The Classical Linear Regression Model is unfortunately not universally valid in econometrics. In this essay,
Peter Lambert examines one of these cases, that of non-stationary data.
In the following report, I intend to set out the results of an attempt to model a non-stationary data series in terms of (mainly) stationary explanatory variables. In the first section, I will outline the results of a conventional analysis of the data ignoring the problem of non-stationarity. The second section will describe the details of the non-stationarity problem. In light of this I will review the conventional results, commenting on their validity. Finally I intend to reach a conclusion as to the applicability of standard procedures to non stationary data in this specific case and in general.
I hoped to model the demand for electricity in Ireland in terms of its main uses. In particular, I looked at industrial use, heating and lighting. Each of these requirements varies over time and I intended to represent the effects of these changes on the demand for electricity. Industrial demand for electricity was represented by an index of industrial production ( 'OUTPUT'). There is an implicit assumption here of a stable relationship between industrial production and the amount of electricity used to produce it. While not true for longer periods as the ratio of capital to labour would change, over a five-year period I hoped that this effect would not be very significant. Meteorological reports of the average temperature ('TEMP') and the average sunlight ('SUN') for each month were used to represent the demand for heat and light respectively. Other uses of electricity such as home entertainment and cooking were not represented due to lack of data. It was assumed therefore, that this 'other domestic' element was constant over the time period in question. Although data was available over quite along period, a sample size of sixty-six months was considered sufficiently large for this analysis without being too cumbersome.
Access to computer resources was limited. Econometric software such as PCGive and Microfit were available while I was testing preliminary models, but the actual model selection procedure and testing were carried out without the benefit of powerful econometric packages. Wallace and Silver's 'HUMMER' programme allowed basic regression to be carried out and provided a limited amount of statistical information. Microsoft Excel spreadsheets provided even less statistical information than HUMMER through their regression command, but it allowed easier calculation of further statistics. Excel also provided greater accuracy, with up to 15 significant figures. For these reasons most of the estimation work was carried out using Excel. Combining the output of its regression command with its calculation abilities allowed most tests to be conducted. I also used Excel's matrix multiplication facilities to solve (XTX)-1XTY manually in some cases (this allowed me to find the t-statistics for the parameter estimates.) HUMMER was used for the less critical and more regression-intensive Dickey-Fuller tests.
The procedure used for model selection was primarily a 'general-to-specific' one. A very general model was first estimated and irrelevant explanatory variables were then excluded. Suspicions that there may be a lag of some time between changes in an explanatory variable and adjustments in the explained variable led to a distributed lag model. A further suspicion of an autoregressive component led to the use of an autoregressive distributed lag model. In keeping with the general-to-specific approach, it was intended that these hypotheses could be dropped should those suspicions prove unfounded.
As the data series were in monthly time series form, it seemed most appropriate to include lags of up to twelve periods on each of the Y variable and the main X variable. With only sixty-six observations it was not feasible to include such lags on the other X variables, though this may have been desirable.
Starting with this general model of 29 explanatory variables, variables were excluded one by one starting with those with the largest t-statistic. Eventually the null hypotheses of a variable's insignificance was rejected for all explanatory variables at the 5% level. Interestingly, SUN was one of the variables excluded in this way. The following model resulted. The reduction from 29 to 8 explanatory variables obviously involved a cost in terms of explanatory power, but this loss was quite small: the R2 figure fell by just 0.014 to 0.9749. The Rbar-squared figure remained constant.1
Variable Coefficient(XTX)ii SE T-Stat ~t49
Constant 81.4394 35.5259 4.2389 19.2124 0.0000
elec-3 -0.3879 0.0030 0.0388 -10.0045 0.0000
elec -7 -0.4326 0.0095 0.0692 -6.2469 0.0000
output 0.0317 0.0002 0.0104 3.0626 0.0036
output-6 -0.0714 0.0002 0.0107 -6.6704 0.0000
output-9 -0.0344 0.0002 0.0108 -3.1844 0.0025
temp -0.5512 0.0072 0.0603 -9.1462 0.0000
t 0.2992 0.0011 0.0234 12.7946 0.0000
The autoregressive component of this model ensures that we cannot assume independence between the disturbance term and the explanatory variables:
E(ytt-) 0, for 0
This is very serious for small samples, but for the purposes of this analysis I will invoke a combination of the Mann-Wald theorem and Cramérs theorem which show that for large samples the OLS distribution theory is still asymptotically valid.
A number of tests were carried out to check the validity of omitting certain variables and for the validity of the model as a whole. The results of these tests are shown below.
