Accurate Asymptotic Approximation in the Optimal GMM Frame. Stochastic Volatility Models

Transcription

1 Accurate Asymptotic Approximation in the Optimal GMM Framework with Application to Stochastic Volatility Models Yixiao Sun UC San Diego January 21, 2014

2 GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing Approximations in the Presence of High Variation Scaling t = ˆβ β q 0 ˆV ( ˆβ) and W = ˆβ β 0 0 ^V 1 ( ˆβ) ˆβ β 0 Quantile regression: Variance depends on the probability density function which is often estimated nonparametrically. Time series regression: Variance depends on the spectral density at the origin, which again is often estimated nonparametrically. Other type of regressions or GMM that involve dependent data: spatial data, spatiotemporal data. Variance heterogeneity: Clustered data with inter-cluster heterogeneity. General nonparametric regression/sieve estimation.

3 GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing Approximations in the Presence of High Variation Scaling Conventionally, we just ignore it so that t = ˆβ β q 0 ˆV ( ˆβ) ˆβ β q 0 V ( ˆβ) W = ˆβ β 0 0 ^V 1 ( ˆβ) ˆβ β 0 ˆβ β 0 0 V 1 ( ˆβ) ˆβ β 0 This amounts to approximating the distribution of ^V( ˆβ) by the degenerate distribution concentrating on V( ˆβ). The randomness in ^V( ˆβ) is completely ignored. This leads to all sorts of size distortion. This paper develops an asymptotic approximation that captures the random variation in ^V( ˆβ).

4 Outline GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing 1 Basic Setting and Problem GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing 2 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 3 Practical Implementation Simulation Empirical Application 4

5 The Estimation Problem GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing The moment conditions hold if and only if θ = θ 0. Ef (v t, θ) = 0, t = 1, 2,..., T f () 2 R m and θ 2 R d. m d! the model may be overidenti ed. ff (v t, θ 0 )g may exhibit autocorrelation of unknown forms. For the IV estimator: f (v t, θ) = Z 0 t (Y t X t θ) = 0. For the MLE: f (v t, θ) = log p(v t, θ)/ θ.

6 An Example GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing Stochastic Volatility Model r t = σ t Z t log σ 2 t = ω + β log σ 2 t 1 ω + σ u u t where r t is the rate of return and (Z t, u t ) is iid N(0, I 2 ). θ = (ω, β, σ u ) Moment conditions: E jr t j` = c`e σ`t for some constant c` E jr t r t j j = 2π 1 E (σ t σ t j ) and Ert 2 rt 2 j = E σt 2 σt 2 j where the rhs are functions of θ.

7 Standard One-step GMM GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing De ne g T (θ) = 1 T T f (v t, θ). t=1 The GMM estimator of θ 0 is given by ˆθ GMM = arg min g T (θ) 0 WT 1 g T (θ) θ2θ where W T is a positive de nite weighting matrix. The rst-step estimator for W o = I m, Z 0 Z /T,... θ T = arg min g T (θ) 0 Wo 1 g T (θ) θ2θ

8 GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing GMM is of central Importance in Economics and Finance

9 Outline GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing 1 Basic Setting and Problem GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing 2 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 3 Practical Implementation Simulation Empirical Application 4

10 GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing Two-step GMM: Optimal Weighting Matrix According to Hansen (1982), the optimal weighting matrix W T is the asymptotic variance matrix of p T g T (θ 0 ). Lars Hansen was awarded the Nobel Memorial Prize because of this paper. hp var T gt (θ 0 )i = E 1 T T t=1 T t=1 f (v t, θ 0 )f 0 (v s, θ 0 ) Want to estimate the long run variance nonparametrically. Di erent names: Heteroskedasticity and Autocorrelation Consistent (HAC) Heteroskedasticity and Autocorrelation Robust (HAR)

11 GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing Two-step GMM: Optimal Weighting Matrix hp var T gt (θ 0 )i = T 1 T t=1 cov(u t, u s ) for u t = f (v t, θ 0 ) In nite samples, not much information is available for Eu t u s = cov(u t, u s ) when jt sj is large. It is reasonable to assume that Eu t u s is small (relative to the variance of u t ) when jt sj is large. Given the above two observations, a good estimator of Eu t u s would be t Q h T, s û t ûs 0 T where û t is an estimated version of u t and Q h (, ) satis es Q h (t/t, s/t )! 0 if jt sj /T! 1; Q h (t/t, s/t )! 1 if jt sj /T! 0.

