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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Formulation

single:

    \[Y_i=b_0+b_1X_i\]

multiple:

    \[Y_i=b_0+b_1X_{1i}+b_2X_{2i}\]

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 nR
  • 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

RapidAid

Application to Real-World Data 

Data Source: OECD

OECD iinflation money supply GDP per hour worked

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

11030316_CFA-Exam-Level-2-Linear-Regression


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

RapidInsight
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.