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Basic Econometrics , Gujarati and Porter

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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 thecovariance between two time periods depends only on the distanceor lag between the two periods and not the actual time at which thecovariance is computed.

21.2 If a time series has to be differenced d times before it becomesstationary, it is integrated of order d , denoted as I ( d ). In itsundifferenced form, such a time series is nonstationary.

21.3 Loosely speaking, the term unit root means that a given timeseries is nonstationary. More technically, the term refers to the

root of the polynomial in the lag operator.21.4 It has to be differenced three times.

21.5 The DF test is a statistical test that can be used to determine if a timeseries is stationary. The ADF is similar to DF except that it takesinto account the possible correlation in the error terms.

21.6 The EG and AEG tests are statistical procedures that can be used toto determine if two time series are cointegrated.

21.7 Two variables are said to be cointegrated if there is a stable long-runrelationship between them, even though individually each variableis nonstationary. In that case the regression of one variable on theother is not spurious.

21.8 Tests of unit roots are performed on individual time series.Cointegration deals with the relationship among a group ofvariables, where (unconditionally) each has a unit root.

21.9 If a nonstationary variable is regressed on another nonstationaryvariable(s), the resulting regression may pass the usual statisticalcriteria (high R2 value, significant t ratios, etc.) even though a prioriwe do not expect any relationship between the two. This isespecially so if the two variables are not cointegrated. However, ifthe two variables are cointegrated, even though individually theyare nonstationary, then such a regression may not be spurious.

21.10 See the answer to the preceding question.

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21.11 Most economic time series exhibit trends. If such trends areperfectly predictable, we call them deterministic. If that is not case,we call them stochastic. A nonstationary time series generallyexhibits a stochastic trend.

21.12 If a time series exhibits a deterministic trend, the residuals fromthe regression of such a time series on the trend variable representswhat is called a trend-stationary process. If a time series isnonstationary but becomes stationary after taking its first (or higher)order differences, we call such a time series a difference-stationaryprocess.

21.13 A random walk is an example of a nonstationary process. If avariable follows a random walk, it means its value today is equal toits value in the previous time period plus a random shock (error

term). In such situations, we may not be able to forecast the courseof such a variable over time. Stock prices or exchange rates aretypical examples of the random walk phenomenon.

21.14 This is true. The proof is given in the chapter.

21.15 Cointegration implies a long term, or equilibrium, relationshipbetween two (or more variables). In the short run, however, theremay be disequilibrium between the two. The ECM brings the twovariables back to long term equilibrium.

Empirical Exercises

21.16 (a) The correlograms for all these time series very much resemblethe log GDP correlogram given in Fig. 21.8. All these correlogramssuggest that these time series are nonstationary.

21.17 The regression results are as follows:

∆ log PCE t = 0.1899 + 0.0002 t − 0.0261 log PCE t − 1τ = (1.80) (1.67) ( − 1.71) *

R 2 = 0.0144

*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.

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∆ log DPI t = 0.1235 + 0.0001 t − 0.0161log DPI t − 1τ = (1.41) (1.15) ( − 1.29) *

R 2 = 0.017

* This tau value is not statistically significant, suggesting that the

log PDI time series contains a unit root, that is, it is nonstationary.

∆ log Profits t = 0.1412 + 0.0010 t − 0.0524Profits t-1τ = (2.69) (2.81) ( − 2.60) *

R2 = 0.0354

* This tau value is not statistically significant, suggesting that thistime series has a unit root.

Dividends t = 0.0411 + 0.0004 t − 0.0180Dividends t-1

τ = (2.09) (1.61) ( − 1.42)*

R 2 = 0.0217

* This tau value is not significant, suggesting that thelog Dividends time series is nonstationary.

Thus, we see that all the given time series are nonstationary. Theresults of the Dickey-Fuller test with no trend and no trend and nointercept did not alter the conclusion.

21.18 If the error terms in the model are serially correlated, ADF is themore appropriate test. The τ statistics for the appropriate coefficientfrom the ADF regressions for the three series are:

log PCE -1.73log DPI -1.44log Profits -3.14log Dividends -1.42

The critical τ values remain the same as in Problem 21.17. Again,the conclusion is the same, namely, that the three time seriesare nonstationary.

21.19 (a) Probably yes, because individually the two time seriesare nonstationary.

(b) The OLS regression of dividends on profits gave the followingresults:Variable Coefficient Std. Error t-StatisticC -1.3339 0.0503 -26.50PROFITS 1.1269 0.0103 109.63

R-squared 0.9803 d = 0.1509

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When the residuals from this regression were subjected to unitroot tests with no constant, constant, and constant and trend,the results showed that the residuals were not stationary, thusleading to the conclusion that dividends and profits are notcointegrated. Since this is the case, the conclusion in ( a ) stays.

(c) There is little point in this exercise, as there is no long runrelationship between the two.

(d ) They both exhibit stochastic trends, which is confirmed bythe unit root tests on each time series.

(e) If log Dividends and log Profits are cointegrated, it does notmatter which is the regressand and which is the regressor. Ofcourse, finance theory could resolve this matter.

