ESTIMATION: ONE DIMENSION (PART 1)

Transcription

1 ESTIMATION: ONE DIMENSION (PART 1) 1

2 Goals of this Unit Examine how we can use real world data to estimate the location of ideal points (and cut points). Interest group scores. Optimal classification. Nominate. Show how some of these models can be estimated in R. Introduce methods for determining the appropriate number of dimensions next week. i.e. how many dimensions explain most of the data?

3 Interest Group Scores U.S. examples: Americans for Democratic Action (ADA), American Conservative Union (ACU), National Taxpayers Union, etc. How they work: Count the number of times a legislator supports the interest group s position on a bill and divide it by the number of votes studied by the interest group. Consider the ADA...

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5 Let s focus on these.

6 Interest Group Scores Place the scores on a line: Ki Gi Gr Ba Br Bi J If you don t like liberals on the right, multiply the score, s, by -1 Or rescale to [0,1] using s+1. Often called Reversed ADA scores in the converted form. Liberal (ADA position)

7 Interest Group Scores Ki J Bi Br Ba Gr Gi Liberal Problems with interest group scores 1. The measure is granular (i.e. a legislator can be -.95 but not ). 2. Interest groups study votes with clear positions for and against their interest, pushing legislators near the extremes of the scale. For example, interest groups are not analyzing whether we should fund relief from Hurricane Katrina.

8 Interest Group Scores Ki J Bi Br Ba Gr Gi Liberal 3. Interest groups often announce which bills they will study, so legislators adjust their votes to get a good score. Why is Paul Broun agreeing with the liberals on 30% of the ADA votes? Why is Jack Kingston never voting the liberal position? 4. A bill cannot be included if you don t know the interest group s side of a bill.

9 Interest Group Scores cut point Ki ADA x ADA q J Bi Br Ba Gr Gi D E 5. They implicitly assume the interest group is exterior to the legislators. a. Cut Point: a point that demarcates the yeas on one side from the nays on the other side. It is half the distance between the proposal x and the status quo q. b. To see this point, suppose the ADA was at -.85 and one of the votes was a choice between x = -1 and q = -.8, as shown above. With perfect spatial voting, two legislators, D = -.89 and E = -.8, would agree with the ADA position (nay) and be counted as more liberal on this vote than J = -.95, who votes against the ADA position. which is weird, because J is more liberal in the latent dimension.

10 Interest Group Scores Interest group scores in disguise. Segal-Cover Scores count how many editorials describe a justice as liberal vs conservative using six major newspapers. McGuire s and Oshfeldt s Pro-Nationalism count the percentage of times a delegate voted a pro-national position among sixteen roll calls. What happens when we scale the runaway slave clause based on pronationalism, but a delegate votes based on their opposition to slavery? We might interpret their no on the runaway slave clause as a vote against nationalism, even though they strongly support strengthening the government. Put differently, there can be problems pre-determining the scale.

11 Optimal Classification A non-parametric scaling technique i.e. it s computational, not statistical. Assumptions Voters have single peaked and symmetric preferences. Choices are binary (e.g., yes-no). Voting is sincere. i.e. for each vote, individuals vote for the alternative closest to themselves without thinking about the consequences for the end of the game. The number of dimensions is fixed (today it s one).

12 Optimal Classification How it works 1. Find an optimal cut point for each vote. 2. Optimally classify the voters in the intervals created by the cut points. 3. Repeat the process until the number of classification errors are minimized (or more specifically cannot be reduced). A classification error occurs if someone on the yea (resp. nay) side of the cut point votes nay (resp. yea).

13 Optimal Classification To illustrate, consider four bills B-2: Sept 1989 amendment to a bill that would cut funding for the B-2 bomber. Cambodia: July 1989 amendment to a bill that would give the President the power to fund non-communist forces in Cambodia. Tower: March 1989 proposal to confirm John Tower as secretary of defense. MLK: May 1989 amendment to a bill that would eliminate funding for a commission establishing MLK day.

14 Optimal Classification Five Senators in an initial (say random) order 1. Find optimal cut points We want to minimize these. Nunn Helms Gore Kerry Dole ERRORS B-2 (B) N N N Y N Cambodia (C) Y Y N N Y Tower (T) N Y N N Y MLK (M) N Y N N N C T M B sum = 5

15 Optimal Classification 2. Move Voters to Reduce Error Nunn Helms Gore Kerry Dole ERRORS B-2 (B) N N N Y N Cambodia (C) Y Y N N Y Tower (T) N Y N N Y MLK (M) N Y N N N C T M B sum = 5 Move Dole to eliminate error on B

16 Optimal Classification 2. Move Voters to Reduce Error Nunn Helms Gore Dole Kerry ERRORS B-2 (B) N N N N Y Cambodia (C) Y Y N Y N Tower (T) N Y N Y N MLK (M) N Y N N N C T M B sum = 4 Technically, B remained fixed. It just appears to have moved in my table. Next, move Dole again to eliminate error on C and T.

17 Optimal Classification 2. Move Voters to Reduce Error Nunn Helms Dole Gore Kerry ERRORS B-2 (B) N N N N Y Cambodia (C) Y Y Y N N Tower (T) N Y Y N N MLK (M) N Y N N N C T M B sum = 2 Technically, C,T, and M remained fixed. They just appear to have moved. Next, move Nunn to eliminate T error.

18 Optimal Classification 2. Move Voters to Reduce Error Helms Dole Nunn Gore Kerry ERRORS B-2 (B) N N N N Y Cambodia (C) Y Y Y N N Tower (T) Y Y N N N MLK (M) Y N N N N C T M B sum = 2 Technically, C,T, and M remained fixed. They just appear to have moved. Note, by moving Nunn we added a classification error.

