Introduction

In the realm of linear regression, hypothesis tests play a vital role in assessing the significance of the relationships between variables. These tests allow us to make informed decisions about the presence and strength of associations. In this chapter, we will delve into the systematic process of conducting a hypothesis test within the context of a linear regression model. We will cover the essential steps and provide clear explanations along with illustrative examples to enhance your understanding.

Hypothesis testing involves making statements about population parameters based on sample data. In linear regression, hypothesis tests are commonly used to assess whether the coefficients of explanatory variables are significantly different from zero. This helps us determine if the variables have a statistically significant impact on the dependent variable.

Hypothesis Testing in a Linear Regression

The steps to perform a hypothesis test in a linear regerssion aregiven below.

1) Formulate the Null and Alternative Hypotheses

• The null hypothesis ($H_0$) states that the coefficient of the explanatory variable is equal to zero (no effect).
• The alternative hypothesis ($H_a$) states that the coefficient is not equal to zero (significant effect).

2) Choose the Significance Level ($\alpha$)

• The significance level ($\alpha$) determines the threshold for considering a result statistically significant. Common values are 0.05 and 0.01.

3) Calculate the Test Statistic

• The test statistic is calculated based on the sample data and follows a t-distribution.
• $$\text{Test Statistic (t)}=\frac{\text{(Coefficient Estimate - Null Hypothesized Value)}}{\text{Standard Error of Coefficient Estimate}}$$

4) Determine the Critical Value or p-Value

• The critical value is determined based on the chosen significance level and degrees of freedom.
• The $p$-value is the probability of observing a test statistic as extreme as the calculated one, assuming the null hypothesis is true.

5) Make a Decision

• If the $p$-value is less than $\alpha$, reject the null hypothesis in favor of the alternative hypothesis.
• If the $p$-value is greater than or equal to $\alpha$, fail to reject the null hypothesis.

6) Interpret the Result

• If the null hypothesis is rejected, it implies that the variable has a statistically significant effect on the dependent variable.

Example: Suppose we are analyzing a linear regression model to predict stock prices based on trading volume. Our null hypothesis $H_0$ is that trading volume has no significant effect on stock prices. The alternative hypothesis $H_a$ is that trading volume does have a significant effect.

Given a significance level ($\alpha$) of 0.05, we calculate a test statistic ($t$) of 2.42. With 50 degrees of freedom, the critical t-value is 1.676. Since the calculated t-value (2.42) is greater than the critical t-value (1.676) and the corresponding $p$-value is below 0.05, we reject the null hypothesis. This suggests that trading volume does indeed have a significant effect on stock prices.