Estimating and Accounting for the Output Gap with Large Bayesian Vector Autoregressions

Transcription

1 Estimating and Accounting for the Output Gap with Large Bayesian Vector Autoregressions James Morley 1 Benjamin Wong 2 1 University of Sydney 2 Reserve Bank of New Zealand The view do not necessarily represent those of the Reserve Bank of New Zealand Frontiers in Macroeconomics and Macroeconometrics Workshop Hitotsubashi University, 3-4 November 2017

2 Multivariate Trend-Cycle Decomposition Most T-C methods are univariate (e.g. Hodrick-Prescott filter) However, Evans and Reichlin (1994) consider multivariate Beveridge-Nelson (BN) decomposition: Additional information mechanically lowers the signal-to-noise ratio Estimates of trend and cycle sensitive to variables included in VAR forecasting model Does possible overfitting overstate the amplitude of the cycle?

3 Estimated U.S. Output Gap from Univariate and Multivariate BN Decompositions 2 variable VAR includes output growth and the unemployment rate. 3 variable VAR includes output growth, CPI inflation, and the federal funds rate. 7 variable VAR includes all of the variables in the 2 and 3 variable systems, as well as capacity utilization, the growth of industrial production, and the growth of real personal consumption expenditure.

4 This Paper We propose using Bayesian shrinkage when estimating VAR model Consider large info sets à la Banbura, Giannone, and Reichlin (2010) But shrink towards low signal-to-noise ratio for target variable, instead of random walk Also, use out-of-sample forecasts of target variable to set shrinkage parameter rather than in-sample fit of a smaller system We show how to account for trend and cycle in terms of different sources of information or structural shocks Empirical application considers BVARs with up to 138 variables to estimate the U.S. output gap Monte Carlo analysis examines effects of shrinkage in large systems and when misspecifying the relevant information set

5 Main Findings BVARs produce plausible/intuitive estimates of the U.S. output gap Unemployment rate, CPI, housing starts, consumption, stock prices, real M1, and federal funds rate are key informational variables Estimates largely robust to including additional variables BVARs have better (pseudo) out-of-sample forecasts of output growth than an AR(1) model Monetary policy shocks play little role in the output gap, while oil price shocks explain about 10% of variance over different horizons Monte Carlo analysis suggests BVAR inferences more robust than MLE in large systems or when misspecifying the number of variables

6 Kamber, Morley, and Wong (forthcoming, REStat) Signal-to-noise ratio, δ, maps into the sum of the autoregressive coefficients in an AR(p) model Interpretation: δ% of the variance of output growth is permanent KMW show ρ = 1 1/ δ, where ρ is sum of AR coefficients KMW find δ 0.25 for U.S. data when maximizing the amplitude-to-noise ratio for the BN cycle given an AR(p) model KMW can be interpreted as a dogmatic prior on δ Our approach: Use δ to inform prior on AR coefficients for target variable in BVAR

7 U.S. Output Gap from KMW s BN Filter, δ = 0.25

8 Shrinkage Priors Let output growth be the s th equation in the BVAR E[ V[ E[β ij l ] = 0 V[β ij l ] = p l=1 p l=1 { λ 2 l, 2 λ 2 l 2 β ss l ] = ρ(δ) β ss l ] = χ 2. σ 2 i σ 2 j i = j, otherwise We require χ << λ, thus we set λ = 10χ One hyperparameter: λ We want λ 0 (i.e., more shrinkage) as more series are added in We optimize λ based on out of sample RMSE Key Advantage No need for MCMC simulation of posterior

9 Estimation using dummy observations Y = Xβ + u 0 k,n J p diag(σ 1... σ N )/λ Y d = diag(σ 1... σ n), X d = 0 n,k 0 1,s 1 ρ/χ 0 1,n s 1 1,n [ ] 0 1,s 1 1/χ 0 1,n s Y = [Y Y d ], X = [X X d ] Posterior Distribution vec(β) Σ, Y N(vec( β, Σ (X X ) 1 ) Σ Y IW ( Σ, T d + T k + 2) where β = (X X ) 1 )X Y and Σ = (Y X β) (Y X β).

