Week 10| Multicollinearity / dummy variables for dependent variables Flashcards
Which assumption does heteroskedasticity violate? What happens to the OL estimators when there is heteroskedasticity? (e.g. what nature are they and how are their variances calculated). What about the t and F statistics based on standard OLS formulas?
Heteroskedasticity violates the 3rd assumption: the conditional variance of the random error ( and that of the dependent variable) is constant.
When heteroskedasticity exists but other classical assumptions satisfied, the OLS estimators are linear, unbiased, consistent and (asymptotically) normally distributed. The variances of the OLS estimators cannot be estimated the same way bc the standard OLS formulas assume homoskedasticity and they are biased when the random error is heteroskedastic.
Consequently, the t and F statistics based o standards errors estimated with the standard OLS formulas can be unreasonably small or large.
What two ways can we test heteroskedasticity? Describe how we can test it using the first way, what pattern would make it violate homoskedasticity?
We can graph.
plot ei or ei^2 against some variable which is suspected to cause the conditional variance to vary or against Y-hat if we do not have such candidate variable.
Discernible pattern would indicate violation of homoskedasticity assumption.
E.g. if squared residual ei^2 is increasing in a linear or nonlinear manner as X is increasing. Then variance of the error variable, depends o X, i.e. there is heteroskedasticity (X on x-axis, ei^2 on y-axis)
Why do we use White’s test and what are the 3 steps for the test?
We use white’s test to back up formal hypothesis tests. It is a test for heteroskedasticity
i. Estimate the model with OLS and obtain the residuals ei
ii. Regress the squared OLS residuals, ei^2 on the original independent variables, X’s the squared X’s and the cross-products of X’s and obtain the unadjusted coefficient of determination, Raux^2 from this auxiliary regression
iii. Under H_0: homoskedasticity for large n, the test static defined as n R^2aux has a chi sqaure distribution with df equal to the number of regressors in the auxiliary regression
What is the second way to test heteroskedasticity?
Either way, since the random error, can never be observed in practice, first we have to
estimate the regression model with OLS assuming that it satisfies all assumptions, including
homoskedasticity, and then study the residuals, e, to see whether the regression model
indeed conforms to the assumptions.
What happens when hetero is detected by OLS coefficients have reasonable signs and magnitudes and OLS t-scores are not used for hypothesis testing and model specification?
No need to do anything bc OLS estimators are unbiased.
Problem can be mitigated through:
1. using transformed variables (logarithms ratios) instead of original one, or
2. by estimating model with the weighted least squares (WLS) instead of OLS,
3. taking hetero into consideration in estimation of standard errors
We rely on whites heteroskedasticity consistent standard errors- obtained with R bc WLS isnt used
What is a condition for the WC test to be used?
The sample size must be reasonably large (n>30) e.g
If the multiple R squared variable was 0.2665, the dependent and independent variable are Son and Father respectively and the relationship of their heights are measured. What does the R squared variable imply?
How is a significant regressor determined?
i.e., less than 27% of the total variation in the sons’ height can be explained
by the variation in their fathers’ height.
It is determined by signs of the estimate (positive means significant)
If the independent is a significant regressor, then the relationship is linear
If there is no relationship on the scatterplot, does that support homo or hetero?
IT MEANSN HETERO IS UNLIKELY AN ISSUE IN THIS REGRESSION
SUPPORTS HOMO
What does coefficient of determination tell you and what does the adjusted R^2 value tell you?
coefficient of determination:
It suggests that only about x% of the total sample variations in dependent variable can be explained by this regression model, i.e. by the sample variations in the independent variable/s
Adjusted R^2
after having taken the sample size and the number of independent variables into consideration, the model explains only about 20.2% of the total sample variations in dependent variable
