## _wp16126

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---

### Abstract and Key Findings
- Title: Did the global financial crisis break the U.S. Phillips Curve?; Authors: Stefan Laseen and Marzie Taheri Sanjani; IMF Working Paper WP/16/126; July 2016.
- Primary empirical finding: The Phillips curve was not broken by the Global Financial Crisis (GFC).
- Main interpretation: Changes in shocks (variances) are a more salient feature of the data than changes in coefficients.
- Regime-switching model result: Allowing for regime switching solely in coefficients of the policy rule maximizes the fit among the variants considered.
- Best-fitting model features:
  - Change in coefficients only of the monetary policy rule (not of the Phillips curve).
  - Most important changes across regimes are the variances of structural disturbances.
- Forecasting/conditioning result: Financial and external variables have the highest forecasting power for inflation and unemployment post-GFC.
- Specific variable highlighted: Excess Bond Premium (EBP) contains rich information reflecting credit supply shocks and is an important driver of macroeconomic dynamics.

### Motivation, Hypotheses, and Methods
- Research question: How does unemployment affect inflation since the GFC; did the GFC break the U.S. Phillips curve?
- Puzzles motivating study:
  - “Missing disinflation”: inflation declined after the GFC but less than anticipated.
  - Inflation has been running below the 2 percent target for an extended period despite low unemployment.
- Three hypotheses examined for low inflation post-GFC:
  - (a) Financial frictions and shocks causing slow recoveries and persistently low inflation.
  - (b) Globalization increasing the role of international factors and decreasing the role of domestic factors in the inflation process.
  - (c) Constraints from the effective lower bound on policy rates preventing stabilization policy from lowering real interest rates enough to restore long-run inflation goals.
- Modeling approach:
  - Use multivariate, possibly time-varying models: Large Bayesian Vector Autoregressions (BVARs), Dynamic Factor Models (DFMs), and Markov-switching (MS)-BVARs.
  - Use conditional forecast analysis and counterfactuals to distinguish variance changes from structural parameter variations.
  - Examine informativeness of data blocks: External, Commodities, Labor, Interest Rates, Credit, Financial Risk, TFP.

### Data and Transformations
- Large cross-section dataset (Model 1):
  - Number of quarterly variables: 45 quarterly variables.
  - Sample period: 1987Q1 to 2015Q2.
  - Categories: Real activity, Prices, Monetary, Financial, TFP, Utilization, etc.
  - Data transformations: models feed on annualized log-levels except variables defined as annualized rates (interest rates, corporate bond yields, TFP, etc.) taken in levels.
- Monthly database for MS-VAR (six-variable MS-VAR):
  - Variables y_t = [Û_t ˆπ_PCE_t π_RPIM_t R_t TS_t EBP_t], monthly sample 1991 : 10 to 2015 : 6, all variables expressed at annual rates.
  - EBP (Excess Bond Premium) included because of strong forecasting power for economic activity.

### Models and Identification
- General nonlinear state-space representation:
  - A′0(sc t) y t = C(sc t) x t + Ξ−1 m(sv t) υ t,
  - x t = A′+(sc t) x t−1 + Ξ−1 s(sv t) ε t,
  - Latent Markov states sc and sv govern coefficient and variance switching respectively.
- Large cross-section specifications:
  - Bayesian Dynamic Factor Model (DFM): ∆y t = z t + ΛF t + Ξ−1 υ t; F t follows VAR dynamics.
  - Large BVAR in levels and in differences: y_t = z_t + A1 y_{t−1} + ... + A_p y_{t−p} + Ξ^{-1} ε_t; Δy_t specification similarly.
- Markov-switching structural VAR (MS-VAR) setup:
  - A′_0(s_{c,t}) y_t = A′_+(s_{c,t}) x_t + Ξ^{-1}_s(s_{v,t}) ε_t; coefficient switching s_{c,t} and variance switching s_{v,t} possible independently.

### Counterfactual and Conditional Forecast Strategy
- Two-sample estimation for conditional forecasts:
  - (A) parameters estimated over 1987Q1 to 2007Q4 (pre-2008) for out-of-sample forecasts.
  - (B) parameters estimated over 1987Q1 to 2015Q2 (pre-2015Q3) for in-sample forecasts.
- Two counterfactuals to disentangle shocks vs propagation:
  - C2015V2007: x_t = A′_+(pre15Q3) x_{t−1} + Ξ^{-1}(pre08Q1) ε_t (tests change in propagation).
  - C2007V2015: x_t = A′_+(pre08Q1) x_{t−1} + Ξ^{-1}(pre15Q3) ε_t (tests change in shocks).
- Empirical finding from RMSFE comparisons (Figures referenced in source):
  - C2007V2015 provides a better fit for unemployment and inflation than C2015V2007, indicating changes in the variance-covariance matrix (shocks) are more important than changes in propagation.

### Informativeness of Conditioning Information Sets
- Conditioning blocks evaluated (exact observables listed in source):
  - 1 External Factors; 2 Commodities; 3 Labor Factors; 4 Interest Rates; 5 Credit; 6 Financial Risk; 7 TFP.
- Key findings on forecast informativeness (RMSFE and forecast comparisons over 2008Q1–2015Q2):
  - External data, financial risk, interest rates, and TFP are the most important conditioning information for inflation (order reported in source).
  - Credit variables are important conditioning variables for unemployment and CPI shelter.
  - Conditioning on all non-price information ("All") provides an RMSFE benchmark; commodity scenarios from 2014 onward tended to put downward pressure on inflation but with wide confidence bands and large RMSFEs.
  - Robustness checks with smaller BVARs confirm importance of external and financial variables.

