STAT 541: Linear Tests And Colinearity

# Statistics 541: Linear Tests And Colinearity

## 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) + (X1-X2)
• 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 non-cheating 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 = SE2(partial)/SE2(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?