Linear Regression for Stock Returns

Easy

Statistics & Probability

CorrelationRegressionCausal Inference

Problem

Business Context

At NorthBridge Capital, an analyst wants to explain how regression analysis is used in financial data by modeling a stock's monthly return as a function of the market's monthly return. This is a standard way to estimate the stock's market sensitivity and test whether it tends to outperform after adjusting for market movement.

Problem Statement

Use simple linear regression to model the relationship between AlphaTech's monthly return and the market index return:

$R_{stock} = \alpha + \beta R_{market} + \varepsilon$

Estimate the regression coefficients, test whether the slope is statistically different from 0, and interpret the results in financial terms.

Given Data

You are given summary statistics from 24 monthly observations.

Metric	Value
Number of months $n$	24
Mean market return $\bar{x}$	0.008
Mean stock return $\bar{y}$	0.011
Sum of squares of market returns $S_{xx} = \sum (x_i-\bar{x})^2$	0.0184
Sum of cross products $S_{xy} = \sum (x_i-\bar{x})(y_i-\bar{y})$	0.0258
Sum of squares of stock returns $S_{yy} = \sum (y_i-\bar{y})^2$	0.0412
Significance level $\alpha$	0.05

Requirements

Estimate the slope $\hat{\beta}$ and intercept $\hat{\alpha}$
Compute $R^2$ and explain what it means
Test $H_0: \beta = 0$ versus $H_1: \beta eq 0$
Construct a 95% confidence interval for $\beta$
Interpret the regression in plain financial language

Assumptions

The linear relationship between market and stock returns is appropriate over this period
Monthly observations are treated as independent
Regression residuals have constant variance and are approximately normal for inference

Problem

Business Context

Problem Statement

Use simple linear regression to model the relationship between AlphaTech's monthly return and the market index return:

$R_{stock} = \alpha + \beta R_{market} + \varepsilon$

Estimate the regression coefficients, test whether the slope is statistically different from 0, and interpret the results in financial terms.

Given Data

You are given summary statistics from 24 monthly observations.

Metric	Value
Number of months $n$	24
Mean market return $\bar{x}$	0.008
Mean stock return $\bar{y}$	0.011
Sum of squares of market returns $S_{xx} = \sum (x_i-\bar{x})^2$	0.0184
Sum of cross products $S_{xy} = \sum (x_i-\bar{x})(y_i-\bar{y})$	0.0258
Sum of squares of stock returns $S_{yy} = \sum (y_i-\bar{y})^2$	0.0412
Significance level $\alpha$	0.05

Requirements

Estimate the slope $\hat{\beta}$ and intercept $\hat{\alpha}$
Compute $R^2$ and explain what it means
Test $H_0: \beta = 0$ versus $H_1: \beta eq 0$
Construct a 95% confidence interval for $\beta$
Interpret the regression in plain financial language

Assumptions

The linear relationship between market and stock returns is appropriate over this period
Monthly observations are treated as independent
Regression residuals have constant variance and are approximately normal for inference

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Business Context

Problem Statement

Use simple linear regression to model the relationship between AlphaTech's monthly return and the market index return:

$R_{stock} = \alpha + \beta R_{market} + \varepsilon$

Estimate the regression coefficients, test whether the slope is statistically different from 0, and interpret the results in financial terms.

Given Data

You are given summary statistics from 24 monthly observations.

Metric	Value
Number of months $n$	24
Mean market return $\bar{x}$	0.008
Mean stock return $\bar{y}$	0.011
Sum of squares of market returns $S_{xx} = \sum (x_i-\bar{x})^2$	0.0184
Sum of cross products $S_{xy} = \sum (x_i-\bar{x})(y_i-\bar{y})$	0.0258
Sum of squares of stock returns $S_{yy} = \sum (y_i-\bar{y})^2$	0.0412
Significance level $\alpha$	0.05

Requirements

Estimate the slope $\hat{\beta}$ and intercept $\hat{\alpha}$
Compute $R^2$ and explain what it means
Test $H_0: \beta = 0$ versus $H_1: \beta eq 0$
Construct a 95% confidence interval for $\beta$
Interpret the regression in plain financial language

Assumptions

The linear relationship between market and stock returns is appropriate over this period
Monthly observations are treated as independent
Regression residuals have constant variance and are approximately normal for inference