### Markov-Switching Model Results and Model Selection
- Notation:
  - #v = number of independent Markov states for variance switching; #c = number for coefficient switching.
  - Example: 2v(Eq6)2c denotes two variance states, coefficient switching restricted to equation 6, two coefficient states.
  - Baseline unrestricted switches in all equations denoted 2v2c.
- Main model-fit findings (marginal data densities, log MDDs, and differences reported exactly as in source):
  - Panel (a) General Models excerpts:
    - 1v1c: 685.45 (Diff. from best: -138.94)
    - 2v1c: 764.27 (Diff. from best: -60.12)
    - 3v1c: 781.36 (Diff. from best: -43.03)
    - 1v2c: 765.48 (Diff. from best: -58.91)
    - 2v2c: 784.84 (Diff. from best: -39.55)
    - 3v2c: 812.11 (Diff. from best: -12.28)
  - Panel (b) Restricted Models excerpts:
    - (Eq6)3v2c: 715.19 (Diff. from best: -109.20)
    - (Eq5−6)3v2c: 716.51 (Diff. from best: -107.88)
    - (Eq4−6)3v2c: 777.78 (Diff. from best: -46.61)
    - (Eq3−6)3v2c: 783.61 (Diff. from best: -40.78)
    - (Eq2−6)3v2c: 809.76 (Diff. from best: -14.63)
  - Panel (c) Restricted Models excerpts:
    - 3v(Eq4&5)2c: 812.72 (Diff. from best: -11.67)
    - 3v(Eq1−3&6)2c: 816.76 (Diff. from best: -7.63)
    - 3v(Eq1−3)2c: 808.76 (Diff. from best: -15.63)
    - 3v(Eq4)2c: 824.39 (Diff. from best: 0) — highest in the reported rows.
  - Alternative specification robustness (Table 4 excerpt, MS-VAR in levels):
    - 3v2c: logMDD 3766.25 (Diff. from best: -30.75)
    - 3v all eq 2c eq4: logMDD 3797.00 (Diff. from best: 0)
- Interpretation of model selection:
  - Constant coefficients and constant shock variances (1v1c) is not favored.
  - Adding states in variances or coefficients improves fit substantially.
  - Best-fitting specifications indicate variance switching across equations and coefficient switching confined to the monetary policy (interest-rate) equation.
  - Coefficient switching in the inflation equation alone (e.g., 3v(Eq2)2c) is not favored — consistent with no break in the Phillips curve.
  - Results are robust to lag length, priors, sample length, ordering choices, and alternative identification strategies.

### Regime Characteristics and Economic Interpretation
- Preferred 3v(Eq4)2c specification (description from source):
  - Coefficient state 1: prevails when policy rate is changing (tightening or loosening phases).
  - Coefficient state 2: prevails in passive or constrained phases (e.g., ZLB period).
  - Variance regime 2: associated with low monetary policy shock variance and high import price variance.
  - End-of-sample (June,2015) estimated to be in variance state 2 and coefficient state 1: monetary policy active with relatively high import price variance and low monetary policy shock variance.
- Role of the Zero Lower Bound (ZLB):
  - ZLB (federal funds rate at zero from December 2008 to December 2015) can be captured by coefficient switching and variance switching; term spread included to capture changes in the relationship between policy rate and spread under nonstandard policy.
- Economic implications for inflation-unemployment dynamics:
  - Allowing variance switching explains changing correlations between inflation and unemployment across regimes:
    - Variance regime dominated by import price (cost-push) shocks can generate a positive correlation between inflation and contemporaneous and lagged unemployment.
    - Variance regime dominated by demand disturbances can generate a negative correlation.
  - Omitting variance switching risks biasing inferences toward spurious coefficient switching.
  - When monetary policy is constrained/passive it stabilizes inflation less effectively; the Phillips curve appears steeper in the passive state than in the active policy state.

### Small New-Keynesian Style Model and Long Sample Robustness
- Small model specification (preserving source equations and notation):
  - απ πt = cπ + απ1 πt−1 + απ2 πt−2 + αy ˆyt + σπ επt,
  - βy ˆyt = cy + βy1 ˆyt−1 + βy2 ˆyt−2 + βr (it − πt) + σy εyt,
  - Monetary policy instrument rule: γi1 it = ci + γi2 it−1 + γy yt + γπ πt + σi εit.
- Sample: 1960Q1 to 2015Q2.
- Marginal data density results (Table A1 excerpts; log MDDs and diffs preserved):
  - Panel (a):
    - 1v1c: 2158.6 (Diff. from best: -128.2)
    - 2v1c: 2258.6 (Diff. from best: -28.2)
    - 3v1c: 2267.9 (Diff. from best: -18.9)
    - 1v2c: 2241.9 (Diff. from best: -44.9)
    - 2v2c: 2269.7 (Diff. from best: -17.1)
    - 3v2c: 2274.4 (Diff. from best: -12.4)
  - Panel (b) restricted rows:
    - 3v(PC Eq)2c: 2262.1
    - 3v(IS Eq)2c: 2254.6
    - 3v(Mon Pol Eq)2c: 2286.8 (best among listed)
- Conclusion from small-model robustness:
  - Confirms main MS-BVAR result: changes in shock variances are more salient than changes in coefficients; coefficient switching confined to the policy rule yields highest marginal data density.

### Policy-Relevant Conclusions
- The Phillips curve did not structurally break during the sample period examined.
- Variation in shock variances (financial, external, import-price, demand) accounts for pre- vs post-GFC differences in inflation and unemployment dynamics more than changes in Phillips-curve coefficients.
- Monetary policy coefficients likely experienced regime shifts (active vs passive/ZLB-like regimes), while Phillips-curve coefficients did not.
- For forecasting and policy conditioning:
  - External variables and financial risk variables are most informative for inflation forecasting in the post-GFC period.
  - Credit variables are most important for explaining the post-GFC unemployment rate.
- Practical implication: central banks should account for time-varying shock variances and the informational value of financial and external indicators when designing responses, especially near constraints like the ZLB.

