Back Matter

Appendix A. DFM: Gibbs Sampler: Drawing Transition Equation Matrix

We need to generate G from the conditional density p(G|Q, Λ, Ψ, R, FT;XT). Note, however, that the dependence of G on the other state-space matrices – except for Q – is exclusively through the factors. This is because given factors Ft, the transition equation (8) is a VAR(1):

Therefore, p(G|Q, Λ, Ψ, R, FT; XT) = p(G| Q, FT).

Rewrite the VAR in matrix notation

$\begin{array}{ccc}Y=X\mathbf{G}+\eta & \phantom{\rule{7.0em}{0ex}}& \left(22\right)\end{array}$

where Y, X and η are the (T–1)×N matrices with rows ${F}_{t}^{\prime },{F}_{t-1}^{\prime }$ and ${\eta }_{t}^{\prime },$ respectively. To specify a prior distribution for the VAR parameters, we follow Lubik and Schorfheide (2005) and use a version of Minnesota Prior (Doan, Litterman, Sims 1984) implemented with T* dummy observations Y* and X*. The likelihood function of dummy observations p(Y*|G, Q) combined with the improper prior distribution |Q|-(N+1)/2 ×1G induces the proper prior for the VAR parameters:

$\begin{array}{ccc}p\left(\mathbf{G},\mathbf{Q}\right)\propto p\left({Y}^{*}|\mathbf{G},\mathbf{Q}\right)|\mathbf{Q}{|}^{-\left(N+1\right)/2}×{\mathbf{1}}_{G},& \phantom{\rule{7.0em}{0ex}}& \left(23\right)\end{array}$

where 1G denotes an indicator function equal to 1 if all eigenvalues of G lie inside unit circle. In actual implementation of Minnesota Prior, we set the hyperparameters as follows τ=5, d=0.5, ι=1, w=1, λ=0, λ=0 to generate Y* and X*. Essentially, our prior is tilting the transition equation (21) to a collection of the univariate random walks.

Combining this prior with the likelihood function P(Y|G, Q), we obtain the posterior density of the VAR parameters:

$\begin{array}{ccc}p\left(\mathbf{G},\mathbf{Q}|Y\right)\propto p\left(Y|\mathbf{G},\mathbf{Q}\right)=p\left(Y|\mathbf{G},\mathbf{Q}\right)p\left({Y}^{*}|\mathbf{G},\mathbf{Q}\right)|\mathbf{Q}{|}^{-\left(N+1\right)/2}×{\mathbf{1}}_{G}.& \phantom{\rule{7.0em}{0ex}}& \left(24\right)\end{array}$

It can be shown (e.g. Del Negro, Schorfheide 2004) that our posterior density P(G, Q|Y)=P(G, Q|FT) is truncated Normal-Inverse-Wishart:

$\begin{array}{ccc}\begin{array}{cc}\mathbf{Q}|Y& ~\mathit{\text{IW}}\left(\stackrel{˜}{\mathbf{Q}}\end{array}\left(T+{T}^{*}-N\right)\right)& \phantom{\rule{7.0em}{0ex}}& \left(25\right)\end{array}$
$\begin{array}{ccc}\mathbf{G}|\mathbf{Q},Y~N\left(\stackrel{˜}{\mathbf{G}},{\mathbf{\Sigma }}_{G}\right)×{\mathbf{1}}_{G}& \phantom{\rule{7.0em}{0ex}}& \left(26\right)\end{array}$

Where

$\begin{array}{l}\stackrel{˜}{\mathbf{G}}={\left({X}^{*\prime }{X}^{*}+{X}^{\prime }X\right)}^{-1}\left({X}^{*\prime }{Y}^{*}+{X}^{\prime }Y\right)\\ \stackrel{˜}{\mathbf{Q}}=\left({Y}^{*\prime }{Y}^{*}+{Y}^{\prime }Y\right)-{\left({X}^{*\prime }{Y}^{*}+{X}^{\prime }Y\right)}^{\prime }{\left({X}^{*\prime }{X}^{*}+{X}^{\prime }X\right)}^{-1}\left({X}^{*\prime }{Y}^{*}+{X}^{\prime }Y\right)\\ {\mathbf{\Sigma }}_{G}=\mathbf{Q}\otimes {\left({X}^{*\prime }{X}^{*}+{X}^{\prime }X\right)}^{-1}.\end{array}$

