1. Regression III

2. The regression model has both constants (, b) and variables (X, Y)
The “fit” of the regression equation to the data is numerically expressed by the r2 statistic.
The b for each indep var can be tested for statistical significance using a t test.
The overall model is tested for statistical significance using the F ratio.

3. Assumptions of the regression model
Like all statistics, the regression model has a number of underlying assumptions.
T-test assumes a t distribution
z scores assumes data is normally distributed
We will discuss some of the more common ones.

4. Multicollinearity
When 2 or more independent variables in the model are highly correlated with one another.
Result: bias in the partial regression coefficients
Test: by correlating each variable with the others
Fix: drop all but one of highly correlated variables or combine into a single variable

5. “Dummy” variables
Regression analysis assumes the use of continuous, interval level data
Two types
dichotomous variables (two possible states)
polychotomous variables
may be nominal or ordinal

6. Dummy variables Dichotomous variable
male/female; Republican/Democrat
yes/no
Essentially, a case has or does not have a particular characteristic
Example: last week’s regression model predicting entry GS grade
field of education
veterans’ preference
minority female

7. Polychotomous variables - a number of possible states
often, sometimes, rarely, never
region of country (South, Midwest, East, West)
When using exclude one of the categories
Include three 0/1 variables; eliminate one category
the excluded variable becomes the reference category

8. Autocorrelation
A nonrandom relationship among a variable’s values at different time periods
consistent patterns such as seasonal data
Often found in time series data

9. Autocorrelation
Result: biased t-ratios, confidence limits, and hypotheses tests
Test: plot the residuals - look for distinctive patterns
Fix: introduce another independent variable that explains some of the unexplained variance
more commonly: use a statistical model other than OLS

10. Nonlinear relationships
OLS assumes a linear relationship (remember the straight line we drew based on the regression equation?)
Some of out data does not provide a linear relationship
economic data
population data
data with built-in growth factor

11. Nonlinear relationships
We test for this using a scatterplot.
Does the relationship appear linear?
Fix: transform one of the variables

12. Nonlinear relationship

13. Nonlinear relationship

14. Heteroskedasticity
When the effect of X on Y is not equal across all ranges of Y
Result: affects size of standard error, thus biasing hypothesis test results.

15. Outliers Extreme values Problem: can bias the regression parameters
when a particular (or number of them) don’t seem to fit in with the other data.
Problem: can bias the regression parameters

16. Outliers (Hong Kong and Singapore)