*Source: _wp16126 (IMF Working Paper WP/16/126), Sections 1–4.*

### Section 1

### _wp16126 - Section 1

### Abstract and Key Findings
- Title: Did the global financial crisis break the U.S. Phillips Curve?; Authors: Stefan Laseen and Marzie Taheri Sanjani; IMF Working Paper WP/16/126; July 2016.
- Primary empirical finding: The Phillips curve was not broken by the Global Financial Crisis (GFC).
- Main interpretation: Changes in shocks (variances) are a more salient feature of the data than changes in coefficients.
- Regime-switching model result: Allowing for regime switching solely in coefficients of the policy rule maximizes the fit among the variants considered.
- Best-fitting model features:
  - Change in coefficients only of the monetary policy rule (not of the Phillips curve).
  - Most important changes across regimes are the variances of structural disturbances.
- Forecasting/conditioning result: Financial and external variables have the highest forecasting power for inflation and unemployment post-GFC.
- Specific variable highlighted: Excess Bond Premium (EBP) contains rich information reflecting credit supply shocks and is an important driver of macroeconomic dynamics.

### Introduction — Motivation and Hypotheses
- Research question: How does unemployment affect inflation since the GFC; did the GFC break the U.S. Phillips curve?
- Puzzles motivating study:
  - “Missing disinflation”: inflation declined after the GFC but less than anticipated.
  - Inflation has been running below the 2 percent target for an extended period despite low unemployment.
- Three hypotheses examined for low inflation post-GFC:
  - (a) Financial frictions and shocks causing slow recoveries and persistently low inflation.
  - (b) Globalization increasing the role of international factors and decreasing the role of domestic factors in the inflation process.
  - (c) Constraints from the effective lower bound on policy rates preventing stabilization policy from lowering real interest rates enough to restore long-run inflation goals.
- Implications emphasized:
  - A changing or nonlinear Phillips curve would affect monetary policy design (e.g., pre-emptive vs symmetric responses).
- Approach overview:
  - Use multivariate, possibly time-varying models: Large Bayesian Vector Autoregressions (BVARs), Dynamic Factor Models (DFMs), and Markov-switching (MS)-BVARs.
  - Use conditional forecast analysis to distinguish variance changes from structural parameter variations.
  - Examine which data categories (macroeconomic, financial, external, TFP, etc.) are informative for inflation and unemployment dynamics.

### Models — General Framework
- General nonlinear vector stochastic dynamic process (as in the source):
  - A′0(sc t) y t = C(sc t) x t + Ξ−1 m(sv t) υ t,
  - x t = A′+(sc t) x t−1 + Ξ−1 s(sv t) ε t,
  - Structural disturbances: πεt|Yt,sm t,A0,A+ ∼ N(0n×1, I n).
- Latent state variables:
  - sc and sv take values in {1,2,...,hm} and evolve according to a first-order Markov process p(sc,v t = i | sc,v t−1 = k) = pc,k ik, i,k = 1,2,..,hm.
- Empirical specifications considered:
  - Three large cross-section specifications: dynamic factor model(s) and large-scale BVAR(s).
  - One Markov-switching structural VAR model.

### Empirical Evaluation — Strategy and Data
- Objective: Use a rich information set to compute conditional forecast scenarios and identify effects of various shocks and structural changes on unemployment and inflation dynamics.
- Rationale for large information sets: Reduce non-fundamentalness problems present in smaller-scale models and allow assessment of which categories (labor, financial, external, commodity, TFP) are informative.
- Conditional forecasts: Mix in-sample and out-of-sample forecasts (following Banbura, Giannone, and Lenza (2015) and Giannone, Lenza, and Reichlin (2008)) to separate variance switching from coefficient switching.
- Relation to policy constraints: Can control for changes or restrictions on monetary policy, including the effective lower bound on interest rates.

### A. Large Cross-Sections VAR’s and DFM’s — Dataset and Model 1
- Dataset summary:
  - Number of quarterly variables: 45 quarterly variables.
  - Sample period: from 1987Q1 to 2015Q2.
  - Variable categories included:
    - Real activity: US GDP and expenditure components, world GDP, industrial production index, consumer sentiment, labor market data, total factor productivity (TFP).
    - Prices: commodity price, CPI, producer price index, GDP deflator, imports deflator.
    - Monetary: short-term and long-term interest rates, money supply.
    - Financial: credit to households and corporations, stock price, real effective exchange rate, Moody’s Aaa and Baa corporate bond yield, number of housing units started, economic uncertainty.
  - Data transformation: models feed on annualized log-levels, except variables defined as annualized rates (interest rates, corporate bond yields, TFP, etc.) which are taken in levels.
  - Data sources: Bureau of Economic Analysis, Federal Reserve Board, Bureau of Labor Statistics, Congressional Budget Office, Census Bureau, IMF database, Haver Analytics, Federal Reserve Bank of San Francisco for TFP, Area Wide Model database for three months EURIBOR (as described in the source).
- Model 1 — Bayesian Dynamic Factor Model (DFM) representation:
  - Observed equation: ∆y t = z t + ΛF t + Ξ−1 υ t, t = 1,2,..,T.
  - Factor dynamics: F t = Φ1 F t−1 + .. + Φ s F t−s + Ξ−1 s ε t.
  - Mapping to general framework:
    - p(sc,v t = i | sc,v t−1 = k) = 0 (i.e., no regime switching in this DFM specification).
    - y t = ∆y t, A′0 = I n, C = (Λ, 0 n×r(s−1), I n).
    - x′ t = [F t, F t−1, ..., F t−s+1, z t].
  - Model dimensions:
    - F t is an r-dimensional vector of common factors with r typically much smaller than n.
    - Λ is an n×r matrix of factor loadings.
    - z t contains exogenous variables or constants.
    - υ t is the idiosyncratic component.

