Technical Appendix for Voting, Speechmaking, and the Dimensions of Conflict in the US Senate

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1 Technica Appendix for Voting, Speechmaking, and the Dimensions of Confict in the US Senate In Song Kim John Londregan Marc Ratkovic Juy 6, 205 Abstract We incude here severa technica appendices. First, we formay derive the expressions for c vote and b vote p that appear in section two of the paper. The next contains some detais on the choice theoretic mode underying the standard topic mode. The remainder contain the detais for impementing the SFA mode. A The Legisator and Proposa Intercepts from the Voting Mode Let s start with the egisator s utiity from the aye : and nay aternatives: U ({r pd } D = 2 D a d (r pd x d 2 + ξ aye p ( U ({q pd } D = 2 D a d (q pd x d 2 + ξ nay p (2 Next, et s cacuate the difference between these expressions to get the egisator s preference intensity for the aye outcome. Substituting from expressions ( and (2 we have: Assistant Professor, Department of Poitica Science, Massachusetts Institute of Technoogy, Cambridge, MA, Emai: insong@mit.edu, URL: Professor of Poitics and Internationa Affairs, Woodrow Wison Schoo, Princeton University, Princeton NJ Phone: , Emai: jb@princeton.edu Assistant Professor, Department of Poitics, Princeton University, Princeton NJ Phone: , Emai: ratkovic@princeton.edu, URL: ratkovic

2 V p = U ({r pd } D U ({q pd } D = D a d (r pd x d 2 aye + ξ 2 p ( D a d (q pd x d = = D a d 2 (q2 pd r2 pd + D c vote a d +b vote p 2 2x pd (r pd q pd g vote pd + ξ aye p ( D a d D 2 (q2 pd r2 aye nay pd + E{ ξ p } E{ ξ p } + ( ξnay p ξ aye ξ nay p nay ξ p a d 2 2x pd (r pd q pd g vote pd p + E{ ξ aye nay p } E{ ξ p } (3 ɛ p Now et: E{ ξ aye p } = πaye + ϕ aye p and E{ ξ nay p } = πnay + ϕ nay p substituting this into our expression for c vote + b vote p we have: D a D d 2 (q2 pd r2 aye nay pd + E{ ξ p } E{ ξ p } = = π aye π nay c vote = c vote a d 2 (q2 pd r2 pd + (πaye + b vote p + + ϕ aye p (π nay + ϕ nay p D a d 2 (q2 pd r2 pd + ϕaye p ϕ nay p b vote p Now et s return to the ast ine of expression (3 and substitute:

3 Vp = U ({r pd } D U ({q pd } D = ( D a d D 2 (q2 pd r2 aye nay pd + E{ ξ p } E{ ξ p } + = c vote + b vote p + c vote +b vote ( ξnay p aye p ξ p D Expression (4 matches expression (4 in the text. + E{ ξ aye nay p } E{ ξ p } ɛ p a d 2 2x pd (r pd q pd g vote pd a d x pd g vote pd ɛ p (4 B Choice Theoretic Underpinnings of Topic Modes Whie we opt for SFA, it is usefu to consider the behavior that woud ead one to adopt a topic mode for egisative speech. One way to do this is to suppose that a egisator s speech is generated by the random arriva of opportunities to speak. At each opportunity the egisator must choose one word from a exicon, which we represent by a W vector ω, with each entry corresponding to a different word. Each word has a spatia ocation, which for the moment we pace on a singe dimension. Legisator j {... V } woud derive utiity u( w x j + η j,t from uttering word j {... W } at time t. Shoud the opportunity to speak at time t actuay arise, the egisator utters the word offering the greatest utiity. To keep things simpe we assume that η j,t and η r,s are independent if either j r or t s. Paraeing the deveopment in Maddaa (983, we operationaize our mode with a distributiona assumption for η j,t R, which we take to foow a type I extreme vaue distribution, with probabiity density: f(η = e (η+e η 2

4 and by concretizing the utiity function u( w x j : u( w x j = 2( w x j 2 (5 where x j is the preferred ideoogica signa that egisator j woud ike to convey, and w is the ideoogica connotation of word i. Again foowing Maddaa (983 we see that the probabiity that at a randomy chosen time t egisator j prefers word i to a other eements of the exicon is: q j = eu( w x j W e u( w k x j k= Let word correspond to a stop word. We can rewrite the probabiity j uses word i if she has the opportunity to speak at t as: q j = eu( w x j u( w x j W e u( w k x j u( w x j k= substituting from equation (5 into our expression for q j we have: e x jg +b q j (x, g, b = (6 + W e x jg k +b k k=2 where g k = w k w and b k = w k+ w 2 for k {2... W }. The probabiity of an observed W vector c of word counts is: W q j (x, g, b cw (7 w= With the right choice of Dirichet priors this turns into the atent Dirichet mode of Bei, Ng and Jordan (2003 if we set x j = g = 0 for a i and j. In the idea point setting, though, x j and 3

5 g i correspond with precisey the preferred outcomes and term ideoogies with which we are most interested. Estimation for these modes are not straightforward, requiring a Metropois agorithm or variationa approximations. We favor SFA on theoretica grounds, as it aows egisators to seect words as a function of their preferred outcomes. We aso favor it because it offers a tractabe Gibbs samping scheme for most of the parameters, which we address in the next section. C Estimation of SFA We now shift to a more condensed notation. Hereafter, we reindex the vote and term outcomes using a common index, j, which fas into two sets: J terms and J votes for whether the observed outcome (now a common Y j is a term outcome or vote outcome, and J = J terms + J votes. We wi aso supress the superscript for the θ terms w ikeihood is given by: and θ votes p whie changing to the joint subscript j. The L ( θ vote, θ term, τ T, L ( P Ṽ = = p= P r{v p = W +P Ṽp } 2P W w= P r{t w = T w } W +P 2W. (8 where: P r{t w = T Φ ( θ terms w τ 0 w } = ( ( Φ θw terms τ Tw Φ θw terms τ Tw P r{v p = Ṽp } = Φ ( (2Ṽp θp vote T w = 0 0 < T w (9 (0 and the prior structure is given by: 4

6 c, b w i.i.d. N (µ, µ N (0, g wd, x d og(β, og(β 2 i.i.d. N (0, 4 i.i.d. N (0, ( Pr(a d = 2λ e λ a d Pr(λ =.78e.78λ Combining the ikeihood and prior gives us the posterior: Pr(θ j, τ, β, β 2 Y = L j J { { } (j {jvotes} Φ (θ j Y j ( Φ (θ j Y j { Φ ( ( } (j {Jterms} τ Yj θ j Φ θj τ Yj } 2π e 2 µ2 d D L L 2π e 2 (c µ 2 2 2π e 8 (x d 2 d D L j J 2π e 2 (b j µ 2 2 2π e 8 (g jd 2 e 2 (og β 2 e 2 (og β 2 2 e.78λ β 2π β 2 2π d D 2λ e λ a d (2 We impement two forms of data augmentation. In the first, for each observation we introduce a norma random variabe Z j as is standard in atent probit modes (Abert and Chib, 993. This transforms the ikeihood into a east squares probem, as: Pr(Y j = k Z j, θ j, τ, β, β 2 = L j J 2π e 2 (Z j θ j 2 (3 The second form of augmentation invoves representing the doube exponentia prior for a d to maintain conjugacy. Foowing Park and Casea (2008, we introduce atent variabes τ, such that: 5

7 d τ 2 N (0 D, D τ (4 D τ = diag( τ 2, τ 2 2,, τ 2 D (5 τ 2, τ 2 2,..., τ 2 D d D λ 2 2 e λ2 τ 2 d /2 d τ 2 d (6 where, after integrating out τ 2, we are eft with the LASSO prior. The proposed method differs from the presentation in Park and Casea (2008 in that we know σ 2 =, by assumption. C. The Gibbs Samper Next, we outine the Gibbs samper. A conditiona posterior densities are conjugate normas except λ, τ 2, β, and β 2. For a derivation of the posterior densities of λ and τ 2, see Park and Casea (2008. We fit β and β 2, which determine τ, using a Hamitonian Monte Caro agorithm, but first we describe the Gibbs updates. The Gibbs updates occur in two steps. First, we pace a data on the atent z scae. Second, we update a of the remaining parameters. For the first step, we sampe as: T N (θ j,, 0, ; Y j =, j j votes T N (θ j,,, 0 ; Y j = 0, j j votes Zj (7 T N (θ j,, τ k, τ k ; Y j = k, j J terms N (θ j, ; Y j missing Note that we have ignored missing vaues up to this point. In the Bayesian framework used here, imputing is straightforward: the truncated norma is repaced with a standard norma, whether term or vote data. Next, we update a of θ j except for τ using a Gibbs samper, as: 6

