Too Logit to Quit

1
Too Logit to Quit

• Why you should (or shouldnt) take POLS 7050

2
Different Dependent Variables

• OLS assumes a continuous, unbounded dependent
  variable (or so Ive said)
• This is not directly stated in the assumptions,
  but it is implicit
• Today
• Discuss OLS as a linear probability model
• Highlight the problems with OLS
• Discuss alternative estimators

3
OLS with a Nominal/Ordinal D.V.

• It doesnt make much sense to think of a 1 unit
  increase in x causing a b unit increase in y
  because increases/decreases dont make sense with
  nominal variables.
• For ordinal variables, a constant b-unit increase
  for changes in x doesnt work out
• Instead, we might conceive of a random utility
  model where we estimate the probability of each
  choice as a function of the independent
  variables.

4
OLS as a Random Utility Model

• Assume 2 categories for the D.V. (e.g. two
  countries either go to war or not). Then if
• The expected value of y can be interpreted as the
  probability that the country chooses option 1
  (war)

5
OLS as a Linear Probability Model

• Todays Data U.S. House Elections
• Dependent Variable 1 if Democrat wins, 0
  otherwise
• Independent Variable District ideology ( for
  Dem. Pres. Candidate in last election)
• Go to Data

6
Whats wrong with this Picture?

• Heteroskedastic
• Out of Bounds Predictions
• Wrong Functional Form
• Non-Normal Errors

7
Solutions

• Patch up the OLS estimator
• Use WLS to fix the heteroskedasticity problem (we
  know the nature of the problem)
• Fudge any out of bound predictions
• Nothing we can do about the functional form. OLS
  is inherently linear and additive
• In short, we have simply reached the limits of
  OLS. These results are not uninterpretable, but
  they are a poor way to model these kinds of data.

8
Solutions

• In some sense, our goal will be to transform the
  data to achieve a suitable functional form.

9
Solution (1 of many)

• Transform the equation using the Logistic
  Cumulative Distribution Function to get the
  correct functional form

10
Consequences of the Model

• Non-linear estimator
• Not a constant b-unit change in y for a 1-unit
  change in x (effect of x depends on value of x)
• Effect of x1 also depends on value of x2
• Numeric value of Coefficient gives us little
  information of itself
• Cannot compare size of coefficients (just like
  OLS)
• What can we do? Look for Signs and Stars

11
Solution 2 Latent DV

• Assume that dichotomous DV is a manifestation of
  a latent variable

t
Liberal Decision (0)
Conservative Decision (1)
Liberalness of Decision