- CFA Exams
- 2024 Level II
- Topic 1. Quantitative Methods
- Learning Module 1. Basics of Multiple Regression and Underlying Assumptions
- Subject 2. A Multiple Linear Regression Example

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##### Subject 2. A Multiple Linear Regression Example PDF Download

The population parameters β

_{0}, β

_{1}, ..., β

_{k}are unknown and are estimated using a sample of T observations on the dependent variable Y and the K independent variables X

_{1t}, X

_{2t}, ... , X

_{kt}. Once we have estimated the parameters β

_{0}, β

_{1}, ..., β

_{k}, we obtain an estimated regression equation, which is called the

*sample regression equation*.

_{t}= b

_{0}+ b

_{1}x

_{1t}+ b

_{2}x

_{2t}+ ... + b

_{0}x

_{kt}

The value b

_{0}is the sample estimate of the population parameter β

_{0}, the value b1 is the sample estimate of the population parameter β

_{1}, and so forth. The value y-hat

_{t}is called the fitted value of Y

_{t}or the predicted value of Y

_{t}.

*Example*

It is reasonable to suspect that gasoline mileage for a car is determined mainly by the car's weight and engine size. We decided to estimate the regression

_{t}= b

_{0}+ b

_{1}X

_{1t}+ b

_{2}X

_{2t}+ e

_{t}

where

- Y
_{t}= the gasoline mileage (in miles per gallon) of the t-th car. - X
_{1t}= the engine size of the t-th car (in hundreds of cubic inches). - X
_{2t}= the weight of the t-th car (in tons).

The following table shows the results of this linear regression using a sample of T = 10 different cars.

Therefore, we obtain the estimated equation (after rounding) of y-hat

_{t}= 54.3182 - 4.0129X

_{1t}- 15.9806X

_{2t}

The predicted mileage for a car that has a 2.4-hundred-cubic-inch engine and weighs 0.9 ton is obtained by substituting the values X

_{1t}= 2.4 and X

_{2t}= -0.9 into the estimated equation. The predicted value is then 54.3182 - 4.0129 (2.4) - 15.9806 (0.9) = 30.3047, or about 30.3 miles per gallon.

The standard error column gives the standard error (the standard deviation) of the estimated regression coefficients.

We have T = 10 observations and k = 2 explanatory variables in the model, so the appropriate degrees of freedom is 10 - 2 - 1 = 7.

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