Use the properties of the least-squares estimators given in...

Use the properties of the least-squares estimators given in Section 11.4 to complete the following. a. Show that under the null hypothesis $H_{0}: \beta_{i}=\beta_{i 0}$ $$T=\frac{\widehat{\beta}_{i}-\beta_{i 0}}{S \sqrt{c_{i i}}}$$ possesses a $t$ distribution with $n-2$ df, where $i=1,2$. b. Derive the confidence intervals for $\beta_{i}$ given in this section.

Use the properties of the least-squares estimators given in Section 11.4 to complete the following.
a. Show that under the null hypothesis $H_{0}: \beta_{i}=\beta_{i 0}$
$$T=\frac{\widehat{\beta}_{i}-\beta_{i 0}}{S \sqrt{c_{i i}}}$$
possesses a $t$ distribution with $n-2$ df, where $i=1,2$.
b. Derive the confidence intervals for $\beta_{i}$ given in this section.

Key Concepts

Degrees of Freedom

The degrees of freedom in regression analysis, particularly in the context of the t distribution for testing regression coefficients and constructing confidence intervals, are typically given by the number of observations minus the number of estimated parameters. This adjustment accounts for the estimation of parameters and is crucial for accurate inference.

Confidence Intervals

Confidence intervals for regression coefficients are derived by taking the point estimate and adding and subtracting a margin based on the critical value from the t distribution times the standard error of the estimate. This interval provides a range of plausible values for the true coefficient, reflecting the uncertainty inherent in the estimation process.

t Distribution

In the context of regression analysis, once the estimator variances are estimated by replacing the error variance with its unbiased estimator, the standardized estimator follows a t distribution with appropriate degrees of freedom. This follows from the fact that the numerator is normally distributed and the denominator is an independent chi-squared variable scaled appropriately, which is central to hypothesis testing for regression coefficients.

Least Squares Estimators

These are the estimators obtained by minimizing the sum of squared residuals in a regression model. They possess properties such as being unbiased and having minimum variance among all linear unbiased estimators under the Gauss?Markov assumptions. Their variance is expressible in terms of the error variance and the design matrix, which is critical for constructing inferential statements about the regression coefficients.

Hypothesis Testing for Regression Coefficients

This involves testing hypotheses about the true values of the regression coefficients, such as whether a coefficient differs from a specified value. The t-statistic is computed by taking the difference between the estimated and hypothesized coefficient values, divided by the estimated standard error. Under the null hypothesis, this statistic follows a t distribution, which allows for significance testing.

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