The obvious way to find evidence of spurious regressions is by using simulations. The first simulation on the topic was by Granger and Newbold (1974) who generated pairs of independent random walks, from (26.2) with a1 = a2 = 1. Each series had 50 terms and 100 repetitions were used. If the regression (26.1) is run, using series that are temporarily uncorrelated, one would expect that roughly 95 percent of values of 111 on b would be less than 2. This original simulation using random walks found 111 < 2 on only 23 occasions, out of the 100, 111 was between 2 and 4 on 24 times, between 4 and 7 on 34 times, and over 7 on the other 19 occasions.
The reaction to these results was to re-assess many of the previously obtained empirical results in applied time series econometrics, which undoubtedly involved highly autocorrelated series but had not previously been concerned by this fact. Just having a high R2 value and an apparently significant value of b was no longer sufficient for a regression to be satisfactory or its interpretations relevant. The immediate questions were how one could easily detect a spurious regression and then correct for it. Granger and Newbold (1974) concentrated on the value of the Durbin-Watson statistic; if the value is too low, it suggests that the regressions results cannot be trusted. Quick fix methods such as using a Cochrane-Orcutt technique to correct autocorrelations in the residuals, or differencing the series used in a regression were inclined to introduce further difficulties and so cannot be recommended. The problem arises because equation (26.1) is misspecified, the proper reaction to having a possible spurious relationship is to add lagged dependent and independent variables, until the errors appear to be white noise, according to the Durbin-Watson statistic. A random walk is an example of an I(1) process, that is a process that needs to be differenced to become stationary. Such processes seem to be common in parts of econometrics, especially in macroeconomics and finance. One approach that is widely recommended is to test if Xt, Yt are I(1) and, if yes, to difference before performing the regression (26.1). There are many tests available, a popular one is due to Dickey – Fuller (1979). However, as will be explained below, even this approach is not without its practical difficulties.
A theoretical investigation of the basic unit root, ordinary least squares, spurious regression case was undertaken by Phillips (1986). He considered the asymptotic
properties of the coefficients and statistics of equation (26.1), a, b, the f-statistic for b, R2 and the Durbin-Watson statistics p. To do this he introduced the link between normed sums of functions of unit root processes and integrals of Wiener processes. For example if a sample Xf of size T is generated from a driftless random walk then
T -3/1 X Xf ^ oe W(f)df
where о3 is the variance of the shock,
T-2 X x2 ^ o2 w2(f)df
and if Xf, Yf are an independent pair of such random walks, then
T-2 X XfYt ^ ОеОл
where V(f), W(f) are independent Wiener processes. As a Wiener process is a continuous time random process on the real line [0, 1], the various sums are converging and can thus be replaced by integrals of a stochastic process. This transformation makes the mathematics of the investigation much easier, once one becomes familiar with the new tools. Phillips is able to show that
1. the distributions of the f-statistics for a and b from (26.1) diverge as f becomes large, so there is no asymptotically correct critical values for these conventional tests.
2. b converges to some random variable whose value changes from sample to sample.
3. Durbin-Watson statistics tend to zero.
4. R2 does not tend to zero but to some random variable.
What is particularly interesting is that not only do these theoretical results completely explain the simulations but also that the theory deals with asymptotics, T ^ ж, whereas the original simulations had only T = 50. It seems that spurious regression occurs at all sample sizes.
Haldrup (1994) has extended Phillips’ result to the case for two independent I(2) variables and obtained similar results. (An I(2) variable is one that needs differencing twice to get to stationarity, or here, difference once to get to random walk.) Marmol (1998) has further extended these results to fractionally integrated, I(d), processes. Unpublished simulation results also exist for various other long – memory processes, including explosive autoregressive processes, (26.2) with a1 = a2 = a > 1. Durlauf and Phillips (1988) regress an I(1) process on deterministic
polynomials in time, thus polynomial trends, and found spurious relationships. Phillips (1998) has recently discussed how all such results can be interpreted by considering a decomposition of an I(1) series in terms of deterministic trends multiplied by stationary series.