Hierarchical Bayes prediction for the 2008 US Presidential election

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1 MPRA Much Persoal RePEc Archve Herarchcal Bayes predcto for the 008 US Presdetal electo Pakaj Sha ad Ashok Basal Faculty of Maagemet Studes, Uversty of Delh, Delh, Uversty of Delh 3. August 008 Ole at MPRA Paper No. 0470, posted 5. September :55 UTC

2 Herarchcal Bayes Predcto for the 008 US Presdetal Electo Pakaj Sha ad Ashok K. Basal Abstract I ths paper a procedure s developed to derve the predctve desty fucto of a future observato for predcto a multple regresso model uder herarchcal prors for the vector parameter. The derved predctve desty fucto s appled for predcto a multple regresso model gve Far (00) to study the effect of fluctuatos ecoomc varables o votg behavor U.S. presdetal electo. Numercal llustratos suggest that the predctve performace of Far s model s good uder herarchcal Bayes setup, except for the 99 electo. Far s model uder herarchcal Bayes setup dcates that the forthcomg 008 US presdetal electo s lkely to be a very close electo slghtly tlted towards Republcas. It s lkely that republcas wll get 50.90% vote wth probablty for w US Presdetal Electo.. Itroducto Cosder a predcto problem where the outcomes x, x,..., x of formatve expermets are depedet wth probablty desty fucto f θ ),,,...,. The ( x outcome x + of a future depedet expermet has p.d.f. f ( x + θ + ), the parameter θ has same parameter space Θ as that of θ (,,..., ). Our objectve s to derve the + predctve desty fucto of x +, gve the outcomes x, x,..., x of formatve expermets for predcto a multple regresso model. Oe approach to deal wth ths predcto problem s to employ herarchcal prors a Bayesa framework. Herarchcal prors are used whe the parameter θ s a vector ( θ, θ,..., θ ) ad t s assumed that θ (,,..., ) are dstrbuted ɶ depedetly wth commo pror dstrbuto g ( θ λ) ad a secod stage pror dstrbuto g( λ ) s placed o t,.e., o λ. A herarchcal Bayesa regresso model has bee foud useful the area of appled ecoometrcs ad statstcs. Ldley & Smth (97) tally developed the geeral Bayesa lear model, whch s also kow as (lear) herarchcal model. Polasek (984) developed a emprcal Bayes estmato of a -stage herarchcal model. Polasek & Potzelberger (988) carred out robust Bayesa aalyss wth a herarchcal tme seres model usg Austra ecoomc data. Berger ad Berler (986) used ε cotamated class of prors to represet the ucertaty both g ( θ λ) ad g( λ) to vestgate the robustess wth respect to herarchcal prors. Atchso & Dusmore (975) llustrates the wde applcablty of Bayes predctve approach. Faculty of Maagemet Studes, Uversty of Delh, Delh. (Emal: pakaj-sha@fms.edu) Departmet of Statstcs, Uversty of Delh, Delh

3 I secto, we demostrate the stadard Bayesa method to fd the predctve desty fucto of a future observato x +, gve the outcomes of a formatve expermet, uder herarchcal prors. I secto 3, the derved predctve desty fucto s modfed for the purpose of predcto a multple regresso model, assumg that θ 's are depedet ad ther pror dstrbutos are descrbed two stages. The expressos for oe perod forward forecast ad predctve terval are obtaed sectos 4 ad 5. I secto 6, to demostrate the herarchcal Bayes approach to forecast the 008 US presdetal electo, the derved results are appled to the multple regresso model ad data gve Far (00) for studyg the effect of fluctuatos ecoomc varables o votg behavor U.S. presdetal electo. Far (978) examed the ecoomc determats of the presdetal popular vote. Far's model has cotrbuted sgfcatly to research to presdetal electo. The more recet works the area are foud Berry ad Harpham (996), Erkso ad Wleze (996), Hbbs (000) ad Far (004). Gleser (99, 005) crtcally exames the Far s model. We deote desty fucto g (.) o parameter space Θ (.e., pror as well as posteror), desty fucto f (.) o the sample observatos ad p (.) as predctve desty fucto to smplfy the otatos.. Predcto Uder Herarchcal Prors Let x, x,..., x be depedet observatos from f ( x θ ),,,...,, where θ s are depedet ad ther pror dstrbuto may be descrbed two stages. Stage: θ s are codtoally depedetly dstrbuted as g( θ λ ) wth a commo parameter λ Λ. Stage : The parameter λ at stage has a proper pror dstrbuto g( λ ). Let the future observato x+ be dstrbuted as f ( x + θ + ) ad θ + has the same parameter space Θ as that of θ (,,..., ). The predctve desty fucto of the future observato x +, gve x { x, x,..., x }, may be obtaed as follows: where, p( x+ x) p( x+ θ ) g( θ x) dθ (.) Θ P( x+ θ ) f ( x+ θ+ ) g( θ + θ ) dθ + (.) Θ g( θ+ θ ) g( θ+ λ) g( λ θ ) dλ (.3) Λ