### Implications and Interpretation
- The empirical evidence across large VAR/DFM and regime-switching models suggests:
  - Variation in the sources of economic disturbances (shocks and their variances) explains pre- vs post-GFC differences more than variation in the Phillips-curve dynamics (coefficients).
  - Structural disturbances’ variances change importantly across regimes.
  - Monetary policy rule coefficients may have experienced regime change, while the Phillips curve coefficients did not show a break in the models that fit best.
- Forecasting implication: Conditioning on external and financial risk variables improves forecasts for inflation and unemployment in the post-GFC period.

*Source: _wp16126 - Section 1 (IMF Working Paper WP/16/126), Stefan Laseen and Marzie Taheri Sanjani.*

### Section 2

### _wp16126 - Section 2

### Data Description and Transformations
- Table 1 enumerates 43 series grouped broadly as Real Activities, TFP, Prices, Monetary, Financial, and Utilization-related series.
- Key series, their mnemonics, descriptions, sources, and exact transformations:
  - 1 world RGDP — YOY (pct change) World RGDP — transformation: level/100
  - 2 rgdp — US Real GDP (SAAR, Bil.Chn.2009) — transformation: log level x 4
  - 3 ip — US Industrial Production Index (SA, 2012=100) — transformation: log level x 4
  - 4 c — US Real Personal Consumption Expenditures (SAAR, Bil.Chn.2009) — transformation: log level x 4
  - 5 g — US Real Government Consumption Expenditures & Gross Investment(SAAR, Bil.Chn.2009) — transformation: log level x 4
  - 6 i — US Real Gross Private Domestic Investment (SAAR, Bil.Chn.2009) — transformation: log level x 4
  - 7 x — US Real Exports of Goods & Services (SAAR, Bil.Chn.2009) — transformation: log level x 4
  - 8 m — US Real Imports of Goods & Services (SAAR, Bil.Chn.2009) — transformation: log level x 4
  - 9 emp — US All Employees: Total Nonfarm Payrolls (SA, Thous) — transformation: log level x 4
  - 10 u — US Unemployment Rate: 16 Years + (SA, %) — transformation: level / 100
  - 11 nairu — US Natural Rate of Unemployment [CBO] (%) — transformation: level / 100
  - 12 cap ut — US Capacity Utilization: Industry (SA, Percent of Capacity) — transformation: level / 100
  - 13 util — Utilization of capital and labor — transformation: log level x 4
  - 14 util-invest — Utilization in producing investment — transformation: log level x 4
  - 15 util-consumption — Utilization in producing non-investment business output ("consumption") — transformation: log level x 4
  - 16 c conf — US University of Michigan: Consumer Sentiment (NSA, Q1-66=100) — transformation: level / 100
  - 17 tfp-util — Utilization-adjusted TFP — transformation: log level x 4
  - 18 tfp-I-util — Utilization-adjusted TFP in producing equipment and consumer durables — transformation: log level x 4
  - 19 tfp-C-util — Utilization-adjusted TFP in producing non-equipment output — transformation: log level x 4
  - 20 oil — Spot Price Idx of UK Brt Lt/Dubai Med/Alaska NS heavy (2010=100) — transformation: log level x 4
  - 21 non oil — Non-fuel Primary Commodities Index (2010=100) — transformation: log level x 4
  - 22 cpi shelter — US CPI-U: Shelter (SA, 1982-84=100) — transformation: log level x 4
  - 23 cpi core — US CPI-U: All Items Less Food & Energy (SA, 1982-84=100) — transformation: log level x 4
  - 24 pce core — US PCE less Food & Energy: Chain Price Index (SA, 2009=100) — transformation: log level x 4
  - 25 ppi — US PPI: Finished Goods (SA, 1982=100) — transformation: log level x 4
  - 26 gdp def — US GDP Implicit Price Deflator (SA, 2009=100) — transformation: log level x 4
  - 27 mxrmd — US Imports Deflator (excluding raw materials) — transformation: log level x 4
  - 28 wages — US Avg Hourly Earnings: Prod & Nonsupervisory: Total Private Industries(SA, /Hour) — transformation: log level x 4
  - 29 euro-stn — Euro Area 11-19: 3-Month EURIBOR (%) — transformation: level / 100
  - 30 libor us — 3-Month London Interbank Offer Rate: Based on US (%) — transformation: level / 100
  - 31 ust3m — 3-Month Treasury Bills, Secondary Market (% p.a.) — transformation: level / 100
  - 32 ust10 — 10-Year Treasury Note Yield at Constant Maturity (% p.a.) — transformation: level / 100
  - 33 m1 — Money Stock: M1 (SA, Bil.) — transformation: log level x 4
  - 34 m2 — Money Stock: M2 (SA, Bil.) — transformation: log level x 4
  - 35 loan hh — US: Household & Nonprofit Outstanding Debt (SA, Bil.US) — transformation: log level x 4
  - 36 loan corp — US: Nonfinancial Corporations Outstanding Debt (SA, Bil.US) — transformation: log level x 4
  - 37 reer — Real Broad Trade-Weighted Exchange Value of the US (Mar-73=100) — transformation: log level x 4
  - 38 sp500 — Stock Price Index: Standard & Poor’s 500 Composite (1941-43=10) — transformation: log level x 4
  - 39 corp Aaa — Moody’s Seasoned Aaa Corporate Bond Yield (% p.a.) — transformation: level / 100
  - 40 corp Baa — Moody’s Seasoned Baa Corporate Bond Yield (% p.a.) — transformation: level / 100
  - 41 pol uncert — Policy-related Economic Uncertainty — transformation: level / 100
  - 42 ebp-oa — Excess Bond Premium — transformation: level
  - 43 gz-spr — Gilchrist and Zakrajsek default risk spread — transformation: level

- Economic uncertainty data is constructed using Nick Bloom’s methodology and can be downloaded from the Economic Policy Uncertainty homepage.

