## APPENDIX A. DFM: GIBBS SAMPLER: DRAWING TRANSITION EQUATION MATRIX

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### I. Introduction — context and models
- Dynamic factor models (DFM) and dynamic stochastic general equilibrium (DSGE) models are widely used for empirical macroeconomic research.
- Traditional applications of DFMs include construction of coincident and leading indicators, forecasting of macro time series, real-time monitoring, monetary policy applications (Factor Augmented VAR), and study of international business cycles.
- DSGE models focus on sources of business cycle fluctuations and nominal rigidities; extended DSGE models are used for projections and policy analysis at major central banks.
- Core research question:
  - How similar or different are latent empirical factors extracted by a DFM versus estimated data-rich DSGE model states when both are estimated on the same rich data set?
  - Specific aims:
    - Determine whether empirical factors and DSGE states span a common factor space.
    - Assess predictability of estimated DFM latent factors from DSGE model states.

### Models: Dynamic Factor Model (DFM) and Data-Rich DSGE
- DFM specification (static stacked form):
  - Equation (1): X_t = Λ F_t + e_t
  - Equation (2): F_t = G F_{t-1} + η_t,  η_t ~ iid N(0,Q)
  - Equation (3): e_t = Ψ e_{t-1} + v_t,  v_t ~ iid N(0,R)
- DFM notation and assumptions:
  - F_t is a 1×N vector of common unobserved factors related to a 1×J large (J N≫) panel X_t.
  - Λ is JN× matrix of factor loadings; e_t are idiosyncratic errors allowed to be serially correlated.
  - G is the NN× matrix governing common factor dynamics; η_t are stochastic innovations.
  - Factors and idiosyncratic errors uncorrelated at all leads and lags: E(F_t e_s) = 0 for all t,s.
  - Following Stock and Watson (2005), Q, R and Ψ are assumed diagonal (exact dynamic factor model).
  - Static vs dynamic: static form loads on contemporaneous factors; a dynamic measurement with lag polynomials can be rewritten in static stacked form.
- Data-rich DSGE state-space (Kryshko (2011) variant):
  - Transition: S_t = G(θ) S_{t-1} + H(θ) ε_t,  ε_t ~ iid N(0,Q(θ))  (Equation (4))
  - Measurement: [X_t^F ; X_t^S] = [Λ(θ)_F ; Λ(θ)_S] S_t + e_t  (Equation (5))
  - Measurement errors: e_t ~ iid N(0, Ψ + R) with Ψ, R diagonal (Equation (6))
  - DSGE features: New Keynesian core with capital, Calvo price rigidities, investment adjustment costs, real money in utility, Taylor-type policy rule.

### Econometric methodology and estimation approach
- Both models estimated on Stock and Watson (2008) large U.S. panel; sample restricted to 1984:Q1 – 2005:Q4 for stability.
- Bayesian MCMC methods used for both:
  - DFM: Gibbs sampler (no Metropolis step because no deep structural θ).
  - Data-rich DSGE: Gibbs sampler with Metropolis step (Kryshko (2011)).
- DFM dimensionality choice:
  - dim(F_t) = N = 6, equal to number of non-redundant model states in the data-rich DSGE model.
  - Justification: first 6 principal components account for about 75 percent of variation; adding the 7th adds 4.4 percent (marginal improvement over 75 percent).
- Identification normalizations (following Geweke and Zhu 1996; Jungbacker and Koopman 2008):
  - (i) set Q = I_N to fix scale of factors;
  - (ii) require one loading in Λ to be positive for each factor (sign restriction);
  - (iii) normalize some factor loadings in Λ to pin down rotation.
- Block-diagonal identification scheme for Λ:
  - Upper NN× block Λ_1 and lower block Λ_2 with four blocks for core series grouping: real output, inflation, nominal interest rate, inverse velocity of money (pattern (12) in source).

### Gibbs sampler for the DFM — algorithmic steps
- Target posterior: p(Γ, F^{1:T} | X^{1:T}) ∝ p(F^{1:T} | Γ) p(X^{1:T} | F^{1:T}, Γ)  (Equations (13)-(14)).
- Main Gibbs steps:
  1. Specify initial values Γ^(0) and F^{(0),1:T}.
  2. For g = 1,..., n_sim:
     - 2.1 Generate latent factors F^{(g),1:T} from p(F^{1:T} | X^{1:T}; Γ^{(g-1)}) using Carter-Kohn (1994) forward-backward algorithm (Kalman filter + smoother).
     - 2.2 Generate state-space parameters Γ^{(g)} from p(Γ | X^{1:T}, F^{(g),1:T}) by drawing from known conditional densities (conjugate priors).
  3. Return {Γ^{(g)}, F^{(g),1:T)} for g = 1,..., n_sim.
- Sampling details:
  - Given F^{1:T} and diagonal R and Ψ, measurement equations decompose to multivariate linear regressions with AR(1) errors; use Chib and Greenberg (1994) for conditional posteriors of Λ, Ψ, R.
  - Given F^{1:T}, transition reduces to VAR(1): F_t = G F_{t-1} + η_t with η_t ~ iid N(0,I) (Equation (16)); draw G conditional on F^{1:T} using Minnesota prior implemented via dummy observations (Doan, Litterman and Sims 1984; Lubik and Schorfheide 2005), truncated to stationarity.

### Appendix A objective: drawing the transition matrix G
- Objective: generate G from conditional density p(G | F, X, Q, Λ, Ψ, R). Dependence on other matrices (except Q) is through factors F_t; transition is a VAR(1) given F_t.
- VAR(1) representation (Equation (21)):
  - Ft = G Ft-1 + ηt,  ηt ~ iid N(0, Q),  t = 1,...,T.
- Minnesota prior via dummy observations:
  - Prior combined with dummy observations Y*, X*; indicator 1_{G1} enforces stationarity (all eigenvalues of G inside unit circle).
  - Hyperparameters used: τ = 5, ι = 0.5, d = 1, μ = 0, λ = 0, w = 1.
  - Prior tilts transition toward univariate random walks.
- Posterior for (G, Q) is truncated Normal-Inverse-Wishart; with Q fixed at I_N (to fix factor scale), posterior reduces to truncated matrix normal for G.
- Drawing G:
  - Q is set to Q = I_N (not estimated).
  - Draw G from posterior; discard draws with any eigenvalue ≥ 1 in absolute value (enforce stationarity).

### Data, panel composition, and preprocessing
- Panel source and sample:
  - Stock and Watson (2008) panel covering 1959:Q1 – 2006:Q4; sample restricted to 1984:Q1 – 2005:Q4.
- Data composition:
  - 12 core series (used for DSGE concepts / DFM normalization).
  - 77 non-core informational series (load on all DSGE states / DFM factors).
- Core series details:
  - Real output: real GDP, index of total industrial production, index of industrial production: manufacturing.
  - Price inflation: GDP deflator inflation, PCE deflator inflation, CPI inflation.
  - Nominal interest rates: federal funds rate, 3-month T-bill rate, yield on AAA-rated corporate bonds.
  - Inverse velocity of money (IVM): IVM based on M1, IVM based on M2, IVM based on adjusted monetary base.
- Transformations:
  - Measurement equations modeled without intercepts ⇒ data demeaned.
  - Each time series standardized to have sample variance equal to unity (core series not scaled when estimating data-rich DSGE model).

