS_SAMPLING_PARAM_PREC_FAC for 

%%%%%% each Gibbs iteration. After each Iteration the results 

%%%%%% are stored into the global calculation data structure 

%%%%%% after ignoring the first input.version.burn_in draws. 

function DO_CALCULATION_GIBBS_SAMPLING (input) 

%function DO_CALCULATION_GIBBS_SAMPLING (input) 


%%%%% set parameters 

K = input.specification.model.K; 

M = input.specification.dim.M; 

N = input.specification.dim.N; 

for Gibbsiteration=1:input.version.nGibbsit %%% Gibbs Start 

GLOG (sprintf(’Gibbsiteration: %d’,Gibbsiteration),2);


end 

[bk_filter] = DO_CALCULATION_GIBBS_SAMPLING_BK_FILTER (input); 

% see Sequence Diagram Block B.3.1 

[bk_smoother] = DO_CALCULATION_GIBBS_SAMPLING_BK_SMOOTHER (input, bk_filter); 


[param_prec_obs] = DO_CALCULATION_GIBBS_SAMPLING_PARAM_PREC_OBS (input, bk_smoother); 


[param_prec_fac] = DO_CALCULATION_GIBBS_SAMPLING_PARAM_PREC_FAC (input, bk_smoother); 


% all gibbs results to be stored here 

if Gibbsiteration > input.version.burn_in 

end 

calculation.Lam_collect (Gibbsiteration-input.version.burn_in,[1:76, 78:120],:)=... 

calculation.stateSpaceStructure.Lam(1:N,1:K+M); 

calculation.Phi_bar_collect (Gibbsiteration-input.version.burn_in,:,:,:)=... 

param_prec_fac.Phi_draw; 

calculation.QQ_bar_collect (Gibbsiteration-input.version.burn_in,:,:)=... 

param_prec_fac.Q_draw; 

calculation.F_bar_collect (Gibbsiteration-input.version.burn_in,:,:)=... 

bk_smoother.Xsi_S(:,1:K); 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO_CALCULATION_GIBBS_SAMPLING_BK_FILTER %%%%% 

%%%%%% Kalman Filter %%%%% 

%%%%%% see Sequence Diagram Block B.3.1 %%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% Bayesian Kalman Filter 

function [bk_filter] = DO_CALCULATION_GIBBS_SAMPLING_BK_FILTER (input) 

%function DO_CALCULATION_GIBBS_SAMPLING_BK_FILTER (input) 



Y_Data = calculation.stateSpaceStructure.XX; 

H_Prior = calculation.stateSpaceStructure.Lam; 

Xsi_Prior = calculation.stateSpaceStructure.Xsi_in; 

P_Prior = calculation.stateSpaceStructure.P_in; 

G_Prior = calculation.stateSpaceStructure.Phi_bar; 

Q_Prior = calculation.stateSpaceStructure.QQ_bar; 

R_Prior = calculation.stateSpaceStructure.RR; 

Xsi_all = calculation.stateSpaceStructure.F_bar; 



d = input.specification.model.d; 

%GLOG (size(Y_Data),1); 

%GLOG (size(H_Prior),1);


%%%%% start kalman filter 

% Setting Dimensions 

[T,var] = size(Y_Data); 

[H_row,H_col] = size(H_Prior); % has to equal size of Lam_bar 

[Xsi_row,Xsi_col] = size(Xsi_Prior); 

[G_row,G_col] = size(G_Prior); 

[Q_row,Q_col] = size(Q_Prior); 

[R_row,R_col] = size(R_Prior); 

km = Xsi_col/13; 

kmd = Xsi_col; 

%Variables for State-Space 

Y_t = Y_Data; 

H_t = H_Prior; 

G = G_Prior; 

R = R_Prior; 

Q = Q_Prior; 

vecQ = reshape(Q,Q_row^2,1); 

% Sequence of draws to be stored in Xsi_all and P_all 

Xsi_all = zeros(T,((K+M)*d)); 

P_all = zeros(((K+M)*d)^2,T); 

