A strong consistency proof for heteroscedasticity and autocorrelation consistent covariance matrix estimators

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1 A strong consistency proof for heteroscedasticity and autocorrelation consistent covariance matrix estimators Robert M. de Jong Department of Economics Michigan State University 215 Marshall Hall East Lansing, Michigan 48824, USA Phone October 21, Introduction This paper considers strong consistency of heteroscedasticity and autocorrelation consistent covariance matrix estimators. Sometimes such estimators in the literature are referred to as Newey-West estimators. Weak consistency proofs for these estimators can be found in White (1984), Newey and West (1987), Gallant and White (1988), Andrews (1991), Hansen (1992a), and De Jong and Davidson (1997). The only results establishing strong consistency, to the best of the author s knowledge, are Corradi (1997) and Altissimo and Corradi (1997). Corradi (1997) improves upon the results in Altissimo and Corradi (1997). This paper improves Corradi s conditions substantially. A strong consistency proof is given that is The author thanks an anonymous referee for comments that lead to finding the error in Bruce Hansen s paper, and Bruce Hansen for discussion and for pointing out a possible way of fixing the problem in the proof. 1

2 based on a corrected version of the proof in Hansen (1992a). In addition, I point out that the proof in Hansen (1992a) is in error as it stands, and a corrected version of Hansen s proof is given. The conditions needed for the strong consistency proof are only slightly stronger than those needed for the corrected version of Hansen s (1992) weak consistency proof. 2 Main results Heteroskedasticity and autocorrelation consistent covariance matrix estimators seek to estimate Ω = lim n 1 Ex t x s n s=1 where x t R m, and it is assumed that all elements of Ω are finite. As in Hansen (1992a) it is assumed that the parameter θ R k has some consistent estimator ˆθ, and in practice we will have to use ˆx t = x t (ˆθ) instead of x t (θ) = x t for the estimation of Ω. Identically to Hansen (1992a), define ˆΩ = n 1 j= n+1 k(j/γ n)ˆγ(j) where ˆΓ(j) = n 1 ˆx tˆx t+j for j 0 and ˆΓ(j) = ˆΓ( j) for j < 0. Also, define Ω = n 1 j= n+1 k(j/γ n) Γ(j) where Γ(j) is defined as ˆΓ(j) except that the variables x t are used instead of ˆx t in the definition. k(.) is called the kernel function, and γ n is called the bandwidth or lag truncation parameter. Our results will be based on the following lemma: Lemma 1 If for some 0 < β <, k=1 E max 2 k n 2 k+1 X n β <, then X n as 0. The proofs of this lemma and of the theorems can be found in Section 3. We will need the following assumption for showing our strong consistency result. Note that. denotes the Euclidean norm in what follows, and for a random vector X we define X p = (E X p ) 1/p. Below, α(m) and φ(m) denote the α- and φ-mixing coefficients for x t. Assumption 1 1. For all x R, k(x) 1 and k(x) = k( x); k(0) = 1; k(x) is continuous at zero and for almost all x R; k(x) dx <. R 2. For some r (2, 4] and for some p r, sup t 1 x t p <, and A = 12 α(m) 2(1/r 1/p) < m=0 2

3 or A = 4 φ(m) 1 2/p <. m=0 3. γ n is nondecreasing in n, γ n as n, and γ n = O(n 1/2 1/r (log(n)) 1/r λ ) for some λ > k(x) l(x), where l(x) is a nonincreasing function such that x l(x) dx <. R Using Assumption 1, the following theorem is easily derived: Theorem 1 Under Assumption 1, Ω Ω as 0. A corrected version of Hansen s (1992a) weak consistency proof requires the following assumptions, that are weaker than Assumptions 1.3 and 1.4: Assumption 1.3 γ n as n, and γ n = o(n 1/2 1/r ). Assumption 1.4 x k(x) dx <. R The corrected version of Theorem 1 in Hansen (1992a) is then as follows: Theorem 2 Under Assumptions 1.1, 1.2, 1.3, and 1.4, Ω Ω p 0. The differences between the corrected version of Hansen s (1992a) weak consistency result of Theorem 2 and the new strong consistency result of Theorem 1 are the requirement that γ n is nondecreasing, a slightly different requirement on the asymptotic behavior of γ n, and the stronger conditions of Assumption 1.4 in comparison to 1.4. But clearly, the strengthening of Assumptions 1.1, 1.2, 1.3 and 1.4 to 1.1, 1.2, 1.3 and 1.4 is only minor. The interested reader is referred to Section 3 for an explanation of the error in Hansen (1992a). The error in Hansen (1992a) is far from innocent, as it slows down the proven convergence rate of Ω Ω from γ n n 2/r 1, as incorrectly reported in Hansen (1992a), to γ 2 nn 2/r 1. In practice we often need a strong consistency result for ˆΩ rather than for Ω. Such a result can be proven under the following assumption. Below, let x a t denote the a-th element of x t, where a = 1,..., m, and let Θ denote the parameter space for θ. Θ is assumed to be convex and compact. Assumption 2 For some λ > 0, γ n ˆθ θ (log(n)) 1+λ = O(1) almost surely, and max sup n 1 (E sup ( / θ)x a t (θ) 2 + E sup x t (θ) 2 ) <. a n 1 3

