disc choice5.tex; April 11, ffl See: King - Unifying Political Methodology ffl See: King/Tomz/Wittenberg (1998, APSA Meeting). ffl See: Alvarez

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1 disc choice5.tex; April 11, Lecture Notes on Discrete Choice Models Copyright, April 11, 2001 Jonathan Nagler 1 Topics 1. Review the Latent Varible Setup For Binary Choice ffl Logit ffl Likelihood for Logit ffl Probabilities 2. RUM: The Random Utility Model formulation of choice. ffl Flexible ffl A strong (or weak) model of behavior. 3. Multinomial Logit ffl Likelihood ffl Probabilities 4. Conditional Logit ffl a More General Systemic Component of Model ffl Probabilities are Calculated the same as MNL 5. Measuring The Effects of Changes in X: ffl Analagous to OLS

2 disc choice5.tex; April 11, ffl See: King - Unifying Political Methodology ffl See: King/Tomz/Wittenberg (1998, APSA Meeting). ffl See: Alvarez and Nagler 1995 ( Perot "; Table 4), first differences 6. Measuring the Effects of Changes in the Choices ffl See: Alvarez and Nagler 1998 ( Collide"; Table 5 - moving Labour Party). 7. Goodness of Fit ffl Percent correctly predicted in 2-choice case. ffl Baseline Prediction in 2-choice case. ffl Percent Correctly predicted in J-choice case: Baseline Prediction Classification Schemes 8. Independence of Irrelevant Alternatives (IIA) ffl What is IIA ffl IIA does not aggregate ffl McFadden test for IIA 9. Multinomial Probit (MNP) ffl Does not impose IIA. ffl Need to put restrictions on ±. ffl Estimation via: Gauss, Gaussx, Limdep ffl Beyond 5 choices?

3 disc choice5.tex; April 11, Some Equivalencies of Logit Models 11. Scobit ffl MNL can be used to recover reduced form CL estimates. ffl MNL is equivalent to Binary Logit (under IIA). ffl What if basic assumption of the shape of the response curve is wrong? 12. Heteroscedastic Probit 13. Selection Bias ffl Heckman ffl Dubin and Rivers

4 disc choice5.tex; April 11, Latent Variable Setup: Binary Probit/Logit y Λ = fi 0 x + ffl (1) y = 1 if y Λ > 0 (2) y = 0 if y Λ» 0 (3) Now assume F is the cumulative distribution for ffl. Prob(y = 1) = Prob(y Λ > 0) (4) = Prob(fi 0 x + ffl > 0) = Prob(ffl > fi 0 x) = 1 Prob(ffl < fi 0 x) = 1 F ( fi 0 x) (5) If F is symmetric about 0, Prob(y = 1) = 1 F ( fi 0 x) = F (fi 0 x) (6) If F is the logistic distribution this gives us logit, if F is the cumulative normal distribution we have probit. In either case, logit or probit would recover consistent estimates of the parameter fi. Logistic distribution looks like: F (x) = e x (7)

5 disc choice5.tex; April 11, So, if F is logistic: Prob(y = 1) = F (fi 0 x) = = e fi 0 x e fi 0 x 1 + e fi 0 x So, if we could estimate fi, then we could compute the quantity of interest (P ). We use maximum likelihood to compute fi: we want to find fi to maximize: Pr(Y j fi; X) In the simple case, we have two possibilities: y i = 1 or y i = 0. Pr(y 1 ;y 2 ; ::::; y N j fi) = Y yi=1 Pr(Y i = 1 j fi; X) Y Pr(Y i = 0 j fi; X) y i =0

6 disc choice5.tex; April 11, Wework with something similar to the above: the likelihood function (L). The following assumes F is symettric, and substitutes F for Pr(Y i = 1): L = Y yi=1 F (X i fi j fi) Y y i =0 (1 F (X i fi j fi)) = Y F (X i fi j fi) y i Y (1 F (Xi fi j fi)) 1 y i We always work with log(l) - or the log-likelihood function. LL = X y i ln(f (X i fi)) + X (1 y i )ln((1 F (X i fi))) So, we take the first derivatives of the above expression with respect to fi, set them equal to 0, and solve for ^fi.

7 disc choice5.tex; April 11, Random Utility Models (RUM) Assume the i th individual's utility of the j th choice is given as: U ij = V ij + ffl ij (8) where V ij is a systemic component of utility and ffl ij is a stochastic component of utility. Assume that the i th individual chooses choice j iff: U ij > U ik 8 k 6= j (9) Notice that ffl ij is subscripted by i and j. We have one disturbance per respondent per choice. Simple setup of V ij : V ij = fi j X ij (10) (11) This model is very flexible: it allows for more than 2 choices. The model is a weak or strong model of behavior.