12 GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing Two-step GMM: Optimal Weighting Matrix Let ũ t = f (v t, θ T ). The optimal Weighting Matrix or the HAR variance estimator T T 1 t W T θ T = T Q h t=1 s=1 T, s! T 1 ũ t T T ũ t τ=1 ũ s 1 T! 0 T ũ τ, τ=1 for some shrinking function Q h (, ). The smoothing parameter h controls for the degree of shrinkage. General quadratic LRV or HAR variance estimator

13 Kernel HAR Variance Estimator GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing kernel LRV estimator: t Q h T, s t s := k T bt where k b (x) = k(x/b). t s = k b T h = 1/b (there is a reason for this parametrization) Q h (t, s) depends on (t, s) only through t s. Lots of research on this. Newey and West (1987, 1994), Andrews (1991),... Note: as b increases, the variance of W T θ T increases and the bias of W T θ T decreases (behave like the number of terms in series regression).

14 GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing b=1 b=0.5 b= Figure: Graph of k(/b) Yixiao Sun: UC weights San Diegoattached Accurateto Asymptotic autocovariances Approximation inof the Optimal GMM Frame

15 Do you HAC? GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing

16 Series HAR Variance Estimator Let Q h ( t T, s T ) = 1 K Series LRV estimator: GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing K t s φ j φ j j=1 T T W T ( θ T ) = 1 K K W Tj ( θ T ) j=1 where W Tj ( θ T ) are the Direct Estimators : W Tj ( θ T ) = 1 T " T t φ j t=1 T with h = K. ũ t ũ # " T s φ j s=1 T φ j (t/t ) is a set of orthonormal bases with energy concentrated mostly at the origin and R 1 0 φ j (x) dx = 0. ũ s ũ # 0

17 Kernel HAR v.s. Series HAR GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing By Fourier series expansion or Mercer s Theorem (K-L expansion) k(t s) = λ j φ j (t) φ j (s), j=1 So for φ j () = φ j (/b), we have k( t s t bt ) = λ j φ j j=1 bt s φ j = bt λ j = k(0) = 1. j=1 Compare (Hard thresholding vs soft thresholding) Q h ( t T, s K T ) = 1 t s j=1 K φ j φ j T T t s λ j φ j φ j j=1 T T

18 Outline GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing 1 Basic Setting and Problem GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing 2 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 3 Practical Implementation Simulation Empirical Application 4

19 Two-step GMM Estimator GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing One-step (First-step) GMM estimator for W o = I m, Z 0 Z /T. θ T = arg min θ2θ g T (θ) 0 W 1 o g T (θ), Two-step (Second-step) GMM estimator ˆθ T = arg min θ2θ g T (θ) 0 W 1 T ( θ T )g T (θ). The di erence lies in the weighting matrix. Two-step (Second-step) GMM estimator is asymptotically more e cient (in what sense?) But there is a cost in estimating the weighting matrix.

20 GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing Reasons for Using a Two-step GMM Estimator E ciency of the point estimator. Power of the associated test. To implement GMM criterion-based tests, we often use the two-step estimator. Numerical stability.

21 Two-step GMM Testing GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing The null hypothesis H 0 : Rθ 0 = r 2 R q and alternative hypothesis H 1 : Rθ 0 6= r. Focus on linear restrictions but nonlinear ones are allowed. The F-test version of the Wald statistic is W T = p T (R ˆθ T r) 0 n R G T ( ˆθ T ) 0 W 1 T ( ˆθ T )G T ( ˆθ T ) 1 R 0 o 1 p T (R ˆθ T r)/p, where G T (θ) = g T (θ) θ 0. Compare with the asymptotic variance in the case of GLS: G T ( ˆθ T ) 0 W 1 T ( ˆθ T ) G T ( ˆθ T ) X 0 var(ujx ) X

22 Two-step GMM Testing GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing Let ˆθ T,R be the restricted second-step GMM estimator: ˆθ T,R = arg min θ2θ g T (θ) 0 W 1 T ( θ T )g T (θ) s.t. Rθ = r, and de ne pd T := Tg T ( ˆθ T ) 0 WT 1( θ T )g T ( ˆθ T ) Tg T ( ˆθ T,R ) 0 WT 1( θ T )g T ( ˆθ T,R ) The score statistic or Lagrange Multiplier (LM) statistic. Let T (θ) = GT 0 (θ) W T 1( θ T )g T (θ) and de ne ps T = T T ( ˆθ T,R ) 0 G 0 T ( ˆθ T,R )W 1 T ( θ T )G T ( ˆθ T,R ) 1 T ( ˆθ T,R )

