i=i /v and running the aggregate time-series regression giving and x..
2.1.3 Pooled estimators
The data could be pooled assuming common slope parameters. Different pooled estimators are obtained depending on whether the intercepts are assumed to be the same or different across groups. In the latter case, two estimators are proposed in the literature, depending on whether the intercepts are treated as fixed unknown parameters or random. These are referred to as the 'fixed-effects' and 'random-effects' estimators. In the case of common intercepts (i.e. the simple pooled estimator) the procedure involves running the ordinary least squares (OLS) regression yjt=f}'xit+oit over ;'=1, 2, TV and t= 1, 2 T. Thus:
Mean group estimators
Finally the data could be used to estimate coefficients for each group separately, which is possible given that T is assumed large, giving
The parameters of interest, the average of A>. can then be calculated explicitly. There are a number of ways such an average can be computed. We refer to these estimators as the 'mean group estimators'. One possibility would be to use a simple average over tin ^efficient estimates for each group. Alternatively, it may be of interest to use a weighted average of A V such as the generalized least squares estimator suggested by Swamy (1971). In the case of a Swamy-type estimator the weights are inversely related to the standard errors of the estimated coefficients (e.g. Hsiao 1986, 1992).
All four procedures give average estimates of pi. The difference is that in the case of the last estimator the averaging is explicit, while in the other cases it is implicit. In applied work, the time-series and cross-section procedures are most commonly used, the pooled estimators less often, and explicit averaging over estimates for each group is rather rare. In the case where the regressors are all strictly exogenous and the parameters are assumed to be random and distributed independently of the regressors, then all four procedures give unbiased estimates of the mean of the parameters, albeit with different variances. Thus applied workers can argue that if they are primarily interested in unbiased estimates of the mean effect, it does not matter which they use. This is essentially the argument of Zellner (1969), who showed that there is no aggregation bias under these conditions.
There also seems to be an implicit assumption that the above argument can be extended to dynamic models with lagged endogenous variables, in the sense that all four estimators will be consistent rather than unbiased. However, it is not true that time-series, cross-section and pooled estimators will generally give consistent estimates in dynamic random coefficient models. We provide a different explanation from those current in the literature of why estimates based on time series and cross-sections tend to provide conflicting results. Bohi (1981) discusses the differences in the estimates of energy demand elasticities for OECD countries. The fact that the time-series and cross-section estimates commonly differ is usually taken to suggest that there is misspecification in one or both of them. Because the pattern of correlation between included and excluded variables will differ in time-series and cross-section regressions, model misspecification will have quite different effects in each (see, for example, Hsiao 1986:212). In the models we examine in Sections 2.2, 2.3 and 2.4, we assume that there is no misspecification in the micro relations. We have thus excluded a number of possible reasons that are given for cross-sections providing different estimates of the long-run relationship than time series. Instead we have adopted an approach which is closer to that of Haavelmo (1947), who posed the issue as two problems of aggregation. We are concerned with the different effects of averaging over groups or over time periods. The properties of time-series regressions on group averages are examined in more detail by Pesaran et al. (1989) and Lee et al. (1990).
DYNAMIC MODELS OF HETEROGENEOUS PANELS
Consider the following simple dynamic generalization of (2.1):
where the coefficients X1 and p1 are assumed to vary randomly across groups. The argument below applies to models with lags of higher order on both y and xit. A random intercept can be allowed for in this model by including unity as an element in xit. Dynamic models for panels where the slope coefficients are assumed to be fixed across groups are surveyed by Sevestre and Trognon (1992). The first issue is that the choice of the long-run parameters of interest and the specification of the random coefficient model is not unambiguous in dynamic heterogeneous panel models. The long-term mean effects of x, ony can he Hefined as being eaual either tn a Inno-rnn estimate calculated from the means of Pi and ki namely P(( I - A) where/* = Iti pjN and^ = X ^/W. or to the mean of the long-run estimates, defined by QJNwhere 9, = Ml-/.,). Corresponding to these two different 'average' long-run concepts, we can model the random parameters either by or by where 81=p1/(1-X1). are the long-run individual group coefficients and ^i =X1/(1-X1), are the mean lags for the individual groups. If we assume, forj=1, 2, that E(n;1)=0, then E(j cannot equal zero, and may not even exist, and vice versa.