### Computational implementation and priors
- Computational speed-up:
  - Gibbs sampler uses Jungbacker-Koopman (2008) speed-up as in Kryshko (2011), Section III.B.
  - “Improved” estimation runs 10.5 times faster than no-speedup estimation.
- DFM priors:
  - Joint Normal-InverseGamma prior for (Λ_k, R_kk).
  - RIGsν with location parameter 0.001 and degrees of freedom 3 (notation in source).
  - Prior mean of factor loadings: Λ_0 = 0; prior covariance involves 0N = M I (as in source notation).
  - Prior for Ψ_kk: (0,1)N (made perfectly tight to keep measurement errors iid mean-zero normal).
  - Minnesota prior for G with hyperparameters: 5τ =, 0.5d =, 1ι =, 1w =, 0λ =, 0μ = (notation preserved from source); prior truncated to stationarity region.
- Data-rich DSGE priors:
  - Measurement equation priors for non-core identical to DFM priors; core measurement equations and deep structural θ priors as in Kryshko (2011), Section V.A.

### Empirical factors, DSGE states, and fit metrics
- Estimated data-rich DSGE model states (posterior means and 90 percent credible intervals reported in figures):
  - Three endogenous variables: model inflation ˆtπ, nominal interest rate ˆtR, real household consumption ˆtX.
  - Three structural AR(1) shocks: government spending t_g, money demand t_χ, neutral technology t_Z.
  - DSGE states are smoother (common components) and structural disturbances exhibit large persistent swings.
- Estimated DFM latent factors (posterior means and 90 percent credible intervals):
  - N = 6 factors; F3–F6 partially interpretable due to normalization scheme.
  - Factors 3 and 5 resemble real output and nominal interest rate; factors 4 and 6 may not retain ex-ante normalized meanings ex-post.
  - Credible intervals for F1 and F2 vary over time; expansions prior to recessions suggest non-constant volatility / regime-switching.
- Fit and variance explained (average fraction of unconditional variance captured):
  - Data-rich DSGE model states:
    - about 75 percent of variance for the core macro series.
    - 72 percent of variance for the non-core series.
  - Latent empirical factors (DFM):
    - 95 percent of variance for the core series.
    - 94 percent of variance for the non-core series.
- Series-specific examples:
  - DSGE states capture about 15 to 25 percent of variance in exchange rate depreciations and stock returns.
  - DSGE states capture about 65 to 85 percent of variance of interest rate spreads and credit stocks.
  - DFM latent factors explain about 97-98 percent of variance of interest spreads and credit stocks; about 65-82 percent of variability in exchange rate depreciations; 80-82 percent of stock returns.
  - Implication: DSGE model potentially misspecified along the “financial” dimension.

### Comparing factor spaces and predictive regressions
- Notation: ()pm_tF and ()pm_tS denote posterior means of empirical factors and data-rich DSGE state variables.
- For each DFM factor ()pm_itF estimate OLS predictive regression:
  - ()() ,0,1,, pmpm itii tit FSuβ ′ = + + β  (equation (17) in source) with mean-zero errors.
  - Predicted empirical factors: ()ˆ pm_tF = ()() 01 ˆˆ pmpm_tt FS = + ββ  (equation (18) in source).
- Results (reported in Tables C6 and C7; Figure C4):
  - Regressing DSGE states on DFM factors: R2 values:
    - Inflation PI_t: R2 0.984
    - Interest Rate R_t: R2 0.991
    - Real Consumption X_t: R2 0.998
    - GOV_t: R2 0.999
    - CHI_t: R2 0.999
    - Z_t: R2 0.990
  - Regressing DFM factors on DSGE states: R2 values:
    - Factor 1: R2 0.979
    - Factor 2: R2 0.924
    - Factor 3: R2 0.949
    - Factor 4: R2 0.981
    - Factor 5: R2 0.989
    - Factor 6: R2 0.991
- Interpretation:
  - DSGE states predict empirical DFM factors very well ⇒ factor spaces spanned by DSGE states and by DFM latent factors are closely aligned.
  - Close alignment implies DSGE captures essential sources of co-movement; differences in fit may reflect restricted factor loadings in DSGE.
  - Proximity enables mapping structural shocks into DFM factor space for wide-panel propagation.

### Propagation of structural shocks through DFM (impulse responses)
- Mapping procedure:
  - Use posterior means of factor loadings (dfm Λ and dfm–dsge Λ) and predicted factors to generate impulse responses for panel X_t via expressions (19) and (20) in source that utilize Sε,+∂∂ from DSGE transition and link (18).
- Shocks analyzed:
  - Monetary policy shock R_t(ε) and technology shock Z_t(ε).
- Monetary policy innovation:
  - A contractionary monetary policy shock corresponds to 0.75 percent (75 basis points) increase in the federal funds rate.
  - Qualitative mechanism from DSGE:
    - Higher nominal policy rate ⇒ lower real money balances and higher saving motive ⇒ lower current consumption ⇒ aggregate demand falls ⇒ firms reduce output and cut prices (limited by nominal rigidities) ⇒ falling aggregate prices and negative inflation rates.
  - Empirical comparisons:
    - DFM mapping helps correct the “price puzzle” present in some DSGE responses for PCE deflator and CPI inflation.
    - Industrial production response to tightening may differ from FAVAR findings due to inertia/identification differences.
    - Real GDP responses: DFM-implied response nearly negligible; data-rich DSGE-implied response about minus 0.5 percent.
  - Non-core responses (reported in Figure C6):
    - Declines in real consumption of durables, real residential investment, housing starts.
    - Falling commodity price inflation and investment deflator inflation.
    - Employment falls and unemployment rises with a lag; unemployment peaks in the 5th quarter after shock (hump-shaped).
    - Interest rate spreads widen; consumer credit contracts; dollar appreciates; deterioration of U.S. terms of trade.
  - Overall: DFM and data-rich DSGE predictions for non-core variables are fairly close; DFM predictions for core series appear more plausible in eliminating the price puzzle.
- Technology innovation (positive TFP shock):
  - Qualitative responses (Figures C7 and C8):
    - Real output broadly increases; some disagreement on real GDP magnitude between DFM and DSGE.
    - Capital investment and household consumption increase; prices face downward pressure due to lower marginal costs and persistent technology ⇒ firms expect lower future inflation ⇒ current prices fall.
    - Under estimated Taylor rule, Fed lowers policy rate in response to lower inflation ⇒ interest rates fall, boosting output further; later Fed raises rate as effects dissipate (hump in interest rate IRF).
    - Money balances rise with a lag (M1S and monetary base more than M2S).
    - Non-core responses: expansion in subcomponents of real GDP (durables consumption, residential investment), increases in employment; commodity and investment deflator inflation fall; interest rate spreads shrink; credit eases; dollar appreciates modestly; real wage increases while average hours decline and remain below steady state for a prolonged period.
  - Comparison to literature: shapes and magnitudes broadly align with regular DSGE results; differences in hours dynamics and persistence reflect model-specific persistence.

