Multiple Regression

INCM 9102Quantitative Methods

Multiple Regression

Previously, we learned that a simple linear equation of a line takes the general form of y=mx+b, where:

• Y is the dependent variable • m is the slope of the line• X is the independent variable or predictor• b is the Y-intercept.

When we discussion regression models, we transform this equation to be:

Y = bo + b1x1

Where bo is the y-intercept and b1 is the slope of the line. The “slope” is also the effect of a one unit change of x on y.

Multiple Regression

This was fine…but typically we don’t have just one predictor – we have lots of predictors.

When we discussion multiple regression models, the general form of the equation is like this:

Y = bo + b1x1 + b2x2 + b3x3 … bnxn

Where bo is still the y-intercept and “bi “ is the effect of a unit change of each of the individual predictors on the y (dependent) variable.

Lets discuss the general form of different hypothetical multiple regression models…

Multiple Regression

The requirements for Multiple Regression are general the same as they were for Linear Regression:

1.The relationship of the dependent and the independent (s)

variables is assumed to be linear.

2.The relationship of the dependent and the independent (s)

variables will have some (hopefully) significant correlation.

3.There should be no extreme values that influence (usually

negatively) the results.

4.Results are homoscedastic.

5.All observations are independent.

Multiple Regression

But…there are some issues in Multiple Regression which are not present in Linear Regression:

1.Multicollinearity amongst predictors

2.“Ingredient” variables

3.Selection Methods/Model Parsimony

Lets explore each of these in turn…

Multiple Regression

Multicollinearity – what is it and what’s the big deal?

Lets look at the temperature data and predict the temp in August…

Correlation Matrix – how is each potential predictor correlated individually with August Temperature?

Now…lets build the regression model…pay particular attention to the beta coefficients and the p-values…

Multicollinearity – what is it and what’s the big deal?

The moral of the story is that caution must be employed in interpreting the individual regression coefficients in a multiple regression analysis. The regression can be used to determine a predicted value of August temperature, even when it is difficult to interpret the sample regression coefficients.

If the individual coefficients are important, the multicollinearity must be removed…

Consider the VIF (Variance Inflation Factor).

VIF = 1/(1-R2)…where the R2 value here is the value when the predictor in question is set as the dependent variable.

Multiple Regression

Consider the VIF (Variance Inflation Factor).

VIF = 1/(1-R2)…where the R2 value here is the value when the predictor in question is set as the dependent variable.

For example, if the VIF = 10, then the respective R2 would be 90%. This would mean that 90% of the variance in the predictor in question can be explained by the other independent variables.

Because so much of the variance is captured elsewhere, removing the predictor in question should not cause a substantive decrease in overall R2.

The rule of thumb is to remove variables with VIF scores greater than 10.

Multiple Regression

Multiple Regression

What is an “ingredient” variable?

If the dependent variable is comprised of one of the predictor variables (or vice versa), the results are not reliable.

One or both of the following will happen:

You will generate an incredibly high R2 value The predictor in question will have a DOMINATING t-statistic

Lets look at the credit data…

Multiple Regression

What are the different selection methods and what are the differences?

“All In”

“Forward”

“Backward”

“Stepwise”

Model Parsimony = less is more. You are better off with an R2 of .75

and 3 predictors than with an R2 of .80 and 10 predictors.

Work from the Correlation Matrix

Manual Process

Multiple Regression

Additional Topics

1.Transformations

Logs

Discretization

Square/Square Root

2.Dummy Coding

K-1 new values

3.Additional Plots

Predicteds versus Actuals

Residuals versus Actuals

Residuals versus Predicteds

Standardized Residuals