For panels where N and T are large, separate regressions can be run for each group and the mean group estimators can be computed for the long-run effects from the individual group estimates. These estimates, whether obtained by simple averaging or by weighted averaging, will yield consistent estimates of X and p under Ha, and of 8 under Hb, for both N and T large.
In the case where x1t are integrated of order 1, namely x1t~I(1) and there is a single cointegrating relationship betweeny1t and x1t, a simple regression ofy1t on x1t will yield super-consistent estimates of 8, the cointegrating vector.2 These could be averaged over groups. In this case the weighted mean group estimator (obtained, for example, using the Swamy procedure) may not be appropriate since the estimated standard errors from the cointegrating regression are incorrect. Estimating the N separate regressions also allows one to test the hypothesis that the coefficients are independent of the regressors, an assumption we maintain for the moment. However, in empirical applications estimating N separate regressions is rare. We shall now consider the application of the other three estimation procedures discussed in Section 2.1 to the dynamic heterogeneous panel model (2.4), under the two random coefficient hypotheses Ha and Hb, and under two assumptions about the process generating the x1t, namely when the x1t are I(0) and when they are I(1) variables.
THE CROSS-SECTION ESTIMATOR
Aggregating (2.4) overtime we obtain ^
where y>_-\ ~ y,.i-1, the growth term \7y, (viT v,o)/T captures the end effects, and h — Consider now the properties of the cross-section estimator under Ha and Hb defined by (2.5) and (2.6).
alternative approaches 19
Short-run coefficients random
The regression defined by (2.8) is the 'between' regression for the dynamic model and will produce biased estimates of /; and X even if the intercept parameter in (2.4) is the onlv parameter that varies across groups and N and 71—. This is due to the fact that v, is correlated with ^¿,-ieven for large T. A more promising route is to rewrite (2.7a) under Ha as a cross-section (rather than 'between' regression):
Thus the coefficients of the level of the explanatory variables in the cross-section regression are in fact the appropriate long-run averages. It may appear that the usual cross-sections, estimated purely from levels, are misspecified to the extent that they omit the growth terms ATy;. However, asymptotically the growth terms such as ATy (and ATx, had x, t-1 appeared in the regression) are uncorrelated with the levels terms, so omitting them does not affect the consistency of the average long-run effects. This also indicates that cross-section estimates of the average long-run effect will be robust to dynamic misspecification of the underlying micro model.
For a finite T, the familiar cross-section estimator of /?/(\~X) or 6 obtained by running the least squares regression of Si on will not be unbiased, and the bias disappears only under Hb and as 71—. Increasing the number of groups, N, while holding T fixed does not eliminate the bias. Pesaran and Smith (1995) give a formal proof and provide some indications of the direction and the size of the bias under certain assumptions.
Long-run coefficients random
We now turn our attention to the hypothesis Hb, under which the random components in the parameters are introduced through the mean lags and the long-run coefficients (see (2.6)). In this case we have
We now turn our attention to the hypothesis Hb, under which the random components in the parameters are introduced through the mean lags and the long-run coefficients (see (2.6)). In this case we have
which under Hb yields and noting that 1+m = 1/(1-/.,)\yc have the following expression for ri,. = (1 +fi + |2i) E, + - A-,yi
In this case, it can also be shown that for a finite T the cross-section estimator yields a biased estimator of the mean of the long-run coefficients (i.e. 0= E\fii/(1-X1)\). But for stationary xits, the bias will be of the order of Tl, which is of a lower order of magnitude than the bias in estimating the average long-run coefficient from the means of the short-run micro coefficients (i.e. J#/(1-A)=E(#i)/\1-E(Ai)]). It is therefore more reasonable to interpret the cross-section estimator of 0 as an estimator of the mean of the long-run coefficients rather than the estimator of the long-run coefficients calculated from the means of the short-run micro coefficients.
In short, the above results indicate that cross-section estimates of dynamic random coefficient models will not produce unbiased estimates of the long-run parameters unless T, the number of time periods used to form the average, is large and the randomness in the parameters is introduced through long-run coefficients and the mean lags. This casts serious doubt on the common practice of basing the estimation of long-run coefficients on cross-sections for a single year or a handful of years.
