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

The population parameters β0, β1, ..., βk are unknown and are estimated using a sample of T observations on the dependent variable Y and the K independent variables X1t, X2t, ... , Xkt. Once we have estimated the parameters β0, β1, ..., βk, we obtain an estimated regression equation, which is called the sample regression equation.

y-hatt = b0 + b1x1t + b2x2t + ... + b0xkt

The value b0 is the sample estimate of the population parameter β0, the value b1 is the sample estimate of the population parameter β1, and so forth. The value y-hatt is called the fitted value of Yt or the predicted value of Yt.

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

Yt = b0 + b1X1t + b2X2t + et

where

• Yt = the gasoline mileage (in miles per gallon) of the t-th car.
• X1t = the engine size of the t-th car (in hundreds of cubic inches).
• X2t = 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-hatt = 54.3182 - 4.0129X1t - 15.9806X2t

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 X1t = 2.4 and X2t = -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.