Conditional on Λ, ∆Λ, and Σ,equation (3.28) is a multivariate linear regression of yd on Zd,such that the posterior distribution is obtained from standard SUR results as
p(Ψ|YT,Σ,Λ,∆Λ)∼N( ¯Ψ,V¯Ψ), (3.29) where V¯Ψ = V−1Ψ +Zd0(IT ⊗Σ−1)Zd−1
and Ψ = ¯¯ VΨ V−1Ψ Ψ +Zd0(IT ⊗Σ−1)yd .
To obtain draws from the posterior distribution, Villani (2009) suggests using a three-block Gibbs sampler that iterates between (3.24), (3.25) and (3.29).
Appendix 3.D Estimation under limited availability of the survey nowcasts
Below, we outline the Bayesian estimation of the augmented VAR when survey nowcasts are available only for a subset of the variables in yt and when the time series foryt starts earlier than the nowcasts. We distinguish the cases of section 3.2.2 and 3.2.3 respectively.
3.D.1 Model without Wright’s democratic steady-state prior (sec-tion 3.2.2)
In what follows, we derive a SUR representation of the model that acknowledges the two particularities of the data: First, we show how equation (3.8) changes when survey nowcasts are observed for a subset of the variables in yt. For that purpose, we make the following definitions
st := partly observedM ×1 vector of survey nowcasts in equation (3.8) s∗t := observed N ×1 vector of survey nowcasts with N < M
R := N ×M selection matrix obtained from IM by deleting the rows that refer to the elements ofyt, for which we do not observe nowcasts, such thats∗t =Rst
Pre-multiplying the part of equation (3.8) that refers to st byR, we obtain
" we can only identify these reduced form parameters.
Next, we show how the SUR representation of equation (3.19) changes when the time series of survey nowcasts st starts later. For that purpose, we denote the periods when only yt is observed by t = 1, . . . , t1. Accordingly, the periods when both yt and st are observed are t = t1+ 1, . . . , T. As an intermediate step to the SUR representation, we consider the matrix representation of equation (3.18). Specifically, the part referring to the survey nowcasts becomes
(other elements are defined as before). By vectorizing, we obtain the SUR representation:
with we adjust the prior distribution specified in section 2.3 by dropping the elements that refer to the unobserved parts of st.
The model specified through (3.32-3.33) has the structure of a multivariate linear seem-ingly unrelated regression (SUR) model, such that we can use (once again) standard results for its Bayesian estimation (see Geweke, 2005, p.162ff). Specifically, the full conditional posterior of β is
One complication arises with the full conditional posterior of the variance-covariance matrix (see Swamy and Mehta, 1975): It is not completely clear, how to use the time series of different length in computing the sample estimate of Σ∗. If we use for each of its elements the maximum number of observations available, Σ∗ is not guaranteed to be invertible (see Schmidt, 1977). We avoid the problem by estimating the parameter of the full conditional posterior of Σ∗ on the overlapping sample (t = t1 + 1, . . . , T), thus throwing away the observations of the earlier sub-sample in its computation. Specifically, as an approximation to the full conditional posterior of Σ∗ we use
p(Σ∗|YT, β)∼IW(T −t1, UT−t1) (3.35)
3.D.2 Model with Wright’s democratic steady-state prior (sec-tion 3.2.3)
In appendix 3.C, we have derived the full conditional posterior distributions of {Λ,∆Λ}, {ψ,∆ψ} and Σ respectively. Here, we adjust the SUR representations leading to the full conditional posteriors of {Λ,∆Λ} (eq. 3.23) and {ψ,∆ψ} (eq. 3.29) to the case, when survey nowcasts are observed for a subset of the variables of the VAR, and their observations start later. Given the SUR representations and an appropriately adjusted prior distribution (see Appendix 3.D.1), the posterior is obtained from standard results
(see Geweke, 2005, p.162ff). The same complications as in Appendix 3.D.1 arise with respect to the posterior of Σ∗.
I. Full conditional posterior of {Λ,∆Λ}:
I.a. Pre-multiplying the part of (3.13) that refers tost byR, we obtain
"
where ψ+∗ is the reduced form ofRψ+ (other elements are defined as before).
I.b. As an intermediate step to the SUR representation, we consider the matrix represen-tation of equation (3.22). Specifically, the part referring to the survey nowcasts becomes
Sψ+
.By vectorizing, we see that the SUR representation (eq. 3.23) becomes
The full conditional posterior is obtained as in (3.34).
II. Full conditional posterior of
ψ,∆∗ψ :
II.a. Pre-multiplying (eq. 3.27) by R, we obtain s∗t −X
II.b. Accordingly, the SUR representation in the spirit of (3.28) - that allows for a standard Bayesian treatment as in (3.34) - becomes
For the posterior of Σ∗, essentially, we use the same approximation as in (3.35).
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Ich versichere hiermit, dass ich Kapitel 1 der vorliegenden Arbeit ohne Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel verfasst habe.
Kapitel 2 entstammt einer gemeinsamen Arbeit mit Xuguang Sheng von der American University in Washington, D.C. und Jingyun Yang vom Rush University Medical Center in Chicago. Meine individuelle Leistung bei der Erstellung des Kapitels beträgt 45 %.
Kapitel 3 entstammt einer gemeinsamen Arbeit mit Christoph Frey von der Universität Konstanz. Meine individuelle Leistung bei der Erstellung des Kapitels beträgt 65 %.