The population parameters β₀, β₁, ..., 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 β₀, β₁, ..., β_k, we obtain an estimated regression equation, which is called the sample regression equation.

y-hat_t = b₀ + b₁x_1t + b₂x_2t + ... + b₀x_kt

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-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

Y_t = b₀ + b₁X_1t + b₂X_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.