 ## B. Definition of the Multiple Regression Model

### Assumptions A0-A3

The multiple linear regression model (MLRM) is a generalization of the simple LRM:

#### Assumption A0 about the LRM

The data consist of N vectors of related variables. where is the value of the variable for the observation. For example, might be income, might be the price of milk, might be number of children, etc. Then equals the number of children in household (or observation) 3. We begin numbering the X variables at 2 to hold a place for the constant term.

#### Assumption A1: The Population Regression Equation (PRE)

The X variables and Y are related by: Each coefficient is an unknown population parameter. Notice that the simple linear regression model is the special case of the multiple regression model, k=2. Since an economic variable like Y is undoubtedly influenced by many potentially observable influences (like price, income) there is perhaps a better way to interpret any linear regression model as a special case of a larger (greater k) model. In particular, we should think of the smaller model as a difference in the observed sample information. If we do not observe all the factors that influence Y, then we have a a sample restricted in the X variables that we can observe. We do not necessarily have the wrong model if we only include one variable in the regression when there should be many.

The sample for a multiple regression model is written: #### The PRE written in Matrix Notation

It is much easier to work with the multiple regression model in matrix notation. To do this we have to define several vectors and matrices: Notice that the subscripts for the matrix X are the reverse of the usual row/column ordering. The reason is that we want to "bind" the variable name first ( refers to variable number 2), but we naturally write an equation on one line (or row), forcing us to have refer to a column rather than a row. With the notation above, all the sample information is represented by one matrix equation. ### Interpretation of in the LRM

Looking at the PRE equation (**), we can take expectations of both sides: Then, using basic calculus we see We can interpret the coefficient on the jth variable to be the partial derivative of the expected value of Y with respect to a change in the jth variable. For example, if is log-consumption of milk and is the log of household income, then is the income elasticity of milk demand.
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