Business Math

MSOR 221 Statistical Inference

Chapter 18 Multiple Regression

Model and Required Conditions

For k independent variables (predicting variables) x1, x2, … , xk, the multiple linear regression model is represented by the following equation:

[pic]

where (1, (2, … , (k are population regression coefficients of x1, x2, … , xk respectively, (0 is the constant term, and ( (the Greek letter epsilon) represents the random term (also called the error variable) – the difference between the actual value of Y and the estimated value of Y based on the values of the independent variables. The random term thus accounts for all other independent variables that are not included in the model.

Required Conditions for the Error Variable:

1. The probability distribution of the error variable ( is normal.

2. The mean of the error variable is 0.

3. The standard deviation of ( is [pic], which is constant for each value of x.

4. The errors are independent.

The general form of the sample regression equation is expressed as follow:

[pic]

where b1, b2, … , bk are sample linear regression coefficients of x1, x2, … , xk respectively and b0 is the constant of the equation.

For k = 2, the sample regression equation is [pic]where b0, b1, and b2 can be found by solving a system of three normal equations:

[pic]

Example 1

| |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |

| |1 |200 |100 |100 |20000 |200 |1 |40000 |82.99 |

| |5 |700 |300 |1500 |210000 |3500 |25 |490000 |305.20 |

| |8 |800 |400 |3200 |320000 |6400 |64 |640000 |394.73 |

| |6 |400 |200 |1200 |80000 |2400 |36 |160000...