As stressed above, in many cases the finding of autocorrelation is an indication that the model is misspecified. If this is the case, the most natural route is notto change the estimator(from OLS to EGLS) but to change themodel. Typically, three (interrelated) types of misspecification may lead to a finding of autocorrelation in the OLS residuals:
dynamic misspecification, omitted variables and functional form misspecification.
If we leave the case where the error term is independent of all explanatory variables, there is another reason why GLS or EGLS may be inappropriate. In particular, it is pos- sible that the GLS estimator is inconsistent because the transformed model does not satisfy the minimal requirements for the OLS estimator to be consistent. This situation can arise even if OLS applied to the original equationisconsistent. Section 4.11 provides an empirical example of this issue.
4.10.1 Misspecification
Let us start with functional form misspecification. Suppose that the true linear relation- ship is betweenytand logxtas
yt=𝛽1+𝛽2logxt+𝜀t,
and suppose, for illustrative purposes, thatxt increases witht. If we nevertheless esti- mate a linear model that explainsyt fromxt, we could find a situation as depicted in Figure 4.5. In this figure, based upon simulated data withxt=tandyt=0.5 logxtplus a small error, the fitted values of a linear model are connected while the actual values are not. Very clearly, residuals of the same sign group together. The Durbin–Watson statistic corresponding to this example is as small as 0.193. The solution in this case is not to re-estimate the linear model using feasible generalized least squares but to change the functional form and include logxtrather thanxt.
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WHAT TO DO WHEN YOU FIND AUTOCORRELATION? 127
2.00
0.10
1 40
Time
y
Figure 4.5 Actual and fitted values when true model isyt=0.5 logt+𝜀t.
As discussed previously, the omission of a relevant explanatory variable may also lead to a finding of autocorrelation. For example, in Section 4.8 we saw that excluding suf- ficient variables that reflect the seasonal variation of ice cream consumption resulted in such a case. In a similar fashion, an incorrect dynamic specification may result in auto- correlation. In such cases, we have to decide whether the model of interest is supposed to be static or dynamic. To illustrate this, start from the (static) model
yt=xt𝛽+𝜀t (4.58)
with first-order autocorrelation 𝜀t=𝜌𝜀t−1+𝑣t. We can interpret the above model as describing E{yt|xt} =xt𝛽. However, we may also be interested in forecasting on the basis of current xt values as well as lagged observations on xt−1 andyt−1, that is, E{yt|xt,xt−1,yt−1}. For the above model, we obtain
E{yt|xt,xt−1,yt−1} =xt𝛽+𝜌(yt−1−xt−1 𝛽) (4.59) and we can write a dynamic model as
yt=xt𝛽+𝜌yt−1−𝜌xt−1 𝛽+𝑣t, (4.60) the error term of which does not exhibit any autocorrelation. The model in (4.60) shows that the inclusion of a lagged dependent variable and lagged exogenous variables results in a specification that does not suffer from autocorrelation. Conversely, we may find autocorrelation in (4.58) if the dynamic specification is similar to (4.60) but includes, for example, only yt−1 or some elements of xt−1. In such cases, the inclusion of these
‘omitted’ variables will resolve the autocorrelation problem.
The static model (4.58) with first-order autocorrelation provides us with E{yt|xt} as well as the dynamic forecast E{yt|xt,xt−1,yt−1} and may be more parsimonious
k k compared with a full dynamic model with several lagged variables included (with
unrestricted coefficients). It is a matter of choice whether we are interested inE{yt|xt} or E{yt|xt,xt−1,yt−1} or both. For example, explaining a person’s wage from his or her wage in the previous year may be fairly easy, but may not provide answers to the questions in which we are interested. In many applications, though, the inclusion of a lagged dependent variable in the model will eliminate the autocorrelation problem.
It should be emphasized, though, that the Durbin–Watson test is inappropriate in a model where a lagged dependent variable is present. In Subsection 5.2.1, particular attention is paid to models with both autocorrelation and a lagged dependent variable.
4.10.2 Heteroskedasticity-and-autocorrelation-consistent Standard Errors for OLS
Let us reconsider our basic model
yt=xt𝛽+𝜀t, (4.61)
where𝜀t is subject to autocorrelation. If this is the model in which we are interested, for example because we want to know the conditional expectation ofyt given a well- specifiedxt, we can choose to apply the GLS approach or apply ordinary least squares while adjusting its standard errors. This last approach is particularly useful when the correlation between𝜀tand𝜀t−scan be argued to be (virtually) zero after some lag length Hand/or when the conditions for consistency of the GLS estimator happen to be violated.
IfE{xt𝜀t} =0 andE{𝜀t𝜀t−s} =0 fors=H,H+1, . . ., the OLS estimator is consistent, and its covariance matrix can be estimated by
V̂∗{b} = ( T
∑
t=1
xtxt )−1
TS∗ ( T
∑
t=1
xtxt )−1
, (4.62)
where
S∗= 1 T
∑T t=1
e2txtxt+ 1 T
H−1∑
j=1
wj
∑T s=j+1
eses−j(xsxs−j +xs−jxs). (4.63) Note that we obtain the White covariance matrix, as discussed in Subsection 4.3.4, if wj=0, so that (4.62) generalizes (4.30). In the standard casewj=1, but this may lead to an estimated covariance matrix in finite samples that is not positive definite. To pre- vent this, it is common to use Bartlett weights, as suggested by Newey and West (1987).
These weights decrease linearly withjaswj=1−j∕H. The use of such a set of weights is compatible with the idea that the impact of the autocorrelation of orderjdiminishes with|j|. Standard errors computed from (4.62) are referred to asheteroskedasticity-and- autocorrelation-consistent (HAC) standard errorsor simplyNewey–West standard errors. Withwj=1 they are referred to as Hansen–White standard errors. HAC standard errors may also be used when the autocorrelation is, strictly speaking, not restricted toH lags, for example with an autoregressive structure. Theoretically, this can be justified by applying an asymptotic argument thatHincreases withTasT goes to infinity (but not as fast asT). Empirically, this may not work very well in small samples. Modern econo- metric software packages provide alternative ways to implement HAC standard errors.
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Either the researcher should specify the maximum lag lengthHa priori, or the package selectsHas a function of the sample size (e.g.H=T1∕4). In some programmes,H+1 is referred to as the ‘bandwidth’. The Bartlett weights guarantee that the estimatorS∗is positive definite in every sample.