Estimating Structural Shocks with DSGE Models

Transcription

1 Estimating Structural Shocks with DSGE Models Michal Andrle Computing in Economics and Finance, June 4 14, Oslo

2 Disclaimer: The views expressed herein are those of the authors and should not be attributed to the International Monetary Fund, its Executive Board, or its management.

3 Outline of the Talk Structural shocks? Model vs. Filter Misspecification Propagation mechanisms, economics Number of shocks, residuals Bliss of Stochastic Singularity Factor analysis of the OECD data Give me my residuals baaack! SVD filtering Evil of Correlated shocks Causes & consequences Testing for misspecification PROCRUSTES filtering Illustrations...

4 Structural Shocks... WE NEED STRUCTURAL SHOCKS TO TELL ECONOMICALLY MEANINGFUL STORIES... Operational definition of structural shock : 1. You can tell reasonable, economically meaningful stories. Shocks are not systematically cross-correlated Misspecification: DSGE models start with uncorrelated (independent) shocks that render IRFs, FEVDs, etc. meaningful but end up with correlated shocks in most cases.

5 Issues & Suggested Solutions: 1. Model vs. Filter Misspecification a) model s economics can be off b) economics is right but the filter is ill-conditioned. How many structural shocks can we hope to identify? factor/dpca analysis of OECD data suggest just few key shocks a few shocks responsible for bulk of dynamics, bulk of shocks contributing little to data dynamics is stochastic singularity actually a blessing? Analysis in the paper: 1. Testing for misspecification: correlated shocks. SVD Filtering: dealing with stochastic singularity 3. Procrustes Filtering: (can we explaining data with uncorrelated shocks?)

6 Misspecification Sign: Correlated Shocks Assume a linear model with structural shocks, ε, Y t = D(L)ε t ε t N(, I) (1) and associated filter F(L) for uncovering structural shocks, ˆε t = F(L)Y t. () Sufficient (but not necessary) condition for misspecification is when estimated shocks, ˆε t are significantly cross- and auto-correlated. Leads to fishy shock decompositions...

7 Cross-Correlated Shocks: Fishy FISH :1 :3 1:1 1:3 :1 :3 3:1 3:3 4:1 4:3

8 Example: Stochastic Rank of US Data since 195 % % % % Real GDP (Y) Real Investment (I) Real Imports (M) Short term Interest rate (IR) percentage explained % % % Real Consumption (C) Real Exports (X) Unemployment Rate (UR) Data Fit with one factor Fit with two factors % 5 Y C I X M UR IR One factor Two factors Three factors Andrle, Brůha, Solmaz (14)

9 Example: Stochastic Rank of US Data (JPT 1) Justiniano, Primiceri, and Tambalotti (JME, 1) (JPT 1) Consumption vs. Investment Growth (q/q, ann.) 4 195:1 196:1 197:1 198:1 199:1 :1 1:1 Output vs. Hours Worked Growth (q/q, ann.) :1 196:1 197:1 198:1 199:1 :1 1:1 Output vs. Investment Growth (q/q, ann.) 1 1 3

10 Typical Investment Shock (JPT 1)... but risk shocks and credit shocks on the same boat, essentially 1. Output.6 Consumption Investment 1 Hours Inflation. Interest

11 1 3 frequency 1 3 frequency Shock Cross-Correlation & Misspecification... not a white noise, really.8 Coherence of Inv. Specific and TFP Coherence of Inv. Specific and Preference frequency Coherence of Price and Wage Markup Shocks frequency Coherence of TFP and Wage Markup Shocks

12 Misspecification Sign: Correlated Shocks JPT-1: correlation of inv. shock(t) and pref. shock(t-1) is.8 4 Diagonal, 15 years 4 Off Diagonal, 15 years Diagonal, years Off Diagonal, years Diagonal, 5 years Off Diagonal, 5 years Diagonal, 1 years Off Diagonal, 1 years Finite sample distributions of N(, I k )

13 Ill-Conditioning and Fragility Even with a great model, Kalman smoothing is a rather fragile exercise that does not aspire for robustness. Shock identification is a signal-extraction exercise, Kalman filter must be viewed as as a filter: ε t = F(L)Y t (Andrle 13a, 13b) The filter F(L) may be very ill-conditioned, i.e. over-sensitive to small changes in the input data, Y. It s a property of the model/filter! The goal of explaining 1% of data dynamics with structural shocks and a stylized model is fraught with hazards What s wrong with having residuals? Your OLS has it...

