Introduction to POL 217

Transcription

1 Introduction to POL 217 Brad Jones 1 1 Department of Political Science University of California, Davis January 9, 2007

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3 Topics of Course Outline Models for Categorical Data.

4 Topics of Course Models for Categorical Data. Models for Events Data.

5 Topics of Course Models for Categorical Data. Models for Events Data. There is a close connection.

6 Outline Several Applied Problem Sets (50 percent).

7 Several Applied Problem Sets (50 percent). One Take-Home Exam...

8 Several Applied Problem Sets (50 percent). One Take-Home Exam... or Paper and Presentation (40 percent).

9 Several Applied Problem Sets (50 percent). One Take-Home Exam... or Paper and Presentation (40 percent). Participation beyond Breathing (10 percent).

10 Pros and Cons Outline Exams may help prepare for comprehensives.

11 Pros and Cons Exams may help prepare for comprehensives. Exams don t (usually) result in articles.

12 Pros and Cons Exams may help prepare for comprehensives. Exams don t (usually) result in articles.... though all of you will need to discuss quantitative work.

13 Categorical Response Variables Binary (logit or probit).

14 Categorical Response Variables Binary (logit or probit). Ordinal (proportional odds/cumulative probit).

15 Categorical Response Variables Binary (logit or probit). Ordinal (proportional odds/cumulative probit). Nominal (baseline category logit).

16 Categorical Response Variables Binary (logit or probit). Ordinal (proportional odds/cumulative probit). Nominal (baseline category logit). (We ll worry about events data in a few weeks).

17 Why Logit or Probit? Outline Suppose y is binary.

18 Why Logit or Probit? Suppose y is binary. Regression assumes y is unbounded and continuous.

19 Why Logit or Probit? Suppose y is binary. Regression assumes y is unbounded and continuous. Hence, y = β 0 + β X + ɛ.

20 Why Logit or Probit? Suppose y is binary. Regression assumes y is unbounded and continuous. Hence, y = β 0 + β X + ɛ. ŷ must be unbounded.

21 Why Logit or Probit? Suppose y is binary. Regression assumes y is unbounded and continuous. Hence, y = β 0 + β X + ɛ. ŷ must be unbounded. Linear change in E(y) x not natural.

22 Why Logit or Probit? Suppose y is binary. Regression assumes y is unbounded and continuous. Hence, y = β 0 + β X + ɛ. ŷ must be unbounded. Linear change in E(y) x not natural. P(y = 1) = P i P(y = 0) = (1 P i ) = Q i E(y) = P i (1) + Q i (0) E(y) = P i (1)

23 Why Logit or Probit? Outline More Trouble.

24 Why Logit or Probit? More Trouble. ˆɛ = y i ˆβ k x i.

25 Why Logit or Probit? More Trouble. ˆɛ = y i ˆβ k x i. y = 1 : 1 ˆβ k x i y = 0 : 0 ˆβ k x i

26 Why Logit or Probit? More Trouble. ˆɛ = y i ˆβ k x i. x i, ɛ assumes two values. y = 1 : 1 ˆβ k x i y = 0 : 0 ˆβ k x i

27 Why Logit or Probit? Outline Heteroskedasticity

28 Why Logit or Probit? Outline Heteroskedasticity E(y) = ˆβ k x i = P i and 1 ˆβ k x i = Q i

29 Why Logit or Probit? Heteroskedasticity E(y) = ˆβ k x i = P i and 1 ˆβ k x i = Q i Noting (without proof) that var(ɛ) = (1 ˆβk x i ) 2 P i + ( ˆβk x i ) 2 Q i var(ɛ) = (Q i ) 2 P i + ( P i ) 2 Q i = Q i P i (Q i P i ) = Q i P i ([1 P i ] + P i ) = Q i P i = (1 ˆβ k x i )( ˆβ k x i )

30 Motivating Logit (or Probit) Suppose E(y) = P(y = 1 x) = β k x ik, and y is binary. Pr(y = 1 x) = exp( β k x ik )

31 Motivating Logit (or Probit) Suppose E(y) = P(y = 1 x) = β k x ik, and y is binary. Pr(y = 1 x) = Let Z = β k x ik, then Pr(y = 1 x) = exp( β k x ik ) exp( Z) = exp(z) 1 + exp(z)

32 Motivating Logit (or Probit) Suppose E(y) = P(y = 1 x) = β k x ik, and y is binary. Pr(y = 1 x) = Let Z = β k x ik, then Pr(y = 1 x) = exp( β k x ik ) exp( Z) = exp(z) 1 + exp(z) This is the c.d.f. for the logistic distribution.

33 Motivating Logit (or Probit) Suppose E(y) = P(y = 1 x) = β k x ik, and y is binary. Pr(y = 1 x) = Let Z = β k x ik, then Pr(y = 1 x) = exp( β k x ik ) exp( Z) = exp(z) 1 + exp(z) This is the c.d.f. for the logistic distribution. Problems Solved: Z is unbounded; P i must stay in unit interval. P i is nonlinearlly related to parameters (though logit is linear!)

34 Logit Model Outline Odds Ratios are given by P i /(1 P i ) = exp(z)

35 Logit Model Odds Ratios are given by P i /(1 P i ) = exp(z) Log-odds are then log[p i /(1 P i )] = Z

36 Logit Model Odds Ratios are given by P i /(1 P i ) = exp(z) Log-odds are then log[p i /(1 P i )] = Z The Logit Model log P i 1 P i = Z = β k x ik

37 Logit Model Odds Ratios are given by P i /(1 P i ) = exp(z) Log-odds are then log[p i /(1 P i )] = Z The Logit Model log P i 1 P i = Z = β k x ik This is the logit transformation and yields the logit model.

38 Logit Model Odds Ratios are given by P i /(1 P i ) = exp(z) Log-odds are then log[p i /(1 P i )] = Z The Logit Model log P i 1 P i = Z = β k x ik This is the logit transformation and yields the logit model. Again, Z unbounded; perfect prediction impossible.