S/F Ekonometri Ch 11.1

If heteroscedasticity is present, it means that the variability of the errors or residuals in a regression model is not constant across all levels of the independent variables. In such cases, the assumptions underlying the conventional t and F tests are violated, leading to invalid results. The conventional tests assume homoscedasticity, where the variability of the errors is constant. Therefore, if heteroscedasticity is present, the conventional t and F tests cannot be relied upon for accurate inference and decision-making.

If a regression model is mis-specified and an important variable is omitted, the OLS residuals will show a distinct pattern. This is because the omitted variable may have a significant impact on the dependent variable, causing the model to incorrectly estimate the relationship between the included variables and the dependent variable. As a result, the residuals will exhibit a systematic pattern, indicating that the model is not capturing all the relevant factors affecting the dependent variable. Therefore, the statement "True" is a correct explanation.

In the presence of heteroscedasticity, OLS estimators are biased but not necessarily inefficient. Heteroscedasticity refers to the situation where the variability of the error term in a regression model is not constant across all levels of the independent variables. While heteroscedasticity can lead to biased OLS estimators, it does not always result in inefficiency. In some cases, heteroscedasticity can actually improve the efficiency of the OLS estimators. Therefore, the statement that OLS estimators are both biased and inefficient in the presence of heteroscedasticity is false.

The statement is true because heteroscedasticity refers to the unequal variance of the error terms in a regression model. In order to test for heteroscedasticity, assumptions need to be made about the correlation between the error term and other variables. Without these assumptions, it is not possible to have a general test of heteroscedasticity that is free of any assumptions. Therefore, the correct answer is true.

In the presence of heteroscedasticity, the usual OLS method does not always overestimate the standard errors of estimators. Heteroscedasticity refers to a situation where the variability of the error term is not constant across different levels of the independent variable. In such cases, the OLS method may either overestimate or underestimate the standard errors of estimators, depending on the nature and extent of heteroscedasticity. Therefore, the statement that the usual OLS method always overestimates the standard errors of estimators in the presence of heteroscedasticity is false.

If residuals estimated from an OLS regression exhibit a systematic pattern, it does not necessarily mean that heteroscedasticity is present in the data. Heteroscedasticity refers to the unequal variance of the residuals across different levels of the independent variable(s). A systematic pattern in the residuals could indicate other issues such as autocorrelation or misspecification of the model, but it does not directly imply heteroscedasticity. Therefore, the given statement is false.

If a regressor with nonconstant variance is omitted from a model, the OLS residuals will not necessarily be heteroscedastic. The omission of a regressor with nonconstant variance may or may not affect the heteroscedasticity of the residuals. It depends on the specific relationship between the omitted regressor and the other variables in the model. Therefore, the statement is false.