23 Overidenti cation Test GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing J Statistic: Series HAR: Kernel HAR: J T = Tg T ˆθ T 0 W 1 T ( θ T ) 1 gt ˆθ T J T = Tg T ˆθ T 0 W 1 T ( ˆθ T ) 1 gt ˆθ T J T! d B q (1) 0 Z 1 d K J T! F (q, K q + 1). K q Z 1 0 Q h (r, s) db q (r)db 0 q (s) 1 B q (1). Details are available in Sun and Kim (2013, JoE).

24 Outline Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 1 Basic Setting and Problem GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing 2 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 3 Practical Implementation Simulation Empirical Application 4

25 Conventional Asymptotic Approximation Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR The fundamental question is how to approximate the distributions of W T, D T and S T. This is a di cult problem as W T ( θ T ) shows up twice. A rst step is to come up with a good approximation to the distribution of W T ( θ T ). Under the conventional asymptotics, we approximate the distribution of W T ( θ T ) by the degenerate distribution that concentrates at its probability limit. Under the conventional asymptotics, all three statistics W T, D T and S T are asymptotically distributed as χ 2 p/p. Numerous studies have shown that the chi-square approximation is very poor.

26 Towards the New Asymptotics Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR The conventional asymptotics is obtained by embedding the nite sample situation in the asymptotic sequence h! as T! but h/t! 0. However, in any given nite sample, h is a given constant. A more accurate approximation may be obtained by embedding the nite sample in a new asymptotic sequence where h is held xed

27 Towards the New Asymptotics Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR For Series HAR variance, K is xed. W T ( θ T ) = 1 K K W Tj ( θ T ) j=1 W T ( θ T ) is an average of a nite number of direct and inconsistent estimators W Tj ( θ T ). The amount of smoothing does not increase with the sample size. For kernel HAR variance, b is xed. In the frequency domain, W T ( θ T ) is a smoothing over nite number of periodograms. Vogelsang and coauthors

28 Towards the New Asymptotics Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR The new asymptotics is called xed-smoothing asymptotics (h is xed) The conventional asymptotics is called increasing-smoothing asymptotics (h! ). Fixed-smoothing asymptotics Increasing-smoothing asymptotics h is xed as T! h! as T! but h/t! 0

29 Towards the New Asymptotics Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR

30 Towards the New Asymptotics Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR To a great extent and from a broad perspective, the idea is line with many other areas of research in econometrics where more accurate distributional approximations are the focus of interest. A common theme for coming up with a new approximation is to embed the nite sample situation in a di erent limiting thought experiment: Strong IV vs. Weak IV vs. Many (weak) IV asymptotics Unit root/normal approximation vs. local to unity approximation Extreme Quantile vs. Intermediate Quantile Approximation Asymptotics under a drifting sequence vs. Asymptotics under a xed parameter.

31 Related Recent Literature Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR Time Series setting: Kiefer, Vogelsang and Bunzel (2000) Kiefer and Vogelsang (2002a, 2002b, 2005), Jansson (2004), Sun, Phillips, Jin (2008), Sun and Phillips (2009), Gonçlaves and Vogelsang (2011), Phillips (2005), Müller (2007), and Sun (2011, 2013). Spatial setting: Bester, Conley, Hansen and Vogelsang (2011) and Sun and Kim (2014). Panel data setting: Gonçalves (2011), Kim and Sun (2013), and Vogelsang (2012). Other related papers: Ibragimov and Müller (2010), Shao (2010) and Bester, Hansen and Conley (2011).

