Samprit Chatterjee - Handbook of Regression Analysis With Applications in R

where is an indicator variable with value if the observation is a member of group and otherwise. The usual interpretation of the slope still applies: is the expected change in associated with a one‐unit change in holding all else fixed. Since only takes on the values or , this is equivalent to saying that the expected target is higher for group members ( ) than nonmembers ( ), holding all else fixed. This has the appealing interpretation of fitting a constant shift model, where the regression relationships for group members and nonmembers are identical, other than being shifted up or down; that is,

for nonmembers and

for members. The ‐test for whether is thus a test of whether a constant shift model (two parallel regression lines, planes, or hyperplanes) is a significant improvement over a pooled model (one common regression line, plane, or hyperplane).

Would two different regression relationships be better still? Say there is only one numerical predictor ; the full model that allows for two different regression lines is

for nonmembers ( ), and

for members ( ). The pooled model and the constant shift model can be made to be special cases of the full model, by creating a new variable that is the product of and . A regression model that includes this variable,

corresponds to the two different regression lines

for nonmembers (since ), implying and above, and

for members (since ), implying and above.

The ‐test for the slope of the product variable ( ) is a test of whether the full model (two different regression lines) is significantly better than the constant shift model (two parallel regression lines); that is, it is a test of parallelism. The restriction defines the pooled model as a special case of the full model, so the partial ‐statistic based on (2.1),

on degrees of freedom, provides a test comparing the pooled model to the full model. This test is often called the Chow test(Chow, 1960) in the economics literature.