which has the same form as (6.29) except for the fact that the disturbance is not stationary, it has variance ta- that increases over time. Nelson and Plosser call the model (6.29) trend-stationary processes (TSP) and model (6.30) difference-stationary processes (DSP). Both the models exhibit a linear trend. But the appropriate method of eUminating the trend differs. To test the hypothesis that a time series belongs to the TSP class against the alternative that it belongs to the DSP class. Nelson and Plosser use a test developed by Dickey and Fuller." This consists of estimating the model

y, = a + py,, + p/ + E, (6.32)

which belongs to the DSP class if p = 1, = 0 and the TSP class if p < 1. Thus we have to test the hypothesis p = I, = 0 against p < 1. The problem here is that we cannot use the usual least squares distribution theory when p = 1. Dickey ad Fuller show that the least squares estimate of p is not distributed around unity under the DSP hypothesis (that is, the true value p = 1) but rather around a value less than one. However, the negative bias diminishes as the number of observations increases. They tabulate the significance points for testing the hypothesis p = 1 against p < 1. Nelson and Plosser applied the Dickey-Fuller test to a wide range of historical time series for the U.S. economy and found that the DSP hypothesis was accepted in all cases, with the exception of the unemployment rate. They conclude that for most economic time series the DSP model is more appropriate, and that the TSP model would be the relevant one only if we assume that the errors u, in (6.29) are highly autocorrelated.

The problem of testing the hypothesis p = 1 in the first order autoregressive equation of the form

y, = a + py,. I + u,

is called "testing for unit roots." There is an enormous literature on this problem but one of the most commonly used tests is the Dickey-Fuller test. The standard expression for the large sample variance of the least squares estimator p is (1 - p")/F which would be zero under the null hypothesis. Hence, one needs to derive the limiting distribution of p under , p = 1 to apply the test.

For testing the hypothesis p = 1, p = 0 in (6.32) Dickey and Fuller" suggest a LR test, derive the limiting distribution and present tables for the test. The F-values are much higher than those in the usual F-tables. For instance, the 5% significance values from the tables presented in Dickey and Fuller, and

"C. R. Nelson and C. I. Plosser, "Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications," Journal of Monetary Economics, Vol. 10, 1982, pp. 139-162.

"D. A. Dickey and W. A. Fuller, "Distribution of the Estimators for Autoregressive Time-Series with a Unit Root," Journal of ttie American Statistical Association, Vol. 74, 1979, pp. 427-431.

"D. A. Dickey and W. A. Fuller, "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Vol. 49, No. 4, 1981, pp. 1057-1072. See tables on p. 1063.

"d f. for denominator = « - 3.

As an illustration consider the example given by Dickey and Fuller."" For the logarithm of the quarterly Federal Reserve Board Production Index 1950-1 through 1977-4 they assume that the time series is adequately represented by the model:

y, = Po + Pi + «1 ,-1 + «2(y,-, - y,-2) + e, (6.33)

where e, are IN(0, tr) random variables. The ordinary least squares estimates are:

y, - y= 0.52 + 0.00120/ - 0.119y, , + 0.498(y, , - y,.)

/- (0 15) (0 00034) (0 033) (0.081)

RSS = 0.056448 y, - y, , = 0.0054 + 0.447(y, , - y,)

(0 0025) (0 083)

RSS = 0.063211

where RSS denotes the residual sum of squares and the numbers in parentheses are the "standard errors" as output from the usual regression program. The F-test for the hypothesis p, = 0, a, = 1 is

(0.063211 - 0.056448)/2 , p - - = 5 34

0.056448/106

If we use the standard F-tables this F-ratio is significant at even the 1% significance level. But the F-ratio tabulated by Dickey and Fuller is 6.49 for the 5% significance level. Thus, the hypothesis that the second order autoregressive process (6.33) has a unit root is accepted at the 5% significance level, though it is rejected at the 10% significance level.

Spurious Trends

If P = 0 in equation (6.30) the model is called a trendless random walk or a random walk with zero drift. However, from equation (6.31) note that even though there is no trend in the mean, there is a trend in the variance. Suppose

"Dickey and Fuller, "Likelihood Ratio Statistics," pp. 1070-1071.

the corresponding F-values from the standard F-tables (when the numerator d.f. is 2 as in this test) are as follows:

"C. R. Nelson and H. Kang, "Pitfalls in the Use of Time as an Explanatory Variable in Regression," Journal of Business and Economic Statistics. Vol. 2. January 1984, pp. 73-82. "C. I. Plosser and G. W. Schwert, "Money, Income and Sunspots: Measuring Economic Relationships and the Effects of Differencing," Journal of Monetary Economics, Vol. 4, 1978, pp. 637-660.

that the true model is of the DSP true with (3 = 0. What happens if we estimated a TSP type model? That is, the true model is one with no trend in the mean but only a trend in the variance, and we estimate a model with a trend in the mean but no trend in the variance. It is intuitively clear that the trend in the variance will be transmitted to the mean and we will find a significant coefficient for t even though in reality there is no trend in the mean. How serious is this problem? Nelson and Kang" analyze this. They conclude that:

1. Regression of a random walk on time by least squares will produce values of around 0.44 regardless of sample size when, in fact, the mean of the variable has no relationship with time whatsoever.

2. In the case of random walks with drift, that is p 5 0, the R will be higher and will increase with the sample size, reaching one in the Umit regardless of the value of p.

3. The residual from the regression on time which we take as the de-trended series, has on the average only about 14% of the true stochastic variance of the original series.

4. The residuals from the regression on time are also autocorrelated being roughly (1 - 10/ at lag one, where N is the sample size.

5. Conventional /-tests to test the significance of some of the regressors are not valid. They tend to reject the null hypothesis of no dependence on time, with very high frequency.

6. Regression of one random walk on another, with time included for trend, is strongly subject to the spurious regression phenomenon. That is, the conventional /-test will tend to indicate a relationship between the variables when none is present.

The main conclusion is that using a regression on time has serious consequences when, in fact, the time series is of the DSP type and, hence, differencing is the appropriate procedure for trend elimination. Plosser and Schwert also argue that with most economic time series it is always best to work with differenced data rather than data in levels. The reason is that if indeed the data series are of the DSP type, the errors in the levels equation will have variances increasing over time. Under these circumstances many of the properties of least squares estimators as well as tests of significance are invalid. On the other hand, suppose that the levels equation is correctly specified. Then all differencing will do is produce a moving average error and at worst ignoring it will give inefficient estimates. For instance, suppose that we have the model

y, = a + px, + 7/ + u,