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

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Bayesian FAVARs with Agnostic Identification 21 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 component. The authors Forni, Hallin, Lippi and Reichlin (2001) tried to combine the ap- proximate DFM of Chamberlain (1983) and Chamberlain and Rothschild (1984), which
22 Bayesian FAVARs with Agnostic Identification 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 contemporaneously. 22 For more details please refer to BBE [2005].
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