Econometrics

1
Econometrics

• Lecture Notes Hayashi, Chapter 6g
• Serial Correlation in GMM

2
The Model

• Consider a single equation GMM modelyt ztd
  et
• The model allows for random regressors, with
  instruments xt.
• The model allows for conditional
  heteroscedasticity and serial correlation.
• Let gt xtet

3
Serial Correlation

• gt, E(gt) 0
• Serial correlation in et and hence in gt Gj
  E(gtgt-j), j0,?1,?2,
• gt satisfies Gordins condition.
• The long-run covariance matrix S ?j-?,,?Gj
  G0 ?j1,,?(Gj Gj) is nonsingular.

4
Serial Correlation

• 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 plimn??W S-1.

5
Serial Correlation

• 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.

6
Estimating S

• Consistently estimate the individual
  autocovariancesGj (1/n) ?tj1,,n gtgt-j
  (j0,1,n-1)where gt xtet and et yt-ztdGMM
• If the lag length q is known, then Gj 0 for
  jgtq. Therefore,S ?j-q,,qGj G0
  ?j1,,q(Gj Gj)

7
Estimating S

• If q is not known, there are several approaches
  of kernel estimation availableS
  ?j-n1,,n-1k(j/q(n))Gj where k(.) is the
  kernel and q(n) is the bandwidth, which increases
  with the sample size.
• Truncated Kernelk(x) 1 for x?1 0 for
  xgt1.This truncated kernel-based S is not
  guaranteed to be positive semidefinite in finite
  sample.

8
Estimating S

• Bartlett Kernelk(x) 1-x for x?1 0 for
  xgt1.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)

9
Estimating S

• Quadratic Spectral (QS) KernelSince k(x)?0 for
  xgt1 in the QS kernel, all the estimated
  autocovariances Gj (j0,1,,n-1) enter the
  calculation of S even if q(n)ltn-1.

10
Conditional Homoscedasticity

• gt, gt xtet ? et
• E(gtgt-j) ? E(etet-j)
• Conditional HomoscedasticityWith serial
  correlation,E(etet-j)xt,xt-j) wj
  (j0,?1,?2,)E(gtgt-j) E(etet-jxtxt-j)
  wjE(xtxt-j) Gj

11
Conditional Homoscedasticity

• Estimating Gj
• Let et be the estimated et or residual wj is
  consistently estimated by (1/n)?tj1,netet-j.
• E(xtxt-j) is consistently estimated by
  (1/n)?tj1,nxtxt-j.
• Gj (1/n)?tj1,netet-j(1/n)?tj1,nxtxt-j
  
• Estimating SS G0 ?j1,,q(Gj Gj)

12
Conditional Homoscedasticity

• Let W is the autocovariance matrix of et

13
Conditional Homoscedasticity

• If the lag length q is known, then Gj 0 for
  jgtq. Therefore,
• If q is not known, for Bartlett kernel, let

14
Conditional Homoscedasticity

• The single equation GMM under conditional
  homoscedasticity and serial correlation is the
  2SLS with serial correlation.
• Let X xt, t1,2,,n. Z and y are the data
  matrices of the regressors and the dependent
  variable.

15
Conditional Homoscedasticity

• d2SLS ZX(XWX)-1XZ-1ZX(XWX)-1Xy
• The consistent estimate of the asymptotic
  variance-covariance matrix of d2SLS is
  ZX(XWX)-1XZ-1
• If Z X, d2SLS ZZ-1Zy dOLS with the
  consistent estimate of variance-covariance matrix
  ZZ(ZWZ)-1ZZ-1

16
Conditional Homoscedasticity

• Given that we have a consistent estimator of W,
  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.

17
GLS for AR(1) Process

• The Model
• yt ztd et
• et fet-1 ut
• Autocovariances
• g0 s2/(1- f2)
• g1 f g0 s2 f/(1- f2)
• gj f gj-1 s2 fj/(1- f2) for jgt1

18
GLS for AR(1) Process

• Var(et) s2/(1- f2) V
• V-1 CC

19
GLS for AR(1) Process

• y Cy, Z CZ, e Ce u N(0,s2I)
• y Z d u

20
GLS for AR(1) Process

• GLS estimator dGLS OLS estimator for the
  transformed model y Z d u
• dGLS (ZZ)-1Zy
• Var(dGLS) s2 (ZZ)-1
• EstVar(dGLS) s2 (ZZ)-1
• s2 (y-ZdGLS)(y-ZdGLS)/(n-L)