STAT 541: Linear Tests And Colinearity
Statistics 541: Linear Tests And Colinearity
Admistrivia
Linear Tests
Consider productivity (Y) as a function of the amount of labor hours
in packing material (X1) and in shipping material (X2). Which
generates more output per hour, X1 or X2?
 Easy case: confidence intervals for betas don't overlap
 Given it takes so much more time to pack than to ship, why
aren't these very different? Its the marginal slope that we
are measuring. Correct economics says they should be the
same.
 What regression would we do if we believed the ecnomic theory
that X1 and X2 were identical? We'd regression Y on (X1+X2).
 Write model as Y ~ (X1+X2) + (X1X2)
 Intepret coefficients
 to test which is larger, use the coefficient on second term
 Notice exact same fits are possible as in Y ~ X1 + X2
 What if we regressed on: Y ~ (X1+X2) + X1? Same answer!
 General policy: Fit reduced model with appropiate variables,
then add enough variables to span space.
EXAMPLE: ETS reciently alleged that there was cheating on SAT exams at some
centers. Suppose we have a Y = test result when arriving on campus, X
= test result taken noncheating centers. Z = test results taken at
cheating centers, I = which center. How would you test their claim?
Answer: Y = (X+Z) is small model. Y = X + Z + I is full model. Now
use partial F.
Colinearity
 The good life: orthogonal design (collected from designed
experiement)
 consider a 2x2 table
 partial coefficient equals marginal coeficient
 there is only "one slope" so easy to understand
 The real world: multicolinearity
 marginal slope is different than partial slope
 draw diagram: Y ~X+Z, X~Z, then Z>Y via two routes
 Signs can change
 intepretation is difficult
 Controlled experiments are really useful here
 Measuring colinearity via the VIF
 VIF = SE^{2}(partial)/SE^{2}(marginal)
 (technically: diagonal of inverse of correlation matrix)
 VIF > 1

 Example:
 Regress stock on "market". But what is the market?
 DOW is what people have heard of, but S&P500 is what is
traded.
 Why not use both?
Last modified: Tue Feb 13 08:53:07 2001