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13But matrix algebra becomes almost a necessity to avoid tedious algebraic manipulations.

CHAPTER TWELVE: AUTOCORRELATION 455

OLS confidence interval, we could accept the hypothesis that true p2 is zero with 95% confidence. But if we were to use the (correct) GLS confidence interval, we could reject the null hypothesis that true p2 is zero, for b2 lies in the region of rejection.

The message is: To establish confidence intervals and to test hypotheses, one should use GLS and not OLS even though the estimators derived from the latter are unbiased and consistent. (However, see Section 12.11 later.)

OLS Estimation Disregarding Autocorrelation

The situation is potentially very serious if we not only use p2 but also continue to use var(p2) = a2/E xt2, which completely disregards the problem of autocorrelation, that is, we mistakenly believe that the usual assumptions of the classical model hold true. Errors will arise for the following reasons:

1. The residual variance a2 = J2 uj /(n — 2) is likely to underestimate the true a2.

2. As a result, we are likely to overestimate R2.

3. Even if a2 is not underestimated, var(p2) may underestimate var(p2)ARi [Eq. (12.2.8)], its variance under (first-order) autocorrelation, even though the latter is inefficient compared to var(p2)GLS.

4. Therefore, the usual t and F tests of significance are no longer valid, and if applied, are likely to give seriously misleading conclusions about the statistical significance of the estimated regression coefficients.

To establish some of these propositions, let us revert to the two-variable model. We know from Chapter 3 that under the classical assumption a 2 =

provides an unbiased estimator of a2, that is, E(a2) = a2. But if there is autocorrelation, given by AR(1), it can be shown that

where r = tn=-11 xtxt-1/ tn=1 xt2 , which can be interpreted as the (sample) correlation coefficient between successive values of the X's.15 If p and r are both positive (not an unlikely assumption for most economic time series), it is apparent from (12.4.1) that E(a2) < a2; that is, the usual residual variance

15See S. M. Goldfeld and R. E. Quandt, Nonlinear Methods in Econometrics, North Holland Publishing Company, Amsterdam, 1972, p. 183. In passing, note that if the errors are positively autocorrelated, the R2 value tends to have an upward bias, that is, it tends to be larger than the R2 in the absence of such correlation.

456 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

formula, on average, will underestimate the true a2. In other words, a2 will be biased downward. Needless to say, this bias in a2 will be transmitted to var (fa) because in practice we estimate the latter by the formula a2/J2 xf.

But even if a2 is not underestimated, var (fa) is a biased estimator of var(fa)AR1, which can be readily seen by comparing (12.2.7) with (12.2.8),16 for the two formulas are not the same. As a matter of fact, if p is positive (which is true of most economic time series) and the X's are positively correlated (also true of most economic time series), then it is clear that var (fa) < var(fa)ARi (12.4.2)

that is, the usual OLS variance of fa underestimates its variance under AR(1) [see Eq. (12.2.9)]. Therefore, if we use var (fa), we shall inflate the precision or accuracy (i.e., underestimate the standard error) of the estimator fa. As a result, in computing the t ratio as t = fa/se (fa) (under the hypothesis that fa = 0), we shall be overestimating the t value and hence the statistical significance of the estimated fa. The situation is likely to get worse if additionally a2 is underestimated, as noted previously.

To see how OLS is likely to underestimate a2 and the variance of fa, let us conduct the following Monte Carlo experiment. Suppose in the two-variable model we "know" that the true fa1 = 1 and fa = 0.8. Therefore, the stochastic PRF is

Hence,

which gives the true population regression line. Let us assume that ut are generated by the first-order autoregressive scheme as follows:

where st satisfy all the OLS assumptions. We assume further for convenience that the st are normally distributed with zero mean and unit ( = 1) variance. Equation (12.4.5) postulates that the successive disturbances are positively correlated, with a coefficient of autocorrelation of +0.7, a rather high degree of dependence.

Now, using a table of random normal numbers with zero mean and unit variance, we generated 10 random numbers shown in Table 12.1 and then by the scheme (12.4.5) we generated ut. To start off the scheme, we need to specify the initial value of u, say, uo = 5.

Plotting the ut generated in Table 12.1, we obtain Figure 12.5, which shows that initially each successive ut is higher than its previous value and

16For a formal proof, see Kmenta, op. cit., p. 281.