CHAPTER 21: TIME SERIES ECONOMETRICS: SOME BASIC CONCEPTS

Transcription

1 CHAPTER 21: TIME SERIES ECONOMETRICS: SOME BASIC CONCEPTS 21.1 A stochastic process is said to be weakly stationary if its mean and variance are constant over time and if the value of the covariance between two time periods depends only on the distance or lag between the two periods and not the actual time at which the covariance is computed If a time series has to be differenced d times before it becomes stationary, it is integrated of order d, denoted as I (d). In its undifferenced form, such a time series is nonstationary Loosely speaking, the term unit root means that a given time series is nonstationary. More technically, the term refers to the root of the polynomial in the lag operator It has to be differenced three times The DF test is a statistical test that can be used to determine if a time series is stationary. The ADF is similar to DF except that it takes into account the possible correlation in the error terms The EG and AEG tests are statistical procedures that can be used to to determine if two time series are cointegrated Two variables are said to be cointegrated if there is a stable long-run relationship between them, even though individually each variable is nonstationary. In that case the regression of one variable on the other is not spurious Tests of unit roots are performed on individual time series. Cointegration deals with the relationship among a group of variables, where (unconditionally) each has a unit root If a nonstationary variable is regressed on another nonstationary variable(s), the resulting regression may pass the usual statistical criteria (high R 2 value, significant t ratios, etc.) even though a priori we do not expect any relationship between the two. This is especially so if the two variables are not cointegrated. However, if the two variables are cointegrated, even though individually they are nonstationary, then such a regression may not be spurious See the answer to the preceding question. 219

2 21.11 Most economic time series exhibit trends. If such trends are perfectly predictable, we call them deterministic. If that is not case, we call them stochastic. A nonstationary time series generally exhibits a stochastic trend If a time series exhibits a deterministic trend, the residuals from the regression of such a time series on the trend variable represents what is called a trend-stationary process. If a time series is nonstationary but becomes stationary after taking its first (or higher) order differences, we call such a time series a difference-stationary process A random walk is an example of a nonstationary process. If a variable follows a random walk, it means its value today is equal to its value in the previous time period plus a random shock (error term). In such situations, we may not be able to forecast the course of such a variable over time. Stock prices or exchange rates are typical examples of the random walk phenomenon This is true. The proof is given in the chapter Cointegration implies a long term, or equilibrium, relationship between two (or more variables). In the short run, however, there may be disequilibrium between the two. The ECM brings the two variables back to long term equilibrium. Empirical Exercises (a) The correlograms for all these time series very much resemble the log GDP correlogram given in Fig All these correlograms suggest that these time series are nonstationary The regression results are as follows: log PCE t = t log PCE t 1 τ =(1.80) (1.67) ( 1.71) * R 2 = *In absolute terms, this tau value is less than the critical tau value, suggesting that there is a unit root in the log PCE time series, that is, this time series is nonstationary. 220

3 log DPI t = t log DPI t 1 τ = (1.41) (1.15) ( 1.29) * R 2 = * This tau value is not statistically significant, suggesting that the log PDI time series contains a unit root, that is, it is nonstationary. log Pr ofits t = t Pr ofits t-1 τ =(2.69) (2.81) ( 2.60) * R 2 = * This tau value is not statistically significant, suggesting that this time series has a unit root. Dividends t = t Dividends t-1 τ =(2.09) (1.61) ( 1.42) * R 2 = * This tau value is not significant, suggesting that the log Dividends time series is nonstationary. Thus, we see that all the given time series are nonstationary. The results of the Dickey-Fuller test with no trend and no trend and no intercept did not alter the conclusion If the error terms in the model are serially correlated, ADF is the more appropriate test. The τ statistics for the appropriate coefficient from the ADF regressions for the three series are: log PCE log DPI log Profits log Dividends The critical τ values remain the same as in Problem Again, the conclusion is the same, namely, that the three time series are nonstationary (a) Probably yes, because individually the two time series are nonstationary. (b) The OLS regression of dividends on profits gave the following results: Variable Coefficient Std. Error t-statistic C PROFITS R-squared d =

4 When the residuals from this regression were subjected to unit root tests with no constant, constant, and constant and trend, the results showed that the residuals were not stationary, thus leading to the conclusion that dividends and profits are not cointegrated. Since this is the case, the conclusion in (a) stays. (c) There is little point in this exercise, as there is no long run relationship between the two. (d) They both exhibit stochastic trends, which is confirmed by the unit root tests on each time series. (e) If log Dividends and log Profits are cointegrated, it does not matter which is the regressand and which is the regressor. Of course, finance theory could resolve this matter The scattergrams of the first differences of log DPI, log Profits, and log Dividends, all show diagrams similar to Fig In the first difference form each of these time series is stationary. This can be confirmed by the ADF test In theory there should not be an intercept in the model. But if there was a trend term in the original model, then an intercept could be included in the regression and the coefficient of that intercept term will indicate the coefficient of the trend variable. This of course assumes that the trend is deterministic and not stochastic. To see this, we first regressed log Dividends on log Profits and the trend variable, which gave the following results: Dependent Variable: log DIVIDEND Variable Coefficient Std. Error t-statistic C Log PROFITS trend R-squared But one should be wary of this regression because this regression assumes that there is a deterministic trend. But we know that the dividend time series has a stochastic trend. Now regressing the first differences of log Dividends on the first differences of log Profits and the intercept, we get the following results: 222