4 The above assumption is needed for showing the following theorem: Theorem 3 Under Assumptions 1 and 2, ˆΩ Ω as 0. Note that typically, minimization estimators satisfy n 1/2 (log log(n)) 1/2 ˆθ θ = O(1) almost surely, implying that for that case, the first assumption of Assumption 2 is satisfied if for some λ > 0, γ n n 1/2 (log(n)) 1+λ = O(1). For example, Sin and White (1995) provide such law of the iterated logarithm type results for minimization estimators. Corradi (1997) showed strong consistency of Ω under the assumption that γ n = o(n 1/8 ), under the existence of higher than 8th moments, and under more strict requirements on the alpha-mixing coefficients. A result for ˆΩ is not given in that paper. Altissimo and Corradi (1997) does provide such a result, although under stronger conditions than needed in this paper. Note that Newey-West type estimators have applications in the unit root literature as well, and Hansen (1992a) provides a weak consistency result for such cases. It is possible to copy his reasoning and the reasoning of Theorem 3 and come up with an analogue of Hansen s Theorem 3 in the unit root setting. However, to the best of the author s knowledge, no almost sure convergence rate of, for example, the least squares estimator in the presence of a unit root has been established, so it would be difficult to judge the content of such a result. 3 Proofs Proof of Lemma 1: as Note that according to Davidson (1994), Theorem 18.3, X n 0 if lim m P (sup n m X n > δ) = 0 for all δ > 0. Next, note that sup n m X n sup k [log(m)/ log(2)] max 2 k n 2 X n, and k+1 therefore P (sup X n > δ) P ( max X n > δ) n m 2 k n 2 k+1 δ β k=[log(m)/ log(2)] and the result follows. k=[log(m)/ log(2)] E max 2 k n 2 k+1 X n β, 4

5 Proof of Theorem 1 and 2: First note that E Ω Ω 0 as in Hansen (1992a) under the stated conditions. Also note that the proof of Lemma 2 in Hansen (1992a) is in error, because in the proof of Hansen s Lemma 2, it is implicitly incorrectly assumed that (in the notation of this paper) x t x t+j = E(x t x t+j x t+m, x t+m 1,...) for 0 m j. The proof can be corrected by noting that x t x t+j satisfies E(x t x t+j F t+j ) = x t x t+j where F t = σ(x i : 1 i t), and proceeding to consider the adapted mixingale {x t x t+j, F t+j }. Then, using Hansen s Lemma 1, for j 0 and γ > β 1 E(x a t x b t+j F t+j m ) Ex a t x b t+j β (12α(m j) 1/β 1/γ I(m > j) + 2I(m j)) x t 2γ x t+j 2γ. (1) Therefore, using Lemma 2 in Hansen (1991) (see also Hansen (1992b)) for the adapted mixingale {x a t x b t+j, F t+j }, max max x a 2 k n 2 k+1 t x b t+j Ex a t x b t+j r/2 ( ) 36 (12α(m j) 1/β 1/γ I(m > j) + 2I(m j)) m=0 2 (k+1)(2/r) 36(r/(r 2)) 3/2 (A + 2j) sup t 1 2 k+1 j (r/(r 2)) 3/2 ( x t r/2 p x t+j r/2 p ) 2/r x t 2 p. (2) As in Hansen (1992a), the same result can be established for the φ-mixing case. The main argument then is as follows: max 2 k n 2 k+1 Ω () E Ω () r/2 n 1 max n 1 k(j/γ n ) (x a 2 k n 2 k+1 t x b t+j Ex a t x b t+j) r/2 2 k+1 1 max n 1 l(j/γ n ) (x a 2 k n 2 k+1 t x b t+j Ex a t x b t+j) r/2 5

