Assumption checking in “normal” multiple regression

with Stata

2

Assumptions in regression analysis•No multi-collinearity

•All relevant predictor variables included•Homoscedasticity: all residuals are from a distribution with the same variance•Linearity: the “true” model should be linear.•Independent errors: having information about the value of a residual should not give you information about the value of other residuals•Errors are distributed normally

3

FIRST THE ONE THAT LEADS TO NOTHING NEW IN STATA (NOTE: SLIDE TAKEN LITERALLY FROM MMBR)

Independent errors: having information about the value of a residual should not give you information about the value of other residuals

Detect: ask yourself whether it is likely that knowledge about one residual would tell you something about the value of another residual.Typical cases: -repeated measures-clustered observations (people within firms / pupils within schools)

Consequences: as for heteroscedasticityUsually, your confidence intervals are estimated too small (think about why that is!).

Cure: use multi-level analyses

In Stata:Example: the Stata “auto.dta” data setsysuse auto

corr (correlation)vif (variance inflation factors)

ovtest (omitted variable test)

hettest (heterogeneity test)

predict e, residswilk (test for normality)

Finding the commands

• “help regress”• “regress postestimation”

and you will find most of them (and more) there

6

Multi-collinearity A strong correlation between two or more of your predictor variables

You don’t want it, because:1. It is more difficult to get higher R’s2. The importance of predictors can be difficult to

establish (b-hats tend to go to zero)3. The estimates for b-hats are unstable under slightly

different regression attempts (“bouncing beta’s”)

Detect: 4. Look at correlation matrix of predictor variables5. calculate VIF-factors while running regression

Cure:Delete variables so that multi-collinearity disappears, for instance by combining them into a single variable

7

Stata: calculating the correlation matrix (“corr”) and VIF statistics (“vif”)

8

Misspecification tests(replaces: all relevant predictor

variables included)

9

Homoscedasticity: all residuals are from a distribution with the same variance

Consequences: Heteroscedasticiy does not necessarily lead to biases in your estimated coefficients (b-hat), but it does lead to biases in the estimate of the width of the confidence interval, and the estimation procedure itself is not efficient.

Testing for heteroscedasticity in Stata

• Your residuals should have the same variance for all values of Y hettest

• Your residuals should have the same variance for all values of X hettest, rhs

11

Errors distributed normally

Errors are distributed normally (just the errors, not the variables themselves!)

Detect: look at the residual plots, test for normality

Consequences: rule of thumb: if n>600, no problem. Otherwise confidence intervals are wrong.

Cure: try to fit a better model, or use more difficult ways of modeling instead (ask an expert).

First calculate the errors:predict e, resid

Then test for normalityswilk e

Errors distributed normally