32 Outline Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 1 Basic Setting and Problem GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing 2 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 3 Practical Implementation Simulation Empirical Application 4

33 Fixed-smoothing Asymptotics Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR Let Q T,h t T, s = Q h ( t T T, s T ) 1 T 1 T T τ=1 Q h ( t T, τ T ) + 1 T T Q h ( τ τ=1 T, s T ) T τ 1 =1 T τ 2 =1 Q h ( τ 1 T, τ 2 T ) = Q h ( t T, s T ) Q h (, s T ) Q h ( t T, ) + Q (, ) be the (doubly) demeaned weighting function. Then W T (θ) = 1 T T T QT,h t=1 s=1 t T, s f (v t, θ)f (v s, θ) 0. T

34 Fixed-smoothing Asymptotics Heuristically where and 1 W T θ T = T ) T t=1 Z 1 Z T s=1 Q T,h W = Λ W Λ 0 and W = Q h (r, s) = Q h (r, s) Z 1 0 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR t T, s f (v t, θ T )f (v s, θ T ) 0. T Q h (r, s)λdb m (r)db m (s) 0 Λ 0 := W Z 1 0 Z 1 Z 1 0 Q h (r, τ 2 ) dτ Q h (τ 1, s) dτ 1 Q h (r, s)db m (r)db m (s) 0 Z 1 Z Q h (τ 1, τ 2 ) dτ 1 dτ 2.

35 Fixed-smoothing Asymptotics Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR Heuristically p T ˆθ T θ 0 = G 0 T W T ( θ T ) 1 G T 1 G 0 T W T ( θ T ) 1p T g T (θ 0 ) d! G 0 W 1 G 1 G 0 W 1 ΛB m (1). where G = G (θ 0 ) and G (θ) = E f (v t, θ) θ 0. ˆθ T is not asymptotically normal but rather asymptotically mixed normal. Conditional on W 1, the limiting distribution is normal with the conditional variance depending on W 1. Due to the asymptotic mixed-normality, it is di cult to show the asymptotic pivotality for W T, D T, and S T.

36 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR Fixed-smoothing Asymptotics: Main Results Theorem W T = hr p T ˆθ i 0 n T θ 0 R G T ( ˆθ T ) 0 WT 1( ˆθ T )G T ( ˆθ T ) o 1 1 R 0 h R p T ˆθ i 0 T θ 0 /p h d! R G 0 W 1 G i 1 0 h G 0 W 1 ΛB m (1) R G 0 W 1 G i 1 1 R 0 h R G 0 W 1 G i 1 G 0 W 1 ΛB m (1) /p d = F

37 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR Fixed-smoothing Asymptotics: Main Results C pp = C qq = Z 1 Z Z 1 Z 1 0 Theorem 0 Q h (r, s)db p (r)db p (s) 0, C pq = Z 1 Z Q h (r, s)db p (r)db q ( Q h (r, s)db q (r)db q (s) 0, D pp = C pp C pq C 1 qq C 0 pq. d F = Bp (1) C pq Cqq 1 B q (1) 0 D 1 pp Bp (1) C pq Cqq 1 B q (1) /p, where p is the number of joint hypotheses q is the degree of overidenti cation q = m d.

38 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR Fixed-smoothing Asymptotics: Main Results F is pivotal (Neat and Elegant). When the model is exactly identi ed, we have q = 0. In this case, F = B p (1) 0 C 1 pp B p (1) /p. This limit is the same as in the one-step GMM framework. The limiting distribution F depends on the degree of over-identi cation q. Typically such a dependence shows up only when the number of moment conditions is assumed to grow with the sample size. Here the number of moment conditions is xed. It remains in the limiting distribution because it contains information on the dimension of the random matrix W.

39 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR Fixed-smoothing Asymptotics: Main Results Theorem Under some assumptions, for a xed h, D T = W T + o p (1) and S T = W T + o p (1), as T!.

40 Outline Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 1 Basic Setting and Problem GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing 2 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 3 Practical Implementation Simulation Empirical Application 4

41 Series Case Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR Note that K = Cpp C pq Cpq 0 C qq K Z 1 j=1 0 Z 1 Z 1 d = Qh (r, s) db p+q (r) dbp+q(s) Z 1 0 Φ j (r) db p+q (r) Φ j (r) db p+q (r) where R 1 0 Φ j (r) db p+q (r) s d iidn(0, I p+q ), we have Cpp C K pq s W p+q (K, I p+q ), C 0 pq C qq 0 a standard Wishart distribution. D pp = C pp C pq Cqq 1 Cpq 0 s W p (K independent of both C pq and C qq. q, I p ) /K and is

42 (Mixed) Noncentral F Approximation Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR This brings F close to Hotelling s T 2 distribution, which is the same as a standard F distribution after some multiplicative adjustment. The only di erence is that B p (1) C pq Cqq 1 B q (1) is not normal and hence Bp (1) C pq Cqq 1 B q (1) 2 does not follow a chi-square distribution. However, conditional on C pq Cqq 1 B q (1), jjb p (1) C pq Cqq 1 B q (1) jj 2 follows a noncentral χ 2 p distribution with noncentrality parameter 2 = jjc pq C 1 qq B q (1) jj 2. It then follows that F is conditionally distributed as a (mixed) noncentral F distribution.