### Principal components, variance breakdowns, and selected numerical results from appendices
- Principal components analysis (sample 1984:Q1–2005:Q4; 88 observations; extracted components, Table C1):
  - Eigenvalues and proportions (first 20 components):
    - 1 | 19.82739 | Proportion 0.2228 | Cumulative Proportion 0.2228
    - 2 | 17.19605 | Proportion 0.1932 | Cumulative Proportion 0.4160
    - 3 | 11.53612 | Proportion 0.1296 | Cumulative Proportion 0.5456
    - 4 | 7.696642 | Proportion 0.0865 | Cumulative Proportion 0.6321
    - 5 | 6.321275 | Proportion 0.0710 | Cumulative Proportion 0.7031
    - 6 | 4.194795 | Proportion 0.0471 | Cumulative Proportion 0.7503
    - 7 | 3.923900 | Proportion 0.0441 | Cumulative Proportion 0.7943
    - 8 | 2.703644 | Proportion 0.0304 | Cumulative Proportion 0.8247
    - 9 | 2.398092 | Proportion 0.0269 | Cumulative Proportion 0.8517
    - 10 | 1.661967 | Proportion 0.0187 | Cumulative Proportion 0.8703
    - 11 | 1.501482 | Proportion 0.0169 | Cumulative Proportion 0.8872
    - 12 | 1.221368 | Proportion 0.0137 | Cumulative Proportion 0.9009
    - 13 | 0.983267 | Proportion 0.0110 | Cumulative Proportion 0.9120
    - 14 | 0.943250 | Proportion 0.0106 | Cumulative Proportion 0.9226
    - 15 | 0.690347 | Proportion 0.0078 | Cumulative Proportion 0.9303
    - 16 | 0.627333 | Proportion 0.0070 | Cumulative Proportion 0.9374
    - 17 | 0.589301 | Proportion 0.0066 | Cumulative Proportion 0.9440
    - 18 | 0.519803 | Proportion 0.0058 | Cumulative Proportion 0.9498
    - 19 | 0.481761 | Proportion 0.0054 | Cumulative Proportion 0.9553
    - 20 | 0.419039 | Proportion 0.0047 | Cumulative Proportion 0.9600
- Fraction of unconditional variance captured (selected entries from Tables C2–C5):
  - Pure DFM (Table C2 / C4; algorithm Jungbacker-Koopman; identification: Block Diagonal):
    - Core Variables: Factors 0.948 | Measurement Error 0.052
    - Real output: Factors 0.993 | Measurement Error 0.007
    - Inflation: Factors 0.896 | Measurement Error 0.104
    - Interest rates: Factors 0.990 | Measurement Error 0.010
    - Money velocities: Factors 0.914 | Measurement Error 0.086
    - Non-Core Variables: Factors 0.941 | Measurement Error 0.059
    - Financial variables: Factors 0.854 | Measurement Error 0.146
  - Data-rich DSGE (Tables C3 / C5; on average, 20K draws, 4K burn-in):
    - Examples from Table C5 decompositions:
      - Real GDP: All 0.081 | GOV 0.000 | CHI 0.040 | MP 0.648 | Shocks 0.770 | Error 0.230
      - IP_Total: All 0.167 | GOV 0.308 | CHI 0.021 | MP 0.395 | Shocks 0.891 | Error 0.110
      - GDP Def inflation: All 0.011 | GOV 0.000 | CHI 0.011 | MP 0.789 | Shocks 0.811 | Error 0.189
      - FedFunds: All 0.004 | GOV 0.000 | CHI 0.135 | MP 0.817 | Shocks 0.956 | Error 0.044
      - U_DURATION: All 0.656 | GOV 0.149 | CHI 0.015 | MP 0.147 | Shocks 0.967 | Error 0.033
- Unconditional variance breakdown by factor (selected rows from Table C4, Pure DFM):
  - Real GDP: F1 0.119 | F2 0.142 | F3 0.301 | F4 0.160 | F5 0.115 | F6 0.148 | Factors 0.984 | Error 0.016
  - IP_Total: F1 0.137 | F2 0.105 | F3 0.343 | F4 0.135 | F5 0.113 | F6 0.164 | Factors 0.996 | Error 0.004
  - GDP Def inflation: F1 0.147 | F2 0.173 | F3 0.166 | F4 0.169 | F5 0.110 | F6 0.142 | Factors 0.907 | Error 0.094
  - FedFunds: F1 0.135 | F2 0.169 | F3 0.185 | F4 0.169 | F5 0.186 | F6 0.148 | Factors 0.993 | Error 0.008

### Figures and impulse-response comparisons (appendix summaries)
- Figures reported:
  - Figure C2: Data-Rich DSGE Model states (posterior means and 90% credible intervals).
  - Figure C3: Pure DFM factors (posterior means and 90% credible intervals).
  - Figure C4: Actual empirical factors vs empirical factors predicted by DSGE states — close alignment for Factors 1–6.
  - Figures C5–C8: Impulse responses (posterior means and 90% credible intervals) comparing:
    - DFM-DSGE mapping (blue line) vs Pure DFM (red line) for monetary policy and technology shocks on core and non-core series across horizons 1–40.
  - Noted features:
    - Monetary policy shock impulse responses include RGDP, IP_total, IP_mfg, PGDP, PCED, CPI_ALL, FedFunds, TBill_3m, AAABond, IVM measures.
    - Non-core responses include RCons_Dur, RResInv, HStarts_WST, Emp_Services, URate_all, commodity prices, investment deflator, consumer credit, exchange rates.
    - Technology shock impulse responses include expansions in real activity components, declines in commodity and investment deflator inflation, credit easing, and modest dollar appreciation.

### Key conclusions and practical implications
- The factor spaces spanned by empirical DFM factors and by data-rich DSGE model states are very closely aligned (high R2 in predictive regressions).
- Implications:
  - DSGE models capture essential sources of co-movement in a large panel; differences in fit vs DFM may result from restricted factor loadings in DSGE.
  - Close alignment increases confidence in propagating structural shocks from DSGE to a wide array of macro and financial series via linear mapping into DFM factor space.
  - Empirical DFM factors can be economically interpreted via isomorphism to DSGE state variables.
- Practical payoff:
  - Enables propagation of monetary policy and technology innovations in a non-structural DFM to obtain predictions for many series (real activity, price indices, labor market indicators, interest rate spreads, money and credit stocks, exchange rates).
  - Non-core variable responses serve as empirical plausibility checks on DSGE specifications.

*Source: APPENDIX A. DFM: GIBBS SAMPLER: DRAWING TRANSITION EQUATION MATRIX; APPENDIX C. TABLES AND FIGURES (content unit: _wp11216).*

### APPENDIX A. DFM: GIBBS SAMPLER: DRAWING TRANSITION EQUATION MATRIX ............... 27

### APPENDIX A. DFM: GIBBS SAMPLER: DRAWING TRANSITION EQUATION MATRIX ............... 27

### I. INTRODUCTION — context and models
- Dynamic factor models (DFM) and dynamic stochastic general equilibrium (DSGE) models are widely used for empirical research in macroeconomics.
- Traditional applications of DFMs include:
  - construction of coincident and leading indicators (Stock and Watson 1989, Altissimo et al. 2001, Matheson 2011),
  - forecasting of macro time series (Stock and Watson 1999, 2002a, b; Forni, Hallin, Lippi and Reichlin 2003; Boivin and Ng 2005),
  - real-time monitoring (Giannone, Reichlin, Small 2008; Aruoba, Diebold, and Scotti 2009; Aruoba, Diebold 2010),
  - monetary policy applications (Factor Augmented VAR approach of Bernanke, Boivin, and Eliasz 2005, Stock and Watson 2005),
  - study of international business cycles (Kose, Otrok, Whiteman 2003, 2008; Del Negro and Otrok 2008; Aruoba, Diebold, Kose, Terrones 2011).
- DSGE models focus on sources of business cycle fluctuations and assessing nominal rigidities and frictions; recent DSGE work replicates many salient data features (Christiano, Eichenbaum, and Evans 2005; Smets and Wouters 2003, 2007).
- DSGE models extended to open economy and multisector contexts are used for projections and policy analysis at major central banks (Adolfson et al. 2007, 2008; Edge, Kiley and Laforte 2009; Coenen, McAdam and Straub 2008).