4 g( λ θ ) g( λ) g( θ λ) g( λ) g( θ λ) dλ Λ (.4) ad g( θ x) g( θ x, λ) g( λ) dλ (.5) Λ g( θ x, λ) Θ f ( x θ ) g( θ λ) f ( x θ ) g( θ λ) dθ (.6) f ( x θ ) g( θ λ), (.7) f ( x θ ) g( θ λ) dθ Θ sce x,..., be depedet. Example., x x are depedet radom varables ad θ, θ,..., θ are also assumed to Let x, x,..., x be depedet observatos from N( θ, r),,,...,.. θ ad kow commo precso r. Let the future observato x + ~ N ( θ, r + ), wth mea. Assume that θ s are depedet ad ther pror dstrbutos are descrbed two stages (c.f. Berger (985). Stage : θ s are depedet ad ormally dstrbuted each wth mea µ ad kow precso τ. We have g τ τ (.8) π ( θ µ ) exp[ ( θ µ ) ] Stage : the commo parameter µ at stage has a ormal pror dstrbuto wth mea a ad precso b ; t s represeted by g b b (.9) π ( µ ) exp[ ( µ a) ] 3

5 Though the MCMC methods freed the aalysts from usg cojugate pror dstrbutos for mathematcal coveece, the advatage of cojugate pror s that t treats the pror formato as f t were a prevous sample of the same process. Let us use the fact that the sample mea provdes the suffcet statstc for the ukow mea of the ormal populato. Let we fd θ θ ad x g( µ θ ) g( µ ) g( θ µ ) g( µ ) g( θ µ ) dµ x, τ ' τ exp[ ( µ c) ] π (.0) g( θ θ ) g( θ µ ) g( µ θ ) dµ + + τ ' exp[ π τ ' ( + θ c) ] (.) p( x θ ) f ( x θ ) g( θ θ ) dθ g( θ x, µ ) τ " τ" exp[ ( x ) + c ] π f ( x θ ) g( θ µ ) f ( x θ ) g( θ µ ) dθ ( r+ τ ) ( r+ τ ) exp[ ( θ µ ') ] π (.) (.3) g( θ x) g( θ x, µ ) g( µ ) dµ 4

6 l l exp[ ( θ g) ] π Thus the predctve desty fucto of a future observato x +, gve x, s gve by (.4) p p( x ( x + x) + θ ) ξ ( θ x) dθ l4 l4 exp[ ( x+ g ) ] π (.5) Where, l " l 4 τ, l3 + l τ l 3 τ" ( ), τ ' l ( r+ τ ) b, l + b l τ r + τ, rx+ τa g, r + τ g ba+ τg) (, τ ' rτ ' τ" r + τ ', ττ' τ ', τ τ τ+ ', τ ' τ+ b, τ ad c ( τ θ + ba) /( τ + b) 3. Predcto the Regresso Model Let the formatve expermets assume ormal regresso of edogeous varable x o m exogeous varables t, t 3... t m. x β+ βt βmtm+ ε,,,..., (3.) where, each ε ~ N(0, σ ) wth mea 0 ad varace σ so that E ( x ) T β β [ β β... β ]' ad T [ t t... t ]. wth m 3 m The formatve expermets yeld observatos x, x,..., x, whch are depedetly dstrbuted havg ormal p.d.f. wth respectve meas θ, θ,..., θ ad commo varace σ. Here θ ( T β ). 5

