Heteroskedasticity and Autocorrelation Consistent Standard Errors

Transcription

1 NBER Summer Institute Minicourse What s New in Econometrics: ime Series Lecture 9 July 6, 008 Heteroskedasticity and Autocorrelation Consistent Standard Errors Lecture 9, July, 008

2 Outline. What are HAC SE s and why are they needed?. Parametric and Nonparametric Estimators 3. Some Estimation Issues (psd, lag choice, etc.) 4. Inconsistent Estimators Lecture 9, July, 008

3 . What are HAC SE s and why are they needed? Linear Regression: y t = x t β + e t ˆ β = xx t t' xy t t' = β + xx t t' xe t t' t= t= t= t= t t t t XX t= t= ˆ ( β β) = x x ' xe ' = S A. d S XX p Σ XX = E(x t x t ), A N (0, Ω ), where Ω = lim Var hus a ˆ β N β, V, where HAC problem : ˆΩ =??? Vˆ = S Ω ˆS XX XX t = xe t t ' Lecture 9 3, July, 008

4 (Same problem arises in IV regression, GMM, ) Notation: Ω = lim Var t = xe t t ' Let w t = x t e t (assume this is a scalar for convenience) Feasible estimator of Ω must use ˆ w = xeˆ, t t t ( ˆ t ) ( w t ) he estimator is Ω ˆ { w }, but most of our discussion uses ˆ { } convenience. Ω for Lecture 9 4, July, 008

5 An expression for Ω: Suppose w t is covariance stationary with autocovariances γ. (Also, for notational convenience, assume w t is a scalar). hen Ω= var( wt) = var( w+ w w) t= = { γ0 + ( )( γ+ γ ) + ( )( γ + γ ) +... ( γ + γ )} = γ ( γ + γ ) = + = If the autocovariances are -summable so that Ω = var( w ) t t= = γ γ < then Recall spectrum of w at frequency ω is S(ω) = i γ e ω, so that Ω = π = π S(0). Ω is called the Long-run variance of w. Lecture 9 5, July, 008

6 II. Estimators (a) Parametric Estimators, w t ~ ARMA φ(l)w t = θ(l)ε t, S(ω) = iω iω θ( e ) θ( e ) σ ε iω iω π φ( e ) φ( e ) Ω = π S(0) = σ ( θ θ θ ) θ () q ε = σ ε φ() ( φ φ φp) Ω= ˆ ˆ σ ε ( ˆ θ ˆ ˆ θ θq) ( ˆ φ ˆ φ ˆ φ ) p Lecture 9 6, July, 008

7 Jargon: VAR-HAC is a version of this. Suppose w t is a vector and the estimated VAR using w t is w =Φ ˆ w Φ ˆ w + ˆ ε, t t p t p t where ˆ Σ ε = ˆˆ' εtεt t= hen Ω= ˆ ( I Φ ˆ... Φ ˆ ) Σ ˆ ( I Φ ˆ... Φ ˆ ) ' p ε p Lecture 9 7, July, 008

8 (b) Nonparametric Estimators Ω= γ = m Ω= ˆ = m K ˆ γ with ˆ γ = ww = ˆ γ ( 0) t t t= + Lecture 9 8, July, 008

9 III. Issues (a) ˆΩ 0? (b) Number of lags (m) in non-parameter estimator or order of ARMA in parametric estimator (c) form of K weights Lecture 9 9, July, 008

10 (a) ˆΩ 0? Ω= ˆ ˆ σ ε ( ˆ θ ˆ ˆ θ θq) ( ˆ φ ˆ φ ˆ φ ) p (yes) m Ω= ˆ K ˆ γ (Not necessarily) = m ˆ ( ˆ ˆ ˆ ) MA() Example: Ω = (γ + γ 0 + γ ) and Ω = γ + γ0 + γ γ γ 0 / but ˆ γ ˆ > γ 0 / is possible. Newey-West (987): Use K = m + (Bartlett Kernel ) Lecture 9 0, July, 008