10 Data Benchmark model includes output growth (target variable) + 22 variables (taking logs as appropriate and differencing until stationary): 1. Oil Prices 2. CPI inflation 3. Unemployment Rate 4. Hourly Earnings 5. Federal Funds Rate 6. Stock Price Index 7. Yield Spread 8. GDP Deflator 9. Employment 10. Income 11. Real PCE 12. Industrial Production 13. Capacity Utilization 14. Housing Starts 15. PPI (all commodities) 16. PCE Deflator 17. Hours 18. Productivity 19. Total Reserves 20. Non Borrowed Reserves 21. Real M1 22. Real M2

11 BN Decomposition Consider companion form of VAR(p) forecasting model: ( x t µ) = F( x t 1 µ) + Hν t BN trend and cycle for x t is (see Morley, 2002): τ t = x t + F(I F) 1 ( x t µ) c t = F(I F) 1 ( x t µ) or, letting Γ i = F i (I F) 1 τ t = x t + Γ 1 ( x t µ) c t = Γ 1 ( x t µ).

12 Cycle can be written as a linear decomposition of all the historical forecast errors... c t = Γ 1 ( x t µ) { t 1 } Γ i +1 Hν t i i=0

13 ...and so can trend growth τ t = {x t + Γ 1 ( x t µ)} {x t 1 + Γ 1 ( x t 1 µ)} = µ + Γ 0 Hν t.

14 Two Decompositions 1. Sources of information Which variables contain the most information for estimating trend and cycle? Which variables should be included in forecasting model? 2. Role of Structural Shocks Given forecast errors and identification restrictions, SVAR analysis straightforward What drives the trend and cycle?

15 U.S. Output Gap (Benchmark Model)

16 Standard Deviations of Informational Contributions

17 Varying the Information Set

18 Omitting Important Information

19 Out of Sample RMSE

20 Causal Determinants of Output Gap and Trend Growth We identify two shocks using standard timing restrictions An oil price shock A monetary policy shock Then we consider a forecast error variance decomposition (FEVD) and a historical decomposition

21 FEVD

22 Historical Decomposition

23 Monte Carlo Analysis DGPs based on application, T = variable 2. 8 variable variable Given simulated and actual data, estimate VAR via maximum likelihood versus Bayes and then estimate cycle using BN decomposition 1. 1 variable 2. 8 variable variable Calculate RMSE of estimates relative to true cycle (based on BN decomposition given true VAR and population values of parameters)

24 (a) Mean and standard deviation of RMSE relative to true cycle Estimated Model 1 variable 8 variables 23 variables 1 variable 8 variables 23 variables 1 variable Shrinkage (0.239) (0.904) (0.530) MLE (0.261) (1.072) (2.357) 8 variables Shrinkage DGP (0.189) (0.392) (0.337) MLE (0.195) (0.546) (0.732) 23 variables Shrinkage (0.192) (0.388) (0.343) MLE (0.202) (0.514) (0.593) (b) Proportion of Monte Carlo Draws RMSE Shrinkage < RMSE MLE Estimated Model 1 variable 8 variables 23 variables 1 variable DGP 8 variables variables

25 Summary Bayesian shrinkage makes application of BN decomposition with large information sets feasible and avoids overfitting Movements in trend and cycle can be accounted for based on different sources of information or structural shocks When estimating the U.S. output gap, it is more important to include key variables than to consider a really large information set This suggests carefully selecting observables is better than using factors when considering large datasets for T-C decomposition

26 Other Applications and Extensions Reliability in real time? Time-varying parameters Sources of trend inflation (Kamber and Wong, 2017) Role of foreign shocks in driving output gap and trend growth for open economies (Morley, Vehbi, and Wong, in progress) Mixed frequency modeling Multiple target variables neutral rates Financial cycles

27 Canadian Output Gap (Morley, Vehbi, and Wong)

28 Historical Decomposition of the Canadian Output Gap

29 Historical Decomposition of Canadian Annual Trend Growth

30 Additional Slide: Why is estimated output gap deeper in 1982 than in 2009?