7 Cosder the data set represeted by T t t3... tm x t t3... t m x, X..... t t3... t x m. The least square estmate of β s gve by ˆ β ( T ' T ) T ' X. βˆ has a multvarate ormal dstrbuto,.e. ˆ ~ (, ( ' ) β N ) m β σ T T dstrbuto,.e., ˆ ~ (, ( ' ) T β N T β σ T T T T ). adtβ ˆ has a ormal Thus T ˆ σ ' ' β ~ N( Tβ, T ( ) ) T T T. Note that x x T ˆ β s a suffcet statstc for θ ( ) θ, where θ Tβ. σ σ We have x ~ N( θ, p ) wth mea θ ad varace p, where p The precso of x s gve by kr where, k ad r V ( x) pσ p σ. T ( T T ) T Thus x ~ N( θ, kr) wth mea θ ad precso kr. Let the outcome x + of future expermet be also detcally dstrbuted wth mea θ + T+ β ) ad precso r,.e., ( + k x ˆ T β ~ N(, k ), where k ( T ( T T ) T. + θ + r + + ) 6

8 Therefore, the predctve desty fucto of future observato x +, whe the herarchcal pror dstrbuto for θ ( T β ) s gve by equatos (.8) ad (.9) ad f r σ s assumed to be kow, may be easly rewrtte as Where, x T ˆ l4 l4 p( x + x) exp[ ( x+ g ) ] π β, ( T ( + ) T T T+ ) k, k, p p T (3.) ( T T ) T l 4 τ, l 3 " l + l τ l 3 τ" ( ), τ ' l ( kr+ τ ) b, l + b l τ kr+ τ ba+ τg) g (, τ ' τ ' kτ + b. g kr x+ τa kr+ τ, krτ ' τ" k r+ τ ', ττ' τ ', τ τ+ τ ', τ 4. Oe -Perod Forward Forecast O the bass of observatos x, x,..., x, the oe -perod forward forecast ca be expressed as Xˆ () E[ x + x, x,... x ] where, g x + p( x+ x) dx+ g ba+ kτg ) kr x+ τ a (, g, x T ˆ β, τ ' k τ + b, τ ' kr+ τ βˆ s the least squares estmate of β ad p T ( T T ) T. k, p (4.) 7

9 5. Predctve Iterval Let us deote x φ ( x + ) exp[ + ] π q ad Φ (q) φ ( x ) + dx. (5.) + The the probablty P[ x+ > q x] s gve by P[ X > q x + ] p( x+ x) dx+ [ Φ( q * )], q where, q * l ( q ). 4 g (5.) 6. Illustrato Cosder the followg modfed model gve by Far (00) for studyg the fluece of fluctuatos ecoomc varables o votg behavor U.S. presdetal electo. E( Vote) β+ βparty+ β3durato+ β4perso+ β5war+ β6growth+ β7iflato+ β8goodews (6.) The otato for the above regresso equato s as follows: Vote Icumbet share of two party vote. Icumbet vote s dvded by the Democratc plus Republca vote Party f there s a Democratc cumbet at the tme of electo ad f there s a Republca cumbet Durato 0 f the cumbet party has bee power for oe term, f the cumbet party has bee power for two cosecutve terms,.5 for three cosecutve terms,.50 for four cosecutve terms, ad so o. Perso f cumbet s rug for electo ad 0 otherwse. War for the electos of 90, 944 ad 948, ad 0 otherwse. Growth growth rate of real per capta GDP the frst three quarters of the electo year (aual rate) Iflato absolute value of the growth rate of the GDP deflator the frst 5 quarters of the admstrato (aual rate) except for 90, 944, 948, where the values are zero. 8