Cross-section estimates with integrated variables
We now examine the impact of non-stationary variables on the cross-section estimates. Consider the disaggregated time series relations where yit and xit are each I(1) and pairwise cointegrated, namely sit is I(0). To allow for an intercept, we can consider yit and xit being measured as deviations from their initial values xi0 and yi0, treating these as fixed and assuming that £¿0=0. Here for simplicity of exposition we also consider the case where xit is a scalar variable.
Averaging the data across time periods and assuming random coefficients as in (2.5), the OLS regression from the cross-section yields
On the assumption that the x„ follow random walks with drifts, namely then Pesaran and Smith (1995) show that as A'—/J.even if T is finite. This result indicates that the spurious correlation problem does not arise in the case of cross-section regressions, even if the underlying variables xjt and y„ contain unit roots. Under the assumption that xjt and sitare independently distributed, is also an unbiased estimator of [i though the usual standard error formulae are no longer valid.
The cross-section disturbance is with variance where m2 is the variance of ni. The first term is likely to be small, particularly if T is large, and the cross-section variance will be dominated by parameter heterogeneity. In general, therefore, the disturbances of the cross-section equation are heteroscedastic, but consistent estimates of the conditional variances of the cross-section coefficients can be obtained using the procedure suggested, for example, by White (1980). It is important to note, however, that the above results are valid under the rather strong assumption that the xit are strictly exogenous, and do not hold in the standard case discussed in the time- series literature of cointegrated variables where possible dependence between xit and eiV (for some t and t') is not ruled out.
The consistent estimate of the long-run effect in the cross-section is in striking contrast to the estimates from the aggregate time-series estimator. Pesaran and Smith (1995) show that, if the xit are serially correlated, then the error in the aggregate equation will also be serially correlated, producing inconsistent estimates if a lagged dependent variable is included; and that if the regressors are I(1) and the micro relationships cointegrate for each group, then the aggregate relationship need not be cointegrated.
With the pooled estimator the same problem arises as with the aggregate time-series estimator. The imposition of a common slope across different micro relations induces serial correlation in the composite disturbances of the pooled regression if the xit are themselves serially correlated, producing inconsistent estimates if a lagged dependent variable is included in the regression. Robertson and Symons (1992) consider the case where the data are generated by
and the estimated model is
The long-run estimator $ = y/O — i)has the probability limit /;*( 1+p)/ (l+ph). Furthermore, as p approaches its upper bound (p= 1) from below, then plimO?) and plim(^) tend to zero and unity respectively.
In more general models similar results are obtained even when xit are serially uncorrelated. Consider the following simple dynamic generalization of (2.10):
yit = Pi\xit+Pi2xi't-\+eit where and ni1 and ni2 have means zero with variances mn and m22 and the covariance m12=m21 Assume further that the xlt are generated as in (2.11) and the estimated model is yit=71Xit+72Xi, t-1+tyi, t-1+Uit For large T and N, Pesaran and Smith (1995) show that plim<?i)-0, (2.15)
in which where Yi>?2 and X are the OLS estimators of v, and X respectively in the pooled regression. Further, the estimator of the long-run coefficient 6 = has the following probability limit:
These results have a number of important features:
1 The 'average' short-run impact of xjt on vjt is consistently estimated by Yi.
2 The pooled regression will generally yield an inconsistent estimator of the 'average' long-run impact of xit on yit. As with the Robertson and Symons example, the extent of the bias depends on the magnitude of p. As p approaches unity from below (|p|<1), X also tends towards unity and the long-run coefficient may become computationally unstable. Letting p—> 1, we have
It is important to note that this result holds only for values of p approaching unity from below and is not valid for p=1.
3 The inconsistency of the pooled estimator vanishes only under the parameter homogeneity assumption. In this case co,y=0, /', /= 1. 2, and Yi and Yi both converge to the fixed coefficients /;, and /;2. respectively. Notice also that unlike in the Robertson and Symons example the inconsistency of the pooled estimators remains even if the xit are serially uncorrelated.