21.20

The scattergrams of the first differences of log DPI, log Profits, andlog Dividends, all show diagrams similar to Fig. 21.9. In the firstdifference form each of these time series is stationary. This can beconfirmed by the ADF test.

21.21 In theory there should not be an intercept in the model. But if therewas a trend term in the original model, then an intercept could beincluded in the regression and the coefficient of that intercept termwill indicate the coefficient of the trend variable. This of courseassumes that the trend is deterministic and not stochastic.

To see this, we first regressed log Dividends on log Profits and thetrend variable, which gave the following results:

Dependent Variable: log DIVIDEND

Variable Coefficient Std. Error t-StatisticC 0.4358 0.1053 4.14Log PROFITS 0.4245 0.0402 10.55trend 0.0127 0.0007 17.71

R-squared 0.9914

But one should be wary of this regression because this regressionassumes that there is a deterministic trend. But we know that thedividend time series has a stochastic trend.

Now regressing the first differences of log Dividends on the firstdifferences of log Profits and the intercept, we get the followingresults:

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Variable Coefficient Std. Error t-StatisticC 0.0193 0.0021 9.24∆ logProfits 0.0585 0.0302 1.94

R-squared 0.01524

In this regression the intercept is significant, but not the slopecoefficient. The intercept value of 0.0193 is in theory equal to thecoefficient of the trend variable in the previous equation; the twovalues are not identical because of rounding errors as well as the factthat the trend in the dividends series is not deterministic.

This exercise shows that one should be very careful in including thetrend variable in a time series regression unless one is sure that the

trend is in fact deterministic. Of course, one can use the DF andADF tests to determine if the trend is stochastic or deterministic.

21.22 From the first difference regression given in the preceding exercise,we can obtain the residuals of this regression ( ˆt u ) and subject them

to unit root tests. We regressed ˆt u∆ on its own lagged value withoutintercept, with intercept, and with intercept and trend. In each casethe null hypothesis was that these residuals are nonstationary, that is,they contain a unit root test. The Dickey-Fuller τ values for thethree options were -17.05, -17.01, and -17.22. In each case thehypothesis was rejected at 5% or better level (i.e., p value lower than5%). In other words, although log Dividends and log Profits werenot cointegrated, they were cointegrated in the first difference form.

21.23 (a ) Since τ is less than the critical τ value, it seems that thehousing start time series is nonstationary. Therefore, there isa unit root in this time series.

(b) Ordinarily, an absolute t value of as much as 2.35 or greaterwould be significant at the 5% level. But because of the unitroot situation, the true t value here is 2.95 and not 2.35. This

example shows why one has to be careful in using the t statisticindiscriminately.

(c) Since the τ of 1t X −∆ is much greater than the corresponding

critical value, we conclude that there is no second unit root in thehousing start time series.

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21.24 This is left for the reader.

21.25 (a ) & ( b)

Y exhibits a linear trend, whereas X represents a quadratictrend.

Here is the graph of the actual and fitted Y values:

From the given regression results you might think that thisis a "good" regression in that it has a high R2 and significantt ratios. But it is a totally spurious relationship, because weare regressing a linearly trended variable ( Y ) on a quadraticallytrended variable ( X ). That something is not right with this modelcan be gleaned from the very low Durbin-Watson d value.The point of this exercise is to warn us against reading too much inthe regression results of two deterministically trended variables withdivergent time paths.

21.26 (a ) Regression (1) shows that the elasticity of M1 with respect toGDP 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

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suspect that there is correlation in the error terms or that thisregression is spurious.

(b) In the first difference form, there is still positive relationshipbetween the two variables, but now the elasticity coefficient has

dropped dramatically. Yes, the d values might suggest that thereis no serial correlation problem now. But the significant dropin the elasticity coefficient suggests that the problem here may beone of lack of cointegration between the two variables.

(c) & ( d ) From regression (3) it seems that the two variables arecointegrated, for the 5% critical τ value is –1.9495 and theestimated tau value is more negative than this. However, the 1%critical tau value is –2.6227, suggesting that the two variablesare not cointegrated. If we allow for intercept and intercept andtrend 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 logsof money and GDP. The equation given here takes into account theerror correction mechanism (ECM), which tries to restore theequilibrium in case the two variables veer from their long-run path.However, the error term in this regression is not statisticallysignificant at the 5% level.

Since, as discussed in ( c) and ( d ) above, the results of thecointegration tests are rather mixed, it is hard to tell whether theregression results presented in (1) are spurious or not.

21.27 (a ) & ( b) The time graph of CPI very much resembles Fig. 21.12.This graph clearly shows that generally there is an upward trend in

the CPI. Therefore, regression (1) and (2) are not worthconsidering. Note that the coefficient of the lagged CPI is positivein 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 istrend 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 theF test.

Using the R2 version of the F test, the R2 value of regression (1)is 0.0703, which is the restricted R2. The R2 value of regression(3), which is 0.507, is the unrestricted R2. Hence the F value is:

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F =(0.507 − 0.0703) / 2

(1 − 0.507) / 45= 19.9305

Referring to the DF F values given in Table D.7 in App. D, you cansee that the observed F value is highly significant (Note: The tabledoes not give the F value for 40 observations, but mentallyinterpolating 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.

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