6 2 k+1 ( l(j/γ 2 k+1) 2 k max (x a 2 k n 2 k+1 t x b t+j Ex a t x b t+j) r/2 ) 2 k+1 ( l(j/γ 2 k+1) 2 k 2 (k+1)(2/r) 36(r/(r 2)) 3/2 (A + 2j) sup t 1 = O(γ 2 2 k+1 2 k(2/r 1) ) x t 2 p where r/2 > 1, and the second and third third inequality use the assumed properties of γ n, k(.) and l(.), the fourth uses the result established in Equation (2), and the last result uses that sup n 1 γn 2 n l(j/γ n) j < if x l(x) dx <. Therefore by Lemma 1 we will R have strong consistency for Ω if k=0 γr 2 k(r/2)(2/r 1) <. If γ 2 k+1 n = O(n 1/2 1/r (log(n)) 1/r λ ) for some λ > 0, the last requirement is satisfied. Proof of Theorem 2: Similarly to the previous proof, note that Ω E Ω 0 as in Hansen (1992a), and using the same mixingale definition it follows that Ω E Ω r/2 n 1 n 1 k(j/γ n ) (x a t x b t+j Ex a t x b t+j) r/2 n 1 n 1 = O(γ 2 nn 2/r 1 ) k(j/γ n ) n 2/r 36((r/(r 2)) 3/2 (A + 2j) sup t 1 x t 2 p under the conditions of the theorem, and the second inequality uses Lemma 2 of Hansen (1991). The result now follows. 6

7 Proof of Theorem 3: First note that the result follows by Theorem 1 if we can show that ˆΩ that (ˆΩ Ω) () n 1 j= n+1 k(j/γ n )n 1 (ˆx a t ˆx b t+j x a t x b t+j) Ω as 0. Next, note n 1 j= n+1 k(j/γ n ) ˆθ θ n 1 sup( ( / θ)x a t (θ) x b t+j(θ) + ( / θ)x b t+j(θ) x a t (θ) ) 2 max a ( n 1 j= n+1 k(j/γ n ) ˆθ θ (n 1 sup ( / θ)x a t (θ) 2 ) 1/2 (n 1 sup x t (θ) 2 ) 1/2 ) C max a ((γ 1 n n 1 j= n+1 k(j/γ n ) )(log(n)) 1 λ (n 1 sup ( / θ)x a t (θ) 2 ) 1/2 (n 1 sup x t (θ) 2 ) 1/2 ) C (log(n)) 1 λ (n 1 sup x t (θ) 2 ) 1/2 max a ((n 1 sup ( / θ)x a t (θ) 2 ) 1/2 ), for constants C > 0 and C > 0, where the second inequality uses Taylor s theorem, the third uses the Cauchy-Schwartz inequality, and the fourth uses the assumed almost sure convergence rate for ˆθ θ. The result now follows from an application of Lemma 1 with β = 1, together with E max (log(n)) 1 λ max 2 k n 2 k+1 a k=1 (n 1 sup ( / θ)x a t (θ) 2 ) 1/2 (n 1 sup x t (θ) 2 ) 1/2 C k=1 k 1 λ max (sup a n 1 n 1 E sup ( / θ)x a t (θ) 2 ) 1/2 (sup n 1 n 1 E sup x t (θ) 2 ) 1/2 < for some constant C > 0. 7

8 References Altissimo, F. and V. Corradi, 1997, A LIL for m-estimators and applications to hypothesis testing with nuisance parameters, IER working paper , University of Pennsylvania. Andrews, D.W.K., 1991, Heteroscedasticity and autocorrelation consistent covariance matrix estimation, Econometrica 59, Corradi, V., 1997, Deciding between I(0) and I(1) via FLIL-based bounds, mimeo, University of Pennsylvania. Davidson, J., 1994, Stochastic limit theory (Oxford University Press, Oxford). De Jong, R.M. and J. Davidson, 1997, Consistency of kernel estimators of heteroscedastic and autocorrelated covariance matrices, mimeo, Michigan State University. Gallant, A.R. and H. White, 1988, A unified theory of estimation and inference for nonlinear dynamic models (Basil Blackwell, New York). Hansen, B.E., 1991, Strong laws for dependent heterogeneous processes, Econometric theory 7, Hansen, B.E., 1992a, Consistent covariance matrix estimation for dependent heterogeneous processes, Econometrica, 60, Hansen, B.E., 1992b, Strong laws for dependent heterogeneous processes. Erratum, Econometric theory 8, Newey, W.K. and K.D. West, 1987, A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix, Econometrica, 55, Sin, C.Y. and H. White, 1995, Information criteria for selecting possibly misspecified parametric models, Journal of Econometrics, 71, White, H., 1984, Asymptotic theory for econometricians (Academic Press, New York). 8