How can we solve the problem of autocorrelation as in (III.I.2-1))?

(III.I.2-1)

We can write (III.I.2-1) in terms of variances

(III.I.2-2)

but since we may assume homoskedasticity it follows that

(III.I.2-3)

Using (III.I.2-1) the covariance with a lag of 1 period is

(III.I.2-4)

In general the covariance with lag k can be written as

(III.I.2-5)

and the covariance matrix of the residuals as

(III.I.2-6)

which therefore yields

(III.I.2-7)

TheGLS solution to this problem, if is assumed to be known, is a tree stage procedure

stage 1

(III.I.2-8)

stage 2

(III.I.2-9)

stage 3

(III.I.2-10)

using equation (III.I-8).

Remark that

(III.I.2-11)

and that

(III.I.2-12)

which implies

(III.I.2-13)

There are two other alternatives to solve the problem of (III.I.2-1): lagged variables, and distributed lags.

Sometimeslagged endogenous and/or exogenousvariables can be introduced in order to reduce the autocorrelation problem. This is a quite flexible tool with however, one important drawback. The introduction of eachadditional variable reduces the degrees of freedom, whereas the use of alagged variable mostly forces the researcher to reduce the sample range which also implies a loss of degrees of freedom ( T - K). Also remember that OLS is not an efficient estimator in the context of autocorrelation. Even worse is the estimation of parameters of lagged endogenous variables in the context of autocorrelation induced by unobserved variables (or wrong specification): in this case the OLS estimates will be biased.

Distributed lags can also be used to solve problem of (III.I.2-1). A huge advantage over the lagged variables method is that the distributed lags do not incur as large a loss of degrees of freedom as the first method. A drawback of this method could be the fact that the estimation procedure is not always straightforward and sometimes even difficult. The method of distributed lags will be discussed in detail in later sections.

In econometrics theDurbin-Watson statistic (c.q. d statistic) is frequently used

(III.I.2-14)

Since

(III.I.2-15)

eq. (III.I.2-14) can be rewritten as

(III.I.2-16)

for large samples, where is the correlation between the residual at period t and the residual at t-1. Note that d = 2 in case there is no (first order) autocorrelation in the residual series.

The Durbin-Watson statistic will fail when time series data are strongly seasonal. In this case an adapted d statistic, for a seasonal period s, can be defined as

(III.I.2-17)

which is in fact an autocorrelation coefficient of order s.

The d statistic will, in general, also fail in case a regression equation under OLS is estimated with the lagged endogenous variable variable used as regressor. In this case Durbin suggests alternatives (the so-called h statistics) such as

(III.I.2-18)

Another way to detect AR(p) autocorrelation is theBreusch-Godfrey test based on a test-equation: the residuals are explained by it�s lagged values (p is the number of lags included in the test-equation) and all exogenous variables of the original equation. The presence of AR(p) autocorrelation is tested by the use of a Chi-square-statistic (c.q. (T-p)*R-square).

Below you �ll find an example of how the Breusch-Godfrey test can be applied to test for AR(p) autocorrelation (this test is applied to our example-equation):

Breusch-Godfrey test for AR(p) residuals by Least Squares:Estimation with OLS:Endogenous variable =  ResidualsVariable                     Parameter                S.E.      T-STATemploy                   -6.574423604           7.235339332     -0.909expend                   +2.286765988           2.796278607     +0.818const                   +82.24312207          278.872104        +0.295resid(-1)                -0.449639967           0.2411178087    -1.86resid(-2)                -0.2887649456          0.194556882     -1.48resid(-3)                -5.403469274e-002      0.2080984616    -0.262-tail-t at 95 percent = 2.0421-tail-t at 95 percent = 1.697R-squared of stationary series = 0.1647233405Durbin-Watson                  = 1.999045701Degrees of freedom             = 30Variance of regression         = 1052262.317Standard Error of regression   = 1025.79838Sum of Squared Residuals       = 31567869.52Correlation matrix of parameters:            +1.00  -0.91  -0.51  +0.67  -0.08  +0.34             -0.91  +1.00  +0.21  -0.61  +0.14  -0.27             -0.51  +0.21  +1.00  -0.31  -0.00  -0.21             +0.67  -0.61  -0.31  +1.00  +0.16  +0.42             -0.08  +0.14  -0.00  +0.16  +1.00  +0.20             +0.34  -0.27  -0.21  +0.42  +0.20  +1.00 (T-p)*R-square =  5.435870235Critical Chi-square value (95%) = 7.81 (degrees of freedom = 3)

Many other tests are available to diagnose autocorrelation problems, many of which belong to the domain of time series analysis (chapter 5).