%invI = inv(eye(size(Xsi_Prior,1)^2) - kron(G,G)); 

%vecP_Prior = invI * vecQ; 

%P_Prior = reshape(vecP_Prior,size(Xsi_row,1),size(Xsi_row,1)); 

% Initialization the state vectors variance-covariance matrix 

%Xsi_Prior = zeros(Xsi_row,1); % could be taken in case of no initial value 

%P_Prior = eye(Xsi_row); % could be taken in case of no initial value 

Xsi_tlag = Xsi_Prior; 

P_tlag = P_Prior; 

% Final Draws to be stored in Xsi_F and P_F 

Xsi_F = zeros(Xsi_row,Xsi_col); 

P_F = zeros(Xsi_row^2,T); 

for t=1:T 

%=======================================================% 

% Updating equations (Kim&Nelson) % 

%*******************************************************% 

Eta_tlag = Y_t(t,:)’ - H_t * Xsi_tlag(1:(K+M)); % 

f_tlag = H_t * P_tlag(1:(K+M),1:(K+M)) * H_t’ + R;% 

if_tlag = inv(f_tlag); % 

%if_tlag = pinv(f_tlag); % 

K_t = P_tlag(:,1:(K+M)) * H_t’ * if_tlag; % 

% % 

Xsi_tt = Xsi_tlag + K_t * Eta_tlag; % 

P_tt = P_tlag - K_t * H_t * P_tlag(1:(K+M),:); % 

%=======================================================% 

%-------------------------------------------------------% 

%=======================================================% 

% Prediction equation (Kim&Nelson) % 

%*******************************************************%


end%for 

% Note that indexp* P_tt * G’ + Q; % 

%=======================================================% 

if t


% Preparing dimension for Transformation 

regDim = [1:(K+M)]; 

[l_star] = length(regDim); 

[l_reg] = length(Q_Prior); 

T = size(Xsi_Tlag,1); 

% Transformation of Q in case of singularity 

Q = Q_Prior; 

Q_star = Q(1:l_star,1:l_star); 

% Transformation of G in case of singular Q 

G = G_Prior; 

%G_star = zeros(l_star,l_reg); 

G_star = G(1:l_star,1:l_reg); 

% Transformation of Xsi in case of singular Q 

Xsi_F = Xsi_F(1,1:l_reg); 

Xsi_Tlag = Xsi_Tlag(:,1:l_reg); 

P_F = P_F(1:l_reg,1:l_reg); 

P_Tlag = P_Tlag(1:l_reg^2,:); 

for t=1:T-1 

end%for 

%=======================================================% 

% Final Updating procedure % 

%*******************************************************% 

Xsi_TP = Xsi_Tlag(T-t+1,1:l_star)’; 

Xsi_TT = Xsi_Tlag(T-t,:)’; 

P_TL = P_Tlag(:,T-t); 

P_TT = reshape(P_TL’,(d*(K+M)),(d*(K+M))); 

%f_ts = pinv(G_star * P_TT * G_star’ + Q_star); 

f_ts = inv(G_star * P_TT * G_star’ + Q_star); 

K_ts = P_TT * G_star’ * f_ts; 

Xsi_TXsi = Xsi_TT + K_ts * (Xsi_TP - G_star * Xsi_TT); 

P_TXsi = P_TT - K_ts * G_star * P_TT; 

%*******************************************************% 

% singles out latent factors 

indexnM=[ones(K,d);zeros(M,d)]; 

indexnM=find(indexnM==1); 

Xsi_Tlag(T-t,:) = Xsi_TXsi’; 

Xsi_Tlag(T-t,indexnM) = mvnrnd(Xsi_Tlag(T-t,indexnM)’,P_TXsi(indexnM,indexnM),1); 

Xsi_S = Xsi_Tlag(:,1:(K+M)); 

P_S = P_TXsi; 

%%%%% set bk_smoother output structure 

bk_smoother.Xsi_S = Xsi_S;