### Model 2 and Model 3: Large Cross-Section Bayesian VAR Models (Representation and Implementation)
- General VAR in levels:
  - y_t = z_t + A1 y_{t−1} + ... + A_p y_{t−p} + Ξ^{-1} ε_t, t = 1,2,..,T
- VAR in differences:
  - Δy_t = z_t + B1 Δy_{t−1} + ... + B_p Δy_{t−p} + Ξ^{-1} ε_t, t = 1,2,..,T
- Casting into the state-space representation requires imposing specific constraints (examples provided in the text), including:
  - p(s_{c,v,t}=i | s_{c,v,t−1}=k) = 0 for the VAR in differences,
  - y_t = Δy_t (for VAR in differences),
  - A′_0 = I_n, C = (I_n, 0_{n×np}), Ξ^{-1}_m = 0,
  - x′_t = [y_t, y_{t−1}, ..., y_{t−s+1}, z_t].
- Advantage: Large BVARs and factor-augmented VARs mitigate non-fundamentalness by expanding the information set, reducing the risk that agents have material private information not in the econometrician’s model.

### Shocks vs Propagation: Conditional Forecasts and Counterfactual Exercises
- Conceptual point:
  - Split between shocks and propagation depends on the conditioning information set; non-fundamentalness can arise with news shocks.
  - Two remedies: increase the information set (factor-augmented VARs, large BVARs) or allow MA representations with roots inside the complex unit circle.
- Methodology:
  - Conditional forecasts: projections of variables of interest assuming known future paths of some other variables (scenario analysis is conditional forecasting for a specific event).
  - Computational approach: use Bańbura, Giannone and Lenza (2015) Kalman-filter-based algorithm for conditional forecasts in large models (computationally viable compared with Waggoner and Zha (1999) for high-dimensional data).
  - Baseline: BVAR on level variables; robustness checks: Dynamic Factor Models (DFMs) and BVAR in first differences.
  - Large BVARs and DFMs estimated in two samples:
    - (A) : x_t = A′_+(pre08Q1) x_{t−1} + Ξ^{-1}(pre08Q1) ε_t (parameters estimated over 1987Q1 to 2007Q4 for out-of-sample forecasts)
    - (B) : x_t = A′_+(pre15Q3) x_{t−1} + Ξ^{-1}(pre15Q3) ε_t (parameters estimated over 1987Q1 to 2015Q2 for in-sample forecasts)
- First counterfactual (role of change in propagation):
  - Counterfactual process: x_t = A′_+(pre15Q3) x_{t−1} + Ξ^{-1}(pre08Q1) ε_t
  - Denoted in charts as C2015V2007.
  - Interpretation: if counterfactual inflation dynamics matches actual post-2007Q4 outcomes (low RMSFE), then changes in propagation explain post-GFC inflation dynamics.
- Second counterfactual (role of change in shocks):
  - Counterfactual process: x_t = A′_+(pre08Q1) x_{t−1} + Ξ^{-1}(pre15Q3) ε_t
  - Denoted in charts as C2007V2015.
  - Interpretation: better fit of C2007V2015 (lower RMSFE) indicates critical role of shocks post-crisis.
- Empirical outcome summarized:
  - Charts (Figure 3 referenced) show RMSFE for unemployment rate and PCE core inflation under both hybrid scenarios.
  - The bottom-panel counterfactual (C2007V2015) provides a better fit for both unemployment and inflation than the top-panel counterfactual (C2015V2007).
  - Conclusion: data favor changes in the variance-covariance matrix (shocks) rather than changes in propagation; implication: the Phillips curve did not change during the sample period.

### Conditioning Information Sets: Scenarios and Forecast Informativeness
- Purpose: assess informativeness of conditioning on blocks of observables (External, Commodities, Labor, Interest Rates, Credit, Financial Risk, TFP) for inflation and unemployment dynamics.
- Conditional forecast setup:
  - Conditional forecasts generated over post-crisis period 2008Q1 to 2015Q2.
  - Use (A) for out-of-sample forecasts (parameters estimated 1987Q1 to 2007Q4) and (B) for in-sample forecasts (1987Q1 to 2015Q2).
  - Compare conditional forecasts to realized data via RMSFE to gauge informativeness.
- Table 2: Scenario observable groupings (exact observables listed)
  - 1 External Factors:
    - World GDP Growth
    - Real Exports of Goods & Services
    - Real Imports of Goods & Services
    - Real Broad Trade-Weighted Exchange Value of the US (Mar-73=100)
  - 2 Commodities:
    - 3-Month EURIBOR
    - Imports Deflator (excluding raw materials)
    - Oil Price Index (Brent/Dubai/WTI)
    - Non-fuel Primary Commodities Index
  - 3 Labor Factors:
    - Unemployment Rate
    - Total Nonfarm Payrolls
    - Natural Rate of Unemployment
    - Avg Hourly Earnings: Total Private Industries
  - 4 Interest Rates:
    - 3-Month LIBOR USD
    - 3-Month Treasury Bill Yield
    - 10-Year Treasury Note Yield
  - 5 Credit:
    - Nonfinancial Corporations Outstanding Debt
    - Household & Nonprofit Outstanding Debt
    - Aaa Corporate Bond Yield
  - 6 Financial Risk:
    - Baa Corporate Bond Yield
    - Policy-related Economic Uncertainty
    - Excess Bond Premium
    - Gilchrist and Zakrajsek default risk spread
  - 7 TFP:
    - Utilization-adjusted TFP
    - Utilization-adjusted TFP in producing equipment and consumer durables
    - Utilization-adjusted TFP in producing non-equipment output
- Key findings on informativeness and forecasting performance:
  - Figures 4 and 5 present RMSFE (point-wise) and forecasts for unemployment rate (level), CPI shelter, CPI Core, and PCE core (prices in annual growth rate).
  - In-sample forecasts outperform out-of-sample forecasts.
  - Substantial out-of-sample inaccuracy for the unemployment rate indicates its dynamics changed substantially post-crisis.
  - During 2008Q1–2009Q3 (unemployment up, inflation down), most conditional scenarios imply lower PCE core inflation than unconditional (naive) forecast.
  - RMSFE results (Figures 4 and 6) indicate relative importance for inflation forecasting:
    - External data, financial risk, interest rates, and TFP are the most important conditioning information for inflation (listed in order reported).
  - For unemployment and CPI shelter:
    - Credit appears to be an important conditioning variable.
  - Scenario "All": conditioning on all non-price information in the model provides an upper/lower RMSFE benchmark along with the unconditional scenario.
  - Commodity scenarios from 2014 onward tended to put downward pressure on inflation but with wide confidence bands and large RMSFEs.
  - Overall conclusion: financial variables (credit and risk) are important for explaining unemployment dynamics; external factors and financial risks are relevant for price inflation.
- Robustness checks:
  - Smaller BVARs estimated per scenario (e.g., 10-variable small BVAR for External factor scenario) confirm importance of external and financial variables.

