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92

Summary

1. Most economic data consist of time series and there is very often a correlation in the errors corresponding to successive time periods. This is the problem of autocorrelation.

2. The Durbin-Watson (DW) test is the most often used to test for the presence of autocorrelation. If this test detects the presence of autocorrelation, it is customary to transform the data on the basis of the estimated first-order autocorrelation and use least squares with the transformed data.

3. There are several limitations to this procedure. These limitations (discussed in Section 6.9) are:

(a) The serial correlation might be of a higher order.

(b) The serial correlation might be due to omitted variables.

(c) The serial correlation might be due to the noninclusion of lagged values of the explained and explanatory variable, that is, due to misspecification of the dynamic process.

4. Very often, simple solutions are suggested for handling the serial correlation problem, such as estimation in first differences. The issue of estimation of equations in levels versus first differences is discussed in Section 6.3 and also in Section 6.10. Other solutions are discussed in Section 6.4.

5. There have been some extensions and further tables prepared for the DW test. These extensions are outlined in Section 6.6.

6. The DW test is not applicable if there are lagged dependent variables in the model. Durbin suggested an alternative test, known as Durbins / -test. This

is similar to the one we discussed in Section 6.10 where a trend in the variance was transmitted to a trend in the mean. In this case a simple test for the ARCH effect, that is, a test for the hypothesis a, = 0 in (6.38) is to get the OLS residuals e, and regress e? on ej, (with a constant term) and testing whether the coefficient of ¸? i is zero. An LM (Lagrangian multiplier test) is to use nR as with one d.f. This would enable us to see whether the significant DW test statistic is due to serial correlation in e, or due to the ARCH effect. Many empirical studies have found significant ARCH effects.

The estimation of the ARCH model can be carried out by an iterative procedure. First, we estimate (6.37). We then get estimates of , and a, in (6.38) by regressing ] on e?,. Now we estimate (6.37) as a heteroskedastic regression model, since we have an estimate of h,. This process can be repeated until convergence.

There are, however, problems with this simple procedure that we have ignored. We might get estimates of a, less than zero or greater than one. These problems as well as the computation of the asymptotic variances and covariances are discussed in Engles paper. The purpose of our discussion is to point out one more source for a significant DW test statistic when, in fact, there is no serial correlation.



test is explained and illustrated in Section 6.7. Some problems with its use are also illustrated there. This test again, is for first-order autocorrelation only.

7. A general test which is, however, asymptotic, is the LM test. This test can be used for any specified order of the autocorrelation. It can be applied whether there are lagged dependent variables or not. It can be used with standard regression packages. It is based on omitted variables. This test is discussed in Section 6.8. It consists of two steps:

(a) First estimate the equation by ordinary least squares and get the residual

(b) Now introduce appropriate lags of u, in the original equation and reestimate it by least squares. Test that the coefficients of the lagged m,s are zero using the standard tests.

The LM test has not been illustrated with an example. This is left as an exercise. Many of the data sets presented in the book are time-series data, and students can use these to apply the LM test.

8. The effect of autocorrelated errors on least squares estimators are:

(a) If there are no lagged dependent variables among the explanatory variables, the estimators are unbiased but inefficient. However, the estimated variances are biased, sometimes substantially. These problems are discussed in Section 6.5 and the biases are presented for some simple cases.

(b) If there are lagged dependent variables among the explanatory variables, the least squares estimators are not even consistent (see Section 6.7). In this case the DW test statistic is biased as well. This is the reason for the use of Durbins A-test.

9. Obtaining a significant DW test statistic does not necessarily mean that we have a serial correlation problem. In fact, we may not have a serial correlation problem and we may be applying the wrong solution. For this purpose Sargan suggested that we first test for common factors and then apply tests for serial correlation if there is a common factor. This argument is explained at the end of Section 6.9 and illustrated with an example.

10. Economic time series can conveniently be classified as belonging to the DSP class or TSP class. The appropriate procedure for trend elimination (whether to use differences or regressions on time) will depend on this classification. One can apply the Dickey-Fuller test (or Sargan-Bhargava test) to test whether the time series is of the DSP type or TSP type. Most economic time series, however, are of the DSP type and, thus, estimation in first differences is appropriate. However, differencing eHminates all information on the long-run properties of the model. One suggestion that has been made is to see whether the time series are cointegrated. If this is so, then both long-run and short-run parameters can be estimated (either separately or jointly).

11. Sometimes even though the errors in the equation are not autocorrelated, the variance of the error term can depend on the past history of errors. In such models, called ARCH models, one can find a significant DW test statistic even though there is no serial correlation in the errors. A test for the ARCH effect will enable us to judge whether the observed serial correlation is spurious.



EXERCISES

Exercises

1. Explain the following.

(a) The Durbin-Watson test.

(b) Estimation with quasi first differences.

(c) The Cochrane-Orcutt procedure.

(d) Durbins / -test.

(e) Serial correlation due to misspecified dynamics.

(f) Estimation in levels versus first differences.

2. Use the Durbin-Watson test to test for serial correlation in the errors in Exercises 17 and 19 at the end of Chapter 4.

3. 1 am estimating an equation in which y, i is also an explanatory variable. I get the following results.

y, = 2.7 + OAx, + 0.9y, , fg

(0 4) (0 06) UW - l.y

1 find that the R is very high and the Durbin-Watson statistic is close to 2, showing that there is no serial correlation in the errors. My friend tells me that even if the R is high, this is a useless equation. Why is this a useless equation?

4. Examine whether the following statements are true or false. Give an explanation.

(a) Serial correlation in the errors w leads to biased estimates and biased standard errors when the regression equation = + uis estimated by ordinary least squares.

(b) The Durbin-Watson test for serial correlation is not appUcable if the errors are heteroskedastic.

(c) The Durbin-Watson test for serial correlation is not applicable if there are lagged dependent variables as explanatory variables.

(d) An investigator estimating a demand function in levels and first differences obtained /?s of 0.90 and 0.80, respectively. He chose the equation in levels because he got a higher R. This is a valid reason for choosing between the two models.

(e) Least squares techniques when applied to economic time-series data usually yield biased estimates because many economic time series are autocorrelated.

(f) The Durbin-Watson test can be used to describe whether the errors in a regression equation based on time-series data are serially independent.

(g) The fact that the Durbin-Watson statistic is significant does not necessarily mean that there is serial correlation in the errors. One has to apply some other tests to come to this conclusion.

(h) Consider the model y, = c(y, , + fix, + u„ where the errors are autoregressive. Even if the OLS method gives inconsistent estimates of



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