### Empirical factor literature and number of factors
- Co-movement of large panels of macroeconomic and financial data can be captured by relatively few common unobserved factors.
- Historical and recent views on the stochastic dimension of the U.S. economy:
  - Sargent and Sims (1977): dynamic index model with two indices fits real variables well.
  - Giannone, Reichlin and Sala (2004): number of common shocks is two.
  - Bai and Ng 2007; Hallin and Liška 2007; Stock and Watson 2005: argue for a higher number of dynamic factors ranging from four to seven.

### DSGE states and data-rich DSGE approach
- DSGE dynamics often governed by a handful of state variables and exogenous processes (preference and/or technology shocks).
- Boivin and Giannoni (2006) combine DSGE and factor models into a data-rich DSGE model where DSGE states are factors and factor dynamics are subject to DSGE-implied restrictions.
  - The data-rich DSGE model can provide better estimates of DSGE states and structural shocks.
  - Boivin and Giannoni (2006) showed, and Kryshko (2011) confirms in related work, that data-rich DSGE estimation delivers different estimates of deep structural parameters compared to standard non-data-rich estimation.

### Research question and motivation
- Core question: How similar or different are the latent empirical factors extracted by a DFM versus the estimated data-rich DSGE model states when both are estimated on the same rich data set?
- Specific aims:
  - Determine whether empirical factors and DSGE states span a common factor space.
  - Assess whether true estimated DFM latent factors can be predicted from DSGE model states with reasonable accuracy.
- Reasons for comparing factor spaces:
  - Evaluation tool for DSGE model specification: if spaces are close, DSGE captures sources of co-movement in the large panel, and differences in fit may be due to restricted factor loadings.
  - Economic interpretation: alignment of factor spaces facilitates interpretation of empirical factors as isomorphic to DSGE state variables with clear economic meaning.
  - Policy propagation: if spaces are close, structural shocks from DSGE can be propagated through a non-structural DFM to obtain predictions for a broad range of macro series.

### Data, models, and estimation approach
- Both a data-rich DSGE model and an empirical dynamic factor model are estimated on a rich panel of U.S. macroeconomic and financial data compiled by Stock and Watson (2008).
- The specific data-rich DSGE variant is taken from Kryshko (2011).
- Estimation uses Bayesian Markov Chain Monte Carlo (MCMC) methods.

### Key findings (as presented)
- The spaces spanned by empirical factors and by data-rich DSGE model states are very close.
- Using a collection of linear regressions, the authors are able to predict the true estimated factors from the DSGE states fairly accurately.
- Given the predictive accuracy, the paper maps impacts of structural DSGE shocks on data-rich DSGE states into empirical factors period by period.
- Method to obtain impulse responses for many series:
  - Multiply responses of empirical factors by DFM factor loadings to generate impulse responses of data indicators to structural shocks.
- Examples of propagated impulse responses (from monetary policy innovations):
  - Contractionary monetary policy leads to a decline in housing starts and in residential investment.
  - Unemployment rate exhibits a hump-shaped positive response peaking in the 5th quarter after the shock before returning to normal.
  - Commodity price inflation shows negative rates of inflation following the shock.
  - Widening of interest rate spreads.
  - Contraction of consumer credit.
  - Appreciation of the dollar.
- Notation regarding coverage: these propagated responses occur despite the DSGE model not explicitly modeling many of these features.

### Relation to prior methods
- The approach is similar in spirit to:
  - Factor Augmented VAR (FAVAR) implemented by Bernanke, Boivin and Eliasz (2005) and Stock and Watson (2005) to study monetary policy impact on a large panel.
  - Structural factor model of Forni, Giannone, Lippi and Reichlin (2009).
- Closest related work: Bäurle (2008) — incorporates prior information from a DSGE model into DFM estimation and analyzes the impact of monetary policy shocks on factors and selected data series.

### Paper organization (overview)
- Section II: presents the variant of a dynamic factor model and snapshot of the data-rich DSGE model used.
- Section III: discusses the econometric methodology to estimate the two models.

*Source: APPENDIX A. DFM: GIBBS SAMPLER: DRAWING TRANSITION EQUATION MATRIX, from the provided content unit.*

### Section  IV  describes  our  data  set  and  transformations.  In  Section  V  we  proceed  by

### _wp11216 - Section  IV  describes  our  data  set  and  transformations.  In  Section  V  we  proceed  by

### Structure and roadmap
- Section IV describes the data set and transformations.  
- Section V: empirical analysis steps
  - discuss the choice of the prior distributions of dynamic factor model’s parameters;
  - analyze estimated empirical factors and the posterior estimates of the DSGE model state variables and explore how well they capture co-movements in the data;
  - compare the spaces spanned by the latent empirical factors and by the data-rich DSGE model state variables;
  - use the proximity of the factor spaces to propagate monetary policy and technology innovations in a non-structural dynamic factor model to obtain predictions for macro series of interest.
- Section VI concludes.

### Two models (Section II)
- Purpose: describe a variant of a dynamic factor model (DFM) and present a data-rich DSGE model with a New Keynesian core estimated on the same large panel of macro and financial series.

A. Dynamic Factor Model
- Model specification (static form):
  - Equation (1): X_t = Λ F_t + e_t
  - Equation (2): F_t = G F_{t-1} + η_t, η_t ~ iid N(0,Q)
  - Equation (3): e_t = Ψ e_{t-1} + v_t, v_t ~ iid N(0,R)
- Notation and assumptions:
  - F_t is a 1×N vector of common unobserved factors related to a 1×J large (J N≫) panel X_t.
  - Λ is JN× matrix of factor loadings; e_t are idiosyncratic errors allowed to be serially correlated.
  - G is the NN× matrix governing common factor dynamics; η_t are stochastic innovations.
  - Factors and idiosyncratic errors are assumed uncorrelated at all leads and lags: E(F_t e_s) = 0 for all t,s.
  - Following Stock and Watson (2005), matrices Q, R and Ψ are assumed diagonal (exact dynamic factor model), implying E(e_it e_js)=0 for i≠j for all t,s.
- Static vs dynamic factors:
  - As written, the model is static because X_t loads only on contemporaneous factors (no lags).
  - Note: a general measurement equation with lag polynomials can be rewritten into the static form by stacking lags; in this paper static and dynamic factors coincide for simplicity.