%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO_CALCULATION_GIBBS_SAMPLING_PARAM_PREC_FAC %%%%% 

%%%%%% %%%%% 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% Inference on State Equation 

function [param_prec_fac] = DO_CALCULATION_GIBBS_SAMPLING_PARAM_PREC_FAC (input, bk_smoother); 

%function [param_prec_fac] = DO_CALCULATION_GIBBS_SAMPLING_PARAM_PREC_FAC (input, bk_smoother); 





T = input.specification.dim.T; 

d = input.specification.model.d; 

Xsi_S = bk_smoother.Xsi_S; 

%****************************% 

% univariate AR OLS % 

%****************************% 

% At this point we are interested in generating the i’th variances: 

% Only Sigma Required 

for i=1:K+M 

end%for 

[Phi_i,Phi_ci,vi,Qi(i),invFYFYi]=estvar(Xsi_S(:,i),d,[]); 

% At this point we only need to save Qi(i) which respectively as 

% diagonal elements forms the Prior Q_0 

Q_0 = diag(Qi); % maybe better diag(Qi(:,:)) 

Q_prior = Q_0; 

Omega_0 = zeros(d*(K+M));%,size(Qmega_0,2)); 

Omega_0 = diag(kron(1./Qi,1./[1:d])); % Qmega_0 = Q_prior.*Omega_0; 

%


end%for 

F_reg = F_reg(:,:); 

F_reg = F_reg(d+1:T,:); 

% For doing inference on the Transition equation one has to first draw Q and the draw vec(Phi) 

% Note that for generalization it is good to write [kappa1_prior,kappa2_prior; kappa1_post and kappa1_post] 

% shortcuts 

VV = V_hat’*V_hat; 

FF_reg = inv(F_reg’*F_reg); 

Phi_FF = inv(Omega_prior + FF_reg); 

% Draw posterior Q 

Q_bar = Q_prior + VV + Phi_hat’*Phi_FF*Phi_hat; 

kappa1_prior = K+M+2; 

kappa1_post = T+kappa1_prior; 

kappa2_prior = Q_prior; 

kappa2_post = Q_bar; % Scale Matrix 

% Inverse Wishart draw 

% df = degrees of freedom 

%QW = wishrnd(inv(kappa2_post),kappa1_post); 

%Q_draw = inv(QW); % =1 % 0.999 

end 

vec_Phi_draw = mvnrnd(vecPhi_posterior,sigma_Phi,1); 

Phi_draw = reshape(vec_Phi_draw’,d*(K+M),(K+M))’; 

calculation.stateSpaceStructure.Phi_bar(1:K+M,:) = Phi_draw; 

Phi_draw = reshape(Phi_draw,K+M,K+M,d); 

%%%%% set param_prec_obs output structure 

param_prec_fac.Q_draw = Q_draw; 

param_prec_fac.Phi_draw = Phi_draw; 

%%%%%%**********************************************************%%%%% 

%%%%%% Bayesian FAVAR Code August 26th %%%%%


%%%%%%**********************************************************%%%%% 

%%%%%% DO_CALCULATION_GIBBS_SAMPLING_PARAM_PREC_OBS %%%%% 

%%%%%% %%%%% 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% Inference on Observation Equation 

function [param_prec_obs] = DO_CALCULATION_GIBBS_SAMPLING_PARAM_PREC_OBS (input, bk_smoother); 

%function [param_prec_obs] = DO_CALCULATION_GIBBS_SAMPLING_PARAM_PREC_OBS (input, bk_smoother); 



XX = calculation.stateSpaceStructure.XX; 



N = input.specification.dim.N; 

T = input.specification.dim.T; 

Xsi_S = bk_smoother.Xsi_S(:,1:K+M); 

% prior distributions for VAR part, need Lam and R 

s0 = 3; 

alpha = 0.001; 

M0 = eye(K+M); % Variance Parameter in prior on i-th coeff 

Param1 = inv( M0 + inv(Xsi_S’*Xsi_S) ) ; 

for i=1:N 

if i K 

%**********************% 

% b) draw Lam_ii % 

%**********************% 

% Given : Factors,Data, and Previously generated R_ii


end%for 

end%if 

% Variables needed 

M_i_bar = inv ( inv(M0) + Xsi_S’*Xsi_S ); 