8 µ N c N b j N ( L J = j= Z j, LJ + L 2 J 2 + ( J j= Z j J +, J 2 + ( L = Z j L +, L 2 + (8 (9 (20 Z j = Z j c b j + µ (2 Update x, w, v from SVD of Z (22 a N X = ( A X vec(z, A where [ ( ( ( ] vec x g : vec x 2 g 2 :... : vec x L g L A = X X + T with T = diag(τ 2 x d N where J j= Z j, da dg j J j= N g j d ( a 2 dg 2 j d + 4J L = Z j, d a dx d L =, J j= (a 2 dg 2 j d + 4J ( a 2 dx 2 d + 4L and, L = (a 2 dx 2 d + 4L (23 (24 (25 τ 2 InvGauss ( λ 2 Gamma Z = Z j, d j x dg jd a d d d ( λ 2 a 2 d L +,, λ 2 L τ 2 / = (26 (27 C.2 The Hamitonian Monte Caro Samper We have no cosed form estimates for the conditiona posterior densities of β and β 2. To estimate these, we impement a Hamitonian Monte Caro scheme adapted directy from Nea (20. We 7

9 adapt the agorithm in one important manner: rather than taking a negative gradient step, we cacuate the numerica Hessian and take a fraction (α of a Newton-Raphson step at each. We seect α so that the acceptance ratio of proposed (β, β 2 is about.4. Specificay, et dev(β, β 2 denote the estimate deviance at the point (β, β 2. Define the numerica gradients, dev(β, β 2 and 2 dev(β, β 2 as the estimated gradient at (β, β 2 and dev(β, β 2, 22 dev(β, β 2, and 2 dev(β, β 2 as the cross derivative. Next, define the empirica Hessian as: dev(β, β 2 2 dev(β, β 2 Ĥ(β, β 2 = (28 2 dev(β, β 2 22 dev(β, β 2 We impement the agorithm in Nea (20 exacty, except instead taking updates of the form: β β 2 we instead do updates of the form: + := β β 2 (β, β 2 α (29 2 (β, β 2 β β 2 + := β β 2 } (β, β 2 α {Ĥ(β, β 2 (30 2 (β, β 2 where the Hessian and gradients are updated every third update of the parameters. The step ength parameter α is adjust every 50 iterations to by a factor of 4/5 if the acceptance rate is beow 0%, 5/4 if the acceptance rate is above 90%, and eft the same otherwise. After the burn-in period, the acceptance rate eves off around 45%. We impement twenty steps in order to produce a proposa. 8

10 C.3 Numerica Approximation of the Deviance Cacuating the gradient and Hessian terms, and assessing the proposa, in the Hamitonian Monte Caro scheme requires evauating functions of the form (a, b = og(φ(a Φ(b. Unfortunatey, for vaues of a and b much arger in magnitude than 5.3 produces returns vaues of or 0, eaving it impossibe to evauate the ogarithm. Extrapoating from the observed vaues yieds the inear approximation: (a, b = a b a 2 b 2 og( a b {og( a b } 2 ab γ (3 where 9

11 γ = We derived the vaues for γ from fitting a mode over the range 4 b < a 8. (32 We get a mean absoute error of 0.065, or 0.08% error as a fraction of the vaue returned by R. We use this approximation in order to extrapoate to vaues where R returns vaues of NA or Inf for f(a, b. D How do Dimensions and Topics Differ? As we noted in the introduction, there is an apparent inconsistency between Congress voting across a broad array of topics (nationa defense, agricuture, socia insurance, etc. whie ro ca voting over this range of substantive topics arranges into a singe dimensiona ideoogica space. This seeming paradox is cosey tied to the distinction between topics, as estimated by topic modes, and dimensions, as estimated by SFA. There is no actua contradiction. Which rendition of Congress a researcher finds more usefu: a coection of substantive egisative topics subject to Congressiona dispensation, or a ow dimensiona radiography of the ideoogica space that organizes Congress, depends on the researcher s objectives. To differentiate topics from dimensions, consider the distinction between surface and source traits. Factor anaysis pioneer Raymond Catte characterized a surface trait as... an obvious going together of this and that whie he defined the source trait as the underying contributor or 0

