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

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Bayesian FAVARs with Agnostic Identification 19 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
20 Bayesian FAVARs with Agnostic Identification 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 factor 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]
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