1. Autocorrelation and Heteroskedasticity

2. Introduction Assess the main ways of remedying autocorrelation
Describe the problem of non-constant error terms
Assess the main test for heteroskedasticity
Examine the main remedies for heteroskedasticity
Introduce the multivariate approach to regression analysis

3. Cochrane-Orcutt methodology
This approach to remedying autocorrelation relies on the generalised difference equation approach
It is an iterative procedure, the estimates of the parameters converge towards their actual value.
It assumes the autocorrelation follows a first order autoregressive process (simplest form)

4. Cochrane-Orcutt Procedure

5. Procedure (Based on previous slide)
Run regression and collect the error term (u).
Run a second regression of u against u(t-1) to obtain an estimate of ρ.
Form the generalised difference equation, to obtain new estimates of the parameters
Re-run this equation, obtain u and repeat process until there is no significant difference from one iteration to the next.

6. Restricted Version of the GDE
The generalised difference equation can be written in the following form, which includes a restriction, the product of the coefficients on y(-1) and x is equal to the negative of the coefficient on x(-1) :

7. Unrestricted version of the GDE
In the unrestricted version of the equation in the previous slide, there are no restrictions on the values of the coefficients:

8. The Common Factor Test
To determine whether the restricted or unrestricted versions are best, we can use the common factor test, under the null hypothesis that the restriction holds, the test statistic is:

9. Heteroskedasticity
This occurs when a Gauss-Markov assumption is broken, it indicates non-constant variance of the error term.
It is often caused by variables which have substantial differences in the values of the observations, i.e. GDP differing between the USA and Cuba
With heteroskedasticity the estimator is no longer BLUE, as it is not the best.

10. Tests for Heteroskedasticity
Goldsfeldt-Quandt test (not used much now)
Whites Test (used in E-views computer software).
LM test (Similar to the LM test for heteroskedasticity)

11. White’s Test
This test follows a similar pattern to the LM test for autocorrelation discussed earlier.
Based on the same first equation as in slide 4, we estimate the model and collect the residual u.
Square the residual to form the variance of the residual
Run a secondary regression of the squared residual on the explanatory variable and this variable squared

12. White’s Test
This secondary regression appears like the following:

13. White’s Test
Collect the R-squared statistic and multiply by the number of observations to obtain the test statistic.
This follows a chi-squared distribution, the d of f equal the number of parameters in the secondary regression (i.e. 2 in above example)
The null hypothesis is no heteroskedasticity.
If there were 2 explanatory variables, you could include the cross product of these variables, in the secondary regression.

14. Remedies for heteroskedasticity
If the standard deviation of the residual is known, the heteroskedasticity can be removed by dividing the regression equation through by the standard deviation of the residual (weighted Least Squares)
If this is not known, as is likely, we need to stipulate what the standard deviation is equal to. We can then divide the regression equation through by this variable to obtain a constant variance residual.

15. Remedying Heteroskedasticity
If we assume the error variance is proportional to x, the explanatory variable and we have the usual model:

16. Remedying Heteroskedasticity
If we divide our model through by the square root of x:

17. Remedying Heteroskedasticity
Taking the new error term, we can show it is no longer suffering from heteroskedasticity by proving it is now constant:

18. Multivariate Regression
Regressions with more than one explanatory variable are similar to the bi-variate case, we can interpret the coefficients and t-statistics in much the same way as before.
We now have a potential problem with multicollinearity, when the explanatory variables are correlated
The formula for the β contains an expression for the covariance between the explanatory variables.

19. Multivariate Regression Analysis
The regression coefficients are more likely to be accurate, the less closely related are the explanatory variables
This allows testing for the significance of groups of variables
The R-squared increases with the more explanatory variables
We need to consider whether all relevant effects are included in the model
When estimating the t-tests, we need to account for the extra degrees of freedom
There is a trade off between the number of explanatory variables and the ease of interpreting the model

20. Conclusion
The common factor test can be used to determine the best solution to the problem of autocorrelation
In the presence of heteroskedasticity, the estimator is no longer BLUE.
White’s test is the most common test for heteroskedasticity
Weighted Least Squares can be used to remedy the heteroskedasticity problem.