%M0 + Xsi_S(:,1:K+M)’*Xsi_S(:,1:K+M); 

Lam_i_bar = M_i_bar *(Xsi_S’*Xsi_S)*Lam_i_hat; 

%inv(M_i_bar) *(Xsi_S(:,1:K+M)’*Xsi_S(:,1:K+M)) * Lam_i_hat; 

Lam_i_hat = Lam_i_bar’ + randn (1, K+M) * chol (R_draw*M_i_bar); 

calculation.stateSpaceStructure.Lam(i,1:K+M) = Lam_i_hat; 

%%%% Alternative Approach 

%Lam_Sigma = calculation.stateSpaceStructure.RR(i,i) * inv(M_i_bar); 

% Draw Lam from Normal Distribution 

%Lam_draw = mvnrnd(Lam_i_bar’, Lam_Sigma,1); 

%calculation.stateSpaceStructure.Lam(i,1:K+M) = Lam_draw; 

% Ldraw(nit,NN,:)=Lam_bar(1:N,1:K); 

% Fdraw(nit,:,:)=Xsi_S(:,1:K); 

%%%%% set param_prec_obs output structure 

%param_prec_obs.Lam_draw = Lam_draw; 

%param_prec_obs.R_draw = R_draw; 

param_prec_obs = 1; 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO _CALCULATION_IRA %%%%% 

%%%%%% Impule-Response-Analysis %%%%% 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% This function starts the DO_CALCULATION_IRA_UHLIG or the 

%%%%%% DO_CALCULATION_IRA_BBE function depending on the value of 

%%%%%% input.version.ira_mode which contains information about 

%%%%%% the selected Impule Response Mode to run. 

function [ira] = DO_CALCULATION_IRA (input) 

%function [ira] = DO_CALCULATION_IRA (input) 



switch input.version.ira_mode 

end 

case 1 

case 2 

[ira.finalresponses] = DO_CALCULATION_IRA_UHLIG (input); 

% See Sequence Diagram Block B.4.1 

[ira.finalresponses] = DO_CALCULATION_IRA_BBE (input); 

% See Sequence Diagram Block B.4.2 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

% clear calculation 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


spec_nBC = length(input.specification.IRA.BC); 

for bc_i = 1:spec_nBC; 

end; 

nsteps = input.specification.IRA.nsteps; 

scale = calculation.IRA.scale; 


specAlphaDraws = input.specification.IRA.alpha_draws; 

ira.finalresponses(bc_i).no = size(ira.finalresponses(bc_i).response,3); 

%GLOG (sprintf(’Accepted Responses for BC-%d: %d’,bc_i, ira.finalresponses(bc_i).no),1); 

% transform back to levels 

for i=1:size(ira.finalresponses(bc_i).response,1); 

end; 

if input.specification.IRA.var_index_sr(i,2)==4 

ira.finalresponses(bc_i).response(i,:,:) =... 

exp(ira.finalresponses(bc_i).response(i,:,:))-ones(1, nsteps,ira.finalresponses(bc_i).no); 

elseif input.specification.IRA.var_index_sr(i,2)==5 

end 

ira.finalresponses(bc_i).response(i,:,:) = ... 

exp(cumsum(ira.finalresponses(bc_i).response(i,:,:),2))-ones(1,... 

