Appendix E. Long-run Covariance Estimation

The long-run (variance) covariance matrix (LRCOV) occupies an important role in modern econometric analysis. This matrix is, for example, central to calculation of efficient GMM weighting matrices (Hansen 1982), heteroskedastic and autocorrelation (HAC) robust standard errors (Newey and West 1987), and is employed in unit root (Phillips and Perron 1988) and cointegration analysis (Phillips and Hansen 1990, Hansen 1992b).

EViews offers tools for computing symmetric LRCOV and the one-sided LRCOV using nonparametric kernel (Newey-West 1987, Andrews 1991), parametric VARHAC (Den Haan and Levin 1997), and prewhitened kernel (Andrews and Monahan 1992) methods. In addition, EViews supports Andrews (1991) and Newey-West (1994) automatic bandwidth selection methods for kernel estimators, and information criteria based lag length selection methods for VARHAC and prewhitening estimation.

Technical Discussion

Our basic discussion and notation follows the framework of Andrews (1991) and Hansen (1992a).

Consider a sequence of mean-zero random p -vectors {Vt(v)} that may depend on a

K -vector of parameters v , and let Vt ∫ Vt(v0) where v0 is the true value of v . We are interested in estimating the LRCOV matrix Q ,

2p times the spectral density matrix of Vt evaluated at frequency zero (Hansen 1982, Andrews 1991, Hamilton 1994).

Closely related to Q are two measures of the one-sided LRCOV matrix:

j = 0

776—Appendix E. Long-run Covariance Estimation

The matrix L1 , which we term the strict one-sided LRCOV, is the sum of the lag covariances, while the L0 also includes the contemporaneous covariance G(0) . The two-sided

LRCOV matrix Q is related to the one-sided matrices through Q = G(0) + L1 + L1¢ and

Q = L0 + L0¢ – G(0).

Despite the important role the one-sided LRCOV matrix plays in the literature, we will focus our attention on Q , since results are generally applicable to all three measures; exception will be made for specific issues that require additional comment.

In the econometric literature, methods for using a consistent estimator ˆ and the corre- v

Vˆ ∫ V (ˆ )

sponding t t v to form a consistent estimate of Q are often referred to as heteroskedasticity and autocorrelation consistent (HAC) covariance matrix estimators.

There have been three primary approaches to estimating Q :

1.The nonparametric kernel approach (Andrews 1991, Newey-West 1987) forms estimates of Q by taking a weighted sum of the sample autocovariances of the observed data.

2.The parametric VARHAC approach (Den Haan and Levin 1997) specifies and fits a parametric time series model to the data, then uses the estimated model to obtain the implied autocovariances and corresponding Q .

3.The prewhitened kernel approach (Andrews and Monahan 1992) is a hybrid method that combines the first two approaches, using a parametric model to obtain residuals that “whiten” the data, and a nonparametric kernel estimator to obtain an estimate of the LRCOV of the whitened data. The estimate of Q is obtained by “recoloring” the prewhitened LRCOV to undo the effects of the whitening transformation.

Below, we offer a brief description of each of these approaches, paying particular attention to issues of kernel choice, bandwidth selection, and lag selection.

Nonparametric Kernel

The class of kernel HAC covariance matrix estimators in Andrews (1991) may be written as:

k is a symmetric kernel (or lag window) function that, among other conditions, is continous at the origin and satisfies k(x) £ 1 for all x with k (0) = 1 , and bT > 0 is a band-

778—Appendix E. Long-run Covariance Estimation

Note that k(x) = 0 for x > 1 for all kernels with the exception of the Daniell and the Quadratic Spectral. The Daniell kernel is presented in truncated form in Neave (1972), but EViews uses the more common untruncated form. The Bartlett kernel is sometimes referred to as the Fejer kernel (Neave 1972).

A wide range of kernels have been employed in HAC estimation. The truncated uniform is used by Hansen (1982) and White (1984), the Bartlett kernel is used by Newey and West (1987), and the Parzen is used by Gallant (1987). The Tukey-Hanning and Quadratic Spectral were introduced to the econometrics literature by Andrews (1991), who shows that the latter is optimal in the sense of minimizing the asymptotic truncated MSE of the estimator (within a particular class of kernels). The remaining kernels are discussed in Parzen (1958, 1961, 1967).

Bandwidth

The bandwidth bT operates in concert with the kernel function to determine the weights for the various sample autocovariances in Equation (E.4). While some authors restrict the bandwidth values to integers, we follow Andrews (1991) who argues in favor of allowing real valued bandwidths.

To construct an operational nonparametric kernel estimator, we must choose a value for the bandwidth bT . Under general conditions (Andrews 1991), consistency of the kernel estimator requires that bT is chosen so that bT Æ • and bT § T Æ 0 as T Æ •. Alternately, Kiefer and Vogelsang (2002) propose setting bT = T in a testing context.