Hence in this model the primary source of the bias is the heterogeneity of the coefficients pa and fii2 that are ignored in the pooled regression. The serial correlation in the xit tends to accentuate the problem. The direction of the bias critically depends on the signs and the magnitudes of m12 and p. When these are positive we have X*>0 and there will be a tendency to underestimate the 'average' impact of xi,t-1 and overestimate the 'average' long-run impact of xit onyit Remarkably, in this example the 'average' short-run impact of xiton yit is consistently estimated by the coefficient of xit in the pooled regression.
The Robertson and Symons result arises because when the xit are serially correlated the errors in the pooled regression associated with the micro relations (2.10) will also be serially correlated, and therefore mistakenly including a lagged dependent variable in the equation will result in inconsistent estimates. A similar kind of problem is present, however, even if the pooled regression is not misspecified. Suppose the micro relations are correctly specified by (2.4) explicitly including an intercept term, with random parameters defined according to Ha in (2.5). The pooled regression is given by
where a, is the intercept term and
Different group-specific fixed or random effects can be included in the pooled regressions through the intercept term ai. It is easily shown that yi,t-1 and xit are correlated with vit, thus rendering the pooled estimators inconsistent, and E(yi,t-1 vit) does not vanish even if the xit are serially uncorrelated. Instrumental variable estimation of the pooled regression faces major difficulties. Given the structure of the composite disturbances vit, it may not be possible to come up with variables that are uncorrelated with vit while at the same time having a non-zero correlation with the variables included in the pooled regression.
Integrated variables also cause difficulty for the pooled regression. Suppose each micro relationship cointegrates with different parameters. The pooled regression by imposing a common parameter on all the micro relations generates a residual which has an I(0) component, the residual from the cointegrating micro relationship, and an I(1) component, the product of the difference between the micro and imposed parameter and the I(1) regressor. Thus the pooled regression will not constitute a cointegrating regression and the parameter estimates will not be consistent.
The traditional argument for pooling is that one obtains more efficient estimates of the average effect by imposing the restriction that the coefficients are the same. What the argument of this section suggests is that pooling is likely to produce inconsistent effects of the average effect in dynamic models. Most of the literature on pooled estimation of dynamic models avoids the difficulties discussed in this section by focusing on the intercept term ai as the main source of parameter heterogeneity in the model and assuming that the slope coefficients Xi and fc are the same across groups. But this is a very strong assumption which is not likely to be met in many energy applications.
MODEL MISSPECIFICATION AND ALTERNATIVE ESTIMATES OF LONGRUN ELASTICITIES
In the discussion above it was assumed that the model was correctly specified, in which case the cross-section if based on relatively long time averages can provide quite reasonable point estimates of the mean of the long-run coefficients, although more efficient estimates might be obtained from a weighted average of the individual time-series estimates—what we have called mean group estimates. However, different types of misspecification of the model are likely to have different effects on the two estimators, and here we briefly discuss the consequences of four types of misspecification on these estimates. In each case, comparison of the mean group estimates and cross-section estimates may be informative about the likelihood and possible form of the misspecification. In many cases, the misspecification can be thought of as inducing a correlation between the estimated coefficients and the regressors, contrary to our maintained assumption so far that they were independent. This correlation can be investigated empirically when time-series equations are estimated for each group separately.
If the estimated equation omits relevant dynamics, the cross-section may be more robust to dynamic misspecification than the averages of the individual time series as in the case Baltagi and Griffin (1984) examined. The estimation of long-run effects using time-series estimates in situations where substantial lags exist between price and income changes, and changes in energy demand, may require very long spans of data. For example, the full impact of price changes on residential energy demand may take up to ten years for it to become complete, as it takes time to get vehicles, central heating systems and household appliances replaced.
Similar considerations also apply to interfuel substitution at the industry level. However, when considering long time periods the assumption of structural stability is less likely to be fulfilled. Despite the recent advances in the theory of cointegrated systems (e.g. Engle and Granger 1991), identification of adjustment dynamics in most time-series applications remains troublesome, and the estimates of the longrun effects can be very sensitive to the particular dynamic specification chosen.
To examine the effects of dynamic misspecification, consider the following random coefficient distributed lag model:
where j are distributed independently of the eitwd xit with means fy and constant variances. Suppose now that the lags are omitted in error and the static time-series regressions ylt = ai+biXit+Uit are estimated. Further assume that
The probability limits of the OLS estimates of bi in the above static regressions are given by3
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