Multiple Regression Model
The multiple regression model is an extension of the simple model discussed in chapter 2. The main difference is that the multiple linear regression model contains more than one explanatory variable. This changes the interpretation of the coefficients slightly and requires another assumption. The general form of the model is shown in equation (5.1) below.
Уі = ві + @2%i2 +— + вкХік + ei i = 1,2, …,N (5.1)
where yi is your dependent variable, xik is the ith observation on the kth independent variable, k = 2,3,…, K, ei is random error, and въ в2,…, вК are the parameters you want to estimate. Just as in the simple linear regression model, each error, ei, has an average value of zero for each value of the independent variables; each has the same variance, a2, and are uncorrelated with any of the other errors. In order to be able to estimate each of the в®, none of the independent variables can be an exact linear combination of the others. This serves the same purpose as the assumption that each independent variable of the simple linear regression take on at least two different values in your dataset. The error assumptions can be summarized as ei|xi2, xi3,… xiK iid (0, a2). Recall from chapter 2 that expression iid means that the errors are statistically independent from one another (and therefore uncorrelated) and each has the same probability distribution. Taking a random sample from a single population accomplishes this.
The parameters в2, вз,…, вк are referred to as slopes and each slope measures the effect of a 1 unit change in xik on the average value of yi, holding all other variables in the equation constant. The conditional interpretation of the coefficient is important to remember when using multiple linear regression.
The example used in this chapter models the sales for Big Andy’s Burger Barn. The model includes two explanatory variables and a constant.
salesi = в1 + в^г^.і + в3adverti + ei i = 1,2,…, N (5.2)
where salesi is monthly sales in a given city and is measured in $1,000 increments, pricei is price of a hamburger measured in dollars, and adverti is the advertising expenditure also measured in thousands of dollars.