For the great majority of supported kernels k (j § bT) = 0 for j > bT so that the bandwidth acts indirectly as a lag truncation parameter. Relating bT to the corresponding integer lag number of included lags m requires, however, examining the properties of the kernel at the endpoints ( j § bT = 1). For kernel functions where k (1) π 0 (e.g., Truncated, Parzen-Geometric, Tukey-Hanning), bT is simply a real-valued truncation lag, with at most m = floor(bT) autocovariances having non-zero weight. Alternately, for kernel functions where k(1) = 0 (e.g., Bartlett, Bohman, Parzen), the relationship is slightly more com-

780—Appendix E. Long-run Covariance Estimation

for the other EViews supported kernels which have q = 2 . The Truncated kernel does not have a theoretically proscribed choice, but Andrews recommends using aˆ (2). The Daniell kernel has q = 2 , though we remind you that it does not satisfy the conditions for Andrews’s theorems. “Kernel Function Properties” on page 785 summarizes the values of ck and q for the various kernel functions.

It is of note that the Andrews and Newey-West estimators both require an estimate of a(q) that requires forming preliminary estimates of Q and the smoothness of Q . Andrews and Newey-West offer alternative methods for forming these estimates.

Andrews Automatic Selection

The Andrews (1991) method estimates a(q) parametrically: fitting a simple parametric time series model to the original data, then deriving the autocovariances G(j) and corresponding a(q) implied by the estimated model.

Andrews derives aˆ (q) formulae for several parametric models, noting that the choice between specifications depends on a tradeoff between simplicity and parsimony on one hand and flexibility on the other. EViews employs the parsimonius approach used by Andrews in his Monte Carlo simulations, estimating p -univariate AR(1) models (one for each element of Vˆ t ), then combining the estimated coefficients into an estimator for a(q).

For the univariate AR(1) approach, we have:

where ˆf (sq ) are parametric estimators of the smoothness of the spectral density for the s -th variable (Parzen’s (1957) q -th generalized spectral derivatives) at frequency zero. Estimators for ˆf (sq ) are given by:

782—Appendix E. Long-run Covariance Estimation

ear combinations of the autocovariance matrices, then use the scalar results to compute nonparametric estimators of the Parzen smoothness measures.

To implement the Newey-West optimal bandwidth selection method we require a value for n , the lag-selection parameter, which governs how many autocovariances to use in forming the nonparametric estimates of f (q) . Newey and West show that n should increase at (less than) a rate that depends on the properties of the kernel. For the Bartlett and the ParzenGeometric kernels, the rate is T 2 § 9 . For the Quadratic Spectral kernel, the rate is T 2 § 25 . For the remaining kernels, the rate is T 4 § 25 (with the exception of the Truncated and the Daniell kernels, for which the Newey-West theorems do not apply).

In addition, one must choose a weight vector w . Newey-West (1987) leave open the choice of w , but follow Andrew’s (1991) suggestion of ws = 1 for all but the intercept in their Monte Carlo simulations. EViews differs from this choice slightly, setting ws = 1 for all s .

Parametric VARHAC

Den Haan and Levin (1997) advocate the use of parametric methods, notably VARs, for LRCOV estimation. The VAR spectral density estimator, which they term VARHAC, involves estimating a parametric VAR model to filter the Vˆ t , computing the contemporaneous covariance of the filtered data, then using the estimates from the VAR model to obtain the implied autocovariances and corresponding LRCOV matrix of the original data.

Suppose we fit a VAR(q ) model to the {Vˆ t}. Let Aˆ j be the p ¥ p matrix of estimated j -th order AR coefficients, j = 1, º, q . Then we may define the innovation (filtered) data and estimated innovation covariance matrix as:

784—Appendix E. Long-run Covariance Estimation

In contrast to the VAR specification in the VARHAC estimator, the prewhitening VAR specification is not necessarily believed to be the true time series model, but is merely a tool for obtaining Vt values that are closer to white-noise. (In addition, Andrews and Monahan adjust their VAR(1) estimates to avoid singularity when the VAR is near unstable, but EViews does not perform this eigenvalue adjustment.)

Next, we obtain an estimate of the LRCOV of the whitened data by applying a kernel estimator to the residuals:

Lastly, we recolor the estimator to obtain the VAR prewhitened kernel LRCOV estimator:

The prewhitened kernel procedure differs from VARHAC only in the computation of the LRCOV of the residuals. The VARHAC estimator in Equation (E.17) assumes that the residu-

als Vt are white noise so that the LRCOV may be estimated using the contemporaneous

variance matrix G , while the prewhitening kernel estimator in Equation (E.21) allows

ˆ (0)

for residual heteroskedasticity and serial dependence through its use of the HAC estimator

ˆ . Accordingly, it may be useful to view the VARHAC procedure as a special case of the

Q

prewhitened kernel with k(0) = 1 and k(x) = 0 for x π 0 .

The recoloring step for one-sided prewhitened kernel estimators is complicated when we

expressions for one-sided LRCOVs are quite involved but the VAR(1) specification may be used to provide insight. Suppose that the VARHAC estimators of the one-sided LRCOV

j = 1

Then the prewhitened kernel one-sided LRCOV estimators are given by: