Question

With regard to regression models, which of the following statements is correct?

i) Linear restrictions on regression parameters cannot be tested using an F-test.

ii) The general-to-specific approach (also called “top-down”) starts with a model containing all explanatory variables. Subsequently, the least significant variables are dropped one by one until all of the variables remaining in the model are statistically significant.

iii) Multicollinearity in a regression results in high t-statistics for individualexplanatory variables and a failure of the F-test to reject the null hypothesis that the explanatory variables are jointly insignificant.

Answer 1

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The correct answer is (ii)

ii) The general-to-specific approach (also called “top-down”) starts with a model containing all explanatory variables. Subsequently, the least significant variables are dropped one by one until all of the variables remaining in the model are statistically significant.

With k regressors, in linear regression, we have a large number of possible model. In general to specific modelling
we start with including all the regressors, making it a general model, and using out stated Hypothesis, eliminate the insignificant regressors, one by one. i.e. we remove one significant regressor, fit again and repeat the process until we have all of the variables remaining in the model statistically significant.

(i) is wrong
F-test is used for testing linear restrictions. An example is given below

Suppose again we have a regression model with two explanatory variables, but we are interested in whether one coefficient is twice the magnitude of another coefficient. The null and alternative hypotheses are:

Then the statistic is F-distributed

URSS - Unrestricted Residual Sum of Squares

RRSS - Unrestricted Residual Sum of Squares

k - Number of parameters in the Unrestricted model

- Number of restrictions in the restricted model as compared to the unrestricted model

(iii) is wrong

MultiCollinearity results in high standard errors NOT t-statistics of the affected coefficients. This results in failure to reject the null hypothesis on coefficient significance.