36J. Durbin, "Testing for Serial Correlation in Least-Squares Regression When Some of the Regressors Are Lagged Dependent Variables,'' Econometrica, vol. 38, 1970, pp. 410-421.

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Since this h value has the standard normal distribution under the null hypothesis, the probability of obtaining such a high h value is very small. Recall that the probability that a standard normal variate exceeds the value of ±3 is extremely small. In the present example our conclusion, then, is that there is (positive) autocorrelation. Of course, bear in mind that h follows the standard normal distribution asymptotically. Our sample of 30 observations may not be necessarily large.

1. It does not matter how many X variables or how many lagged values of Y are included in the regression model. To compute h, we need consider only the variance of the coefficient of lagged Yt-1.

2. The test is not applicable if [n var (a2)] exceeds 1. (Why?) In practice, though, this does not usually happen.

3. Since the test is a large-sample test, its application in small samples is not strictly justified, as shown by Inder37 and Kiviet.38 It has been suggested that the Breusch-Godfrey (BG) test, also known as the Lagrange multiplier test, discussed in Chapter 12 is statistically more powerful not only in the large samples but also in finite, or small, samples and is therefore preferable to the h test.39

17.11 A NUMERICAL EXAMPLE: THE DEMAND FOR MONEY IN CANADA, 1979-I TO 1988-IV

To illustrate the use of the models we have discussed thus far, consider one of the earlier empirical applications, namely, the demand for money (or real cash balances). In particular, consider the following model.40

where M* = desired, or long-run, demand for money (real cash balances) Rt = long-term interest rate, % Yt = aggregate real national income

For statistical estimation, (17.11.1) may be expressed conveniently in log form as ln M* = ln j0 + j ln Rt + j2ln Yt + u (17.11.2)

37B. Inder, "An Approximation to the Null Distribution of the Durbin-Watson Statistic in Models Containing Lagged Dependent Variables,'' Econometric Theory, vol. 2, no. 3, 1986, pp. 413-428.

38J. F. Kiviet, "On the Vigour of Some Misspecification Tests for Modelling Dynamic Relationships," Review of Economic Studies, vol. 53, no. 173, 1986, pp. 241-262.

39Gabor Korosi, Laszlo Matyas, and Istvan P. Szekely, Practical Econometrics, Ashgate Publishing Company, Brookfield, Vermont, 1992, p. 92.

40For a similar model, see Gregory C. Chow, "On the Long-Run and Short-Run Demand for Money,'' Journal of Political Economy, vol. 74, no. 2, 1966, pp. 111-131. Note that one advantage of the multiplicative function is that the exponents of the variables give direct estimates of elasticities (see Chap. 6).

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Since the desired demand variable is not directly observable, let us assume the stock adjustment hypothesis, namely,

Equation (17.11.3) states that a constant percentage (why?) of the discrepancy between the actual and desired real cash balances is eliminated within a single period (year). In log form, Eq. (17.11.3) may be expressed as ln Mt - ln Mt-1 = S(ln M* - ln Mt-1) (17.11.4)

Substituting ln M* from (17.11.2) into Eq. (17.11.4) and rearranging, we obtain ln Mt = S ln fa + faS ln Rt + faS ln Yt + (1 - S)ln Mt-1 + Sut (17.11.5)41

which may be called the short-run demand function for money. (Why?)

As an illustration of the short-term and long-term demand for real cash balances, consider the data given in Table 17.3. These quarterly data pertain to Canada for the period 1979 to 1988. The variables are defined as follows: M [as defined by M1 money supply, Canadian dollars (C$), millions], P (implicit price deflator, 1981 = 100), GDP at constant 1981 prices (C$, millions) and R (90-day prime corporate rate of interest, %).42 M1 was deflated by P to obtain figures for real cash balances. A priori, real money demand is expected to be positively related to GDP (positive income effect) and negatively related to R (the higher the interest rate, the higher the opportunity cost of holding money, as M1 money pays very little interest, if any).

The regression results were as follows43:

lnMt = 0.8561 - 0.0634 lnRt - 0.0237 ln GDPt + 0.9607 lnMt-1 se = (0.5101) (0.0131) (0.0366) (0.0414)

41In passing, note that this model is essentially nonlinear in the parameters. Therefore, although OLS may give an unbiased estimate of, say, fa S taken together, it may not give unbiased estimates of fa and S individually, especially if the sample is small.

42These data are obtained from B. Bhaskar Rao, ed., Cointegration for the Applied Economist, St. Martin's Press, New York, 1994, pp. 210-213. The original data is from 1956-Ito 1988-IV, but for illustration purposes we begin our analysis from the first quarter of 1979.

43Note this feature of the estimated standard errors. The standard error of, say, the coefficient of ln Rt refers to the standard error of faS, an estimator of P1S. There is nosimple way to obtain the standard errors of fa andS individually from the standard error of faS, especially if the sample is relatively small. For large samples, however, individual standard errors of fa and S can be obtained approximately, but the computations are involved. See Jan Kmenta, Elements of Econometrics, Macmillan, New York, 1971, p. 444.

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