14 Robustness: Roughly Right or Precisely Wrong? Ill-conditioning and fragility: [ ] [ ] [ ] y1 1 + φ 1 ε1 = θ y for θ the model is over-sensitive to changes in y. ε (3) [ ε1 ε ] = [.1.3 ] [ ] A.4 =.4 [ ] y1 y [ y1 y ] = [ ] [ ].4 A ε ε = [ ].6. Even with the right model, a small perturbation to input data leads to important changes in the estimates of structural shocks... Below it is demonstrated that Kalman smoother has the form Y = A E as above

15 State-Space Setup The linear state-space model is given by: Y t = ZX t + Hε t (4) X t = TX t 1 + Rε t with ε t N(, I) (5) The model is said to be stochastically singular if rank S y (ω) < rank S ε (ω). For singularity it is sufficient that n ε < n Y. Today, we ll stay in time domain but frequency domain would serve just fine as well...

16 Kalman Smoother as a Least-Squares Problem (1) min X,{ε} Λ = X P 1 X + + N [Y t ZX t ] (HH ) 1 [Y t ZX t ] t=1 N [X t TX t 1 ] (RR ) 1 [X t TX t 1 ]. t=1 The genius of Kalman was to make the problem recursive!... and I m kind of doing the opposite

17 Kalman Smoother as a Least-Squares Problem () Denoting Y = [Y 1 Y... Y N ], E = [ε 1 ε... ε N ], and Z = [X E], the least-squares problem is stated as follows: Z = argmin vecy A vec Z, (6) ZT ZR + H... ZT ZT R ZTR ZR + H... A = = [O H]. (7) ZT N ZT N 1 R ZT N R ZR + H

18 SVD Filtering (1): Handles Singular Models SVD Filter shock estimates for both regular and singular problems A = [ ] [ ] [ ] S U 1 U 1 V 1 = V r i=1 where r = rank(a), U U = I, V V = I and S 1 is diagonal, s i u i v i. (8) S 1 = diag(s 1,..., s p), where p = min{m, n}. When the matrix A is singular, then rank of A is r, then s r+1 = = s p =. The solution to vec Z is then obtained as vec Z = V 1 S 1 1 U 1 vec Y (9) = r u i vec Y v i. s i (1) i=1

19 SVD Filtering () SVD Filter is an extension of the Kalman filter: (i) For regular models is identical to Kalman smoother/filter (ii) Handles singular models (iii) Makes inspecting of ill-conditioning a cinch (SVD of A) (iv) Implementation in frequency domain is more efficient (memory, speed)

20 SVD Filter Example (1) Core Inflation (π) 5 est obs obs+noise 5 :1 5:1 3:1 35:1 4:1 45:1 5:1 55:1 Output Cycle (y) 5 5 :1 5:1 3:1 35:1 4:1 45:1 5:1 55:1

21 SVD Filter Example () Core Inflation (π) 4 demand contrib est :1 5:1 3:1 35:1 4:1 45:1 5:1 55:1 Output Cycle (y) 4 :1 5:1 3:1 35:1 4:1 45:1 5:1 55:1

22 Procrustes Filtering We minimize distance of the model and data in a mean-square sense (Kalman smoother) Using SVD filter we can choose any number of shocks we want to test and distinguish residuals and structural shocks So how about choosing the shocks only from a set of reasonably uncorrelated shocks? Question asked: How big residuals I get when I fit the data with shocks that can t be too correlated? If I cannot fit much, am I in BIG trouble?

23 Intuition Orthogonal Procrustes Problem Given A and B, find orthogonal rotation matrix R such that R = argmin R A B F s.t. R R = I (11) There is an analytical solution to orthogonal Procrustes...

24 Procrustes Filter : Uncorrelated Shocks Imposed Purpose: Given the model, estimate structural shocks to match the data dynamics as best as possible, while keeping the shocks uncorrelated... Z λ = argmin { vecy A vec Z + λ N 1 E E I M }, Here,. M, is a suitable matrix measure L distance penalty function reflecting finite-sample variation Acknowledging the finite-sample distribution of shocks is important.

25 Procrustes Filter : Weak & Strong Version 1. Weak PF: penalizes contemporaneous cross-cov ONLY. Strong PF: penalizes the ACGF/spectrum profile

26 Example: Weak Procrustes using JPT-1 GDP Growth Consumption Growth 1954:4 1964:4 1974:4 1984:4 1994:4 4:4 Investment Growth :4 1964:4 1974:4 1984:4 1994:4 4:4 Real Wages Growth :4 1964:4 1974:4 1984:4 1994:4 4: :4 1964:4 1974:4 1984:4 1994:4 4:4 Hours Worked :4 1964:4 1974:4 1984:4 1994:4 4:4 Inflation (Deflator) :4 1964:4 1974:4 1984:4 1994:4 4:4

27 Example: Strong Procrustes using JPT-1

28 Thank you for your patience...