### Markov-switching Structural Bayesian VAR (Model 4)
- Model form (following Sims and Zha (2006), Sims, Waggoner, and Zha (2008), Hubrich and Tetlow (2015)):
  - A′_0(s_{c,t}) y_t = A′_+(s_{c,t}) x_t + Ξ^{-1}_s(s_{v,t}) ε_t,   t = 1,...,T
- Casting into state-space representation with C = (I_n, 0_{n×np}), Ξ^{-1}_s = 0, and x′_t = [y_{t−1}, y_{t−2}, ..., y_{t−s+1}, z_t].
- Reduced form:
  - y_t = B(s_{c,t}) x_t + u_t(s_{v,t}, s_{c,t}),   t = 1,...,T
  - B(s_{c,t}) = A′_0(s_{c,t})^{-1} A′_+(s_{c,t})
  - u_t(s_{v,t}, s_{c,t}) = A′_0(s_{c,t})^{-1} Ξ^{-1}_s(s_{v,t}) ε_t
  - E(u_t u′_t) = (A_0(s_{c,t}) Ξ^2_s(s_{v,t}) A′_0(s_{c,t}))^{-1}
- Notable modeling implication:
  - Switching in coefficients s_{c,t} implies switching in reduced-form residuals.
  - Allowing switching only in variances does not impose switching in reduced-form coefficients, which remain fixed.

*Source: _wp16126 - Section 2*

### Section 3

### _wp16126 - Section 3

### Notation and Research Questions
- Notation for Markov states:
  - #v,# = 1,2,3 indicate the number of independent Markov states governing variance switching.
  - #c to indicate the number of states governing coefficient switching.
  - When shifts in structural parameters are constrained to a specific equation(s), the restriction is indicated by prefixing - in parenthesis - the letter of the variable, (Eq#), # = 1 : 6. Example: 2v(Eq6)2c denotes an MS-VAR with two Markov states in the variances and two in coefficients with the latter restricted to the last equation.
  - Baseline unrestricted switches in all equations denoted 2v2c.
- Three primary questions:
  - Whether there are periods (e.g., high financial stress or constraints on monetary policy such as the zero lower bound) characterized by different inflation-unemployment dynamics than regular times.
  - If regime switching exists, whether it is confined to variance switching (as Sims and Zha (2006) find) or whether coefficient switching better explains inflation and data dynamics (as Hubrich and Tetlow (2015) find).
  - Whether regime switching is confined to specific equations (e.g., the monetary policy equation alone, or the inflation response to stress) or applies to all equations.

### The Monthly Database (B.2)
- Six-variable MS-VAR identified with Cholesky decomposition. Let y_t = [Û_t ˆπ_PCE_t π_RPIM_t R_t TS_t EBP_t] where:
  - Û_t = monthly seasonally adjusted civilian unemployment rate less the Congressional Budget Office (CBO) historical series for the long-run natural rate.
  - ˆπ_PCE_t = PCE inflation, excluding food and energy prices less expected long-run inflation expectations proxied using the median forecast of long-run PCE or CPI inflation reported in the Survey of Professional Forecasters, with a constant adjustment of 40 basis point before2007.
  - π_RPIM_t = annualized growth rate of the price index for core imported goods (defined to exclude petroleum), less the lagged four-quarter change in core PCE inflation, all multiplied by the share of nominal core imported goods in nominal GDP.
  - R_t = Wu-Xia (2016) shadow nominal federal funds rate.
  - TS_t = 3-Month Treasury Constant Maturity Minus 10-Year Treasury Constant Maturity.
  - EBP_t = excess bond premium.
- All variables are monthly and expressed at annual rates.
- Sample runs from 1991 : 10 to 2015 : 6.
- Choice of excess bond premium (EBP_t) justified by:
  - Financial variables are important for explaining unemployment and inflation dynamics since the GFC.
  - Gilchrist and Zakrajsek (2012) show the excess bond premium has strong forecasting ability for economic activity, outperforming every other financial indicator.
  - Faust et al. (2013) confirm excess bond premium has considerable marginal predictive power for economic activity.

