Introduction to Hypothesis Testing in Multiple Regression

Study Notes

The session focuses on hypothesis testing in multiple regression models.
The text builds upon previous knowledge of OLS (Ordinary Least Squares) regression.

Assumptions are made about the distribution of the error term "u" to understand the distribution of the coefficients.
The key assumption is that the population error term "u," is normally distributed with mean 0 and variance σ2.
Since the coefficients b1, b2,...bj are linear functions of "u," they are also normally distributed.

The given diagram shows the assumed distribution of the error term, assuming a normal distribution.

The text explains that due to the unknown population variance σ2, we use its estimator, leading to the distribution of the coefficients following the t-distribution instead of the standard normal distribution.
The t-distribution is similar to the normal distribution and is used for hypothesis testing when the population variance is unknown.

The text introduces three methods for testing hypotheses in multiple regression using STATA output
- Confidence Intervals: This method involves calculating confidence intervals for the coefficients and determining if the hypothesized value falls within the interval.
- Comparing t-values: This method compares the calculated t-value for the coefficient with the critical t-value based on the degrees of freedom and the chosen significance level. If the calculated t-value exceeds the critical t-value, the null hypothesis is rejected.
- p-values: The p-value represents the probability of obtaining a sample statistic as extreme as the observed value, assuming the null hypothesis is true. A small p-value (typically less than the chosen significance level) suggests evidence against the null hypothesis and leads to its rejection.

The session focuses on hypothesis testing in multiple regression models.
Hypothesis tests can be performed on individual coefficients, the overall fit of the model, and the joint significance of multiple variables.
Methods used to test the hypotheses are similar to those used in two-variable models.

The error term (u) is assumed to be normally distributed with a mean of 0 and a variance of σ2.
Since the error term is unobservable, the assumption of normality is crucial for hypothesis testing.
This assumption allows us to derive the distribution of the sample estimators (b1, b2, ... bj).
Sample estimators are linear functions of the error term and are also normally distributed.

Confidence intervals provide a range of values within which the true population parameter is likely to lie.
If the hypothesized value falls outside the confidence interval, the null hypothesis is rejected.

The t-statistic is calculated by dividing the estimated coefficient (b) by its standard error.
The t-statistic is compared to the critical value from the t-distribution with appropriate degrees of freedom.
If the absolute value of the t-statistic exceeds the critical value, the null hypothesis is rejected.

The p-value represents the probability of observing the estimated coefficient if the null hypothesis is true.
If the p-value is less than the significance level (typically 0.05), the null hypothesis is rejected.