Discussion of "Forecasting in Dynamic Factor Models Subject to Structural Instability, by James Stock and Mark Watson

Transcription

1 Discussion of "Forecasting in Dynamic Factor Models Subject to Structural Instability, by James Stock and Mark Watson Paolo Giordani December 2007 Stock Watson comment () Stock Watson comment December / 12

2 Main theoretical result The principal component estimator of the factors asymptotically spans the space of the true factors even if factor loadings are time-varying (subject to certain conditions). The conditions are less restrictive than what the Introduction may suggest. For example, a fraction π of the series can share a break (time and size). The breaks don t need to average out. Potentially very useful. "Given well-estimated factors, forecasts can be made by standard TVP or rolling regression methods". Computational load for multiple change-point models also much reduced. Stock Watson comment (Sveriges Riksbank) Stock Watson comment December / 12

3 The alternative, setting up a state-space model with breaks, is much more expensive X t (N,1) = c t + Λ t F t + e t, Σ = Cov(e t ) diagonal (k,1) F t = m t + Φ(F t 1 m t ) + ɛ t, Ω = Cov(ɛ t ) diagonal Question: How do we check the conditions? SW check whether the pre and post-break space spanned by the common factors estimated on the full sample is nearly the same as if factors were estimated on separate sub-samples. In this case it works well, but what if, say, the break date was 2004 rather than 1984? Stock Watson comment (Sveriges Riksbank) Stock Watson comment December / 12

4 Main empirical results Some evidence of instability in factor loadings. Strong evidence of instability in the forecasting regressions, which combine: 1 factor loadings Λ 2 factor dynamics Φ 3 means c and m "Consistent with the dynamics of the factor process changing between the two subsamples". Makes sense: changes are likely to be shared by many variables (eg great moderation may have a ected mean, persistence and volatility of most nominal variables). Stock Watson comment (Sveriges Riksbank) Stock Watson comment December / 12

5 Main comments 1 In a small simulation, I nd that SW s key result often (approximately?) holds though breaks do not respect "Condition TV". 2 Overdi erencing may not be necessary or useful. SW di erence every variable suspect of I(1) behavior. In some cases, these variables are arguably piece-wise stationary. Example: in ation. Stock Watson comment (Sveriges Riksbank) Stock Watson comment December / 12

6 tock Watson comment (Sveriges Riksbank) Stock Watson comment December / 12

7 Simulation Factor loadings constant, factor dynamics change. Two factors, real and nominal: One break at t = T /2 +1. F r,t = m r + ϕ r F r,t+1 + ω r ɛ r,t F n,t = m n + ϕ n F n,t+1 + ω n ɛ n,t. Factor dynamics very roughly calibrated on quarterly US GDP growth and in ation. Real factor: constant m r = 0 and ϕ r = 0.4, ω r from 0.6 to 0.3. Nominal factor: m n from 10 to 5, ϕ n from 0.9 to 0.5, ω n from 1 to 0.5. N = , T = 100, Σ = I. Stock Watson comment (Sveriges Riksbank) Stock Watson comment December / 12

8 2 Λ = 6 4 λ λ Nr 0 0 λ Nr λ N 3, where λ i N(1, σ 2 Λ 7 ). 5 Stock Watson comment (Sveriges Riksbank) Stock Watson comment December / 12

9 I assume that SW di erence the nominal variables. SW would do a great job at recovering F t (or F r,t and F n,t ), whether di erencing the nominal variables or not (see Tables). Hence, in this case, no need to overdi erence. Interesting since it does not seem that all these examples are covered by the theorem. Stock Watson comment (Sveriges Riksbank) Stock Watson comment December / 12

10 Over-di. Levels T = 400, N = 40 Canonical correlations with true 1st and 2nd factor σ 2 Λ = 0, no breaks 0.997, , σ 2 Λ = 1, no breaks 0.998, , σ 2 Λ = 0, m n,2 = m n, , , σ 2 Λ = 1, m n,2 = m n, , , , σ 2 Λ = 0, m n,2 = m n, , , , σ 2 Λ = 1, m n,2 = m n, , , , Stock Watson comment (Sveriges Riksbank) Stock Watson comment December / 12

11 Levels Comment σ 2 Λ = 1, m n,2 = m n, , Large break σ 2 Λ = 1, m n,2 = m n,1 15 Large break 0.988, N r = 50, N n = 150 almost everywhere σ 2 Λ = 1, m n,2 = m n,1 5, T = , Very short sample σ 2 Λ = 1, m n,2 = m n,1 5, T = , Very long sample tock Watson comment (Sveriges Riksbank) Stock Watson comment December / 12

12 I also added shifts in loading parameters: extreme shifts required for canonical correlations to fall substantially. When I added shifts in loading parameters: 1 Overdi erencing more fragile than levels. 2 If canonical correlation for nominal factor was "low" (say 0.93), including 4 rather than 2 empirical factors took it to Stock Watson comment (Sveriges Riksbank) Stock Watson comment December / 12