11 An alternative expression for ˆ m Ω= = m K ˆ γ Let W = (w w w ) where W ~ (0, Σ WW ), U = HW, U ~ (0, Σ UU ) with Σ UU = HΣ WW H. Choose H so that Σ UU = D a diagonal matrix. A useful result: with W covariance stationary (Σ WW is special ), a particular H matrix yields (approximately) Σ UU = D (diagonal). (he u s are the coefficients from the discrete Fourier transform of the w s.) Variable π Variance (D ii ) u S(0) u and u 3 S( π/) u 4 and u 5 S( π/) u 6 and u 7 S(3 π/) u and u ( odd, similar for even) S([( )/] π/) Lecture 9, July, 008

12 Some algebra shows that m Ω= ˆ K ˆ γ = = m ku 0 ( )/ u + u + + k = where the k-weights are functions (Fourier transforms) of the K-weights. he algebraic details are unimportant for our purposes but, for those interested, they are sketched on the next slide. Jargon: the term u + u + is called the th periodogram ordinates. Because π Eu ( ) = Eu ( + ) = π S, the th periodogram ordinate is an estimate of the spectrum at frequency (π/). he important point: Evidently, ˆΩ 0 requires k 0 for all. Lecture 9, July, 008

13 he algebra linking the data w, the u s, the periodogram ordinates and the sample covariances parallels the expressions the spectrum presented in lecture. (Ref: Fuller (976), or Priestly (98), or Brockwell and Davis (99)). In particular, let c = iω t we t where ω = π/ t= hen u where ˆp γ = + u + = c ˆ = γ p p p= t= p+ cos( ω ), ( w w)( w w) t t p hus ω. u + u + = c is a natural estimator of the spectrum at frequency Lecture 9 3, July, 008

14 m Ω= ˆ K ˆ γ = = m ku 0 ( )/ u + u + + k = is then a weighted average (with weights k of the estimated spectra at difference frequencies. Because the goal is to estimate at ω = 0, the weights should concentrate around this frequency as gets large. Jargon: Kernel (K-weights, sometimes k-weights), Lag Window (Kweights, RAS follows this notation and calls these lwindow in the code), Spectral Window (k-weights) Lecture 9 4, July, 008

15 runcated Kernel: K = ( m) Spectral Weight (k) (m =, =50) Lecture 9 5, July, 008

16 Newey-West (Bartlet Kernel): K = m + Spectral Weight (k) (m =, =50) Lecture 9 6, July, 008

17 Implementing Estimator (lag order, kernel choice and so forth) Parametric Estimators: AR/VAR approximations: Berk (974) Ng and Perron (00), lag lengths in ADF tests. Related, but different issues. Shorter lags: Larger Bias, Smaller Variance Longer lags: Smaller Bias, Larger Variance Practical advice: Choose lags a little longer than you might otherwise (rational given below). Lecture 9 7, July, 008

18 How Should m be chosen? Andrews (99) and Newey and West (994): minimize mse( ˆΩ Ω) MSE = Variance + Bias ( )/ ˆ u + u + Ω= ku 0 + k = Variance: Spread the weight out over many of the squared u s: Make the spectral weights flat. Bias: E( ˆΩ) u + u depends on the values E + S( π/). So the bias depends on how flat the spectrum is around ω = 0. he more flat it is, the smaller is the bias. he more curved it is, the higher is the bias. Lecture 9 8, July, 008

19 Spectral Window (k ) for N-W/Bartlett Lag Window (K = m = 5 m = 0 m ), = m = 0 m = 50 Lecture 9 9, July, 008

20 MSE minimizing values for w t = φw t + e t, Bartlett Kernel (note the higher is φ, more serial correlation, more curved spectrum around ω = 0, more bias for given value of m.) m* = /3 (φ ) /3 /3 =.8 φ /3 /3 φ m* / / / / /3 8 7 Lecture 9 0, July, 008

21 Kernel Choice: Some Kernels (lag windows) Lecture 9, July, 008

22 Does Kernel Choice Matter? In theory, yes. (QS is optimal) In practice (for psd kernels), not really. Lecture 9, July, 008

23 Does lag length matter? Yes, quite a bit y t = β + w t, w t = φw t + e t, = 50. Reection of -sided 0% test φ m ˆ * AR m ˆ *(PW) m* m* (More Simulations: den Haan and Levin (997) available on den Haan s web page and other papers listed throughout these slides) Lecture 9 3, July, 008