8 disc choice5.tex; April 11, RUM Example 1: Multinomial Logit V ij = ψ j X i (12) V ij = ψ j0 + ψ j1 pid i + ψ j2 educ i + ψ j3 ideology i (13) V i1 = ψ 10 + ψ 11 pid i + ψ 12 educ i + ψ 13 ideology i V i2 = ψ 20 + ψ 21 pid i + ψ 22 educ i + ψ 23 ideology i V i3 = ψ 30 + ψ 31 pid i + ψ 32 educ i + ψ 33 ideology i So: U i1 = ψ 10 + ψ 11 pid i + ψ 12 educ i + ψ 13 ideology i + ffl i1 U i2 = ψ 20 + ψ 21 pid i + ψ 22 educ i + ψ 23 ideology i + ffl i2 U i3 = ψ 30 + ψ 31 pid i + ψ 32 educ i + ψ 33 ideology i + ffl i3 ffl are iid, Type I Extreme Value. P i1 = Pr[(U i1 > U i2 ) & (U i1 > U i3 )] = Pr[(V i1 + " i1 > V i2 + " i2 ) & (V i1 + " i1 > V i3 + " i3 )] = Pr[(" i2 " i1 < V i1 V i2 ) & (" i3 " i1 < V i1 V i3 )] Notice: ψ is indexed by j.

9 disc choice5.tex; April 11, Normalization of One Set of ψ's U i1 = ψ 1 A i + " i1 U i2 = ψ 2 A i + " i2 U i3 = ψ 3 A i + " i3 P i1 = Pr[(U i1 > U i2 ) & (U i1 > U i3 )] = Pr[(ψ 1 A i + " i1 > ψ 2 A i + " i2 ) & (ψ 1 A i + " i1 > ψ 3 A i + " i3 )] = Pr[(" i2 " i1 < (ψ 1 ψ 2 )A i ) & (" i3 " i1 < (ψ 1 ψ 3 )A i )] P i2 = Pr[(" i1 " i2 < (ψ 2 ψ 1 )A i ) & (" i3 " i2 < (ψ 2 ψ 3 )A i )] P i3 = Pr[(" i1 " i3 < (ψ 3 ψ 1 )A i ) & (" i2 " i3 < (ψ 3 ψ 2 )A i )] 3 Quantities: Ψ 1 Ψ 2 = X Ψ 1 Ψ 3 = Y Ψ2 Ψ 3 = Z But: Z = Y X

10 disc choice5.tex; April 11, Example: Ψ 1 = 7 Ψ 2 = 4 Ψ 3 = 0 Yields: Ψ 1 Ψ 2 = 3 Ψ 1 Ψ 3 = 7 Ψ2 Ψ 3 = 4 Same Result if: Ψ 1 = 24 Ψ 2 = 21 Ψ 3 = 17 Yields: Ψ 1 Ψ 2 = 3 Ψ 1 Ψ 3 = 7 Ψ2 Ψ 3 = 4

11 disc choice5.tex; April 11, Probabilities: We do not prove the following here; but it is true. P ij = e V ij PJ k=1 ev ik (14) Simple 2-Choice Case: Pr(Y i = 1) = = This looks like binary logit: F (x) = = = = e fi 0 1 X i e fi 0 1 X i e fi 0 1 X i + e fi 0 2 X i e fi 0 1 X i e x e x 1 e x + 1 e x e x e x + 1 (15) (16)

12 disc choice5.tex; April 11, Table 2: Multinomial Logit and Binomial Logit Estimates British Election (Alvarez and Nagler 1998) Conservative/Alliance Labour/Alliance MNL BL MNL BL Intercept -4.33* -4.40* 4.55* 5.26* (.74) (.76) (.81) (.86) Defense.14*.17* -.17* -.19* (.03) (.03) (.03) (.03) Phillips Curve.08*.10* (.02) (.03) (.03) (.03) Taxation.13*.14* -.06** -.08* (.03) (.03) (.03) (.04) National..16*.16* -.16* -.20* (.03) (.03) (.03) (.03) Redist..07*.06* -.08* -.09* (.02) (.02) (.03) (.03) Crime.08*.08* (.03) (.03) (.02) (.02) Welfare.11*.12* -.11* -.10* (.02) (.02) (.03) (.03) South * -.45* (.16) (.17) (.21) (.22) Midlands (.17) (.17) (.21) (.21) North *.61* (.17) (.18) (.19) (.20) Wales * 1.46* (.35) (.36) (.31) (.33) Scot **.68*.61* (.25) (.26) (.25) (.26) Union *.37*.35* (.16) (.16) (.16) (.17) Public Employee (.15 (.15 (.16 (.16 Blue Collar *.80* (.15) (.16) (.17) (.17) Gender.29*.33* (.14) (.14) (.15) (.16) Age * -.24* (.05) (.05) (.05) (.05) Homeowner.31** * -52* (.18) (.18) (.17) (.17) Income.07*.07* * (.03) (.03) (.03) (.03) Education -.81* -.92* ** (.31) (.31) (.35) (.36) Inflation.28*.31* (.10) (.11) (.12) (.12) Taxes * (.06) (.07) (.07) (.07) Unempl..30* (.06) (.06) (.07) (.08) Number of Observations Log Likelihood Standard Errors in parenthesis. Λ indicates significance at 95% level; ΛΛ indicates significance at 90% level.

