where , are independent with mean zero and common variance u. If we difference this equation, we get

, = • , + 7 + V,

where the error v, = , = , - ,„, is a moving average, and, hence, not serially independent. But estimating the first difference equation by least squares still gives us consistent estimates. Thus, the consequences of differencing when it is not needed are much less serious than those of failing to difference when it is appropriate (when the true model is of the DSP type).

In practice, it is best to use the Dickey-Fuller test to check whether the data are of DSP or TSP type. Otherwise, it is better to use differencing and regressions in first differences, rather than regressions in levels with time as an extra explanatory variable.

Differencing and Long-Run Effects: The Concept of Cointegration

One drawback of the procedure of differencing is that it results in a loss of valuable "long-run information" in the data. Recently, the concept of cointe-grated series has been suggested as one solution to this problem.First, we need to define the term "cointegration." Although we do not need the assumption of normality and independence, we will define the terms under this assumption.

If e, are IN(0, cr) we say e, are 1(0) that is, integrated of order zero. (More generally, e, is a stationary process. This is discussed in Chapter 14 ~ If y, follow a random walk model, that is,

y, = y, , + e,

then we get by successive substitution,

y,= Y, Er-j if = 0

,/ = 0

Thus, y, is a summation of e, and

, = e,

which is 1(0). We say in this case that y, is 1(1) [integrated to order one]. If y, is 1(1) and we add to this z, which is 1(0), then y, + z, will be 1(1). When we specify regression models in time series, we have to make sure that the different variables are integrated to the same degree. Otherwise, the equation does not make sense. For instance, if we specify the regression model:

y, = px, + (6.34)

"A reference which is, however, quite technical for our purpose, is R. F. Engle and C. W. J Granger, "Co-Integration and Error Correction: Representation, Estimation and Testing," Econometnca, Vol. 55, March 1987, pp. 251-276.

""Since , are estimated residuals, the Dickey-Fuller tables have to be adjusted. An alternative test that is also often suggested for testing unit roots is the Sargan-Bhargava test. See J. D. Sargan and A. Bhargava, "Testing Residuals from Least Squares Regression for Being Generated by the Gaussian Random Walk," Econometrica. Vol. 51, 1983. pp. 153-174. This is a Durbin-Watson type test with significance levels corrected.

"R. A. Bewley, "The Direct Estimation of the Equilibrium Response in a Linear Dynamic Model," Economics Letters, Vol. 3, 1979, pp. 357-361. M. R. Wickens and T. S. Breusch, "Dynamic Specification, the Long Run and the Estimation of Transformed Regression Models," Economic Journal. 1988 (supplement), pp. 189-205.

and we say that u, ~ IN(0, cr), that is u, is 1(0), we have to make sure that y, and X, are integrated to the same order. For instance, if y, is 1(1) and x, is 1(0) there will not be any p that will satisfy the relationship (6.34). Suppose y, is 1(1) and X, is 1(1); then if there is a nonzero p such that y, - x, is 1(0) then y, and X, are said to be cointegrated.

Suppose that y, and jc, are both random walks, so that they are both 1(1). Then an equation in first differences of the form

Ay, = a Ax, + Uy, - Px,) + V, (6.35)

is a valid equation. Since Ay„ Ax„ (y, - px,) and v, are all 1(0). Equation (6.34) is considered a long-run relationship between y, and x, and equation (6.35) describes short-run dynamics. Engle and Granger suggest estimating (6.34) by ordinary least squares, obtaining the estimator p of p and substituting it in equation (6.35) to estimate the parameters a and \. This two-step estimation procedure, however, rests on the assumption that y, and x, are cointegrated. It is, therefore, important to test for cointegration. Engle and Granger suggest estimating (6.34) by ordinary least squares, getting the residual u, and then applying the Dickey-Fuller test (or some other test"") based on ,. What this test amounts to is testing the hypothesis p = 1 in

u, = ,„, + e,

that is, testing the hypothesis

: , is 1(1)

In essence, we are testing the null hypothesis that y, and x, are not co-integrated. Note that y, is 1(1) and x, is 1(1), so we are trying to see that u, is not 1(1).

As shown by Bewley and also by Wickens and Breusch," the two-step estimation procedure suggested by Engle and Granger of first estimating the long-run parameter p and then estimating the short-run parameters a and in equation (6.35) is unnecessary. They argue that one should estimate both the long-run and short-run parameters simultaneously and one would get more efficient estimates of the long-run parameter p by this procedure. Dividing (6.35) by X and rearranging we get

y, = px, + Ay, - , - (6.36)

Since Ay, will be correlated with the error u„ equation (6.36) has to be estimated by the instrumental variable method. The coefficients of , and Ax, describe the short-run dynamics. Note that if y, and x, are 1(1), then , and , are like w„ 1(0), that is, they are of a lower order. Wickens and Breusch show that mis specification of the short-run dynamics does not have much of an effect on the estimation of the long-run parameters. For instance, in equation (6.36) even if , is omitted, the estimate of the parameter p will still be consistent. The intuitive reasoning behind this is that , and , are of a lower order than y, and x,.

*6.l I ARCH Models and Serial Correlation

We saw in Section 6.9 that a significant DW statistic can arise through a number of mjsspecifications. We will now discuss one other source. This is the ARCH model suggested by Engle" which has, in recent years, been found useful in the analysis of speculative prices. ARCH stands for "autoregressive conditional heteroskedasticity." When we write the simple autoregressive model

y, = y,-i + e, e, ~ IN(0, a")

we are saying that the conditional mean £(y,y, i) = Xy,-, depends on t but the conditional variance var(y,y,-,) = cr is a constant. The unconditional mean of y, is zero and the unconditional variance is aV{l - X).

The ARCH model is a generaUzation of this, in that the conditional variance is also made a function of the past. If the conditional density/(y,z, ,) is normal, a general expression for the ARCH model is

, 1 ~ Af(g(z, ,), h{z,d)

To make this operational, Engle specifies the conditional mean g(z, ,) as a linear function of the variables z,-, and h as

h, = CLo + a,e? , + « - + • • • + ape? p where e, = y, - g,. In the simplest case we can consider the model

y, = Xy, , + px, + e, e, ~ IN(0, h,) (6.37)

h, = var e, = tto + aie?„, (6.38)

Note that e, are not autocorrelated. But the fact that the variance of e, depends on , gives a misleading impression of there being a serial correlation. For instance, suppose that in (6.37) X = 0, that is, we do not have y, , as an explanatory variable. If we estimate that equation by OLS we will find a significant DW statistic because of the ARCH effect given by (6.38). The situation

*R. F. Engle, "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of U. K. Inflation," Econometrica, Vol. 50, 1982, pp. 987-1007.