Question

Which of the following can cause the usual OLS t statistics to be invalid (that is, not to have t distributions under the null hypothesis H0)? Explain.(i) Heteroskedasticity.(ii) A sample correlation coefficient of 0.95 between two independent variables that are in the model.(iii) Omitting an important explanatory variable

Answer 1

The usual OLS t statistics can be invalid due to heteroskedasticity, high correlation between independent variables, and omitting important explanatory variables.

Step-by-step explanation:

The usual OLS (Ordinary Least Squares) t statistics can be invalid under certain conditions:

1. Heteroskedasticity: Heteroskedasticity occurs when the variance of the error terms in a regression model is not constant. When heteroskedasticity is present, the usual OLS t statistics can be biased and inconsistent, leading to incorrect inference.
2. High correlation between two independent variables: When two independent variables in a regression model have a high correlation (such as a correlation coefficient of 0.95), it can lead to multicollinearity. In this case, the usual OLS t statistics may become sensitive to small changes in the data and may not have the t-distribution under the null hypothesis.
3. Omitting an important explanatory variable: When an important explanatory variable is omitted from the regression model, it can result in omitted variable bias. Omitted variable bias can affect the validity of the OLS t statistics and lead to incorrect inference.

Answer 2

All three scenarios can cause the usual OLS t-statistics to be invalid.

Heteroskedasticity affects the efficiency of the standard errors, high correlation between independent variables can lead to multicollinearity issues, and omitting important variables can introduce bias and affect the validity of hypothesis tests.

When heteroskedasticity is present, the OLS estimator remains unbiased, but the estimated standard errors are incorrect. As a result, the calculated t-statistics may not follow the standard t-distribution under the null hypothesis, leading to unreliable hypothesis tests.

Perfect multicollinearity makes it impossible to obtain unique coefficient estimates because the variance of the estimates becomes infinite. In such cases, OLS cannot be applied, and the t-statistics for the affected coefficients are undefined.