% FINAL RESPONSES 

nsteps,ira.finalresponses(bc_i).no); 

ira.finalresponses(bc_i).response = sort(ira.finalresponses(bc_i).response,3); 

ira.finalresponses(bc_i).fimpResponse = median(ira.finalresponses(bc_i).response,3); 

if ira.finalresponses(bc_i).no > 0 

end; 

% ERROR BANDS 

ira.finalresponses(bc_i).lowerErrorBand = ira.finalresponses(bc_i).response(:,:,floor(0.16*ira.finalresponses(bc_i).no)); 

ira.finalresponses(bc_i).upperErrorBand = ira.finalresponses(bc_i).response(:,:,floor(0.84*ira.finalresponses(bc_i).no)); 

% concatenate the estimate and confidence bounds 

ira.finalresponses(bc_i).collRespMat =... 

cat(3,ira.finalresponses(bc_i).lowerErrorBand, ... 

ira.finalresponses(bc_i).fimpResponse, ... 

ira.finalresponses(bc_i).upperErrorBand); 

% transform scale to std 

ira.finalresponses(bc_i).collRespMat =... 

ira.finalresponses(bc_i).collRespMat ./ repmat(scale’,[1 nsteps 3]) ; 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%%


%%%%%% DO _CALCULATION_IRA_UHLIG %%%%% 

%%%%%% Uhlig (2005) - Sign Restriction %%%%% 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% Impulse Response Analysis with Uhlig (2005) Sign 

%%%%%% Restrictions. Returns finalresponse which is a vector 

%%%%%% with the length of nBlockCriteria. Responses are 

%%%%%% checked to satisfy each block criteria, which are 

%%%%%% set in input.specification.IRA.BC. 

%%%%%% Accepted Responses are added to 

%%%%%% finalresponses(bc_i).response 

%%%%%% where bc_i is the block criteria satistied by 

%%%%%% the responses. 

function [finalresponses] = DO_CALCULATION_IRA_UHLIG (input) 

%function [finalresponses] = DO_CALCULATION_IRA_UHLIG (input) 



GLOG (’Starting Impulse Responses Ulhig (2005)’,2); 




specNSteps = input.specification.IRA.nsteps; 


specAlphaDraws = input.specification.IRA.alpha_draws; 

specZ = input.specification.IRA.sr_horizon; 

%prepare finalresponses 

fr_current_length = zeros (spec_nBC,1); 

% fr_current_length is current length of finalresponses matrix and also the initial size of it 

fr_add_length = zeros (spec_nBC,1); 

% if current length of finalresponses is not enough add fr_add_length more slices 

last_slice = zeros (spec_nBC,1); 


end; 

fr_current_length (bc_i) = ceil(0.03*(specDraws*specAlphaDraws)); 

% set fr_current_length according to your block criteria. 

% the more restrictive a block criteria is, the smaller the 

% impulse responses satisfying the restriction, the smaller 

% the initial value of ft_current_length 

fr_add_length (bc_i) = ceil(0.01*(specDraws*specAlphaDraws)); 

last_slice (bc_i) = 0; 

% initialise finalresponses(bc_i).response with initial size 

finalresponses (bc_i).response (:,:,fr_current_length (bc_i)) = zeros(size(input.specification.IRA.var_index_sr,1), specNSteps); 

% /todo CHECK SCLE FOR SR!!! 

calculation.IRA.scale =... 

std(input.data(:,input.specification.IRA.var_index_sr(:,1)));


calculation.IRA.choleskyRespMat = zeros(specDraws,specK+specM,specK+specM,specNSteps); 

% vector of initial impulse SHOCK: 25 basis points of FFR 

% "CONTRACTIONARY MONETARY POLICY" 

% /TODO : initial impulse vector to be premultiplied 

calculation.IRA.initialImpulse = diag([zeros(1,specK+specM-1) .25]); 

% customize calculation.Lam_collect 

calculation.Lam_collect =... 

calculation.Lam_collect(:,input.specification.IRA.var_index_sr(:,1),:); 

% SELECT VARIABLES FOR IRA 

calculation.Lam_collect = permute(calculation.Lam_collect,[2 3 1]); 

%%%% this [X * F * draws] 

alpha_tilde = zeros(specK+specM,1); 

alpha = zeros(specK+specM,1); 

%each draw of alpha has size (nvar,1) and norm=1 

for i=1:specDraws 

GLOG (sprintf(’Impulse-Responses for Draw: %d’,i),2); 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Begin Calculation of Cholesky Responses Matrix 

