To motivate multiple regression, let's consider the demand for milk by households.
To eliminate the supply curve we will assume that milk is produced with a
constant Marginal Cost technology so that quantities do not affect
Let m be the quantity of milk purchased (say each week) and let
income. Suppose the household has the utility function
where p is the price of milk faced by the household and U is the utility function for the household defined in terms of milk and all other goods. The Marshallian demand function in terms of log milk consumption takes the form:
In other words, if the Cobb-Douglas utility function is correct, the among of milk purchased by the household should be log-linear, with coefficients on income and the price of milk equal to 1 in absolute value. We might specify the following regression equation:
The disturbance term u accounts for different tastes for milk across households associated with different values of . Households with the same income and facing the same prices might have different milk consumptions because of differences in tastes: for example, one household might have young children which would push up total milk consumption.
Even with tastes differing across households, the Cobb-Douglas model implies two hypotheses about the MLRM: = 1 and = -1. These hypotheses might be interesting to test given data on milk prices, household income, and household consumption of milk.
But notice that this demand function has two variables: income and milk prices. We could certainly estimate a simple linear regression model including only one variable or the other (either price or income). But since both variables can be measured much less would be left to the residual term if both were included in the analysis. The Multiple LRM is generalizes the simple LRM to allow two or more variables on the right hand side of the PRE.