CFA charterholders understand the importance of satisfying the assumptions underlying simple or multiple linear regressions to make reliable estimates and forecasts in their investment recommendations, decisions and actions.

This RapidDigest only includes what is covered in the 2021 CFA® Curriculum Readings (Readings 4 and 5).

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Contents

#### Formulation

 single: multiple: #### Assumptions, Breach Detection and Correction

 Assumptions If Breached Tested and Corrected 1. linear relation of X and Y transform X and/or Y or use another method 2. non-random X can still use the method if X not correlated with ε (unconditional heteroskedasticity) or X's (no perfect collinearity) multicollinearity degree assessed exclude one or more X variables 3. error term a. normal distribution b. mean = 0 lagged Y as an X differencing lagged Y c. variance = constant ε correlated with X: conditional heteroskedasticity     omitted variable bias Breusch-Pagan: χ2 test of nR2  robust standard errors generalized least squares Hansen method Newey and West method d. correlation = 0 serial correlation Durbin-Watson test Hansen method Newey and West method #### Application to Real-World Data

Data Source: OECD #### Regression Assumptions In Action Can you see where the regression assumptions are in action?

 Assumptions In Action 1. linear relation of X and Y upward sloping line 2. non-random X Vertical base of the bell curve corresponds to a particular X value. 3. error term Each observation has a bell curve. a. normal distribution Distance of actual value from predicted value of each observation represents the distribution of errors, which is normal; hence, the bell curve. b. mean = 0 The center of each bell curve is the mean of prediction error so that each observation is predicted to fall on the regression line. c. variance = constant Each bell curve, being normal distribution shape, has variability defined by standard deviation, which if squared is the variance. Bell curves, being same size and shape, are identical across actual observations. d. correlation = 0 not shown
Key to Learning

#### Understand the Why

For valid estimation and forecast, the regression assumptions must be held and observed; else the regression model is unreliable because statistical tests are invalid.

#### Real-World Practical Application

https://www.fool.com/investing/2019/05/22/does-esg-investing-produce-better-stock-returns.aspx #### RapidInsight: Scatter Plot Before Regression

• Residuals appear to grow larger as ROIC increases (i.e. conditional heteroskedastic?)
• Understated regression standard errors inflate the t-statistic that increases Type I error (i.e. ROIC is significant predictor of EV/IC when it is actually not).
• Further Reading Digest: Linear Regression (Analysis)

#### RapidInsight: Established Equations Have High Linear Relationship ROIC reflects Earnings Per Share (EPS); whereas EV/IC, as described, is Price-to-Book ratio.  The denominator of both ratios pertains to equity so that they cancel out leaving the relationship of Earnings and Price: either E/P as earnings yield or P/E ratio. Since earnings drive stock prices and intrinsic values then their R2 is necessarily high.