43 (Mixed) Noncentral F Approximation Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR Let κ = K (K p q + 1), δ2 = E 2 = pq K q 1. Theorem For OS HAR variance estimation, we have κ 1 F d = Fp,K p q+1 2, a mixed noncentral F random variable with random noncentrality parameter 2. Theorem P(κ 1 F < z) = F p,k p q+1 z, δ 2 + o K 1 as K!.

44 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 8 p = 1 8 p = K K p = 3 F, q = 0 F, q = 1 F, q = p = 4 F, q = 0 F, q = 1 F, q = F, q = F, q = NCF, q = 0 NCF, q = 1 NCF, q = 2 NCF, q = NCF, q = 0 NCF, q = 1 NCF, q = 2 NCF, q = K K

45 Outline Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 1 Basic Setting and Problem GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing 2 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 3 Practical Implementation Simulation Empirical Application 4

46 Kernel case Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR Theorem For kernel HAR variance estimation, we have where χ 2 p is independent of η 2, pf d = χ 2 p /η 2 η 2 d = e 0 p e 0 p Ip + C pq C 1 D 1 pp Ip + C pq C 1 qq C 1 qq C 0 pq qq Cqq 1 Cpq 0 ep Ip + C pq C 1 qq C 1 qq C 0 pq ep and e p = (1, 0,..., 0) 0 2 R p.

47 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR A Major Breakthrough: High-order re nement The xed-smoothing approximation is high order correct under the conventional asymptotics. Let F α and χ α be the critical values from the xed-smoothing approximation and chi-square approximation, then as h! such that h/t! 0, P(W T < F α ) = α + o(h 1 ) + other terms P(W T < χ α /p) = α + O(h 1 ) + other terms Culmination of multiple-year s research e ort. Tedious proof (90 pages) in a separate paper. Too technical to be presented even in an econometric seminar.

48 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR New Asymptotics under the Local Alternatives Under H 1 : Rθ 0 = r δ 0 / p T where W T ( ˆθ T ) d! F (kδ 0 k 2 V ), h i 0 F (kδ 0 k 2 V ) = B p (1) C pq Cqq 1 B q (1) + V 1/2 δ 0 D 1 pp h i B p (1) C pq Cqq 1 B q (1) + V 1/2 δ 0 /p and V = R G 0 Ω 1 G 1 R 0.

49 Outline Practical Implementation Simulation Empirical Application 1 Basic Setting and Problem GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing 2 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 3 Practical Implementation Simulation Empirical Application 4

50 Basic Framework Practical Implementation Simulation Empirical Application Sample moment conditions: g T (θ) = 1 T T t=1 f (v t, θ 0 ) One-step and two-step GMM estimators θ T = arg min g T (θ) 0 g T (θ) and θ2θ ˆθ T = arg min g T (θ) 0 WT 1 θ T ; h g T (θ) θ2θ where for ũ t := f (v t, θ T ) W T θ T ; h = 1 T T T t=1 s=1 Q h t T, s T ũ t 1 T! 2 T ũ τ. τ=1 If you prefer, the Newey-West estimator can be used. The optimal smoothing parameter h can be estimated by h T := h T ( θ T ) using Andrews (1991) plug-in approach.

51 Wald, LM and LR Practical Implementation Simulation Empirical Application Wald Statistic W T = T (R ˆθ T r) 0 n R Ĝ 0 T W 1 T ĜT 1 R 0 o 1 (R ˆθ T r)/p GMM Criterion-based LR type statistic D T = T ĝ 0 T W 1 T ĝt Tg T ( ˆθ T,R ) 0 W 1 T g T ( ˆθ T,R ) /p Score or LM type of statistic S T = T T ( ˆθ T,R ) 0 G 0 T ( ˆθ T,R ) W T 1 G T ( ˆθ T,R ) 1 T ( ˆθ T,R )/p

