Spurious Regressions with Stationary Processes

Spurious regressions in econometrics are usually associated with I(1) processes, which was explored in Phillips’ well known theory and in the best known simulations. What is less appreciated is that the problem can just also occur, although less clearly, with stationary processes. Table 26.1, taken from Granger, Hyung, and Jeon (1998), shows simulation results from independent series generated by (26.2) with 0 < a1 = a2 = a < 1 and ext, eyt both Gaussian white noise series, using regression (26.1) estimated using OLS with sample sizes varying between 100 and 10,000.

It is seen that sample size has little impact on the percentage of spurious regressions found (apparent significance of the b coefficient in (26.1)). Fluctua­tions down columns do not change significantly with the number of iterations used. Thus, the spurious regression problem is not a small sample property. It is also seen to be a serious problem with pairs of autoregressive series which are not unit root processes. If a = 0.75 for example, then 30 percent of regressions will give spurious implications. Further results are available in the original paper but will not be reported in detail. The Gaussian error assumption can be replaced by other distributions with little or no change in the simulation results, except for an exceptional distribution such as the Cauchy. Spurious regressions also occur if a1 Ф a ъ although less frequently, and particularly if the smaller of the two a values is at least 0.5 in magnitude.

The obvious implications of these results is that applied econometricians should not worry about spurious regressions only when dealing with I(1), unit root, processes. Thus, a strategy of first testing if a series contains a unit root before entering into a regression is not relevant. The results suggest that many more simple regressions need to be interpreted with care, when the series involved are strongly serially correlated. Again, the correct response is to move to a better specification, using lags of all variables.

Table 26.1 Regression between independent AR(1) series

Sample series

a = 0

a = 0.25

a = 0.5

a = 0.75

a = 0.9

a = 1.0

100

4.9

6.8

13.0

29.9

51.9

89.1

500

5.5

7.5

16.1

31.6

51.1

93.7

2,000

5.6

7.1

13.6

29.1

52.9

96.2

10,000

4.1

6.4

12.3

30.5

52.0

98.3

a1 = a2 = a percentage of 111 > 2

4 Related Processes

The final generalization would take variables generated by (26.2) but now allow­ing ext, eyt to be correlated, say p = corr(ext, eyt). Now the series are related and any relationship found in (26.2) will not be spurious, although the extent of the relationship is over-emphasized if the residual achieved is not white noise. The natural generalization is a bivariate vector autoregression or, if p is quite high, and if a = a2 = 1, the series will be cointegrated (as described in Chapter 30 on Cointegration), in which case an error-correction model is appropriate. In all these models, spurious regressions should not be a problem.

References

Dickey, D. A., and W. A. Fuller (1979). Distribution of the estimates for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427-31.

Durlauf, S. N., and P. C.B. Phillips (1988). Trends versus random walks in time series ana­lysis. Econometrica 56, 1333-54.

Granger, C. W.J., N. Hyung, and Y. Jeon (1998). Spurious regression with stationary series.

Working Paper, UCSD Department of Economics.

Granger, C. W.J., and P. Newbold (1974). Spurious regressions in econometrics. Journal of Econometrics 2, 111-20.

Haldrup, N. (1994). The asymptotics of single-equation cointegration regressions with I(1) and I(2) variables. Journal of Econometrics 63, 153 -81.

Kendall, M. G. (1954). Exercises in Theoretical Statistics. London, Griffin.

Marmol, F. (1998). Spurious regression theory with non-stationary fractionally integrated processes. Journal of Econometrics 84, 233-50.

Phillips, P. C.B. (1986). Understanding spurious regressions in econometrics. Journal of Econometrics 33, 311-40.

Phillips, P. C.B. (1998). New tools for understanding spurious regressions. Econometrica 66, 1299-325.

Yule, G. U. (1926). Why do we sometimes get nonsense correlations between time series? Journal of the Royal Statistical Society 89, 1-64.

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