CFA® charterholders know how to interpret regression output to conclude whether their estimates and forecasts are reliable to support their investment recommendations, decisions or actions.
This RapidDigest only includes what is covered in the 2021 CFA® Curriculum Readings (Readings 4 and 5).
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Regression Analysis
Refer to Linear Regression (Assumptions) for the sample data underlying this regression analysis.
Microsoft Excelgenerated regression output of the Application to RealWorld Data
Top Panel: Simple Linear Regression  Bottom Panel: Multiple Linear Regression
Simple Linear Regression
Y = inflation  X = M3 money supply
Model Specification 

Slope Coefficient 

Predicted 2018 inflation 

Prediction Error 

Observed Values 

Model Hypothesis 

ttest statistics on regression coefficients (intercept and slope) 

ttest decision
at 5% significance level t_{α/2,n2} = t_{0.05/2,82}= t_{0.025,6} = 2.447 
reject b_{0} = 0:  3.67  > 2.447 reject b_{1} = 0:  4.21  > 2.447 
Confidence Interval at 5% significance level
critical t_{α/2,n2 }= t_{0.05/2,82}= t_{0.025,6 }= 2.447 
12.42268596 /+ (2.447)(3.382329853) = 20.69894895 to 4.146422975
0.121941844 /+ (2.447)(0.028943608) = 0.051119385 to 0.192764302

pvalue test 
reject b_{0} = 0: 0.010419245 reject b_{1} = 0.005603983 
Coefficient of Determination (R^{2}) 

Correlation Coefficient (R) 

ANOVA Ftest statistic (only has onetail) 
17.75007386 = (4.213083652)^{2}

Ftest decision at 5% significance level
F_{α,k,n2} = F_{0.05,1, 82}= F_{0.05,1,6} = 5.99 
reject b_{1} = 0:  17.75  > 5.99 
Significance F at 5% alpha 

Standard Error of Regression 

Model Conclusion 

Manual Regression 

Manual ANOVA  
Manual Standard Errors (optional) 

Multiple Linear Regression
Y = inflation  X_{1} = M3 money supply  X_{2} = GDP per hour worked
Model Specification 

Slope Coefficients 

Predicted 2018 inflation 

Prediction Error 

Observed Values 

Model Hypothesis (Ftest) 

Ftest statistic (only has onetail) 

Ftest decision
at 5% significance level F_{α,k,n(k+1)} = F_{0.05,2, 8(2+1)}= F_{0.05,2,5} = 5.79 
reject b_{1} = b_{2} = 0:  8.17  > 5.79 
Significance F at 5% alpha 
0.026545006 pvalue 
Model Hypotheses (ttest) 

ttest decision
at 5% significance level t_{α/2,n2} = t_{0.05/2,82}= t_{0.025,6} = 2.447 

Coefficient of Determination (R^{2}) 

Adjusted R^{2} 

Standard Error of Regression 

Model Conclusion 

Understand the Why
The regression model and its estimators must be statistically significant to rely on them for estimation and forecasting in business and investment.
RealWorld Practical Application
https://blog.thinknewfound.com/2016/07/alphasmeasurementproblem/
Legal Notice: no copyright (public domain) or copyright exception (free use) under fair dealing/fair use laws (i.e. educational use, critique, not substantial quote) and proper attribution with link to source
RapidInsight: First Thing First
 In multiple linear regression, the first thing people should look at should be the Significance F or pvalue without having to obtain the Fstatistic. The Ftest evaluates the overall reliability of the regression model.
 In simple linear regression, the Significance F is the square of the tstatistic of the slope coefficient so the Ftest is redundant to the ttest. Therefore, the first thing to look at is the pvalue of the slope coefficient without the need to calculate the tstatistic or specifying the significance level. The pvalue is 0 so the model is impossibly reliable (i.e. cannot reject the null hypothesis that there is zero alpha for value stocks at whatever level of confidence: 95%, 90% or 99%, for example). This conclusion terminates the regression analysis.
 Notwithstanding the regression model is not reliable, the intercept is only significant at below 90% confidence level (i.e. 1  0.1115836).
More from: https://blog.thinknewfound.com/2016/07/alphasmeasurementproblem/
Legal Notice: no copyright (public domain) or copyright exception (free use) under fair dealing/fair use laws (i.e. educational use, critique, not substantial quote) and proper attribution with link to source
RapidInsight: Know Thy Number
 The article mentions two regression assumptions: uncorrelated error term (independent from month to month) and zeromean errorterm. See Linear Regression (Assumptions).
 It further mentions alpha (intercept) is a constant that can be regarded as a random variable. While the intercept is a constant in the regression formula (Y_{i} = α + βX_{i}), it is still an estimate with its own standard error (i.e. 0.0011569 or 11 bps as the article put it), not the standard error of the residuals (i.e. 0.03773 or 377 bps).
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