L15 - Heteroscedasticity

Question 1

What is Heteroscedaticity?

Answer 1

• When finding the OLS estimator we assumed that the variance of the error terms was constant
• Heteroscedasticity is present if the variance of the error term is not a constant.
• This is most commonly a problem when dealing with cross-section data.
• A common situation is where the variance is related to one of the RHS variables (formula below) although this is not the only form of heteroscedasticity

Question 2

What does heteroscedasticity look like on a graph if the variance in x was increasing?

Answer 2

Question 3

What are the implications of Heteroscedasticity?

Answer 3

• Heteroscedasticity does not, in itself, mean that OLS will be biased.
• However, it does mean that OLS will be inefficient since one of the Gauss-Markov assumptions is no longer valid.
• If the variance of the error term is positively correlated with one of the RHS variables then the OLS estimates of the standard errors will be biased downwards. –> similar to serial correlation
• If heteroscedasticity is a symptom of some other kind of misspecification (e.g. omitted variables) then it is possible that OLS will be biased.
• IF the standard errors are biased due to heteroscedasticity then we cannot rely on any hypothesis tests based on the SE

Question 4

How do we deal with heteroscedasticity?

Answer 4

• The problem is that we often dont know the form of the heteroscedasticity prior to estimation - so we can only scale the data if we know what the problem is
• when you have scaled your data it is normal for R² to be low

Question 5

What are the two tests for Heteroscedasticity?

Answer 5

• Goldfeld-Quandt test
• White’s test

Question 6

How is Goldfeld-Quandt test performed?

Answer 6

The Goldfeld-Quandt test provides a basic test fo heteroscedasticity. It is not very common anymore and quite old-fashioned. This is constructed as follows:

1. Order the data according to the size of the exogenous variable we believe is related to the variance of the error term.
2. Divide the sample into three sections of size N₁, N-2N₁ and N₁ respectively. N₁ should be approximately equal to 3N/8.
3. Estimate separate regressions for the first and last N1 observations and generate the residual sum of squares. Perform an F test for the equality of these sums of squares
4. if the lower and upper RSS are the same or close to each other there is little/ no heteroscedasticity, if not then it is present

Question 7

What is the formula for the F-test performed for the Goldfeld-Quandt test?

Answer 7

• Where RSS₁ and RSS₂ are from the bottom and top half of the data respectively
• After calculating the F statstics it is compared with the 5% critcal value

Question 8

What are the Problems with Goldfeld-Quandt test?

Answer 8

1. We need to know which variable to use to order the data before we perform the test.
2. If the number of observations is small it may be impractical to divide the sample into three sections and discard the middle section.

Question 9

What is White’s test for hetroscedasticity?

Answer 9

• This one is quite common, compared to Goldfeld-Quandt’s test

White’s test uses the residuals from OLS estimation to construct a test statistic. It not only helps to detect hetreoscedacity, but also helps identify the variable causing it. The procedure is as follows:

1. Estimate the model by OLS and save the residuals.
2. Regress the squared residuals on the original regressors as well as their squared values and (possibly) their cross-products.
3. Perform either an F-test or a Chi-squared test for the significance of the regressors in the stage 2 regression.

If we had more than one X variable, the auxilary regression would include the cross-product - best to always includ it unless the number of regressors becomes so large that we have insufficient degrees of freedom

When performing the regression, the one with the highest statistic is the one causing the problem

Question 10

What is White’s heteroscedasticity consistent convariance matrix?

Answer 10

• We can use the matrix to scale up the SE not the regression model - this leads to a more accurate standard error
• This does not deal with heteroscedacity just the effect caused by it (larger SE)
• when SE are higher and the t-statistics lowers than when we use the OLS variance

Question 11

Why is White’s heteroscedasticity consistent convariance matrix useful?

Answer 11

• The advantage of this approach is that it allows us to get consistent estimates of the standard errors of the regression parameters in cases where the form of the heteroscedasticity is not obvious.
• Note that OLS will still be inefficient when we use the White covariances (in fact the parameter estimates don’t change at all).
• However, we can conduct reliable statistical inference even when heteroscedasticity is present.

Question 12

What is ARCH?

Answer 12

• Autoregressive Conditional Heteroscedasticity ARCH
• So far the type of heteroscedasticity we have considered has been applicable to cross-section models.
• ARCH is a type of heteroscedasticity which is relevant for time series models and particularly for financial time-series models.
• ARCH models the variance of the error as a function of the size of random shocks hitting the model and its own past values.
• The effects of ARCH are that periods of volatility can last for some time and the OLS residuals may not follow a normal distribution.

Question 13

What can misspecification result in?

Answer 13

• Mutiple test failures