12 Data Generating Process for Comparing SFA and Topic Modes: Likeihood of Voting Yes For Each Legisator by Vote Legisator Vote Vote 2 Vote 3 Vote 4 Vote 5 Vote 6 Legisator 2 Legisator 3 Legisator 4 Legisator 5 Legisator 6 Legisator 7 Legisator 8 Legisator 9 Legisator 0 Darker Coor Means More Likey to Vote Yay Figure : Simuated data setup. Legisators are arrayed across rows and votes across coumns. The darker the square, the more ikey the egisator to vote Aye on that particuar vote. determiner of the observed surface traits (?, p. 45. As an exampe, he offered common symptoms of schizophrenia as surface traits, whie the underying syndrome itsef is the source trait. We provide an exampe to iustrate the point. Assume ten egisators facing six votes. For simpicity, the true underying probabiity of voting Aye comes from an underying process with one ideoogica dimension, as presented in Figure. Legisators are arrayed across rows and votes across coumns. The darker the square, the more ikey the egisator to vote Aye on that particuar proposa. Legisators 5 are more ikey to vote Aye on the first 3 votes and more ikey to vote Nay on the ast 3. Legisators 6 0 are more ikey to vote Nay on the first 3 votes, and Aye on the ast 3. Legisators 5 and 6 are reative moderates, whie bis 3 and 4 are reativey noncontroversia. We fit both SFA and a topic mode to a draw of the vote data. In order to fit a topic mode, we assume each egisator uttered six terms representing their vote and the bi number. For exampe, Specificay, et s i = { 4.5, 3.5,..., 4.5} and w j = { 2.5,.5,..., 2.5}. We drew Y j Bern (Φ(s iw j/2.

13 SFA Resuts Topic Mode Resuts Dimension Most Preferred Bi Dispacements Outcomes Weights Topic Topic 2 Topic Nay on 2 Nay on 4 Aye on Nay on Nay on 6 Nay on Aye on 4 Aye on 2 Aye on Aye on 6 Aye on 3 Aye on Tabe : Resuts from SFA and a Topic Mode on the Simuated Dataset. if the egisator voted Nay on vote 4, we assume that they said Nay on 4, and enter that into a topic mode as a unique word. Each egisator then uttered six eements from the set { Aye on, Nay on, Aye on 2, Nay on 2,..., Aye on 6, Nay on 6 }, one of either Aye or Nay on votes -6. We impemented the EM version of SFA and aso gave the same data to the a Structura Topic Mode, as impemented in stm. We fit a three-topic mode to the data. Four-, five-, and six-topic modes returned quaitativey simiar resuts. The eft three coumns of Tabe contain the resuts from SFA. The first coumn contains the estimated posterior mode, and ony the first dimension has a non-zero mode. The next two coumns contain each egisator s idea points and the bi estimates. SFA returns estimates of the underying structure, correcty recovering the unidimensiona structure of the data generating process, and identifying egisators -4 and 7-0 as reative extremists at opposite ends of the spectrum. SFA aso successfuy identifies the reativey moderate egisators, 5 and 6, and correcty notes which proposas wi draw support from which egisators. The rightmost three coumns of Tabe report the topic mode estimates, presenting the first four terms of the three fitted topics. Consider the first topic. Legisators that vote Nay on votes and 2 are ikey to vote Aye on votes 4 and 6. Simiary, considering the second topic, egisators 2

14 who vote Nay on votes 4 and 6 are ikey to vote Aye on votes 2 and 3. The topic mode is returning surface traits symptoms of an underying ideoogica dimension, but not the dimension itsef. Topic modes provide an exceent too for summarizing word co-occurrence if the goa is primariy descriptive (e.g.,?. If, on the other hand, the researcher seeks to identify the underying structure of the source of the observed behavior, whether it is speech or voting, then SFA provides a cearer picture. 3

15 References Abert, James H. and Siddhartha Chib Bayesian Anaysis of Binary and Poychotomous Response Data. Journa of the American Statistica Association 88: Bei, David M., Andrew Y. Ng and Michae I. Jordan Latent dirichet aocation. J. Mach. Learn. Res. 3: Maddaa, Gangadharrao Soundayarao Limited Dependent and Quaitative Variabes in Econometrics. New York: Cambridge. Nea, Radford. 20. MCMC Using Hamitonian Dynamics. In Handbook of Markov Chain Monte Caro, ed. Steve Brooks, Andrew Geman, Gain Jones and Xiao-Li Meng. CRC Handbooks of Modern Statistica Method Chapman and Ha. Park, Trevor and George Casea The bayesian asso. Journa of the American Statistica Association 03(482: