[DATA] Bear Markets (Refer to Problem 34, Section 4.1) A bear market is a market condition in which the price of the security falls. A bear market in the stock market is defined as a condition in which the market declines by 20% or more over the course of at least two months. The following data represent the number of months and percentage change in the S&P500 (a group of 500 stocks).

b. Assuming the residuals are normally distributed, test whether a linear relation exists between the number of months of a bear market and percent change at the alpha = 0.05 level of significance.

Verified step by step guidance

1. Step 1: Define the hypotheses for the linear regression test. The null hypothesis \(H_0\) states that there is no linear relationship between the number of months of a bear market (independent variable \(X\)) and the percent change in the S\&P 500 (dependent variable \(Y\)), i.e., the slope \(\beta_1 = 0\). The alternative hypothesis \(H_a\) states that there is a linear relationship, i.e., \(\beta_1 \neq 0\).
2. Step 2: Fit a simple linear regression model to the data, where the model is \(Y = \beta_0 + \beta_1 X + \epsilon\), with \(\epsilon\) representing the residuals. Calculate the estimates of the intercept \(\hat{\beta}_0\) and slope \(\hat{\beta}_1\) using the least squares method.
3. Step 3: Calculate the test statistic for the slope \(\hat{\beta}_1\). This involves computing the standard error of the slope \(SE_{\hat{\beta}_1}\) and then the t-statistic as \(t = \frac{\hat{\beta}_1}{SE_{\hat{\beta}_1}}\). The degrees of freedom for this test is \(n - 2\), where \(n\) is the number of data points.
4. Step 4: Determine the critical t-value from the t-distribution table at the significance level \(\alpha = 0.05\) for a two-tailed test with \(n - 2\) degrees of freedom. Compare the absolute value of the calculated t-statistic to this critical value.
5. Step 5: Make a decision based on the comparison: if \(|t|\) is greater than the critical value, reject the null hypothesis \(H_0\) and conclude that there is evidence of a linear relationship between the number of months and percent change. Otherwise, fail to reject \(H_0\) and conclude that there is insufficient evidence to support a linear relationship.

Key Concepts

Linear Regression

Linear regression models the relationship between a dependent variable and one or more independent variables by fitting a linear equation. In this context, it helps determine if the number of months in a bear market predicts the percent change in the S&P 500. The slope indicates the direction and strength of this relationship.

Hypothesis Testing for Regression Slope

This involves testing whether the slope of the regression line is significantly different from zero, indicating a linear relationship. The null hypothesis states no linear relationship (slope = 0), while the alternative suggests a relationship exists (slope ≠ 0). The test uses a t-statistic and compares it to a critical value at a chosen significance level (alpha = 0.05).

Normality of Residuals

Residuals are the differences between observed and predicted values in regression. Assuming residuals are normally distributed is crucial for valid hypothesis testing and confidence intervals. This assumption ensures the reliability of p-values and test statistics used to assess the linear relationship.