B. Data-Rich DSGE Model
- Source: specific version taken from Kryshko (2011), Section II.
- Core features:
  - New Keynesian business cycle core with capital as factor of production, nominal price rigidities a la Calvo (1983), investment adjustment costs.
  - Real money stock enters households’ utility additively separable.
  - Agents: representative household, final and intermediate goods-producing firms, central bank.
  - Intermediate goods: monopolistic competition, hire labor and capital, face Calvo price rigidity and possible indexation to lagged inflation.
  - Monetary policy: central bank sets one-period nominal interest rate via a Taylor-type rule; supplies nominal money balances to meet equilibrium demand.
- State-space summary:
  - Transition equation for non-redundant DSGE state variables S_t:
    - S_t = G(θ) S_{t-1} + H(θ) ε_t, ε_t ~ iid N(0,Q(θ))  (Equation (4))
  - Measurement equations linking core macro series X_t^F and non-core informational macro series X_t^S to DSGE states:
    - [X_t^F ; X_t^S] = [Λ(θ)_F ; Λ(θ)_S] [S_t] + e_t  (Equation (5))
  - Measurement errors e_t may be serially correlated but are uncorrelated across different data indicators (Ψ, R diagonal): e_t ~ iid N(0, Ψ + R) (Equation (6))
- Observation: the data-rich DSGE model (4)-(6) resembles the DFM (1)-(3) with the transition of unobserved factors governed by a DSGE solution and some factor loadings restricted by economic structure.

### Econometric methodology (Section III)
- Estimation approach: Bayesian MCMC methods used for both models; refer to Kryshko (2011) for details on MCMC for the data-rich DSGE model and prior choices for factor loadings.
- Dynamic Factor Model estimation specifics:
  - Reuse the state-space form:
    - X_t = Λ F_t + e_t  (Equation (7))
    - F_t = G F_{t-1} + η_t, η_t ~ iid N(0,Q)  (Equation (8))
    - e_t = Ψ e_{t-1} + v_t, v_t ~ iid N(0,R)  (Equation (9))
  - Collect state-space matrices Γ = {Λ, Ψ, R, G} and latent factors F^{1:T} = {F_1, ..., F_T}.
  - Dimensionality choice:
    - Set dim(F_t) = N = 6, equal to the number of non-redundant model states in the data-rich DSGE model.
    - Justification: literature finds seven dynamic and seven static factors in similar panels; principal components analysis of the data reveals the first 6 principal components account for about 75 percent of variation; adding the 7th would add 4.4 percent to explained variance (marginal improvement over 75 percent).
  - Identification issues:
    - Scales and signs of F_t and Λ are not separately identified; any invertible NN× matrix P yields observational equivalence (Equations (10)-(11)); signs are ambiguous (factor sign flip).
    - Normalizations adopted (following Geweke and Zhu 1996; Jungbacker and Koopman 2008):
      (i) set N = Q I to fix scale of factors;
      (ii) require one loading in Λ to be positive for each factor (sign restrictions);
      (iii) normalize some factor loadings in Λ to pin down factor rotation.
    - Implementation: denote Λ_1 as the upper NN× block of Λ with Λ = [Λ_1; Λ_2]. Rather than lower-triangular restriction, choose a block-diagonal scheme exploiting group structure of core series (four blocks: real output, inflation, nominal interest rate, inverse velocity of money), where each block contains several measures of the same concept (examples given for output and inflation series).
      - The block-diagonal scheme is represented as pattern (12) with 1s indicating non-zero elements in Λ.
    - Acknowledge that this scheme imposes overidentifying restrictions but can be interpreted as a special case of Moench, Ng, and Potter (2008) dynamic hierarchical factor model which introduces intermediate block factors.

### Gibbs sampler for the DFM
- Target posterior: p(Γ, F^{1:T} | X^{1:T}) ∝ p(F^{1:T} | Γ) p(X^{1:T} | F^{1:T}, Γ)  (Equations (13)-(14))
- Main Gibbs steps:
  1. Specify initial values Γ^(0) and F^{(0),1:T}.
  2. For g = 1,..., n_sim:
     2.1 Generate latent factors F^{(g),1:T} from p(F^{1:T} | X^{1:T}; Γ^{(g-1)}) using Carter-Kohn (1994) forward-backward algorithm.
     2.2 Generate state-space parameters Γ^{(g)} from p(Γ | X^{1:T}, F^{(g),1:T}) by drawing from known conditional densities.
  3. Return {Γ^{(g)}, F^{(g),1:T)} for g = 1,..., n_sim.
- Differences vs data-rich DSGE MCMC:
  - No Metropolis step because there are no deep structural parameters θ.
  - Must draw matrix G from the transition equation of factors (in DSGE model G is pinned down by solving DSGE given θ).
- Sampling F^{1:T}:
  - Apply Kalman filter to (7)-(9) to get filtered factors F̂_{t|t}.
  - Use Kalman smoother recursions to sample full path F^{1:T} recursively.
- Sampling Γ | F^{1:T}, X^{1:T}:
  - With diagonal R and Ψ and conditional on factors, measurement equations become a set of standard multivariate linear regressions with AR(1) errors and Gaussian innovations (Equation (15)).
  - Under conjugate prior p(Λ, Ψ, R), use Chib and Greenberg (1994) to derive conditional posteriors for R|Λ,Ψ,G; Λ|R,Ψ,G; Ψ|Λ,R,G and sample accordingly.
- Sampling G:
  - Given factors F^{1:T}, the transition reduces to a VAR(1): F_t = G F_{t-1} + η_t with η_t ~ iid N(0,I) (Equation (16)).
  - Prior on G: Minnesota prior (Doan, Litterman and Sims, 1984; specific version from Lubik and Schorfheide, 2005), truncated to the region consistent with stationarity of (16).
  - Implementation of the prior via dummy observations that tilt the VAR towards univariate random walks.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11216.pdf*

### Appendix A).

### _wp11216 - Appendix A)

### Data
- Panel source: Stock and Watson (2008) large panel of U.S. quarterly macroeconomic and financial time series covering 1959:Q1 – 2006:Q4; sample restricted to 1984:Q1 – 2005:Q4 to avoid the Great Moderation and focus on a relatively stable monetary policy regime.
- Data composition:
  - 12 core series (measure specific DSGE concepts or used in the DFM normalization scheme).
  - 77 non-core informational series (load on all DSGE states / DFM factors).
- Core series include:
  - Three measures of real output: real GDP, index of total industrial production, index of industrial production: manufacturing.
  - Three measures of price inflation: GDP deflator inflation, PCE deflator inflation, CPI inflation.
  - Three indicators of nominal interest rates: federal funds rate, 3-month T-bill rate, yield on AAA-rated corporate bonds.
  - Three series measuring inverse velocity of money (IVM based on M1, IVM based on M2, IVM based on adjusted monetary base).
- Transformations and preprocessing:
  - Measurement equations modeled without intercepts ⇒ models estimated on demeaned data.
  - Each time series standardized to have sample variance equal to unity (core series not scaled when estimating the data-rich DSGE model).

### Computational implementation and speed
- Gibbs sampler implementation uses Jungbacker-Koopman (2008) computational speed-up as presented in Kryshko (2011), Section III.B.
- “Improved” estimation of the empirical DFM runs 10.5 times faster than the no-speedup estimation.

### Priors and posteriors (DFM and data-rich DSGE)
- DFM priors:
  - Joint Normal-InverseGamma prior for (Λk, Rkk).
  - ~ (,) RIGsν with location parameter 0 0.001s and degrees of freedom 0 3.
  - Prior mean of factor loadings: |~ RkΛ1,00 (,) N RkΛM with ,0kΛ0 and 0NMI.
  - Prior for measurement-equation autocorrelation Ψkk: (0,1)N (made perfectly tight to keep measurement errors iid mean-zero normal).
  - Minnesota prior for factor transition matrix G (tilt to univariate random walks).
  - Minnesota hyperparameters in actual implementation: 5, 0.5d, 1, 1w, 0, 0. Prior truncated to region consistent with stationarity of factor transition equation.
- Data-rich DSGE priors:
  - State-space matrices (Λ, Ψ, R): priors for non-core measurement equations identical to DFM priors; core measurement equations follow Kryshko (2011), Section V.A.
  - Deep structural parameters θ: priors identical to Kryshko (2011), Section V.A.
- Estimation algorithms:
  - DFM: Gibbs sampler (Section III.B).
  - Data-rich DSGE: Gibbs sampler with Metropolis step (Kryshko (2011), Section III.A).
- Posterior parameter estimates are collected in the technical appendix (available upon request); paper focuses on factor spaces rather than detailed posterior DFM parameters.