5 Variable Coefficient Std. Error t-statistic C logprofits R-squared In this regression the intercept is significant, but not the slope coefficient. The intercept value of is in theory equal to the coefficient of the trend variable in the previous equation; the two values are not identical because of rounding errors as well as the fact that the trend in the dividends series is not deterministic. This exercise shows that one should be very careful in including the trend variable in a time series regression unless one is sure that the trend is in fact deterministic. Of course, one can use the DF and ADF tests to determine if the trend is stochastic or deterministic From the first difference regression given in the preceding exercise, we can obtain the residuals of this regression ( u ˆt ) and subject them to unit root tests. We regressed uˆt on its own lagged value without intercept, with intercept, and with intercept and trend. In each case the null hypothesis was that these residuals are nonstationary, that is, they contain a unit root test. The Dickey-Fuller τ values for the three options were , , and In each case the hypothesis was rejected at 5% or better level (i.e., p value lower than 5%). In other words, although log Dividends and log Profits were not cointegrated, they were cointegrated in the first difference form (a) Since τ is less than the critical τ value, it seems that the housing start time series is nonstationary. Therefore, there is a unit root in this time series. (b) Ordinarily, an absolute t value of as much as 2.35 or greater would be significant at the 5% level. But because of the unit root situation, the true t value here is 2.95 and not This example shows why one has to be careful in using the t statistic indiscriminately. (c) Since the τ of X t 1 is much greater than the corresponding critical value, we conclude that there is no second unit root in the housing start time series. 223

6 21.24 This is left for the reader (a) & (b) 40 Y Y exhibits a linear trend, whereas X represents a quadratic trend. Here is the graph of the actual and fitted Y values: X Residual Actual Fitted From the given regression results you might think that this is a "good" regression in that it has a high R 2 and significant t ratios. But it is a totally spurious relationship, because we are regressing a linearly trended variable (Y) on a quadratically trended variable (X). That something is not right with this model can be gleaned from the very low Durbin-Watson d value. The point of this exercise is to warn us against reading too much in the regression results of two deterministically trended variables with divergent time paths (a) Regression (1) shows that the elasticity of M1 with respect to GDP is about 1.60, which seems statistically significant, as the t value of this coefficient is very high. But looking at the d value, we 224

7 suspect that there is correlation in the error terms or that this regression is spurious. (b) In the first difference form, there is still positive relationship between the two variables, but now the elasticity coefficient has dropped dramatically. Yes, the d values might suggest that there is no serial correlation problem now. But the significant drop in the elasticity coefficient suggests that the problem here may be one of lack of cointegration between the two variables. (c) & (d) From regression (3) it seems that the two variables are cointegrated, for the 5% critical τ value is and the estimated tau value is more negative than this. However, the 1% critical tau value is , suggesting that the two variables are not cointegrated. If we allow for intercept and intercept and trend in equation (3), then the DF test will show that the two variables are not cointegrated. (e) Equation (2) gives the short-run relationship between the logs of money and GDP. The equation given here takes into account the error correction mechanism (ECM), which tries to restore the equilibrium in case the two variables veer from their long-run path. However, the error term in this regression is not statistically significant at the 5% level. Since, as discussed in (c) and (d) above, the results of the cointegration tests are rather mixed, it is hard to tell whether the regression results presented in (1) are spurious or not (a) & (b) The time graph of CPI very much resembles Fig This graph clearly shows that generally there is an upward trend in the CPI. Therefore, regression (1) and (2) are not worth considering. Note that the coefficient of the lagged CPI is positive in both cases. For stationarity, we require this value to be negative. Therefore, the more meaningful equation here is regression (3). The DF unit root tests suggest that the CPI time series is trend stationary. That is, the values of the CPI around its trend value (which is statistically significant) are stationary. (c) Since Equation (1) omits two variables, we have to use the F test. Using the R 2 version of the F test, the R 2 value of regression (1) is , which is the restricted R 2. The R 2 value of regression (3), which is 0.507, is the unrestricted R 2. Hence the F value is: 225

8 ( ) / 2 F = = ( ) / 45 Referring to the DF F values given in Table D.7 in App. D, you can see that the observed F value is highly significant (Note: The table does not give the F value for 40 observations, but mentally interpolating the given F values, you will reach this conclusion.). Hence, the conclusion is that the restrictions imposed by regression (1) are invalid. More positively, it is regression (3) that seems valid. 226