F-Tests
Null DoF 1 DoF 2 URSS RSS F-Stat Prob.>F
Hypothesis
(H0)
Zero Slopes 7 49 24.7831 986.833 271.7319 0.0000
No Lags 4 49 24.7831 80.9449 27.7602 0.0000
General ardl 21 25 11.3367 24.7831 1.4120 0.2034
reset (Y2) 1 48 1.7977 24.7831 613.7357 0.0000
In the first test, the null hypothesis that all of the variables were irrelevant (have coefficients of zero) was tested against the alternative that they do not. This hypothesis is rejected as there is only a tiny probability of its validity. Similarly in the second test, the null hypothesis that all of the lagged variables are irrelevant is rejected. For the third test the simplified model is tested against the alternative of the original general ARDL model with all of the lags. This test does not reject the simple model at the 5% level of significance. The RESET test tests for the validity of including Y2 as an explanatory variable. This hypothesis is definitely not rejected at the 5% significance level. This suggests that there exists an alternative model which would better explain the Y variable. Unfortunately I was unable to find such a model in the time available.
Based on asymptotic theory, we can test the for non-normality by looking at the standard third and fourth moments of the residuals. Looking individually at the statistics for skewness
Normality Test Statistic Distribution Value Prob. > Skewness Root b1 ~AN(0,9.5) 0.10 0.83 Kurtosis b2 ~AN(3,0.42) 2.47 0.10 Normality N ~2 0.78 0.68
and kurtosis we find that in neither case is the null hypothesis of normality rejected. Similarly, a test based on a standardised combination of the two statistics ('N') does not reject it either.
Graphical techniques show few signs of heteroscedasticity. There were no noticeable trends in the error terms graphed against time or any of the other explanatory variables, nor against the explained variables. A Goldfeld-Quandt test gives a more subjective answer. This showed only a small difference between the R2 statistics for separate regressions for the first 19 observations and the last 19. The F-statistic of 1.860911 is less than the 5% critical value for (11, 11) degrees of freedom (2.817927), so the null hypothesis of no heteroscedasticity is not rejected. Both the graphical techniques and the Goldfeld-Quandt test used here test only for time-related heteroscedasticity. The Breusch-Pagan test is more general, testing for possible heteroscedasticity involving the original regressors. Again this does not reject the null hypothesis of no heteroscedasticity.
Finally, the Durbin Watson-statistic for this regression is 2.30, well above the upper bound for this test when k=8 and n=54. This indicates rejection of the hypothesis of positive serial correlation. This is supported by the first-order serial correlation coefficient of only -0.17.
The model outlined above has good explanatory power (high R2) and passes most of the usual diagnostic tests (except of course the RESET test). This would appear to be a successful attempt at modelling explaining the demand for electricity in terms of an index of industrial production temperature and time. The RESET result indicates that a better model could be found which would be far more powerful, but in its absence this model appears to be a good alternative.
Implicit in the above analysis are certain assumptions which I have neglected until now to prove. Here I will show that these assumptions are not in fact valid and examine the implications of this result for the conclusions of the preceding analysis.
Looking at a correlogram for electricity demand in figure 1, it is quite clear that the sample autocorrelations do not 'decline relatively rapidly' as the order of autocorrelation is increased. In fact it follows a clear trend, rising close to unity for orders which are multiples of twelve and then falling again in between.
Failure of the autocorrelations to collapse is an indicator of non-stationarity in the data. It is not proof. To prove the existence of this problem Dickey-Fuller tests were carried out on the data.
As we can see the null hypothesis of a unit root is only rejected for the output variable. Both ELEC and TEMP may have unit roots.
Implications
The implications of this problem are devastating. Asymptotic theory has been shown to be completely different in cases of non-stationarity from the usual textbook asymptotic theory. R2 has a non-degenerating limiting distribution while the Durbin-Watson statistic converges to zero (one would expect therefore that the R2 values which I found might have been accompanied by a lower Durbin-Watson statistic.) More specifically, a non-stationary regressor cannot fulfil the condition
plim (n-1X ij 2) = constant
This assumption is crucial to all of the statistical inference made above. It embodies the 'standard regularity conditions'. These allow us to show that the OLS estimator is asymptotically normal. Without this condition we can say nothing about the distribution of the OLS estimator or that of the OLS residuals. All of the tests reported in section one are therefore invalid. Each of them made implicit assumptions about the distribution of the OLS estimator or its residuals. These assumptions cannot be shown to be valid and in the absence of any proven valid theory we cannot proceed with the tests.
Conclusion
Non-stationarity of the explained variable means that no valid statistical inference can be made about any autoregressive model, as these include at least one non-stationary regressor. A number of alternatives exist. It is possible that TEMP and ELEC are co-integrated, this would allow us to model some linear function of them which was stationary. Alternatively, as both TEMP and ELEC appear to be seasonal it is possible that the seasonal difference version of them would be stationary.3 Unfortunately I do not have the resources to re-estimate the model taking into account the problem of non-stationarity. I must conclude therefore that the non-stationarity of the data used in this model means that it is not possible to estimate except in special cases such as co-integration.
Notes
1 An alternative model was also estimated which maximised the level of Rbar-squared by only excluding a variable if its inclusion would increase the Rbar-squared figure. This model had seventeen explanatory variables, though, and achieved a gain of only 0.01 in the adjusted R2
2 The 5% critical value here are in fact those for a sample size of 100. Those for smaller samples are smaller in absolute value.
3 I have checked this possibility and found that it does give a stationary Y variable, but TEMP still has a possible unit root.