Data Analytics for Management (Subject) / Linear Regression (Lesson)
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- Splitting Variability into Model and Residual SST : Total variability between Y variable values and the mean value of Y.SSR : Residual/Error variabilitySSM : Model variability; "What the model Explains" SST = SSM + SSR R Square = SSM / SST "Proportion of total variability of Y which is explained by the model"
- Regression & CAPM Fundamental idea in finance is that investors need financials incentives to take on risk. Thus, the expected return R on a risky investment, e.g., a stock, should exceed the riskfree return Rf. The excess return (R – Rf) should be positive The Capital Asset Market Pricing Model (CAPM) says that the expected excess return ona stock is proportional to the excess return on a portfolio of all available assets (the‘market’ portfolio’), namely and R – Rf = b * (Rm – Rf)Return Stock = a + β * Return Market + error• ‘a’ = “excess return” of a stockaccording to the CAPM, the “excess return” is the reward for taking on the “specificrisk” of the stock. As this risk can be eliminated through holding a diversifiedportfolio, CAPM says the excess return should be close to zero• ‘β’ = “market risk” of a stockthis is an indication of how sensitive the stock is to movements in the market as awhole; for a 1% market movement, b = 1 implies the stock tends to move 1%, b < 1means the stock tends to move by less than 1%, b > 1 means the stock tends tomove more than 1%Look at the relationship between returns on a stock against returns on the marketindex, e.g., the S&P500– use regression to estimate the a and b parameters10
- Splitting volatility in MARKET (systematic) and SPECIFIC (unsystematic) risk Systematic Risk = Undiversifiable Market Risk Explained by SP500 Unsystematic Risk = Diversifiable Risk Specific to AAPL (SD of residuals) Total Risk = Systematic Risk + Unsystematic Risk Variance (AAPL) = b2* VAR(SP500) + SDResiduals2
- Checking the residuals Residuals are randomly scattered with no clear pattern --> good!Curved pattern; the relationship is not linear Change in variability (size of residuals) across plot --> σ not equal for all values of x.(HETEROscedasticity)
- Estimating the Error Variance The larger the error variance, s2, the larger the variance of the slope estimate The larger the variability in the xi, the smaller the variance of the slope estimate As a result, a larger sample size should decrease the variance of the slope estimate Problem that the error variance is unknown- We don’t know what the error variance, s2, is, because we don’t observe the errors, ui What we observe are the residuals, ûi We can use the residuals to form an estimate of the error variance