Phi_v = squeeze(calculation.Phi_bar_collect(i,:,:,:)); 

Q_v = squeeze(calculation.QQ_bar_collect(i,:,:)); 

% 

chol_Q = chol(Q_v); 

norm_mat = diag(diag(chol_Q)); 

chol_Q = inv(norm_mat)*chol_Q; % SMAT 

% chol_Q is upper triangular decomposition of omega; 

% gives matrix of initial shocks with 1’s on the diagonal 

chol_Q = calculation.IRA.initialImpulse*chol_Q; 

calculation.IRA.choleskyRespMat(i,:,:,:) = impulsdtrf(Phi_v,chol_Q,specNSteps); 

%%%% End Calculation of Cholesky Responses Matrix 

for nalpha = 1:input.specification.IRA.alpha_draws; 

%for each Cholesky, draw alpha vectors of norm unity 

alpha_tilde=randn(specK+specM,1); % standard Gaussian draws 

alpha=(1/norm(alpha_tilde)) * alpha_tilde; 

candidateA = zeros(specK+specM,1,specNSteps); 

for j=1:specNSteps; %creating structural impulse responses 

candidateA(:,:,j) =... 

squeeze(calculation.IRA.choleskyRespMat(i,:,:,j)) * alpha; 

end; %structural responses combine Cholesky with alpha draws 

candidateB = calculation.Lam_collect(:,:,i) * squeeze(candidateA); 


check_sign_restriction_result = 0;


end; 

end; 

check_sign_restriction_result =... 

DO_CALCULATION_IRA_UHLIG_CHECK_SIGNRESTRICTION (input,candidateB,bc_i); 

if abs(check_sign_restriction_result) == 1 

else 

end; 

last_slice (bc_i) = last_slice (bc_i) + 1; 

GLOG (sprintf(’Response accepted. This is the 

%dth accepted Response for BC-%d’, last_slice (bc_i), bc_i),1); 

finalresponses (bc_i).response (:,:,last_slice (bc_i)) =... 

check_sign_restriction_result * candidateB; 

if fr_current_length (bc_i) == last_slice (bc_i) 

%increase length 

end; 

end; %alphadraws 


end; 

fr_current_length (bc_i) =... 

fr_current_length (bc_i) + fr_add_length (bc_i); 

finalresponses (bc_i).response (:,:,fr_current_length (bc_i)) = zeros(size(input.specification.IRA.var_index_sr,1), input.specification.I 

% CAN NOT ACCEPT RESPONSE 

GLOG (sprintf(’Number of accepted Responses for BC-%d: %d’,bc_i, last_slice (bc_i)),2); 

finalresponses (bc_i).response = finalresponses (bc_i).response (:,:,1:last_slice (bc_i)); 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO_CALCULATION_IRA_UHLIG_CHECK_SIGNRESTRICTION %%%%% 

%%%%%% %%%%% 

%%%%%% see Sequence Diagram Block B.4.1.1 %%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%% This Function checks if a given Response satisfies the 

%%%%%% block criteria bc_i. In that case the result 

%%%%%% is 1/-1. Otherwise 0. 

function [check_result] = DO_CALCULATION_IRA_UHLIG_CHECK_SIGNRESTRICTION (input,candidate,bc_i); 

%function [check_result] = DO_CALCULATION_IRA_UHLIG_CHECK_SIGNRESTRICTION (input,candidate,bc_i); 


specZ = input.specification.IRA.sr_horizon; 

P_INDEX = input.specification.IRA.BC(bc_i).priceIndex; 

M_INDEX = input.specification.IRA.BC(bc_i).moneyIndex; 

I_INDEX = input.specification.IRA.BC(bc_i).interestIndex; 

if all(candidate(P_INDEX (1:(length(P_INDEX))),1:specZ) 0)


% Price - Money - Interestrate 

check_result = 1; 

elseif all(candidate(P_INDEX (1:(length(P_INDEX))),1:specZ) > 0) &... 

all(candidate(M_INDEX (1:(length(M_INDEX))),1:specZ) > 0) &... 

all(candidate(I_INDEX (1:(length(I_INDEX))),1:specZ) < 0); 

else 

end; 

check_result = -1; %According to B. Mackowiak 

check_result = 0; 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% GLOG %%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%% This Function is used to log an output string depending 

%%%%% on its log level glog_type. The global variable 

%%%%% GLOG_MODE signifies the global minimum level for 

%%%%% outputs and is set directly in BAYESIAN_FAVAR. 