13 disc choice5.tex; April 11, Consider just a 2-choice comparison: U i1 = fi 0 1 X i + ffl i1 (17) U i2 = fi 0 2 X i + ffl i2 (18) Pr(U i2 > U i1 ) = Pr(fi 0 1 X i + ffl i1 < fi 0 2 X i + ffl i2 ) = Pr(ffl i1 ffl i2 < (fi 0 2 fi 0 1 )X i) (19) Back to latent variable model: Pr(Y i = 1) = Pr(u i < fi 0 X i ) (20) So: u i ß ffl i1 ffl i2 fi ß fi 2 fi 1 (21) If we assume ffl ij are independent, identically distributed with Type 1 Extreme Value distribution, then ffl i1 ffl i2 is logistically distributed. F (ffl ij ) = exp(e ffl ij ) (22) Assume fi 2 = 0. This is just a normalization, we could assume fi 2 = 17. The only thing that matters is U i1 U i2.

14 disc choice5.tex; April 11, Conditional Logit U ij = ψ 0 j A i + fi 0 X ij + ffl ij (23) where: U ij = utility of the i th respondent for the j th alternative. A i = characteristics of the i th respondent. X ij = characteristics of the j th alternative relative to the i th respondent. ψ j = a vector of parameters relating the characteristics of a respondent to the respondent's utility for the j th alternative. fi = a vector of parameters relating the relationship between the respondent and the alternative (X ij ) to the respondent's utility for the alternative. ffl ij = random disturbance for the i th respondent for the j th alternative; iid, Type I Extreme Value. Notice: ψ j varies across choices. Both conditional logit and multinomial logit models assume that the disturbances, ffl ij, are independent across alternatives.

15 disc choice5.tex; April 11, RUM Example 2: Conditional Logit V ij = fi 1 X ij + ψ j A i V ij = fi 1 issuedist ij + ψ j0 + ψ j1 pid i + ψ j2 educ i V i1 = fi 1 issuedist i1 + ψ 10 + ψ 11 pid i + ψ 12 educ i V i2 = fi 1 issuedist i2 + ψ 20 + ψ 21 pid i + ψ 22 educ i V i3 = fi 1 issuedist i3 + ψ 30 + ψ 31 pid i + ψ 32 educ i So: U i1 = fi 1 issuedist i1 + ψ 10 + ψ 11 pid i + ψ 12 educ i + ffl i1 U i2 = fi 1 issuedist i2 + ψ 20 + ψ 21 pid i + ψ 22 educ i + ffl i2 U i3 = fi 1 issuedist i3 + ψ 30 + ψ 31 pid i + ψ 32 educ i + ffl i3

16 disc choice5.tex; April 11, Table 4 Conditional Logit Estimates British Election Conservative/Alliance Labour/Alliance Defense a -.18* (.02) Phillips Curve -.11* (.02) Taxation -.16* (.02) National. -.18* (.02) Redist. -.08* (.02) Crime -.10* (.05) Welfare -.14* (.02) Intercept * (.69) (.75) South * (.17) (.21) Midlands -.29**.19 (.17) (.20) North * (.18) (.19 Wales * (.36) (.31) Scot * (.25) (.25) Union -.50*.37* (.16) (.16) Public Employee (.15) (.16) Blue Collar.11.70* (.15) (.16) Gender.28*.00 (.14) (.15) Age * (.05) (.05) Homeowner.37* -.54* (.18) (.16) Income.07* -.06 (.03) (.03) Education -.82* -.61** (.32) (.35) Inflation.28* -.03 (.10) (.11) Taxes (.07) (.07) Unempl..28* (.01 (.06) (.07) N 2131 Log Likelihood a The seven issues represent distance absolute value from the respondent tothe mean of the party position. Standard Errors in parenthesis. Λ indicates significance at 95% level; ΛΛ indicates significance at 90% level.

17 disc choice5.tex; April 11, Probabilities We do not prove the following here; but it is true. P ij = e fi0 X ij + ψj 0 A i P J X ik + ψ 0 k=1 efi0 k A i = e V ij P J k=1 ev ik Same probabilities as MNL. Just another form of MNL.

