### Linear Statistical Models: Regression

### Regression Assumptions

**Assumptions in Regression Analysis**

**Independence**
- The residuals are serially independent (no autocorrelation).
- The residuals are not correlated with any of the independent (predictor) variables.
**Linearity**
- The relationship between the dependent variable and each of the independent variables is linear.
**Mean of Residuals**
- The mean of the residuals is zero.
**Homogeneity of Variance**
- The variance of the residuals at all levels of the independent variables is constant.
**Errors in Variables**
- The independent (predictor) variables are measured without error.
**Model Specification**
- All relevant variables are included in the model.
- No irrelevant variables are included in the model.
**Normality**
- The residuals are normally distributed.
This assumption is needed for valid tests of significance but not for estimation of the regression
coefficients.

**Violations of Assumptions**

Regression analysis is generally robust to violations of assumptions
Except for:

- Measurement Errors
- Specification Errors

**Measurement Error**

In the dependent variable:

- Does not bias estimates of the regression coefficient, b.
- Increases standard error of estimate thus weakening tests of significance.

In the independent variable:

- Leads to under estimation of the regressions coefficient, b.

**Specification Errors**

Omission of relevant variables.
Inclusion of irrelevant variables.
Using linear regression when the relationship is no linear.

Linear Statistical Models Course

Phil Ender, 29dec99