Simple Linear Regression Notes


Introduction to Regression Analysis

  • Definition: Regression analysis predicts a dependent variable (response/outcome) using one or more independent variables (predictor/explanatory).

  • Types based on:

    1. Number of independent variables (simple vs. multiple)

    2. Relationship (linear vs. nonlinear)

  • Focus: Simple linear regression - analysis with one independent variable.

Simple Linear Regression Model

  • Equation: Y = α + βX + ϵ

  • Y: Dependent variable (response)

  • X: Independent variable (predictor)

  • α: Intercept

  • β: Coefficient (slope)

  • ϵ: Error term

  • Goal: Estimate α and β that best describe the relationship between X and Y.

Visualizing Simple Linear Regression

  • Data Plotting: Start by graphing the data points (X, Y) to spot any linear trend.

  • Fitting a Line: Multiple methods exist for fitting lines; the objective is to determine the line that best captures this trend using regression analysis.

The Mechanism of Least Squares

  • Distance Measurement: Calculate vertical distances from the fitted line to each data point:

  • For data point (x1, y1): distance = a - y1

  • Sum of Squared Residuals (SSR):

  • SSR = Σ(a - yi)² (negative values complicate analysis)

  • Redefine to minimize SSR, leading to the least squares approach.

  • Goal: Find (α, β) to minimize SSR:

    • SSR = Σ[(α + βxi) - yi]²

Application in Business Context

  • Risk Analysis Case Study: Analyze investment risk for Tesla, Inc. through its financial beta calculation.

  • Beta Definition: Measure of stock volatility relative to the market.

    • Beta > 1: Higher risk/volatility

    • Beta < 1: Lower risk/volatility

  • Practical Task: Calculate Tesla's beta to inform investment strategies based on portfolio risk tolerance.

Data Preparation for Beta Calculation

  • Regression Model to estimate Tesla’s beta:

  • RetTSLA,t = α + β Retmkt,t + ϵ

  • Variables:

    • RetTSLA,t: Tesla's monthly returns

    • Retmkt,t: Market returns (S&P 500 index)

  • Data Source: Extract from Yahoo Finance for selected months.

R Coding for Analysis

  • Loading Data:

  library(tidyverse)
  df1 <- read.csv("TSLA_mly_2010_2013.csv")
  df1 %>% glimpse()

  • Variables in Dataset:

  • date, volTSLA, adjCloseTSLA, volSP500, adjCloseSP500

  • Focus on Stock Prices:

  df1 <- df1 %>% select(date, adjCloseTSLA, adjCloseSP500)

  • Calculating Monthly Returns:

  df1 <- df1 %>% na.omit() %>% mutate(
    lagTSLA = lag(adjCloseTSLA),
    lagSP = lag(adjCloseSP500),
    rtrTSLA = (adjCloseTSLA - lagTSLA) / lagTSLA,
    rtrSP500 = (adjCloseSP500 - lagSP) / lagSP
  )

Visualization of Returns

  • Scatter Plot: Visualize relationship between Tesla’s and S&P 500 monthly returns.

  • Could be Created with:

  ggplot(df1, aes(x = rtrSP500, y = rtrTSLA)) + geom_point()

Performing the Regression in R

  • Using lm() Function:

  betaTSLA <- lm(rtrTSLA ~ rtrSP500, df1)
  summary(betaTSLA)

  • Interpreting Output: The summary gives information on intercept (α) and slope (β):

  • Example Equation: rtrTSLA = 0.06931 + 0.69077 * rtrSP500

Tutorial Seminar Objectives

  1. Replicate the regression analysis learned to deepen understanding.

  2. Create scatter plots using the ggplot2 package in R.

  3. Estimate betas for other stocks (Coca-Cola, Johnson & Johnson).

ggplot2 for Visualization

  • Scatter Plot Example:

  ggplot(df1, aes(x = rtrSP500, y = rtrTSLA)) + 
    geom_point(color = "steelblue", alpha = 0.7, size = 1.5) +
    geom_smooth(method = "lm", se = FALSE, color = "orangered") + 
    labs(title = "Tesla and S&P 500 Monthly Returns", x = "S&P 500 Monthly Returns", y = "Tesla Monthly Returns") + 
    theme_bw()

  • Options for Customization: Color, size, and transparency can be adjusted to improve interpretability of the plot.

Conclusion

  • Key Takeaway: Encouragement to practically apply regression analysis using R to inform business investment decisions based on risk assessment and visualization techniques.