### Empirical factors and estimated DSGE model states
- Estimated data-rich DSGE model states (posterior means and 90 percent credible intervals shown in Figure C2):
  - Three endogenous variables: model inflation ˆtπ, nominal interest rate ˆtR, real household consumption ˆtX.
  - Three structural AR(1) shocks: government spending tg, money demand tχ, neutral technology tZ.
- Observations on DSGE states:
  - Structural disturbances exhibit large swings and prolonged deviations from zero (persistent low-frequency movements).
  - Data-rich DSGE states are much smoother than regular DSGE states (they are common components of a large panel).
  - Money demand shock tχ differs markedly between data-rich and regular DSGE estimations (helps explain housing, consumer credit, non-GDP output measures at cost of poorer fit for IVM_M2S).
  - Counterfactual behavior: government spending shock and real consumption during recessions tended to fall and rise, respectively, in the data-rich DSGE estimation.
- Estimated DFM latent factors (posterior means and 90 percent credible intervals shown in Figure C3):
  - Factors generally lack a direct economic interpretation; exceptions: F3-F6 due to normalization scheme (12).
  - Factors 3 and 5 resemble real output and nominal interest rate; factors 4 and 6 do not necessarily retain their ex-ante normalized meanings ex-post.
  - Credible intervals for F1 and F2 (common to all series) vary over time; expansion of F1 credible bands prior to 1990–91 recession and expansion of F2 prior to 2001 recession may indicate non-constant volatility / regime-switching dynamics.

### How well factors trace data (fit and variance decomposition)
- Average fraction of unconditional variance of data series captured:
  - Data-rich DSGE model states:
    - about 75 percent of variance for the core macro series.
    - 72 percent of variance for the non-core series.
  - Latent empirical factors (DFM):
    - 95 percent of variance for the core series.
    - 94 percent of variance for the non-core series.
- Series-specific findings:
  - Within core series: measures of inflation and inverse money velocities traced more poorly than real output and nominal interest rates in both models.
  - Within non-core series: price and wage inflation measures and financial variables have higher fractions of variance due to measurement errors in both models.
  - Data-rich DSGE model states capture:
    - about 15 to 25 percent of variance in exchange rate depreciations and stock returns.
    - about 65 to 85 percent of variance of interest rate spreads and credit stocks.
  - DFM latent factors explain:
    - about 97-98 percent of variance of interest spreads and credit stocks.
    - about 65-82 percent of variability in exchange rate depreciations.
    - 80-82 percent of stock returns.
- Implication: DSGE model potentially misspecified along the “financial” dimension.

### Comparing factor spaces
- Notation: ()pm tF and ()pm tS denote posterior means of empirical factors and data-rich DSGE state variables.
- For each latent factor ()pm itF, estimated OLS predictive regression:
  - ()() ,0,1,, pmpm itii tit FSuβ ′ β  (equation (17)) with mean-zero homoscedastic errors uit.
  - Construct predicted empirical factors ()ˆ pm tF = ()() 01 ˆˆ pmpm tt FSββ (equation (18)).
- Results (Table C7 and Figure C4): DSGE states predict empirical DFM factors very well ⇒ factor spaces spanned by DSGE state variables and by DFM latent factors are very closely aligned.
- Implications:
  - DSGE model captures essential sources of co-movement in the large panel (core).
  - Differences in fit between data-rich DSGE and DFM may stem from restricted factor loadings in the DSGE model.
  - Proximity of factor spaces supports propagation of structural shocks to a wide array of macro and financial series and facilitates economic interpretation of empirical factors via isomorphism to DSGE states.

### Propagation of monetary policy and technology innovations
- Mapping structural shocks from DSGE into DFM prediction space:
  - Using posterior means of factor loadings dfm Λ and dfm–dsge Λ (data-rich DSGE and empirical DFM), generate impulse responses for panel tX via expressions (19) and (20) that use Sε,+∂∂ computed from DSGE transition equation and link (18).
- Focused shocks:
  - Monetary policy shock R t() ε and technology shock Z t() ε .
- Monetary policy innovation:
  - A contractionary monetary policy shock corresponds to 0.75 percent (or 75 basis points) increase in the federal funds rate.
  - Mechanisms and qualitative responses:
    - Higher nominal policy rate ⇒ lower real money balances (strong liquidity effect) and higher saving motive ⇒ households reduce current consumption ⇒ aggregate demand falls ⇒ firms reduce output and cut prices (limited by nominal rigidities).
    - Firms cut marginal costs via lower return on capital and lower real wages (labor market perfectly competitive; no wage rigidities).
    - Result: falling aggregate prices and negative rates of inflation (consistent with New Keynesian Phillips curve).
  - Empirical comparison:
    - Channeling monetary shock through DFM helps correct the “price puzzle” observed in data-rich DSGE responses of PCE deflator inflation and CPI inflation.
    - Industrial production (IP) response to tightening may appear counterfactual vis-à-vis FAVAR findings (inertia and identification differences may explain).
    - Real GDP responses differ: DFM-implied response is almost negligible; data-rich DSGE-implied response is about minus 0.5 percent.
  - Non-core variable responses (Figure C6): declines in real consumption of durables, real residential investment, housing starts; falling commodity price inflation and investment deflator inflation; employment falls and unemployment rises with a lag; interest rate spreads widen; consumer credit contracts; dollar appreciates; deterioration of U.S. terms of trade.
  - Overall: DFM and data-rich DSGE predictions for non-core variables are fairly close; DFM predictions for core series appear more plausible in eliminating the price puzzle.
- Technology innovation (positive TFP shock):
  - Qualitative responses (Figures C7 and C8):
    - Real output broadly increases (disagreement between DFM and DSGE on real GDP response magnitude).
    - Increased capital investment and household consumption; downward pressure on prices due to lower marginal costs and persistent technology ⇒ firms expect lower future inflation ⇒ current prices fall.
    - Under estimated Taylor rule, Fed lowers policy rate in response to lower inflation ⇒ interest rates fall, boosting output further ⇒ later, as technology effect dissipates, inflationary expectations rise and Fed increases policy rate (hump in interest rate IRF).
    - Money balances increase with a lag (notably for M1S and monetary base more than M2S).
  - Non-core responses: expansion in subcomponents of real GDP (durables consumption, residential investment), increases in employment measures; commodity and investment deflator inflation fall; interest rate spreads shrink; credit conditions ease; dollar appreciates modestly; real wage increases while average hours decline and remain below steady state for a prolonged period.
  - Comparison to literature: shapes and magnitudes align fairly closely with regular DSGE results (Smets and Wouters, Aruoba and Schorfheide, DSSW), though some differences in hours dynamics and persistence likely due to model-specific persistence in technology.

