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):

Ft=GFt1+ηt,ηt~iid N(0,Q),t=1,T.(21)

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

Rewrite the VAR in matrix notation

Y=XG+η(22)

where Y, X and η are the (T–1)×N matrices with rows Ft,Ft1 and ηt, 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:

p(G,Q)p(Y*|G,Q)|Q|(N+1)/2×1G,(23)

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:

p(G,Q|Y)p(Y|G,Q)=p(Y|G,Q)p(Y*|G,Q)|Q|(N+1)/2×1G.(24)

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:

Q|Y~IW(Q˜(T+T*N))(25)
G|Q,Y~N(G˜,ΣG)×1G(26)

Where

G˜=(X*X*+XX)1(X*Y*+XY)Q˜=(Y*Y*+YY)(X*Y*+XY)(X*X*+XX)1(X*Y*+XY)ΣG=Q(X*X*+XX)1.

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

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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

Figure C1.
Figure C1.

DFM: Principal Components Analysis

Data set: DFM3.TXT (standardized)

Citation: IMF Working Papers 2011, 216; 10.5089/9781463903497.001.A999

Table C1.

DFM: Principal Components Analysis

Sample: 1984Q1 2005Q4

Included observations: 88

Computed using: Ordinary correlations

Extracting 20 of 89 possible components

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Table C2.

Pure DFM: Fraction of Unconditional Variance Captured by Factors

iid Measurement Errors; Dataset = DFM3.txt

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

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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

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Table C4.

Pure DFM: Unconditional Variance Captured by Factors

iid Measurement Errors; Dataset = DFM3.txt

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

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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

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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

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Notes: Each line reports the R2 from predictive linear regression: Si,t(pm)=α0,i+α1,iFt(pm)+vi,t, where Si,t(pm) is the posterior mean of the ith data-rich DSGE model state variable and Ft(pm) is the posterior mean of the empirical factors extracted by DFM.
Table C7.

Regressing DFM Factors on Data-Rich DSGE Model States

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Notes: Each line reports the R2 from predictive linear regression (see (17) in the main text): Fi,t(pm)=β0,i+β1,iSt(pm)+ui,t, where Fi,t(pm) is the posterior mean of the ith empirical factor extracted by DFM and St(pm) is the posterior mean of the datarich DSGE model state variables.
Figure C2.
Figure C2.

Data-Rich DSGE Model (iid errors): Estimated Model States

Citation: IMF Working Papers 2011, 216; 10.5089/9781463903497.001.A999

Notes: Source – Kryshko (2011). Figure depicts the posterior means and 90% credible intervals of the data-rich DSGE model state variables (blue line & bands): inflation (PI_T, πt), nominal interest rate (R_T, Rt), real consumption (X_T, xt), government spending shock (GOV_T, gt), money demand shock (CHI_T, χt), and neutral technology shock (Z_T, Zt). Red line corresponds to the smoothed versions of the same variables in a regular DSGE model estimation derived by Kalman smoother at posterior mean of deep structural parameters (see notes to Table D3 in Kryshko (2011) for definition of “regular DSGE estimation”).
Figure C3.
Figure C3.

Pure DFM (iid errors): Estimated Factors

Citation: IMF Working Papers 2011, 216; 10.5089/9781463903497.001.A999

Notes: The figure plots the posterior means and 90% credible intervals of the latent empirical factors extracted by the empirical DFM (7)-(9). Normalization: block diagonal. Algorithm: Jungbacker-Koopman (2008).
Figure C4.
Figure C4.

Do Empirical Factors and DSGE Model State Variables Span the Same Space?

Pure DFM (iid errors): Estimated and Predicted FACTORS

Citation: IMF Working Papers 2011, 216; 10.5089/9781463903497.001.A999

Notes: The figure plots the actual empirical factors extracted by the DFM (7)-(9) (blue line) and the empirical factors predicted by the data-rich DSGE model state variables using (18) in the main text (red line).
Figure C5.
Figure C5.