### The Zero Lower Bound (B.3)
- Timeline: In December 2008, the Federal Reserve lowered the federal funds rate to the zero lower bound (ZLB) where it remained until December2015.
- How MS-BVAR handles ZLB (following Hubrich and Tetlow (2015)):
  - ZLB can be treated as another regime that the model can identify if preferred by the data. If ZLB (or a negative shadow funds rate) is obtained, the perception that the funds rate reacts differently (e.g., can fall no further) would be captured by switching in coefficients plus switching in shock variances, yielding adverse shocks to the shadow funds rate.
  - There could be a change in the relationship between the federal funds rate and the term spread either directly because of negative shadow rates, or because of nonstandard monetary policy measures that stand in for conventional monetary policy. This motivates including the term spread in the model.
  - Hence the model can, in principle, pick out new states to capture the ZLB period.

### Regime Switching Analysis (C)
- Estimation and priors:
  - Follow Hubrich and Tetlow (2015) and Sims and Zha (2006).
  - Minnesota prior for VAR parameters; Dirichlet prior for state transition matrix.
  - Key prior: prior probability of remaining in the same state in the next period as in the current period; chosen so as not to a priori promote more switching in one part of the model over another.
- Model evaluation:
  - Marginal data densities (MDDs) of candidate specifications compared on a log-likelihood scale. Differences of 1 or 2 mean little; differences of 10 or more imply extreme odds ratios for the higher-MDD model.

- Main model-fit findings (summary of Table 3, panel (a) and restricted panels (b) and (c)):
  - Constant coefficients and constant shock variances model 1v1c is not favored.
  - Adding a second state in variances or coefficients improves fit substantially.
  - Best model in panel (a) is 3v2c (three variance states, two coefficient states); same best-fit model as Hubrich and Tetlow (2015).
  - Panel (b) and (c) investigate restrictions of switching to specific equations:
    - Data favor variance switching in all equations, but coefficient switching only in the interest rate equation (monetary policy equation).
    - Coefficient switching in the inflation equation alone, 3v(Eq2)2c, is not favored — implying the Phillips curve did not change significantly since the beginning of the 1990’s.
  - These results align with section 1 findings that changes in shock variances are the most important factor behind changes in inflation and unemployment dynamics, and with Stock and Watson (2012a) and Sims and Zha (2006) who find changes in shocks are more salient than coefficient changes.
  - Robustness: Result of changes in only the monetary policy equation is robust to changes in lag length, priors and changes in data.

- Selected MDD numeric results (Table 3 excerpts, log MDD values and diffs):
  - Panel (a) General Models (Model −→ columns: 1v1c 2v1c 3v1c)
    - MDDs: 685.45 764.27 781.36
    - Diff. from best: -138.94 -60.12 -43.03
  - Panel (a) continued (Model −→ columns: 1v2c 2v2c 3v2c)
    - MDDs: 765.48 784.84 812.11
    - Diff. from best: -58.91 -39.55 -12.28
  - Panel (b) Restricted Models (Model −→ (Eq6)3v2c (Eq5−6)3v2c (Eq4−6)3v2c (Eq3−6)3v2c)
    - MDDs: 715.19 716.51 777.78 783.61
    - Diff. from best: -109.20 -107.88 -46.61 -40.78
  - Panel (b) additional (Model −→ (Eq2−6)3v2c (Eq1,3−5)3v2c (Eq4−5)3v2c)
    - MDDs: 809.76 790.06 738.00
    - Diff. from best: -14.63 -34.33 -86.39
  - Panel (c) Restricted Models (Model −→ 3v(Eq4&5)2c 3v(Eq1−3&6)2c 3v(Eq1−3)2c 3v(eq4−6)2c)
    - MDDs: 812.72 816.76 808.76 791.25
    - All equations 3 states variance switching diffs: -11.67 -7.63 -15.63 -33.14
  - Panel (c) additional rows (Model −→ 3v(Eq1−4)2c 3v(Eq4)2c 3v(Eq2&4)2c 3v(Eq2)2c)
    - MDDs: 813.41 824.39 816.08 791.25
    - Diff. from best: -10.98 0 -8.31 -33.14

- Alternative specification robustness (Table 4 excerpt, MS-VAR in levels estimation results):
  - Model −→ 3v2c 3v all eq 2c eq4
  - logMDD: 3766.25 3797.00
  - Diff. from best: -30.75 0

- Additional notes:
  - Figures (referenced) for preferred 3v(Eq4)2c specification:
    - Coefficient state 1 prevails when policy rate is changing (tightening or loosening phases).
    - Coefficient state 2 prevails in passive or constrained phases (e.g., ZLB period).
    - Variance regime two associated with low monetary policy shock variance and high import price variance.
    - At the end of sample (June,2015) U.S. estimated to be in variance state 2 and coefficient state 1: monetary policy active in this regime with relatively high import price variance and low monetary policy shock variance.
  - Ordering sensitivity: Changes in ordering do not affect the results generally. If EBP_t is ordered before policy rate R_t, models 2v3c and 3v(Eq4)2v are difficult to separate; MDDs for that specification are lower than baseline.

### Implications for the Phillips Curve
- Main inference:
  - Data favor variance switching in all equations, but only coefficient switching in the interest rate equation, implying the Phillips curve did not change during the sample period.
- Cross-correlation and scatter findings:
  - Vector cross-correlation functions (Figure10) for different coefficient and variance regimes show correlations change depending on regime.
  - Variance regime 2 dominated by import price shocks (cost-push) implies a positive correlation between inflation and contemporaneous and lagged unemployment.
  - Variance regime 1 dominated by demand disturbances implies a negative correlation between inflation and unemployment.
  - Allowing for variance switching is important to avoid biasing results toward erroneously finding coefficient switching.
  - Scatter plots (Figure11) show when monetary policy is constrained/passive it does not stabilize inflation shocks as well as when unconstrained; Phillips curve appears seemingly steeper in the passive state than in the active monetary policy state.

