Autocorrelation MS management

Autocorrelation Econometric problems

Autocorrelation

Autocorrelation What is meant by autocorrelation The error terms are not independent from observation to observation – ut depends on one or more past values of u What are its consequences? The least squares estimators are no longer “efficient” (i.e. they don’t have the lowest variance). More seriously autocorrelation may be a symptom of model misspecification How can you detect the problem? Plot the residuals against time or their own lagged values, calculate the Durbin-Watson statistic or use some other tests of autocorrelation such as the Breusch-Godfrey test How can you remedy the problem? Consider possible model re-specification of the model: a different functional form, missing variables, lags etc. If all else fails you could correct for autocorrelation by using the Cochrane-Orcutt procedure or Autoregressive Least Squares

Autocorrelation Sources of Autocorrelation Omitted explaintory variable Misspecification of the mathematical form Interpolation in statistical observation Misspecification of true random error

Autocorrelation First-order autocorrelation

Autocorrelation consequences of autocorrelation

Autocorrelation Detection of autocorrelation

The BG test: We use the two-variable regression model to illustrate the test, although many explanatory can be added to the model. Also, lagged values of the response can be added to the model. Let

Assume that the error term follows the pth-order autoregressive, AR(p), scheme as follows: The null hypothesis H0 to be tested is that H0: ρ1 = ρ2 = ·· · = ρp = 0

The BG test involves the following steps: 1. Estimate (1) by OLS and obtain the residuals, 2. Regress on the original Xt (if there is more than one X variable in the original model, include them also) and , ,… where the latter are the lagged values of the estimated residuals in step 1

and obtain R2 from this (auxiliary) regression. 3. Breusch and Godfrey have shown that

It follows the chi-square distribution with p df It follows the chi-square distribution with p df. If in an application, (n − p)R2 exceeds the critical chi-square value at the chosen level of significance, we reject the null hypothesis, in which case at least one rho in (2) is statistically significantly different from zero.

Alternative Test First apply OLS to the sample observation and obtain the values of the regression residuals “et”s. Since we cannot be a priori certain the values about the existence of autocorrelation or about its pattern. We may experiment various forms of autoregressive structures,

Autocorrelation is judged in the light of statistical significance of the and the overall fit of the above regression. That is we may carry out any one of the standard tests of statistical significance for the estimate of the autocorrelation relationship ( Z or T test).

If the are found statistically significant we accept that ‘s are autocorrelated and there exists autocorrelation.

Durbin’s h Cannot use DW d if there is a lagged endogenous variable in the model sc2 is the estimated variance of the Yt-1 term h has a standard normal distribution

Tests for higher order autocorreltaion Ljung-Box Q (χ2 statistic)

Autocorrelation: Remedies Cochran-Orcutt method (1) Estimate model using OLS and obtain the residuals, ut. (2) Using the residuals run the following regression.

Autocorrelation: Remedies (cont.) Cochran-Orcutt method (cont.) (3) using the p obtained, perform the regression on the generalized differences (4) Substitute the values of B1 and B2 into the original regression to obtain new estimates of the residuals. (5) Return to step 2 and repeat – until p no longer changes.