Impact of Monetary Policy Innovation on Core Macro Series

Citation: IMF Working Papers 2011, 216; 10.5089/9781463903497.001.A999

Notes: The figure plots the impulse responses of data indicators to a 1-standard-deviation monetary policy innovation (εR, t) computed in the data-rich DSGE model (blue line, “DFM-DSGE”) and in empirical pure DFM (red line, “PDFM: all periods”) according to (19) and (20), respectively.The impact of structural shock is mapped from data-rich DSGE model into empirical DFM every period.Data indicators are real GDP (RGDP), industrial production: total (IP_total), industrial production: manufacturing (IP_mfg), GDP deflator inflation (PGDP), PCE deflator inflation (PCED), CPI inflation (CPI_ALL), Federal Funds rate (FedFunds), 3-month T-Bill rate (TBill_3m), yield on AAA rated corporate bonds (AAABond), real money balances based on M1S aggregate (IVM_M1S_det), on M2S aggregate (IVM_M2S), and on adjusted monetary base (IVM_MBase_bar). See the corresponding mnemonics in Appendix B, p. 29.
Figure C6.
Figure C6.

Impact of Monetary Policy Innovation on Non-Core Macro Series

Citation: IMF Working Papers 2011, 216; 10.5089/9781463903497.001.A999

Notes: The figure plots the impulse responses of data indicators to a 1-standard-deviation monetary policy innovation (εR, t) computed in the data-rich DSGE model (blue line, “DFM-DSGE”) and in empirical pure DFM (red line, “PDFM: all periods”) according to (19) and (20), respectively.The impact of structural shock is mapped from data-rich DSGE model into empirical DFM every period.Data indicators are real consumption of durables (RCons_Dur), real residential investment (RResInv), housing starts: West (HStarts_WST), employment in services sector (Emp_Services), unemployment rate (URate_all), commodity price inflation (P_COM), investment deflator inflation (PInv_GDP), consumer credit outstanding (Cons_Credit), 6-month over 3-month T-Bill rate spread (SFYGM6), US effective exchange rate depreciation (DLOG_EXR_US), exports price index (PExports), imports price index (PImports). See the corresponding mnemonics in Appendix B, p. 29.
Figure C7.
Figure C7.

Impact of Technology Innovation on Core Macro Series

Citation: IMF Working Papers 2011, 216; 10.5089/9781463903497.001.A999

Notes: The figure plots the impulse responses of data indicators to a 1-standard-deviation technology innovation (εZ, t) computed in the data-rich DSGE model (blue line, “DFM-DSGE”) and in empirical pure DFM (red line, “PDFM: all periods”) according to (19) and (20), respectively. The impact of structural shock is mapped from data-rich DSGE model into empirical DFM every period. Data indicators are real GDP (RGDP), industrial production: total (IP_total), industrial production: manufacturing (IP_mfg), GDP deflator inflation (PGDP), PCE deflator inflation (PCED), CPI inflation (CPI_ALL), Federal Funds rate (FedFunds), 3-month T-Bill rate (TBill_3m), yield on AAA rated corporate bonds (AAABond), real money balances based on M1S aggregate (IVM_M1S_det), on M2S aggregate (IVM_M2S), and on adjusted monetary base (IVM_MBase_bar). See the corresponding mnemonics in Appendix B, p. 29.
Figure C8.
Figure C8.

Impact of Technology Innovation on Non-Core Macro Series

Citation: IMF Working Papers 2011, 216; 10.5089/9781463903497.001.A999

Notes: The figure plots the impulse responses of data indicators to a 1-standard-deviation technology innovation (εZ, t) computed in the data-rich DSGE model (blue line, “DFM-DSGE”) and in empirical pure DFM (red line, “PDFM: all periods”) according to (19) and (20), respectively.The impact of structural shock is mapped from data-rich DSGE model into empirical DFM every period.Data indicators are real consumption of durables (RCons_Dur1), real residential investment (RResInv1), industrial production: business equipment (IP_BUS_eqpt), employment in services sector (Emp_Services), persons unemployed less than 5 weeks (U_l5wks), commodity price inflation (P_COM), investment deflator inflation (PInv_GDP), commercial and industrial loans (BUS_LOANS), 6-month over 3-month T-Bill rate spread (SFYGM6), US effective exchange rate depreciation (DLOG_EXR_US), real compensation per hour (RComp_Hour), average weekly hours worked (Hours_AVG). See the corresponding mnemonics in Appendix B, p. 29.

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