Multiple Regression (12)

12. Multiple Regression

Simple regression (see chapter 11) can predict a dependent variable as a function of a single independent variable. But often there are multiple variables at play. In order to determine the simultaneous effect of multiple independent variables on a dependent variable, multiple regression is used. The least squares principle fit the model.

12.1. The model

As with simple regression, the first step in the model development is model specification, the selection of the model variables and functional form of the model. This is influenced by the model objectives, namely: (1) predicting the dependent variable, and/or (2) estimating the marginal effect of each independent variable. The second objective is hard to achieve, however, in a model with multiple independent variables, because these variables are not only related to the dependent variable but also to each other. This leaves a web of effects that is not easily untangled.

To make multiple regression models more accurate an error termε” is added, as a way to recognize that none of the described relationships in the model will hold exactly and there are likely to be variables that affect the dependent variable, but are not included in the model.

12.2. Estimating Coefficients

Multiple regression coefficients are calculated with the least squares procedure. However, again this is more complicated than with simple regression, as the independent variables not only affect the dependent variable but also each other. It is not possible to identify the unique effect of each independent variable on the dependent variable. This means that the higher the correlations between two or more of the independent variables in a model are, the less reliable the estimated regression coefficients are.

There are 5 assumptions to standard multiple regression. The first 4 are the same as are made for simple regression (see chapter 11). The 5th states that it is not possible to find a set of nonzero numbers such that the sum of the coefficients equals 0. This assumption excludes the cases in which there is a linear relationship between a pair of independent variables. In most cases this assumption will not be violated if the model is properly specified.

Whereas in simple regression the least squares procedure finds a line that best represents the set of points in space, multiple regression finds a plane that best represents these points (as each variable is represented with its own dimension).

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