14.2 Linear Regression Analysis

Linear regression is a powerful tool for analyzing relationships between variables in finance. It helps predict stock prices based on factors like interest rates, using a simple equation with a slope and y-intercept. Understanding these components is crucial for interpreting financial data.

The method of least squares finds the best-fitting line through data points, minimizing errors. This technique, along with key assumptions and evaluation methods, allows financial analysts to make informed predictions and assess the reliability of their models.

Linear Regression Analysis

Slope and y-intercept interpretation

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• Linear regression model represented by equation $y = \beta_0 + \beta_1x + \epsilon$
  • $y$ dependent variable (stock price)
  • $x$ independent variable (interest rate)
  • $\beta_0$ y-intercept (stock price when interest rate is zero)
  • $\beta_1$ slope (change in stock price for one-unit change in interest rate)
  • $\epsilon$ error term accounts for unexplained variation
• Slope $\beta_1$ represents change in $y$ for one-unit change in $x$
  • Positive slope indicates positive relationship between $x$ and $y$ (higher interest rates associated with higher stock prices)
  • Negative slope indicates negative relationship between $x$ and $y$ (higher interest rates associated with lower stock prices)
• Y-intercept $\beta_0$ represents value of $y$ when $x$ is zero
  • Point where regression line crosses y-axis (stock price when interest rate is zero)
• Slope and y-intercept calculated using formulas:
  • $\beta_1 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i - \bar{x})^2}$
  • $\beta_0 = \bar{y} - \beta_1\bar{x}$
    • $\bar{x}$ and $\bar{y}$ means of $x$ and $y$ (average interest rate and stock price)
    • $n$ number of observations (data points)
• The strength of the linear relationship is measured by the correlation coefficient

Method of least squares

• Statistical approach to find line of best fit for set of data points
• Minimizes sum of squared differences between observed and predicted values from regression line
  • Differences between observed and predicted values called residuals
• Line of best fit minimizes sum of squared residuals (SSR):
  • $SSR = \sum_{i=1}^{n}(y_i - \hat{y}_i)^2$
    • $y_i$ observed value of $y$ for $i$-th observation (actual stock price)
    • $\hat{y}_i$ predicted value of $y$ for $i$-th observation based on regression line (estimated stock price)
• Slope and y-intercept of line of best fit calculated using formulas mentioned in previous objective
• Line of best fit provides best linear approximation of relationship between independent and dependent variables (interest rates and stock prices)
• The standard error of the regression measures the average distance between the observed values and the regression line

Linear regression assumptions

• Linear regression relies on several assumptions that must be met for model to be valid and reliable:
  1. Linearity: Relationship between independent and dependent variables should be linear
     • Assessed visually using scatterplot of data points (stock prices vs interest rates)
  2. Independence: Observations should be independent of each other
     • Violations can occur with time series data (stock prices over time) or clustered data (stock prices within industries)
  3. Homoscedasticity: Variance of residuals should be constant across all levels of independent variable
     • Assessed visually using residual plot (residuals vs predicted values)
  4. Normality: Residuals should be normally distributed
     • Assessed using histogram or normal probability plot of residuals
  5. No multicollinearity: If multiple independent variables, they should not be highly correlated with each other
     • Assessed using correlation matrices or variance inflation factors (VIF)
• If assumptions violated, results of linear regression may be unreliable or misleading
  • Alternative models or transformations of variables may be necessary (logarithmic transformation for non-linear relationships)

Model evaluation and inference

• R-squared measures the proportion of variance in the dependent variable explained by the independent variable(s)
• The F-statistic tests the overall significance of the regression model
• P-values indicate the statistical significance of individual coefficients
• Confidence intervals provide a range of plausible values for the population parameters