18 disc choice5.tex; April 11, Goodness of Fit, Predicted Values, Classification There is no R 2. Pseudo-R 2 "! even worse than R 2. Compute ^P ij. ^P ij! ^Y i Classification Rule (binomial): ^Y i = 1 if ^P i > :5 Percent Correctly Predicted: correct prediction": ^Yi = 1 and Y i = 1 or, ^Y i = 0 and Y i = 0. PCP = 100(# of Correct Predictions) N PMC = Percent in Modal Category

19 disc choice5.tex; April 11, PRE [Proportional Reduction in Error]: PRE = PCP PMC 1 PMC Example 1: PCP =.85 PMC =.80 Example 2: PRE = PCP =.75 PMC =.50 :85 :80 1 :80 = :05 :20 = :25 PRE = :75 :50 1 :50 = :25 :50 = :50 Classification Rule (multinomial): If you require ^P ij > :5 for ^y i = j, then you may not classify some observations. So: ^y i = j if ^Pij > ^P ik 8k 6= j.

20 disc choice5.tex; April 11, Unevenly Distributed Data: If the data has a very skewed distribution (90% 0's; 10% 1's), you may never predict 1 as an outcome. Common in multinomial cases where one case may be relatively rare. A Useful Table (hypothetical numbers): Pred Observed Cons Labour All d Total d Cons d Labour d Alliance Total This table has all the information (except uncertainty). We can see that we do not predict votes for Alliance very well.

21 disc choice5.tex; April 11, cpcp (Herron ) Problem: We treat ^P i = :51 and Y i = 1 the same as ^P i = :95 and Y i = 1. But, we should give more credit for the latter prediction: it is a `better' prediction. Solution: Expected PCP". epcp = 1 N X Yi=1 ^P i + X (1 ^P i ) 1 A Y i =0 Example: Y i ^Pi epcp = 1/3 ( ) =.6

22 disc choice5.tex; April 11, epcp - Multinomial epcp = 1 N NX JX ^Pij (y i == j) i=1 j=1 epcp = 1 3 X Yi=1 ^P i1 + X Yi=2 ( ^P i2 ) + X Yi=3( ^P i3 ) 1 A

23 disc choice5.tex; April 11, Computing Effects of Changes in Characteristics of a Respondent P ij = e fi0 X ij + ψj 0 A i P J X ik + ψ 0 k=1 efi0 k A i (24) Say: we want to know what happens if a i were to change to ~a i ; say a i were to increase by z units. [a i is a particular element of the vector A i.] 1. Compute : ~a i = a i + z 2. Compute: Vij ~ = fi 0 X ij + ψ ~ 0 j A i 3. Compute P ~ ij 4. Compute: Pij ~ ^P ij The last difference is the quantity of interest. We could also compute this for everyone in the sample, and then compute the mean of Pij ~ ^P ij ; and get the effect of all respondents changing their taste on characteristic a by z units.

24 disc choice5.tex; April 11, Table 4 (A/N AJPS) Effects of Economics, Issues, and Anger in the 1992 Election Probability of Voting For: Bush Clinton Perot Personal Finances Better Worse Difference National Economy Better Worse Difference Voter Ideology a Near Far Difference Minorities Assist No Assist Difference Abortion Pro-Life Pro-Choice Difference Term Limits For Against Difference Note: Table entries are the predicted probabilities of a hypothetical individual voting for Clinton, Bush or Perot based on different values of the row-variable. a Probabilities for each of the candidates in the voter-ideology row are based on the ideological distance between the voter and the particular candidate.

25 disc choice5.tex; April 11, Computing Effects of Changes in Characteristics of an Alternative ^P ij = e ^fi 0 X ij + ^ψ 0 j A i PJ k=1 e ^fi 0 X ik + ^ψ 0 k A i (25) We want to alter A i, and recompute ^P ij. Say: X ij = (resp i choice j ) (26) We want to know what happens if the j th choices `moves' z units to the right. 1. Set: g choicej = choice j + z 2. Compute: ~ Xij = (resp i g choicej ) 3. Compute: Vij ~ = fi ~ 0 Xij + ψ 0 j A i 4. Compute P ~ ij 5. Compute: Pij ~ ^P ij The last difference is the quantity of interest.

26 disc choice5.tex; April 11, Table 5 (A/N AJPS) Conditional Logit Estimates of Effect of Movement By the Labour Party +/- 1 Standard Deviation 2 - British Election Conservatives Labour Alliance Baseline Defense 1 ff ff Difference Phillips 1 ff ff Difference Taxation 1 ff ff Difference Nationalization 1 ff ff Difference All Issues 1 ff ff Difference Note: Estimated impact of the Labour party moving from one half a standard deviation to the left of its mean perceived position to one half a standard deviation to the right of its mean perceived position on each of seven issues. Column entries are estimated aggregate vote-shares.