%%%%% If the glog_type of an output string is less than 

%%%%% GLOG_MODE the output is ignored. 

%%%%% Otherwise GLOG uses the disp() function to 

%%%%% display the output string. 

function GLOG (glog_text, glog_type) 

%function GLOG (glog_text, glog_type) 

global GLOG_MODE; 

if glog_type >= GLOG_MODE 

end 

disp (glog_text); 

%%%%%%**********************************************************%%%%% 


%%%%%%**********************************************************%%%%% 

%%%%%% DO_RESULTS %%%%% 

%%%%%% see Sequence Diagram Block C %%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%% Plots the results of all finalresponses containing more 

%%%%% than zero elements. 

function DO_RESULTS (input,results) 

%function DO_RESULTS (input,results) 



if results.ira.finalresponses(bc_i).no > 0 

%%% BBE 

figure(bc_i ) 

for i=1:20 

subplot(5,4,i) 

plot(input.specification.IRA.tstep,... 

input.specification.IRA.zeroline,’-k’, ... 

input.specification.IRA.tstep,...


end; 

end; 

end; 

%%% SR 

squeeze(results.ira.finalresponses(bc_i).collRespMat(i,:,1)),’b--’, ... 

input.specification.IRA.tstep,... 

squeeze(results.ira.finalresponses(bc_i).collRespMat(i,:,2)),’r-’,... 

input.specification.IRA.tstep,... 

squeeze(results.ira.finalresponses(bc_i).collRespMat(i,:,3)),’b--’, ... 

’LineWidth’,2); 

set(gca,’XLim’,[0 input.specification.IRA.nsteps],’XTick’,... 

[0 input.specification.IRA.nsteps],’FontSize’,10); 

title(input.specification.VARNAMES_BBE( i)); 

% input.specification.IRA.var_index_sr(i,1))); 

%figure( (bc_i-1)+1 ) 

%for i=1:20 

% subplot(5,4,i) 

% plot(input.specification.IRA.tstep,input.specification.IRA.zeroline,... 

%’-k’,input.specification.IRA.tstep,... 

%squeeze(results.ira.finalresponses(bc_i).collRespMat(i,:,1)),’b--’, ... 

% input.specification.IRA.tstep,... 

%squeeze(results.ira.finalresponses(bc_i).collRespMat(i,:,2)),’r-’,... 


%squeeze(results.ira.finalresponses(bc_i).collRespMat(i,:,3)),... 

%’b--’,’LineWidth’,2); axis tight;grid on; 

%title(input.specification.ALL_VARNAMES... 

%( input.specification.IRA.var_index_sr(i,1))); 

%end; 

%figure((bc_i -1) +2) 

%for i=21:40 

% subplot(5,4,i-20) 

% plot(input.specification.IRA.tstep,input.specification.IRA.zeroline,... 

%’-k’,input.specification.IRA.tstep,... 

%squeeze(results.ira.finalresponses(bc_i).collRespMat(i,:,1)),’b--’, ... 



% input.specification.IRA.tstep,.squeeze(results.ira.finalresponses(bc_i).. 

%.collRespMat(i,:,3)),’b--’,’LineWidth’,2); axis tight;grid on; 

% title(input.specification.ALL_VARNAMES( input.specification.IRA.var_index_sr(i,1))); 

%end; 

%figure((bc_i -1) +3 ) 

%for i=41:52 

% subplot(5,4,i-40) 

% plot(input.specification.IRA.tstep,input.specification.IRA.zeroline,.. 

%’-k’,input.specification.IRA.tstep,squeeze(results... 