24 Why use more lags than MSE Optimal (Sun, Phillips, and Jin (008) ) Intuition: z ~ N(0,σ ), ˆ σ an estimator of σ (assumed independent of z) z Pr < c Pr z c E ( z c) E g( ) ˆ σ = < = < = ( ) E g σ + E ˆ σ σ g σ + E( ˆ σ σ ) g σ = F () c + Bias( ˆ σ ) g ' + MSE( ˆ σ ) g '' χ ( ) ( ) ( ˆ σ ˆ σ ˆ σ ) ( ) ( ) '( ) ( ) ''( ) MSE Bias + Variance his formula Bias and MSE hus m should be bigger Lecture 9 4, July, 008

25 IV. Making m really big m = b where b is fixed. (Kiefer, Vogelsang and Bunzel (000), Kiefer and Vogelsang (00), Kiefer and Vogelsang (005)) Bartlett m = ( ): ˆ Ω ( wˆ ) = ( ) ˆ γ = + ˆ γ = ww ˆ ˆ = ˆ γ, and where wˆ t = w t w. t t t= + KV (00) show that some rearanging yields : t ˆ / Ω ( wˆ) = wˆ t= = [ s ] d / / Ω = [ ] / ˆ () = But, w W() s, w ξ [ s ] d / / Ω = wˆ ( W( s) sw()), and, where ξ(s) = Ω / (W(s) sw()). Lecture 9 5, July, 008

26 hus Ω ˆ ( wˆ ): t d / t= = 0 Ω ˆ ( wˆ) = wˆ Ω ( W( s) sw()) ds So that Ω ˆ ( wˆ ) is a robust, but inconsistent estimator of Ω. Distributions of t-statistics (or other statistics using this ˆΩ) will not be asymptotically normal, but large sample distributions easily tabulated (see KV (00)). Lecture 9 6, July, 008

27 able of t cricitical values F-critical values.. See KVB (DIVIDE BY!) Lecture 9 7, July, 008

28 Power loss (From KVB) Lecture 9 8, July, 008

29 Size Control (Finite Sample AR()) Lecture 9 9, July, 008

30 (a) Advantages... (i) Some what better size control (in Monte Carlos and in theory (Jansson (004)). (ii) More Robust to serial correlation (Müller (007). Müller proposes estimators that yield t-statistics with t-distributions. (b) Disadvantages... (i) Wider confidence intervals... In the one-dimensional regression problem a value of β β 0 chosen so that β is contained in the (true, not size-distorted) 90% HAC confidence interval with probability 0.0. he value of β is contained in the KVB 90% CI with probability 0.3. (Based on an asymptotic calculation) (ii) Robustness to volatility breaks... No... Lecture 9 30, July, 008

31 KVB Cousins: (Some related work) (a) (Müller (007): Just focus on low frequency observations. How many do you have? Periodogram ordinates at π, =,, (-)/ Let p denote a low frequency cutoff frequencies with periods greater than p are ω π. Solving number of periodogram ordinates with p periods p: p 60 years of data, p = 6 years, number of ordinates = 0. (Does not depend on sampling frequency). Müller : do inference based on these obs. Appropriately weighted these yield t-distributions for t-statisics. Lecture 9 3, July, 008

32 (b) Panel Data: Hansen (007). Clustered SEs (over ) when is large and n is small ( and n fixed). his too yields t-statistics being distributed t (with n df) and appropriately constructed F-stats having Hotelling t-squared distribution. (c) Ibragimov and Müller (007). Allows heterogeneity. Lecture 9 3, July, 008

33 HAC Bottom line(s) () Parametric Estimators are easy and tend to perform reasonably well. In some applications a parametric model is natural (MA(q) model for forecasting q+ periods ahead). In other circumstances VAR-HAC is sensible. (den Haan and Levin (997).) hink about changes in volatility. () Why are you estimating Ω? (a) Optimal Weighting matrix in GMM: Minimum MSE ˆΩ seems like a good idea. (Analytic results on this?) (b) Inference: Use more lags than you otherwise would think you should in Newey-West (or other non-parametric estimators). Worried? Use KVB estimators (and their critical values for tests) or Müller (007) versions (with t or F critical values). Lecture 9 33, July, 008