Anon

Multiple Linear Regression 43

The regression coefficient of each independent variable given in equa-
tion (3.5) represents the average change in the dependent variable per unit
change in the independent variable with the other independent variables
held constant.

Assumptions of the Multiple Linear Regression Model

For the multiple linear regression model, we make the following three
assumptions about the error terms:

Assumption 1. The regression errors are normally distributed with zero

mean.

Assumption 2. The variance of the regression errors (σε^2 ) is constant.

Assumption 3. The error terms from different points in time are inde-

pendent such that εt ≠ εt+d for any d ≠ 0 are independent for all t.

Formally, we can restate the above assumptions in a concise way as

εσi ~0N(),

ii..d.

(^2)
Furthermore, the residuals are assumed to be uncorrelated with the indepen-
dent variables. In the next chapter, we describe how to deal with situations
when these assumptions are violated.

Estimation of the Model Parameters

Since the model is not generally known for the population, we need to esti-
mate it from some sample. Thus, the estimated regression is

(^) ybˆ=+ 0112 bx++bxbx 2 + kk (3.7)
The matrix notation analogue of equation (3.7) is
yy=+ˆ eX=+be (3.8)
which is similar to equation (3.2) except the model’s parameters and error
terms are replaced by their corresponding estimates, b and e.
The independent variables x 1 ,... , xk are thought to form a space of
dimension k. Then, with the y-values, we have an additional dimension such