10 Goodews umber of quarters the frst 5 quarters of the admstrato whch the growth rate of real per capta GDP s greater tha 3. percet at a aual rate except for 90, 944, ad 948, where the values are zero. Table 6.4 gves Far s data o quadreal presdetal electos the Uted States from 96 to 004. Quarterly data o omal GDP, real GDP ad populato are used to costruct the varables Growth, Iflato ad Goodews. The ecoomc data ad formulato for costructo of data o the varables are explaed Far (00, 004). Let us deote the varable Vote by x, ad varables Party, Durato, Perso. War, Growth, Iflato ad Goodews by t, t3, t4, t5, t6, t 7 ad t 8, respectvely. Sce each electo year s uque ad ts result s depedet of ts prevous ad ext electo results, the equato (6.) ca be wrtte the form of equato (3.) ad the results derved equatos (3.), (4.) ad (5.) ca be easly appled for obtag predctve desty fucto, oe perod forward forecast ad probablty for w P[ x+ > 50.0 x]. We recursvely estmate the model ad evaluate the outof-sample oe perod ahead probablty forecast. The parametersβ ( β, β, β 8) of the model are estmated by the least squares method from the data set gve Table 6.4.These estmates are summarzed Table 6.3. The precso r σ s assumed to be kow, we take 8 r as a true value, where ˆ σ RSS RSS ( X T ˆ) β ( X T ˆ β ). The estmates of parameters of pror dstrbuto are made o the bass of results of the formatve expermets. We take the frst stage pror for the ukow mea θ as N ( µ, τ ), where τ Settg a ( x x) ad s the umber of sample observatos. The secod stage pror o µ s dstrbuted as N( a, b) wth mea a ad precso b. T + β ad b r ( T ( T T ) T + + ), the oe perod forward forecast values, ˆ predcto errors ad P[ x+ > 50.0 x] are summarzed Tables 6.0, 6. ad 6.. We fd that the predctve performace of the model s very good wth the above values of the parameters. For the sample perod ( ), the root mea square error of oe perod forward forecast s 3.8 ad the Thel equalty coeffcet s ear zero (0.004). The Thel equalty coeffcet for all other sample perods ( to ) s also ear zero. Root mea square error of oe perod forward forecast s 3.96 ad for the sample perods 9

11 ad , respectvely. It s below. for all other sample perods. Ths suggests the predctve performace of the model s good. For the 000 electo usg sample observatos , the model predcted vctory for Democratc Party caddate Mr. Al Gore by a arrow marg (50.948) wth probablty For the 004 electo usg sample observatos , t predcted vctory for Presdet Bush by a farly comfortable marg (54.463) wth probablty Though Presdet Bush wo both the electos, the marg 000 electo was arrow (50.65). The model predcto was good for the 996 electo whe t predcted vctory for Presdet Clto (5.633) wth probablty usg sample observatos 96-99, Presdet Clto could secure percetage of vote share. The model predctos are also true for the 988, 984 ad 980 electos. The model predcted vctory for the cumbet the 988 ad 984 electos, wth oe perod forward forecasts (probablty to w 0.596) ad 60.3 (probablty to w 0.99), respectvely. Usg sample observatos , the model predcted defeat of the cumbet the electo of 980 wth oe perod forward forecast ad probablty for vctory The 99 electo s the most problematc electo for the model. It predcted vctory for Presdet Bush (54.04) wth a probablty but he lost to Mr. Clto by a farly large marg (46.545). Far (996) tres to expla ths error predcto. 008 US presdetal electo Table 6. gves the herarchcal Bayes forecast o Far s vote model for the 008 electo. It suggests that the 008 presdetal electo s lkely to be a close electo slghtly tlted towards the republcas f the GDP, flato ad Goodews rema at the curret level (July 008) of.0%, 3.0% ad 3 respectvely. At ths level of GDP ad flato, t s lkely that republcas wll get 50.90% vote wth probablty for w

