Question

Suppose you want to test whether you can solely rely on assessment to predict house price, that is, knowing housing characteristics will not help you predict housing price, once assessment is included in the model. Using a sample of 125 houses, you have estimated

Price= α+ β1 assess+ β2 lotsize+ β3 sqrft+ β4 bdrms+ u

and you decide to do a test at the 5% significance level. Then your best approach to answering the question is to

a) check each of the p-values for β2 ,β3 , and β4 . If none of them is < 0.05, you decide you don’t need the characteristics.

b) check the overall F statistic for the model. If the corresponding p-value is < 0.05, you decide you need the characteristics.

c) Test the joint significance of β2 ,β3 , and β4 .That is, estimate Price= α+ βassess+ u , and calculate an F statistic that is equal to (( Rur− Rr)/3)/((1− Rur)/120) , where Rur is the R2 from the first regression and Rr is the R2 from this second regression. If this F statistic is > 2.68 you decide that you need the characteristics (The 5% critical value in an F distribution with df (3, 120) is 2.68).

d) do the same as in d, but the decision rule is that if the F statistic is > 2.68, don’t need the characteristics.

Answer 1

(a) The p-value for each term tests the null hypothesis that the coefficient is equal to zero (no effect). A low p-value (< 0.05) indicates that you can reject the null hypothesis. In other words, a predictor that has a low p-value is likely to be a meaningful addition to your model because changes in the predictor's value are related to changes in the response variable. Since it gives value for only 1 variable, and we have more than one, we do not use this approach.

(b) In general, an F-test in regression compares the fits of different linear models. The F-test can assess multiple coefficients simultaneously.

The F-test of the overall significance is a specific form of the F-test. It compares a model with no predictors to the model that you specify. A regression model that contains no predictors is also known as an intercept-only model.

The hypotheses for the F-test of the overall significance are as follows:

• Null hypothesis: The fit of the intercept-only model and your model are equal.
• Alternative hypothesis: The fit of the intercept-only model is significantly reduced compared to your model.

Since we don't have have values for no variable model, we dont use this option

(c) To perform the hypothesis test, compare the computed value of F to the critical value Fq,N-k for a particular significance level (5% for example), where q is the number of restrictions imposed on the regression equation (here, q = 3) and N-k is the number of observations minus the number of variables in the regression (including the constant – here, k = 5). If the computed value exceeds the critical value, we can reject the null hypothesis, i.e. we need the characteristics. Thus (c) is correct.