### Conclusion (IV)
- Methods: large BVARs, DFM’s and MS-VAR models used to investigate non-linearity in post-crisis dynamics of inflation and unemployment in U.S. data.
- Key findings:
  - Changes in shocks are a more salient feature of the data than changes in coefficients.
  - A model with time-varying coefficients in the policy rule fits better than all other models that allow a change in coefficients.
  - The model with coefficient switching in the simple instrument rule combined with variance switching in all equations attains the highest marginal data density.
  - Conditional forecasts that condition on external variables and financial risk variables come closest to describing inflation dynamics.
  - Credit variables are the most important conditioning variables for the post-GFC unemployment rate.

*Source: _wp16126 - Section 3 (PDF).*

### Section 4

### _wp16126 - Section 4

### Model Specification and Data
- Small empirical New-Keynesian style model used to examine sensitivity to identifying assumptions and sample size.
- Equations (as presented in source):
  - απ πt = cπ + απ1 πt−1 + απ2 πt−2 + αy ˆyt + σπ επt,
  - βy ˆyt = cy + βy1 ˆyt−1 + βy2 ˆyt−2 + βr (it − πt) + σy εyt,
  - Monetary policy instrument rule: γi1 it = ci + γi2 it−1 + γy yt + γπ πt + σi εit.
- Variable definitions (as in source):
  - πt is quarterly inflation in the GDP chain-weighted price index (pt) at an annual rate, πt = 4 (pt/pt−1 − 1).
  - it is the quarterly average federal funds rate in percent at an annual rate.
  - ˆyt is the percentage gap between actual real GDP (qt) and potential GDP (q∗t), ˆyt = (ln qt − ln q∗t).
  - Real potential GDP is the CBO’s estimate.
- Contemporaneous and lag matrices (as in source):
  - A0 = [ [απ βr γi1] [αy βy γy] [0 −βr γπ] ],
  - A1 = [ [απ1 0 0] [0 βy1 0] [0 0 γi2] ],
  - A2 = [ [απ2 0 0] [0 βy2 0] [0 0 0] ].
- Sample: 1960Q1 to 2015Q2.
- Identification: "One can show, using the theorem of Rubio-Ramirez, Waggoner, & Zha (2010), that the model is globally identified."

### Estimation Results and Model Comparison (Table A1)
- Panel (a) General Models (marginal data densities reported as in source):
  - 1v1c 2v1c 2v?3v1c: 2158.6 2258.6 2267.9
  - Diff. from best: -128.2 -28.2 -18.9
  - 1v2c 2v2c 3v2c: 2241.9 2269.7 2274.4
  - Diff. from best: -44.9 -17.1 -12.4
- Panel (b) Restricted Models (marginal data densities reported as in source):
  - Model−→ 3v(PC Eq)2c 3v(IS Eq)2c 3v(Mon Pol Eq)2c: 2262.1 2254.6 2286.8
  - All equations 3states variance switching: -24.7 -32.20
- Key model-evaluation takeaway (preserving source phrasing): "The results confirm the main result from the MS-BVAR estimated on monthly data since the 1990′s. Changes in shocks is a more salient feature of the data than changes in coefficients and a model with time-varying coefficients in the policy rule fits better than all other models that allow change in coefficients. The model with coefficient switching in the simple instrument rule with variance switching in all equations attain the highest marginal data density."

### Regimes, State Probabilities, and Interpretation
- Coefficient switching interpretation:
  - Coefficient states 1 and 2 capture tightening and loosening phases of monetary policy, consistent with the model estimated on monthly data since the beginning of the 1990′s.
  - FigureA5 (described in source): state 1 prevails when the Fed is lowering the federal funds rate; this state is independent of who was chair of the Federal Reserve.
  - Visual regime indicators (as described in source): "The light green color shows Greenspan’s time as chair whereas the red light shows when Burns was chair."
- Comparison with Sims and Zha (2006) (preserving source language):
  - "Note that our results differ slightly from Sims and Zha (2006). While our results point in the same overall direction to those in Sims and Zha our estimates of when changes in monetary policy took place differ. Sims and Zha found e.g. one dominating regime and three less frequent regimes. The dominating regime was named the Greenspan regime since it prevailed during much of Greenspans time as chair. One should, however, bear in mind that this regime was also dominating much of the 1960s and 1970s. We also find a change between chairs, but this mostly occurs after 1987 when we find a clearer pattern of changes in regimes pertaining to time periods when the federal funds rate was lowered, i.e. when the fed loosened the monetary policy stance."
  - Possible reason for differences: inclusion of M2 in Sims and Zha dataset affecting contemporaneous coefficient on M2 in two of four regimes.

### Robustness and Additional Diagnostics
- Robustness checks summarized (preserving source phrasing and figures referenced):
  - Figure A5 shows implied state-probabilities over time for the 3v(Mon Pol Eq)2c model.
  - Figures A1–A4 present In-Sample and Out-of-Sample Root Mean Square Forecast Errors and Conditional Forecasts using alternative conditioning information sets for all models (notes: prices in percentage annual growth rates; unemployment in level).
  - Figure A6 shows probabilities of coefficient state, smoothed estimates, 3v (Mon Pol Eq) 2c model specification.
  - Figure A5 (robustness check using smaller database for conditional forecast) presents top panel: Timeseries of Root Mean Square Forecast Error; bottom panel: average of RMSFEs over forecast horizon (2008Q1-2015Q3). Note descriptions preserved as in source.

### Principal Conclusions
- Changes in shock variances are more prominent in the data than changes in structural coefficients overall.
- A model that allows coefficient switching only in the monetary policy rule (with variance switching in all equations) attains the highest marginal data density among models considered.
- Coefficient-state dynamics align with observable policy loosening and tightening episodes (state 1 associated with lowering of federal funds rate).
- The main result—coefficient switching concentrated in monetary policy rather than in the Phillips curve—is robust to a longer sample (1960Q1–2015Q2) and to an identified structural VAR with a well-defined Phillips curve.

*Source: _wp16126 - Section 4*

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