READING 23: HYPOTHESIS TESTS AND CONFIDENCE INTERVALS IN MULTIPLE REGRESSION

Hypothesis Testing In Multiple Linear Regression

        A t-test is used for hypothesis testing of regression parameter estimates:

        Testing for statistical significance means testing H₀: B_j = 0 vs. H_A: B_j ≠ 0.

Interpreting p-Values

• If the p-value is less than significance level, the null hypothesis can be rejected.
  
• If the p-value is greater than the significance level, the null hypothesis cannot be rejected.

Confidence Intervals for a Regression Coefficient

The confidence interval for regression coefficient is:
         Estimated regression coefficient ± (critical t-value)(coefficient standard error)
The value of dependent variable Y is predicted as:

F-statistic

        The F-distributed test statistic can be used to test the significance of all (or any subset of) the independent variables (i.e., the overall fit of the model) using a one-tailed test:

Restriction

Hypothesis tests of single restrictions involving multiple coefficients requires the use of statistical software packages.

ANOVA

        The ANOVA table outputs the standard errors, t-statistics, probability values (p-values), and confidence intervals for the estimated coefficients.
         Upper and lower limits of the confidence interval can be found in the ANOVA results.

[b₂ − t_α/2× se(b₂)] < B₂ < [b₂ + t_α/2 × se(b₂)]

        The statistics in the ANOVA table also allow for the testing of the joint hypothesis that both slope coefficients equal zero.

H₀: B₁ = B₂ = 0
H_A: B₁ ≠ 0 or B₂ ≠ 0

        The test statistic in this case is the F-statistic.

Examples Of Omitted Variable Bias In Multiple Regressions

        Omitting a relevant independent variable in a multiple regression results in regression coefficients that are biased and inconsistent, which means we would not have any confidence in our hypothesis tests of the coefficients or in the predictions of the model.

The R2 and adjusted R2 in a Multiple Regression

        Restricted least squares models restrict one or more of the coefficients to equal a given value and compare the R2 of the restricted model to that of the unrestricted model where the coefficients are not restricted. An F-statistic can test if there is a significant difference between the restricted and unrestricted R2.