52 Reference Distribution Practical Implementation Simulation Empirical Application Let e t := (e 0 t,p, e 0 t,d p, e0 t,q) 0 s iidn(0, I m ), and C p,t = p 1 T T e t,p, C q,t = 1 T p t=1 T e t,q t=1 C pp,t = 1 T C pq,t = 1 T C qq,t = 1 T T T t=1 τ=1 T T t=1 τ=1 T T t=1 τ=1 where ẽ i,j = e i,j 1 T T s=1 e s,j. Q h T ( t T, τ T )ẽ t,pẽ 0 τ,p, Q h T ( t T, τ T )ẽ t,pẽ 0 τ,q Q h T ( t T, τ T )ẽ t,qẽ 0 τ,q

53 Reference Distribution Practical Implementation Simulation Empirical Application De ne D pp,t = C pp,t C pq,t C 1 qq,t C 0 pq,t. Use F et as the reference distribution for W T, D T, and S T : F et = h C p,t C pq,t C 1 qq,t C q,t i 0 h i D 1 pp,t C p,t C pq,t Cqq,T 1 C q,t /p For the t-test, use t et as the reference distribution: h i t et = C p,t C pq,t Cqq,T 1 C q,t / p D pp,t for p = 1 F et and t et are nite sample versions of F and t. Easy to simulate

54 Practical Implementation Simulation Empirical Application Reference Distribution in the Case of OS HAR estimation De ne δ 2 T = pq h T q 1. Let F 1 α ( δ 2 p, h T p q+1 T ) be the (1 α) quantile of the noncentral F distribution F p, h T p q+1 ( δ T 2 ) with degrees of freedom (p, h T p q + 1) and noncentrality parameter δ T 2. We can use h T p q + 1 h T F 1 α p, h T p q+1 ( δ 2 T ) as the α-level critical value for W T, D T, and S T. Critical values for t-tests can be similarly approximated.

55 Outline Practical Implementation Simulation Empirical Application 1 Basic Setting and Problem GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing 2 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 3 Practical Implementation Simulation Empirical Application 4

56 DGP Practical Implementation Simulation Empirical Application DGP y t = x 0,t α + x 1,t β 1 + x 2,t β 2 + x 3,t β 3 + ε y,t where x 0,t 1 and x 1,t, x 2,t and x 3,t are scalar regressors that are correlated with ε y,t. θ = (α, β 1, β 2, β 3 ) 0 We have m instruments z 0,t, z 1,t,..., z m 1,t with z 0,t 1. The reduced-form equations for x 1,t, x 2,t and x 3,t are given by x j,t = z j,t + m 1 i=d 1 z i,t + ε xj,t for j = 1, 2, 3.

57 DGP Practical Implementation Simulation Empirical Application We assume that z i,t for i 1 follows either an AR(1) process q z i,t = ρz i,t ρ 2 e zi,t, or an MA(1) process q z i,t = ρe zi,t ρ 2 e zi,t, where e zi,t = ei zt + ezt p 0 2 The DGP for ε t = (ε yt, ε x1 t, ε x2 t, ε x3 t) 0 is the same as that for (z 1,t,..., z m 1,t ) except the dimensionality di erence.

58 A Tale of Four Tests Practical Implementation Simulation Empirical Application Null Hypotheses H 01 : β 1 = 0, H 02 : β 1 = β 2 = 0, Four tests with critical values χ 1 α H 03 : β 1 = β 2 = β 3 = 0 pp, K K p + 1 F p,k 1 α p+1, K K p q + 1 F p,k 1 α p q+1 δ2, F 1 α

59 Simulation Results Practical Implementation Simulation Empirical Application Table: Empirical size of the χ 2 test, F test, noncentral F test and nonstandard F test based on the orthonormal series HAR variance estimator for the AR(1) case with T = 100, number of joint hypotheses p and number of overidentifying restrictions q ρ χ 2 CF NCF F χ 2 CF NCF F p = 1, q = 0 p = 3, q =

60 Simulation Results Practical Implementation Simulation Empirical Application Table: Empirical size of the χ 2 test, F test, noncentral F test and nonstandard F test based on the orthonormal series HAR variance estimator for the AR(1) case with T = 100, number of joint hypotheses p and number of overidentifying restrictions q ρ χ 2 CF NCF F χ 2 CF NCF F p = 1, q = 1 p = 3, q =

61 Simulation Results Practical Implementation Simulation Empirical Application Table: Empirical size of the χ 2 test, F test, noncentral F test and nonstandard F test based on the orthonormal series HAR variance estimator for the AR(1) case with T = 100, number of joint hypotheses p and number of overidentifying restrictions q ρ χ 2 CF NCF F χ 2 CF NCF F p = 1, q = 2 p = 3, q =

