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Econometrics

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Econometrics

Lecture Notes Hayashi, Chapter 6g

Serial Correlation in GMM

- Consider a single equation GMM model:yt = ztd + et
- The model allows for random regressors, with instruments xt.
- The model allows for conditional heteroscedasticity and serial correlation.
- Let gt = xtet

- {gt}, E(gt) = 0
- Serial correlation in {et} and hence in {gt}: Gj = E(gtgt-j’), j=0,1,2,…
- {gt} satisfies Gordin’s condition.
- The long-run covariance matrix S = j=-,…,Gj = G0 + j=1,…,(Gj + Gj’) is nonsingular.

- Given a positive definite weighting matrix W, GMM estimator dGMMW of d is consistent and asympototically normal.
- With the consistent estimate of S, the asymptotic variance of dGMMW is heteroscedasticity and autocorrelation consistent (HAC).
- The GMM estimator achieves the minimum variance when plimnW* = S-1.

- Given the consistent estimate of S, statistics for GMM model specification tests such as t, W, J, C, and LR remains valid and retain the same asymptotic distributions in the presence of serial correlation.
- What we need is to be able consistently estimate the long-run variance matrix S.

- Consistently estimate the individual autocovariances:Gj = (1/n) t=j+1,…,nĝtĝt-j’ (j=0,1,…n-1)where ĝt = xtet and et = yt-ztdGMM
- If the lag length q is known, then Gj = 0 for j>q. Therefore,S = j=-q,…,qGj = G0 + j=1,…,q(Gj + Gj’)

- If q is not known, there are several approaches of kernel estimation available:S = j=-n+1,…,n-1k(j/q(n))Gjwhere k(.) is the kernel and q(n) is the bandwidth, which increases with the sample size.
- Truncated Kernel:k(x) = 1 for |x|1; 0 for |x|>1.This truncated kernel-based S is not guaranteed to be positive semidefinite in finite sample.

- Bartlett Kernelk(x) = 1-|x| for |x|1; 0 for |x|>1.The Bartlett kernel-based S is called the Newey-West estimator.S can be made nonnegative definite in finite sample. For example, for q(n)=3, we haveS = G0 + (2/3)(G1 + G1’) + (1/3)(G2 + G2’)

- Quadratic Spectral (QS) KernelSince k(x)0 for |x|>1 in the QS kernel, all the estimated autocovariances Gj (j=0,1,…,n-1) enter the calculation of S even if q(n)<n-1.

- {gt}, gt = xtet et
- E(gtgt-j’) E(etet-j)
- Conditional HomoscedasticityWith serial correlation,E(etet-j)|xt,xt-j) = wj (j=0,1,2,…)E(gtgt-j’) = E(etet-jxtxt-j’) = wjE(xtxt-j’) = Gj

- Estimating Gj
- Let et be the estimated et or residual; wj is consistently estimated by (1/n)t=j+1,…netet-j.
- E(xtxt-j’) is consistently estimated by (1/n)t=j+1,…nxtxt-j’.
- Gj = [(1/n)t=j+1,…netet-j][(1/n)t=j+1,…nxtxt-j’]

- Estimating SS = G0 + j=1,…,q(Gj + Gj’)

- Let W is the autocovariance matrix of {et}

- If the lag length q is known, then Gj = 0 for j>q. Therefore,
- If q is not known, for Bartlett kernel, let

- The single equation GMM under conditional homoscedasticity and serial correlation is the 2SLS with serial correlation.
- Let X = [xt, t=1,2,…,n]’. Z and y are the data matrices of the regressors and the dependent variable.

- d2SLS = [Z’X(X’WX)-1X’Z]-1Z’X(X’WX)-1X’y
- The consistent estimate of the asymptotic variance-covariance matrix of d2SLS is [Z’X(X’WX)-1X’Z]-1
- If Z = X, d2SLS = [Z’Z]-1Z’y = dOLS with the consistent estimate of variance-covariance matrix [Z’Z(Z’WZ)-1Z’Z]-1

- Given that we have a consistentestimator ofW, dGLS = [Z’ W-1Z]-1Z’ W-1y
- The consistency of GLS estimator is not guaranteed.
- However, there is one important special case where GLS is consistent, and that is when the error is a finite-order autoregressive process.
- GLS estimation for the AR(p) error process.

- The Model
- yt = ztd + et
- et = fet-1 + ut

- Autocovariances
- g0 = s2/(1- f2)
- g1 = fg0 = s2 f/(1- f2)
- gj = fgj-1 = s2 fj/(1- f2)for j>1

- Var(et) = s2/(1- f2)V
- V-1 = C’C

- y* = Cy, Z* = CZ, e* = Ce = u ~ N(0,s2I)
- y* = Z* d + u

- GLS estimator dGLS = OLS estimator for the transformed model: y* = Z* d + u
- dGLS = (Z*’Z*)-1Z*’y*
- Var(dGLS) = s2 (Z*’Z*)-1
- Est[Var(dGLS)] = s2 (Z*’Z*)-1
- s2 = (y*-Z*dGLS)’(y*-Z*dGLS)/(n-L)