19 Optimal Classification 2. Move Voters to Reduce Error Helms Dole Nunn Gore Kerry ERRORS B-2 (B) N N N N Y Cambodia (C) Y Y Y N N Tower (T) Y Y N N N MLK (M) Y N N N N C T M B sum = 2 If the cut points are fixed, we cannot get rid of the classification error for M. Hence, this step is done.

20 Optimal Classification 1. Repeat step 1: find optimal cut points given voter positions. Helms Dole Nunn Gore Kerry ERRORS B-2 (B) N N N N Y Cambodia (C) Y Y Y N N Tower (T) Y Y N N N MLK (M) Y N N N N C T M C B sum = 21

21 Optimal Classification 1. Repeat step 1: find optimal cut points given voter positions. Helms Dole Nunn Gore Kerry ERRORS B-2 (B) N N N N Y Cambodia (C) Y Y Y N N Tower (T) Y Y N N N MLK (M) Y N N N N M T C B M sum = 01

22 Optimal Classification 1. Repeat step 1: find optimal cut points given voter positions. Helms Dole Nunn Gore Kerry ERRORS B-2 (B) N N N N Y 0 Cambodia (C) Y Y Y N N 0 Tower (T) Y Y N N N 0 MLK (M) Y N N N N 0 M T C B sum = 0 No more errors, we re done! In general, the process repeats until it cannot reduce the error (which is usually more than zero errors).

23 Optimal Classification Advantages Does not require knowing the interest group s position on a bill (e.g., whether the bill is liberal or conservative). Does not push legislators to the extreme of the scale. Does not require the assumption that the interest group is exterior of the ideal points. Disadvantages Produces an ordering, not cardinal distances. Because it is non-parametric, there is no statistical error for the ideal points. Hence, there are no confidence intervals.

24 NOMINATE NOMINATE (Nominal Three-Step Estimation) Multidimensional scaling technique based on pseudo-maximum likelihood estimation. Two flavors and a groovy application W-NOMINATE: scales fixed ideal points from a single chamber. DW-NOMINATE: allows for linear changes in ideal points over time. Common Space Scores: applies DW-NOMINATE to a roll call matrix of all congress people throughout U.S. history. Thus making the House and Senate comparable throughout time. Does not allow movement of ideal points over time.

25 W-NOMINATE Assumptions Voters have single peaked and symmetric preferences. Choices are binary (e.g., yes-no). Voting is sincere. i.e. for each vote, individuals vote for the alternative closest to themselves without thinking about the consequences for the end of the game. The number of dimensions is fixed (today it s one). Individual utility contains a deterministic component and a random component, both of which are normally distributed this one s new Note, each voter has the same standard deviation for their normal distribution.

26 W-NOMINATE How it works Assume: p legislators, indexed i = 1,, p. q roll calls, indexed j = 1,, q. s dimensions, indexed k = 1,, s. Today s = 1. Legislator i s utility for the yea outcome on roll call j is: deterministic random where y indicates yea.

27 W-NOMINATE Without error, a legislator would vote yea if u i,j,y > u i,j,n, or u i,j,y u i,j,n > 0. With error, a legislator would vote yea if U i,j,y U i,j,n > 0, or u i,j,y u i,j,n + ε i,j,y ε i,j,n > 0. And the probabilities a legislator votes yea or nay are: P(legislator i votes yea) = P(u i,j,y u i,j,n > ε i,j,n ε i,j,y ) P(legislator i votes nay) = P(u i,j,y u i,j,n < ε i,j,n ε i,j,y )

28 W-NOMINATE This allows us to assume the difference in utility is distributed: We then find parameter values that maximize the likelihood of observing the roll call data, using the likelihood function: where τ is the index for yea and nay, P i,j,τ is the probability of voting for choice τ, and C i,j,τ = 1 if the legislator s actual choice is τ; 0 otherwise. The log of the likelihood is:

29 W-NOMINATE And the regression model is: where is the logistic distribution function, x i are the legislators ideal points, Y j and N j are the locations of the yea and nay positions on roll call j, and w and β are scaling parameters. As a further restriction, the outcome points (i.e., alternatives) are estimated in terms of the midpoint, m j, between Y j and N j, and the distance d j between each of those outcomes and the midpoint.

30 W-NOMINATE Normally, we maximize the log-likelihood function by taking the first derivatives w.r.t. each of the parameters, setting the equations equal to zero, and solving. But there are too many parameters and the r.h.s data is missing. So W-NOMINATE estimates in three steps: 1. Estimates m j and d j, conditional on x i, β, and w. 2. Estimates x i, conditional on m j, d j, β, and w. 3. Estimates β and w, conditional on m j, d j, and x i. The procedure repeats until the m j, d j, and x i parameters all correlate at 0.99 or better with the set estimated in the previous estimation.

31 W-NOMINATE Advantages Does not require knowing the interest group s position on a bill (e.g., whether the bill is liberal or conservative) same as before. Does not push legislators to the extreme of the scale same as before. Does not require the assumption that the interest group is exterior of the ideal points same as before. Produces a cardinal ranking based on the likelihood of each ideal point location. Allows for statistical error (with a special boot strap program). Hence, it can provide confidence intervals. Disadvantages Does not work if there is perfect spatial voting (i.e. no error). By assuming everyone has the same utility distribution (just a different mean), it implicitly assumes interpersonal comparability of utility. OC may be better for small samples.

32 Where to Find Existing Ideal Point Estimates Optimal Classification W-NOMINATE DW_NOMINATE Common Space Scores

33 Estimation using R Install TextPad (optional) Install R (click downloads on first page) Install some R packages polc, oc, wnominate, gdata Code for Basic Estimation n_1d.r