### Key conclusions
- The factor spaces spanned by empirical DFM factors and by data-rich DSGE model states are very closely aligned.
- Implications:
  - DSGE models capture essential co-movement sources in a large panel of data; differences in fit vs DFM may result from restricted factor loadings in DSGE.
  - Close alignment increases confidence in propagating structural shocks from DSGE to a wide array of macro and financial series.
  - Empirical DFM factors can be economically interpreted via isomorphism to DSGE state variables through linear mapping.
- Practical payoff:
  - Proximity enables propagation of monetary policy and technology innovations in an otherwise non-structural DFM to obtain predictions for many more series (real activity, price indices, labor market indicators, interest rate spreads, money and credit stocks, exchange rates).
  - Non-core variable responses enrich assessment of policy shocks and can serve as an empirical plausibility check on DSGE specifications.

*Source: _wp11216 - Appendix A).*

### APPENDIX A. DFM: GIBBS SAMPLER: DRAWING TRANSITION EQUATION MATRIX

### APPENDIX A. DFM: GIBBS SAMPLER: DRAWING TRANSITION EQUATION MATRIX

### Objective
- Generate G from the conditional density p(G | F, X, Q, Λ, Ψ, R).  
- Dependence of G on other state-space matrices (except for Q) is exclusively through the factors Ft because, given factors Ft, the transition equation is a VAR(1).

### VAR(1) representation (given factors)
- Transition equation (8) reduces to a VAR(1):
  - Ft = G Ft-1 + ηt,  ηt ~ iid N(0, Q), t = 1,...,T.  (Equation (21))
- Conditional factor likelihood separation:
  - p(G | X, T, T, Q, Λ, Ψ, R) = p(G | F, Q).  
- Matrix notation for the VAR(1):
  - Y = X G' + η  (written in the appendix as Y X η = G + η; equivalently YXη = G in Equation (22))
  - Y, X and η are (T × N) matrices with rows Ft′, Ft-1′ and ηt′, respectively.

### Prior for VAR parameters (Minnesota prior via dummy observations)
- Use a version of Minnesota Prior (Doan, Litterman, Sims 1984) implemented with T* dummy observations Y* and X* (following Lubik and Schorfheide 2005).
- Combine likelihood of dummy observations p(Y* | G, Q) with an improper prior proportional to Q^{-(1)2 N^-+?} to induce a proper prior for VAR parameters (Equation (23)):
  - p(G, Q) ∝ p(Y* | G, Q) × 1_{G1}
  - 1_{G1} is an indicator equal to 1 if all eigenvalues of G lie inside the unit circle.
- Hyperparameters set in implementation (Minnesota Prior):
  - τ = 5, ι = 0.5, d = 1
  - μ = 0, λ = 0, w = 1
- Interpretation: prior tilts transition equation toward a collection of univariate random walks.

### Posterior for VAR parameters
- Combine prior with likelihood p(Y | G, Q) to obtain posterior (Equation (24)):
  - p(G, Q | Y) ∝ p(Y | G, Q) p(G, Q)
- The posterior density p(G, Q | Y) is truncated Normal-Inverse-Wishart (Del Negro, Schorfheide 2004):
  - Q | Y ~ Inverse-Wishart(·) and G | Q, Y ~ Matrix Normal (Equation (25) and (26) summarized)
- Specific posterior forms (as shown):
  - (G, Q | Y)* ~ TN-IW (notation in text)
  - G | Q, Y ~ Matrix Normal: N_{Y×G}( · , · )  (text: |, ~ (,())YIWTTN˜Q)
  - Q | · ~ IW(·)  (Equation (26) style expressions)
- Posterior sufficient statistics (as presented):
  - Definitions involving X, Y and dummy observations:
    - X̃′X̃ = (XXXX)^{-1} (notation in source)
    - Ỹ′Ỹ and cross-products YX, XY etc. appear in the expressions for posterior moments (preserved symbolic forms as in source).

### Drawing G and handling Q
- To fix the scale of factors Ft in estimation, Q is not estimated and is instead set to Q = I_N.
- Given Q, draw G using the posterior distribution given in Equation (26) (the truncated Normal-Inverse-Wishart conditional reducing to a truncated matrix normal for G when Q is fixed).
- Enforce stationarity by discarding any draw of G that has at least one eigenvalue greater than or equal to one in absolute value (explosive eigenvalues).

*Source: APPENDIX A. DFM: GIBBS SAMPLER: DRAWING TRANSITION EQUATION MATRIX*

### APPENDIX C. TABLES AND FIGURES

### _wp11216 - APPENDIX C. TABLES AND FIGURES

### Principal components analysis (DFM)
- Sample: 1984Q1 2005Q4; Included observations: 88; Computed using: Ordinary correlations; Extracting 20 of 89 possible components; Eigenvalues: (Sum = 89, Average = 1)
- Table C1 — ordered components (Number | Value | Difference | Proportion | Cumulative Value | Cumulative Proportion):
  - 1 | 19.82739 | 2.631345 | 0.2228 | 19.82739 | 0.2228
  - 2 | 17.19605 | 5.659930 | 0.1932 | 37.02344 | 0.4160
  - 3 | 11.53612 | 3.839474 | 0.1296 | 48.55955 | 0.5456
  - 4 | 7.696642 | 1.375366 | 0.0865 | 56.25619 | 0.6321
  - 5 | 6.321275 | 2.126480 | 0.0710 | 62.57747 | 0.7031
  - 6 | 4.194795 | 0.270895 | 0.0471 | 66.77227 | 0.7503
  - 7 | 3.923900 | 1.220256 | 0.0441 | 70.69617 | 0.7943
  - 8 | 2.703644 | 0.305552 | 0.0304 | 73.39981 | 0.8247
  - 9 | 2.398092 | 0.736125 | 0.0269 | 75.79790 | 0.8517
  - 10 | 1.661967 | 0.160485 | 0.0187 | 77.45987 | 0.8703
  - 11 | 1.501482 | 0.280114 | 0.0169 | 78.96135 | 0.8872
  - 12 | 1.221368 | 0.238101 | 0.0137 | 80.18272 | 0.9009
  - 13 | 0.983267 | 0.040017 | 0.0110 | 81.16598 | 0.9120
  - 14 | 0.943250 | 0.252902 | 0.0106 | 82.10923 | 0.9226
  - 15 | 0.690347 | 0.063015 | 0.0078 | 82.79958 | 0.9303
  - 16 | 0.627333 | 0.038032 | 0.0070 | 83.42691 | 0.9374
  - 17 | 0.589301 | 0.069497 | 0.0066 | 84.01621 | 0.9440
  - 18 | 0.519803 | 0.038042 | 0.0058 | 84.53602 | 0.9498
  - 19 | 0.481761 | 0.062722 | 0.0054 | 85.01778 | 0.9553
  - 20 | 0.419039 | 0.054135 | 0.0047 | 85.43682 | 0.9600