62 Simulation Results Practical Implementation Simulation Empirical Application Table: Empirical size of the χ 2 test, F test, noncentral F test and nonstandard F test based on the kernel HAR variance estimator for the AR(1) case with T = 100, number of joint hypotheses p and number of overidentifying restrictions q ρ χ 2 F [0] F [q] χ 2 F [0] F [q] χ 2 F [0] F [q] p = 1, q = 0 p = 2, q = 0 p = 3, q =

63 Simulation Results Practical Implementation Simulation Empirical Application Table: Empirical size of the χ 2 test, the F (0) test and the F (q) test based on the Parzen kernel LRV estimator for the AR(1) case with T = 100, number of joint hypotheses p and number of overidentifying restrictions q ρ χ 2 F [0] F [q] χ 2 F [0] F [q] χ 2 F [0] F [q] p = 1, q = 1 p = 2, q = 1 p = 3, q =

64 Simulation Results Practical Implementation Simulation Empirical Application Table: Empirical size of the χ 2 test, the F (0) test and the F (q) test based on the Parzen kernel LRV estimator for the AR(1) case with T = 100, number of joint hypotheses p and number of overidentifying restrictions q ρ χ 2 F [0] F [q] χ 2 F [0] F [q] χ 2 F [0] F [q] p = 1, q = 2 p = 2, q = 2 p = 3, q =

65 Outline Practical Implementation Simulation Empirical Application 1 Basic Setting and Problem GMM Estimation Optimal Weighting Matrix Two-step Estimation and Testing 2 Big Picture Main Results Representation and Approximation: Series HAR Representation and Approximation: Kernel HAR 3 Practical Implementation Simulation Empirical Application 4

66 Practical Implementation Simulation Empirical Application Estimation and Inference in SVM: the Model SVM r t = σ t Z t log σ 2 t = ω + β log σ 2 t 1 ω + σ u u t where r t is the rate of return and (Z t, u t ) is iid N(0, I 2 ). The parameter vector is θ = (ω, β, σ u ). We impose the restriction that β 2 (0, 1), which is an empirically relevant range. The model and the parameter restriction are the same as those considered by Andersen and Sorensen (1996), which gives a detailed discussion on the motivation of the stochastic volatility models and the GMM approach. Alternative to ARCH and GARCH.

67 Practical Implementation Simulation Empirical Application Estimation and Inference in SVM: Data We employ the GMM to estimate the log-normal stochastic volatility model; MLE is too complicated. The data are weekly returns of S&P 500 stocks, which are constructed by compounding daily returns with dividends from a CRSP index le. We consider both value-weighted returns (vwretd) and equal-weighted returns (ewretd). The weekly returns range from the rst week of 2001 to the last week of 2012 with sample size T = 627. We use weekly data in order to minimize problems associated with daily data such as asynchronous trading and bid-ask bounce. This is consistent with Jacquier, Polson and Rossi (1994).

68 Practical Implementation Simulation Empirical Application Estimation and Inference in SVM: the Moments The GMM approach relies on functions of the time series fr t g to identify the parameters of the model. For the simple log-normal stochastic volatility model, it is easy to obtain the following moment conditions: E jr t j` = c`e σ`t for ` = 1, 2, 3, 4 with constants (c 1, c 2, c 3, c 4 ) E jr t r t j j = 2π 1 E (σ t σ t j ) and Er 2 t r 2 t j = E σ 2 t σ 2 t j for j = 1, 2, where E E ω` σ`t = exp 2 + `2σ u 2 σ`1 t σ`2 t j = E σ`1 t E σ`2 t 8(1 β 2 ) σ 2 exp u `1`2β j 4(1 β 2 )

69 Practical Implementation Simulation Empirical Application Estimation and Inference in SVM: the Moments Higher order moments can be computed but we choose to focus on a subset of lower order moments. Andersen and Sorensen (1996) points out that it is generally not optimal to include too many moment conditions when the sample is limited. On the other hand, it is not advisable to include just as many moment conditions as the number of parameters. When T = 500 and θ = ( 7.36, 0.90,.363), which is an empirically relevant parameter vector, Table 1 in Andersen and Sorensen (1996, Table 1) shows that it is MSE-optimal to employ nine moment conditions.