%ira.finalresponses(bc_i).collRespMat(i,:,1)),’b--’, ... 

% input.specification.IRA.tstep,.. 



%queeze(results.ira.finalresponses(bc_i).collRespMat(i,:,3)),... 

%’b--’,’LineWidth’,2); axis tight;grid on; 

% title(input.specification... 

%ALL_VARNAMES( input.specification.IRA.var_index_sr(i,1))); 

%end;


function DO_RESULTS_PLF 

clc; 

k = [2,5,7]; 

for i = 1:length (k) 

end; 

clear calculation; 

if k(i) == 2 

load DATA_050824_D5000_B1500_K2_C; 

elseif k(i) == 5 

load DATA_050824_D3500_B500_K5_C; 

calculation.F_bar_collect = calculation.F_bar_collect (2001:3000,:,:) 


end 

load DATA_050826_D3000_B2000_K7_C; 

disp (’data loaded’); 

t = 1959.1667:1/12:2001.6667; 

l = size(calculation.F_bar_collect,1); 

lh = l/2 

figure (k(i)) 

for j = 1:k(i) 

end 

meanA = mean(calculation.F_bar_collect (1:lh,:,j)); 

meanB = mean(calculation.F_bar_collect (lh+1:l,:,j)); 

subplot(k(i),1,j); plot (t, meanA,’g-’,t,meanA,’r--’, ’LineWidth’,2); 

grid on; 

axis tight; 

title ( sprintf(’Convergence Plot for Factor %d’,j) ); 

if j == 1 

legend(’First half’,’Second half’);legend(’First half’,’Second half’,0); 

end 

saveas(gcf, sprintf ( ’FIG_K%dD%d’ ,k(i),l) , ’fig’); 

saveas(gcf, sprintf ( ’FIG_K%dD%d’ ,k(i),l) , ’jpg’); 

function DO_RESULTS_PLM 

clc; 

clear; 

k = [2,5,7]; 

for i = 1:length (k) 

clear input; 

clear results; 

disp (sprintf(’ready to load data for K%d’,k(i))); 

if k(i) == 2 

load FOR_MESH_K2_RESULTS; %name of the mat file where input and results are stored 

%better use this code as a function


end 

%directly called from DO_RESULTS 

d = size(squeeze( (results.ira.finalresponses(1).response(1,:,:))),2); 

draws = 1:10:d; %show every xth accepted draw response 


load FOR_MESH_K5_RESULTS; 


draws = 1:10:d; 


end 

load FOR_MESH_K7_RESULTS; 


draws = 1:d; 

disp (sprintf(’K%d data ready’,k(i))); 

n = input.specification.IRA.nsteps; 

srs = 1:size(squeeze( (results.ira.finalresponses(1).response(1,:,draws))),1); 

smax = size(squeeze( (results.ira.finalresponses(1).response(1,:,draws))),1); 

figure(k(i)) 

axis normal; 

grid on; 

axis tight; 

subplot(2,2,1); 

mesh (squeeze(results.ira.finalresponses(1).response(1,srs,draws))); 

ylabel(’Horizon’); 

xlabel(’Accepted Response Draws’); 

zlabel(’Response Scale’); 

title (’FFR’); 




title (’COMMODITY PRICE INDEX’); 




title ( ’UNEMPLOYMENT’); 




title ( ’CAPACITY UTIL RATE’); 

saveas(gcf, sprintf ( ’FIG_MESH_K%d’ ,k(i)) , ’fig’); 

saveas(gcf, sprintf ( ’FIG_MESH_K%d’ ,k(i)) , ’jpg’);


Erklärung zur Urheberschaft 

Hiermit erkläre ich, Pooyan Amir Ahmadi, dass ich die vorliegende Arbeit 

allein und nur unter Verwendung der aufgeführten Quellen und Hilfsmittel 

angefertigt habe. 

Pooyan Amir Ahmadi 

Berlin, den 26. August 2005

Measuring the Effects of a Shock to Monetary Policy - Humboldt ...

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