### Fraction of unconditional variance captured by factors (Pure DFM) and by DSGE states
- Table C2 / Table C4 (Pure DFM: fraction of unconditional variance captured by factors; iid Measurement Errors; Dataset = DFM3.txt; on average, 100K draws, 20K burn-in; Algorithm: Jungbacker-Koopman; Identification: Scheme 2 - Block Diagonal)
  - Core Variables: Factors 0.948 | Measurement Error 0.052
  - Real output: Factors 0.993 | Measurement Error 0.007
  - Inflation: Factors 0.896 | Measurement Error 0.104
  - Interest rates: Factors 0.990 | Measurement Error 0.010
  - Money velocities: Factors 0.914 | Measurement Error 0.086
  - Non-Core Variables: Factors 0.941 | Measurement Error 0.059
  - Output and components: Factors 0.982 | Measurement Error 0.018
  - Labor market: Factors 0.981 | Measurement Error 0.019
  - Investment, inventories, orders: Factors 0.986 | Measurement Error 0.014
  - Housing: Factors 0.970 | Measurement Error 0.030
  - Prices and wages: Factors 0.908 | Measurement Error 0.092
  - Financial variables: Factors 0.854 | Measurement Error 0.146
  - Other: Factors 0.973 | Measurement Error 0.027
- Table C3 / 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)
  - Selected series (All | GOV | CHI | MP | Z | Shocks | Error) — Table C3 reports fractions for groups; Table C5 reports detailed decomposition for series. Examples from Table C5:
    - Real GDP: All 0.081 | GOV 0.000 | CHI 0.040 | MP 0.648 | Shocks 0.770 | Error 0.230
    - IP_Total: All 0.167 | GOV 0.308 | CHI 0.021 | MP 0.395 | Shocks 0.891 | Error 0.110
    - IP_MFG: All 0.166 | GOV 0.317 | CHI 0.020 | MP 0.392 | Shocks 0.894 | Error 0.106
    - GDP Def inflation: All 0.011 | GOV 0.000 | CHI 0.011 | MP 0.789 | Shocks 0.811 | Error 0.189
    - PCE Def inflation: All 0.004 | GOV 0.035 | CHI 0.003 | MP 0.703 | Shocks 0.745 | Error 0.255
    - CPI ALL Inflation: All 0.005 | GOV 0.031 | CHI 0.006 | MP 0.600 | Shocks 0.642 | Error 0.358
    - FedFunds: All 0.004 | GOV 0.000 | CHI 0.135 | MP 0.817 | Shocks 0.956 | Error 0.044
    - U_DURATION: All 0.656 | GOV 0.149 | CHI 0.015 | MP 0.147 | Shocks 0.967 | Error 0.033
    - U_M15WKS: All 0.575 | GOV 0.099 | CHI 0.018 | MP 0.284 | Shocks 0.977 | Error 0.023
    - (See Table C5 for the full list of series and corresponding six-column decompositions.)

### Unconditional variance breakdown by factor (selected series, Pure DFM)
- Table C4 (iid Measurement Errors; on average, 100K draws, 20K burn-in; Algorithm: Jungbacker-Koopman)
  - Each series shows contributions of Factors F1–F6 and Measurement Error. Selected rows:
    - Real GDP: F1 0.119 | F2 0.142 | F3 0.301 | F4 0.160 | F5 0.115 | F6 0.148 | Factors 0.984 | Error 0.016
    - IP_Total: F1 0.137 | F2 0.105 | F3 0.343 | F4 0.135 | F5 0.113 | F6 0.164 | Factors 0.996 | Error 0.004
    - IP_MFG: F1 0.131 | F2 0.105 | F3 0.350 | F4 0.136 | F5 0.114 | F6 0.162 | Factors 0.997 | Error 0.003
    - GDP Def inflation: F1 0.147 | F2 0.173 | F3 0.166 | F4 0.169 | F5 0.110 | F6 0.142 | Factors 0.907 | Error 0.094
    - PCE Def inflation: F1 0.148 | F2 0.177 | F3 0.168 | F4 0.173 | F5 0.110 | F6 0.145 | Factors 0.921 | Error 0.079
    - FedFunds: F1 0.135 | F2 0.169 | F3 0.185 | F4 0.169 | F5 0.186 | F6 0.148 | Factors 0.993 | Error 0.008
    - (Table C4 contains the full matrix for all series and F1–F6.)

### Regressions between DFM factors and data-rich DSGE model states
- Table C6 — Regressing Data-Rich DSGE Model States on DFM Factors (predictive linear regression R2)
  - Inflation PI_t: R2 0.984
  - Interest Rate R_t: R2 0.991
  - Real Consumption X_t: R2 0.998
  - Govt Spending shock GOV_t: R2 0.999
  - Money Demand shock CHI_t: R2 0.999
  - Technology shock Z_t: R2 0.990
- Table C7 — Regressing DFM Factors on Data-Rich DSGE Model States (predictive linear regression R2)
  - Factor 1: R2 0.979
  - Factor 2: R2 0.924
  - Factor 3: R2 0.949
  - Factor 4: R2 0.981
  - Factor 5: R2 0.989
  - Factor 6: R2 0.991

### Figures — estimated states, factors, and spanning
- Figure C2 (Data-Rich DSGE Model, iid errors): plots posterior means and 90% credible intervals for DSGE model state variables: PI_T (inflation), R_T (nominal interest rate), X_T (real consumption), GOV_T (government spending shock), CHI_T (money demand shock), Z_T (neutral technology shock). Notes indicate red line = smoothed versions from regular DSGE estimation; blue line & bands = posterior mean and 90% credible interval of data-rich DSGE state variables.
- Figure C3 (Pure DFM, iid errors): plots posterior means and 90% credible intervals of the six empirical factors extracted by the empirical DFM. Normalization: block diagonal. Algorithm: Jungbacker-Koopman (2008).
- Figure C4: compares actual empirical factors (DFM) and empirical factors predicted by data-rich DSGE model states (posterior means), showing close alignment for Factors 1–6 (posterior means plotted).

### Impulse response comparisons: DFM-DSGE vs pure DFM
- Figures C5–C8 present impulse responses (posterior means) and 90% credible intervals comparing:
  - DFM-DSGE (data-rich DSGE mapped to empirical DFM; blue line)
  - PDFM: all periods (pure empirical DFM; red line)
- Figure C5 — Impact of monetary policy innovation (Rt ε) on core macro series. Data indicators include: RGDP, IP_total, IP_mfg, PGDP, PCED, CPI_ALL, FedFunds, TBill_3m, AAABond, IVM_M1S_det, IVM_M2S, IVM_MBase_bar. Plots show impulse responses for horizons 1–40 (periods shown as 5,10,15,...,40 on x-axes).
- Figure C6 — Impact of monetary policy innovation (Rt ε) on non-core macro series. Data indicators include: RCons_Dur, RResInv, HStarts_WST, Emp_Services, URate_all, P_COM, PInv_GDP, Cons_Credit, SFYGM6, DLOG_EXR_US, PExports, PImports.
- Figure C7 — Impact of technology innovation (Zt ε) on core macro series. Data indicators same as Figure C5 (RGDP, IP_total, IP_mfg, PGDP, PCED, CPI_ALL, FedFunds, TBill_3m, AAABond, IVM_M1S_det, IVM_M2S, IVM_MBase_bar).
- Figure C8 — Impact of technology innovation (Zt ε) on non-core macro series. Data indicators include: RCons_Dur1, RResInv1, IP_BUS_eqpt, Emp_Services, U_l5wks, P_COM, PInv_GDP, BUS_LOANS, SFYGM6, DLOG_EXR_US, RComp_Hour, Hours_AVG.

*Source: APPENDIX C. TABLES AND FIGURES (content unit: _wp11216 - APPENDIX C. TABLES AND FIGURES).*

### REFERENCES

### REFERENCES

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### Dynamic factor models, nowcasting, and business-cycle indicators
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### Statistical methods, factor determination, and estimation techniques
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*Source: _wp11216 - REFERENCES*

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