70 Practical Implementation Simulation Empirical Application Estimation and Inference in SVM: Baseline Moments For this reason, we employ two sets of nine moment conditions given in the appendix of Andersen and Sorensen (1996). The baseline set of the nine moment conditions are Ef i (r t, θ) = 0 for i = 1,..., 9 with f i (r t, θ) = jr t j i c`e σ`t, i = 1,..., 4 f 5 (r t, θ) = jr t r t 1 j 2π 1 E (σ t σ t 1 ) f 6 (r t, θ) = jr t r t 3 j 2π 1 E (σ t σ t 3 ) f 7 (r t, θ) = jr t r t 5 j 2π 1 E (σ t σ t 5 ) f 8 (r t, θ) = rt 2 rt 2 2 E σt 2 σt 2 2 f 9 (r t, θ) = rt 2 rt 2 4 E σt 2 σt 2 4 (1)

71 Practical Implementation Simulation Empirical Application Estimation and Inference in SVM: Alternative Moments The alternative set of the nine moment conditions are Ef i (r t, θ) = 0 for i = 1,..., 9 with c`e, i = 1,..., 4 f i (r t, θ) = jr t j i σ`t f 5 (r t, θ) = jr t r t 2 j 2π 1 E (σ t σ t 2 ) f 6 (r t, θ) = jr t r t 4 j 2π 1 E (σ t σ t 4 ) f 7 (r t, θ) = jr t r t 6 j 2π 1 E (σ t σ t 6 ) f 8 (r t, θ) = rt 2 rt 2 1 E σt 2 σt 2 1 f 9 (r t, θ) = rt 2 rt 2 3 E σt 2 σt 2 3 (2) In each case m = 9 and d = 3, and so the degree of overidenti cation is q = m d = 6.

72 Practical Implementation Simulation Empirical Application Estimation and Inference in SVM: CI Construction We focus on constructing 90% and 95% con dence intervals (CI s) for β. Given the high nonlinearity of the moment conditions, we invert the GMM distance statistics to obtain the CI s, which are invariant to the reparametrization of model parameters. For example, a 95% con dence interval is the set of β values in (0,1) that the D T test does not reject at the 5% level. We search over the grid from 0.01 to 0.99 with increment 0.01 to invert the D T test. As in the simulation study, we employ three di erent critical values: χ 1 p α /p, F 1 α [0], and F 1 α [q], which correspond to three di erent asymptotic approximations. For the series LRV estimator, F 1 α [0] is a critical value from the corresponding F approximation.

73 Estimation and Inference in SVM Practical Implementation Simulation Empirical Application Equally-weighted Return Value-weighted Return Baseline Alternative Baseline Alternative Series χ Series F [0] Series F [q] K (140) (138) (140) (140) Bartlett χ Bartlett F [0] Bartlett F [q] b (.0155) (.0157) (0.155) (.0155)

74 Estimation and Inference in SVM Practical Implementation Simulation Empirical Application Equally-weighted Return Value-weighted Return Baseline Alternative Baseline Alternative Parzen χ Parzen F [0] Parzen F [q] b (.0136) (.0139) (.0136) (.0136) QS χ QS F [0] QS F [q] b (.0068) (.0069) (.0067) (.0067)

75 Estimation and Inference in SVM Practical Implementation Simulation Empirical Application There is a noticeable di erence between the CI s based on the χ 2 approximation and the nonstandard F [q] approximation. This is especially true for the case with baseline moment conditions, the Bartlett kernel, and equally-weighted returns. Taking into account the randomness of the estimated weighting matrix leads to wider CI s. The log-volatility may not be as persistent as previously thought.

76 Conclusion The paper has developed new, more accurate, and convenient asymptotic approximations for heteroskedasticity and autocorrelation robust inference in a two-step GMM framework. It is more accurate than the chi-square or normal approximation, especially true when the number of joint hypotheses being tested is large the degree of over-identi cation is high the underlying processes are persistent. I hope this line of research can help deliver more reliable statistical inference.

77 Possible Extensions The results of the paper can be extended easily to the spatial setting, spatial-temporal setting or panel data setting. See Kim and Sun (2014) for a paper in the spatial setting. The general results of the paper can be extended to a nonparametric sieve GMM framework. See Chen, Liao and Sun (2014) for a recent development on autocorrelation robust sieve inference for time series models. Now that we have a new and more accurate asymptotic framework, we can give a more honest assessment of the relative merits of one-step and two-step GMM procedures. Preliminary results are available in Hwang and Sun (2014).

78 Thank you!