As discussed in section III.B, to fix the scale of factors Ft in estimation, we do not estimate Q and instead set Q=IN. Given Q, we then only draw G using the posterior distribution (26). Finally, we enforce the stationarity of factors by discarding those draws of matrix G that have at least one eigenvalue greater than or equal to one in absolute value (explosive eigenvalues).

Appendix B. Data: Description And Transformations

Notes: Transformation codes: 0 – nothing; 1 – log(); 2 – dlog(); 3 – log of the ratio of subaggregate to aggregate; 4 – transformation described in Kryshko (2011), Section IV. Asterisk (*) indicates the transformed variable has been further linearly detrended.Source of data: Stock and Watson (2008), “Forecasting in Dynamic Factor Models Subject to Structural Instability,” available online at: http://www.princeton.edu/~mwatson/ddisk/hendryfestschrift_replicationfiles_April28_2008.zipFull sample available: 1959:Q1-2006:Q4. Sample used in estimation: 1984:Q1-2005:Q4.All series available at monthly frequency have been converted to quarterly by simple averaging in native units.

Appendix C. Tables and Figures

Table C1.

DFM: Principal Components Analysis

Sample: 1984Q1 2005Q4

Included observations: 88

Computed using: Ordinary correlations

Extracting 20 of 89 possible components

Table C2.

Pure DFM: Fraction of Unconditional Variance Captured by Factors

iid Measurement Errors; Dataset = DFM3.txt

on average, 100K draws, 20K burn-in

Table C3.

Data-Rich DSGE Model: Fraction of Unconditional Variance Captured by DSGE Model States

iid Measurement Errors; Dataset = DFM3.txt

on average, 20K draws, 4K burn-in

Table C4.

Pure DFM: Unconditional Variance Captured by Factors

iid Measurement Errors; Dataset = DFM3.txt

on average, 100K draws, 20K burn-in

Notes: Please see Appendix B, p. 29 for the corresponding mnemonics of data indicators reported here.
Table C5.

Data-Rich DSGE Model: Fraction of Unconditional Variance Captured by DSGE Model States

iid Measurement Errors; Dataset = DFM3.txt

on average, 20K draws, 4K burn-in

Notes: Structural shocks are GOV – government spending, CHI – money demand, MP – monetary policy and Z – neutral technology. Please see Appendix B, p. 29 for the corresponding mnemonics of data indicators reported here.
Table C6.

Regressing Data-Rich DSGE Model States on DFM Factors

Notes: Each line reports the R2 from predictive linear regression: ${S}_{i,t}^{\left(\mathit{\text{pm}}\right)}={\alpha }_{0,i}+{\mathbf{\alpha }}_{1,i}^{\prime }{F}_{t}^{\left(\mathit{\text{pm}}\right)}+{v}_{i,t},$ where ${S}_{i,t}^{\left(\mathit{\text{pm}}\right)}$ is the posterior mean of the ith data-rich DSGE model state variable and ${F}_{t}^{\left(\mathit{\text{pm}}\right)}$ is the posterior mean of the empirical factors extracted by DFM.
Table C7.

Regressing DFM Factors on Data-Rich DSGE Model States

Notes: Each line reports the R2 from predictive linear regression (see (17) in the main text): ${F}_{i,t}^{\left(\mathit{\text{pm}}\right)}={\beta }_{0,i}+{\mathbf{\beta }}_{1,i}^{\prime }{S}_{t}^{\left(\mathit{\text{pm}}\right)}+{u}_{i,t},$ where ${F}_{i,t}^{\left(\mathit{\text{pm}}\right)}$ is the posterior mean of the ith empirical factor extracted by DFM and ${S}_{t}^{\left(\mathit{\text{pm}}\right)}$ is the posterior mean of the datarich DSGE model state variables.

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