- CFA Exams
- 2023 Level I
- Topic 1. Quantitative Methods
- Learning Module 2. Multiple Regression
- Subject 4. The Standard Error of Estimate in Multiple Linear Regression Model

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##### Subject 4. The Standard Error of Estimate in Multiple Linear Regression Model PDF Download

The SSE = ∑(y ∑e-hat

**least-squares estimates**of the population parameters β_{0}, β_{1}, ..., β_{k}are the values b_{0}, b_{1}, ..., b_{k}that minimize the residual sum of squares SSE, where

_{t}- y-hat

_{t})

^{2}= ∑(y

_{t}- b

_{0}- b

_{1}x

_{1t}- b

_{2}x

_{2t}- ... - b

_{k}x

_{kt})

^{2}

If we estimate the multiple regression model by the method of least squares and if the model contains a constant term, then we obtain

_{t}= 0

that is, the sum of the residuals is always 0 provided the sample regression model contains a constant term b

_{0}.To test hypotheses and to construct confidence intervals, we need an estimate of σ

_{e}^{2}. Recall that in the simple leaner regression model an unbiased estimator of σ_{e}^{2}is s_{e}^{2}= SSE/(t-2), where t - 2 is the degrees of freedom of the model. An unbiased estimator of σ_{e}^{2}is defined in an analogous manner in the multiple regression model. In the formula for s_{e}^{2}, the denominator is t minus the number of coefficients that have to be estimated. In the multiple regression model, there are (k + 1) coefficients β_{0}, β_{1}, ..., β_{k}to be estimated, and so the denominator in the formula for se2 is t - (k + 1).The

**standard error of estimate (SEE)**of the regression is:

The value t - k - 1 is the degrees of the freedom of the regression equation, in a sense we use up (k + 1) degrees of freedom before calculating SSE because we first estimate the (k + 1) coefficients β

_{0}, β_{1}, ..., β_{k}.Let's continue with the example in los a.

The ANOVA output reports quantities related to the overall explanatory power of the regression.

An unbiased estimate of the variance of the errors is s

_{e}^{2}= SSE /(t - k - 1) = 20.8958/(10 - 3) = 2.9851, and the standard error of estimate is s_{e}= (2.9851)^{1/2}= 1.7277.Based on the empirical rule, for a normal distribution, approximately 95% of the error terms should be less than 2s

_{e}(or 3.4554) units from the estimated plane and the approximately 95% of the sample observations should lie within 3.4554 units of the estimated plane.

**Learning Outcome Statements**

CFA® 2023 Level I Curriculum, Volume 1, Module 2

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**User Contributed Comments**
1

User |
Comment |
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katybo |
t-k-1 = residual degrees of freedom |

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