12 Referece Atchso, J. ad Dusmore, I. R. (975) Statstcal Predcto Aalyss, Cambrdge, Cambrdge Uversty Press. Berger, J. O. (985) Statstcal Decso Theory ad Bayesa Aalyss Sprger, New York. Berger, J. O. & Berler, M. (986): Robust Bayes ad emprcal Bayes aalyss wth - cotamated prors, The Aals of Statstcs, 4 (), pp Berry, B., Ellot, E., ad Harpham, E. J. (996) The yeld curve as a electoral bellwether, Techcal forecastg ad socal chage, 5, pp Erkso, R. S., ad Wleze, C. (996) Of tme ad presdetal electo forecasts PS: Poltcal Scece ad poltcs, 3, pp Far, R. C. (978) The effect of ecoomc evets o votes for presdet, Revew of Ecoomcs ad Statstcs, 60, pp Far, R. C. (996) The effect of ecoomc evets o votes for presdet: 99 update, Poltcal Behavor, 8, pp Far, R. C. (00) Predctg Presdetal Electos ad Other Thgs, Staford: Staford Uversty Press. Far, R. C. (004) A vote equato ad the004 electo, Webste: farmodel.eco.yale.edu/vote004 Gleser, R. F. (99) Ecoomc developmets of presdetal electo: The Far model, Poltcal Behavor, 4, pp Gleser, R. F. (005) Commets for presetato at the roudtable o Far s presdetal vote equato Iteratoal Symposum o forecastg, Sa Atoo, Jue 4, 005 Hastgs, C. (955) Approxmato for Dgtal Computers, Prceto, NJ, Prceto Uversty Press. Hbbs, D. A. (000) Bread ad peace votg U.S. presdetal electo, Publc Choce, 04, pp Ldley, D. V. ad Smth, A. F. M.(97) Bayes estmates for the lear model, (wth dscusso). Joural of Royal Statstcal Socety, B 34, pp. -4. Polasek, W. (984) Multvarate regresso systems: Estmato ad sestvty aalyss for two - dmesoal data. Robustess Bayesa Statstcs, (J. Kadae, ed.) Amsterdam: North- Hollad, pp. -4. Polasek, W. ad Potzelberger, K. (988) Robust Bayesa Aalyss Herarchcal Models, Bayesa Statstcs 3, Oxford Uversty Press, pp

13 Table 6.0 Oe Perod Forward Herarchcal Bayes Forecast Estmates for Far s Vote model Year Sample Forecast Vote share of Icumbet % Actual Vote share of Icumbet % Forecast Error r. m. s. Error Prob. for w* P [ x 50 x] > (96-000) 000 (96-996) (96-99) 99 9 (96-988) (96-984) (96-980) (96-976)

14 Table 6. Oe Perod Forward Herarchcal Bayes Forecast Estmates for Far s Vote model Year Sample Pror Parameters a b τ r σˆ Forecast Vote share of Icumbet % Actual Vote share of Icumbet % Thel Iequalty Coeff. Prob. for w P[ x x 50.0] + > 004 (96-000) 000 (96-996) (96-99) 99 9 (96-988) (96-984) (96-980) (96-976)

15 Table- 6. Herarchcal Bayes Forecast o Far s Vote Model for the 008 Electo Sample Number of observatos 3 Year Growth Iflato Goodews Pror Parameters a b τ r Forecast Vote share of Icumbet % Probablty for W Aprl July

16 Table 6.3 Least Squares Estmates of Far s Vote Model Electo Year Sample costat Party Durato Perso War Growth Iflato ˆβ ˆβ 3 ˆβ 4 ˆβ 5 ˆβ 6 ˆβ 7 ˆβ ˆβ 8 Good News (96-004) (96-000) (96-996) (96-99) (96-988) (96-984) (96-980) (96-976)

17 TABLE- 6.4 Far (00) Data o U.S. Presdetal Electos, Year Vote Party Durato Perso War Growth Iflato Good ews

18 TABLE- 6.5 Far (007) Revsed Data o U.S. Presdetal Electos, Year Vote Party Durato Perso War Growth Iflato Good ews Ja Aprl 007 July

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