A MATLAB Implementation of a Halton Sequence-Based GHK Simulator for Multinomial Probit Models *

Transcription

1 A MATLAB Implementation of a Halton Sequene-Based GHK Simulato fo Multinomial Pobit Models * Jose Miguel M. Abito Depatment of Eonomis National Uvesity of Singapoe Apil 005 * I would like to thank Kut Shmidheiny fo aknowledging my equest fo his MATLAB Eonometis Toolbox and Kenneth Tain fo his online ouse in UC Bekeley. Nonetheless, all eos ae mine.

2 Intodution The populaity of logit-based models is due to the availability of losed-fom expessions fo thei hoie pobabilities. Unfotunately, Logit-based models have impotant limitations patiulaly the inability to handle andom taste vaiation, geneal foms of substitution pattens, and epeated hoie situations with seially oelated eos (Tain, 003. To emedy these defiiees, the Multinomial Pobit (MNP model is often used. Howeve losed-fom hoie pobabilities fo MNP ae not eadily available. Numeial integation methods an be utilized to solve these integals. Howeve, most numeial integation methods ae only feasible and patial to use when the dimension of integation is small enough (< 5 beause of limitations in omputing powe. Theefoe simulation methods an be employed to addess this omputational limitation. This is the pime motivation fo studying simulation-based methods applied to MNP and othe Disete Choie models with analytially intatable hoie pobabilities. In using simulation to estimate the hoie pobabilities of a high-dimensional MNP model, the fist step is to find an appopiate algoithm to use that would povide an estimate with mimal vaiane elative to the tue value of the integal. In geneal, when estimating an integal, vaiane edution tehques suh as Antitheti Monte Calo and Impotane Sampling ae used to intodue some negative oelation o othe e popeties among the pseudo-andom daws fo the simulation. Cude o Non-weighted Monte Calo tehques that ely solely on pseudo-andom geneatos often pefom pooly due to thei low equidistibution popety (Sando and Andas, 004 and high disepany (Bandimate, 00. With this in mind, diffeent simulatos have been used in the MNP ontext. Among these diffeent methods, the GHK (Geweke, 989, 99; Hajivassiliou and MFadden, 998; Keane, 990, 994 simulato is the most widely utilized (Tain, 003 and has been studied and obseved to pefom bette than othe simulatos (Geweke et al., 994; Boh-Supan and Hajivassiliou, 993; Hajivassilious et al., 996. Aside fom Monte Calo based vaiane edution tehques, Quasi-Monte Calo methods an be employed to impove estimation. The main idea behind this is that sine the geneation of pseudo-andom numbes is atually detemisti, the eseahe an mapulate and devise altenative detemisti sequenes of numbes that display bette equidistibution and low disepany popeties. Sando and Tain (00 obseve that (t, m, s-nets and Halton daws pefom bette than independent daws in an appliation to maximum simulated likelihood estimation of a mixed logit model. Moe eently, Sando and Andas (004 show that quasi-monte Calo samples and samples based on othogonal aays have bette pefomane in geneal, ompaed to Cude Monte Calo and Antitheti Monte Calo samples when used with GHK simulatos Howeve, the authos do not examine the quality of these GHK simulatos by applying it in a MNP model and examing whethe estimates using GHK simulatos with diffeent samples diffe sigfiantly. Thus this pape fouses on the patiality of using simulatos based on a patiula Quasi-Monte Calo method.

3 This pape povides a simple and patial inopoation of a quasi-monte Calo sample into Kut Shmidheiny s MNP outine in MATLAB (based on the toolbox used in Shmidheiny [003]. The quasi-monte Calo sample used is the Halton sequene whih is the simplest. Speifially, a andomized Halton sequene is inopoated in the GHK simulato of the toolbox. Though moe sophistiated and effetive andomization tehques fo Halton sequenes ae available (Tuffin, 996; Wang and Hikenell, 000, a simple andomization poedue suggested by Tain (003 is used. Lastly, ompaisons between a pseudo-andom geneato based GHK simulato and a andomized Halton sequene-based GHK simulato is aied out using the tavel hoie data studied in Geene (003 and Geene and Henshe (997. Both GHK simulatos povide simila estimates even with only few simulation uns. This shows that fistly, the GHK simulato itself pefoms well enough even with few simulation uns (given that the numbe of obsevations is suffiiently lage, and seondly, only maginal impovements, if any, an be ahieved with a (ude andomized Halton sequene-based GHK simulato that tend to expend moe estimation time. The next setion povides a bief bakgound on how the GHK simulato is onstuted in geneal settings. In Setion 3, a non-tehal disussion of Halton sequenes is given. Setion 4 biefly disusses the data used followed by the main esults of ou expeiment. The last setion onludes. GHK Simulato The GHK Simulato is based on Impotane Sampling with a eusively tunated multivaiate nomal as its Impotane pdf. The basi idea is to dietly appoximate a etangula pobability (Gouieoux and Monfot, 996. As mentioned, the GHK simulato is the most popula simulato fo MNP (Tain, 003 and has been obseved to pefom bette than othe simulatos (Geweke et al., 994; Boh-Supan and Hajivassiliou, 993; Hajivassilious et al., 996. Conside a geneal model with J altenatives. The utility of peson n fom hoosing altenative j is given by U = + ε ( nj V nj nj fo j =,, J. Afte applying the nomalization (fo identifiation method suggested by Tain (003 and expessing ( as diffeenes against altenative i, we have the following: The main issue is the estimation of hoie pobabilities using simulation. Issues egading the full estimation of the MNP model using simulation (whethe and how to implement Maximum Simulated Likelihood Estimation o Method of Simulated Moments ae not dealt with in this shot pape. 3

4 U = + ε ( nji V nji nji whee ε (,..., = ε ε nji ' N(0, Ω i and is (J- x. The pobability that peson n hooses altenative i will then be given as P = P( V + ε < 0 j i (. nji nji Note that via Choleski deomposition, Ω i = L i L i whee 3 L i = M J 0 M J L L O L 0 0 M JJ and onsequently ( an be eintepeted as U = V + Lη ( i n whee U = ( U n i,..., U ', V = ( Vn i,..., V ' and η n = ( η n,..., η nj '. Note also that η nj nji nji N(0,. The above tansfomation allows us to poeed with simulations using independent daws. onveently as Using Bayes Rule eusively, the hoie pobability fo altenative i an then be ewitten P Vn i = P( η < ( V + η Vn i P( η < η < ( Vn3i + 3η + 3η ( V + P( η3 < η < L 33 η η < V (. The GHK simulato is then alulated as follows (Tain, 003: Vn i Vn i. Calulate P( η < = Φ(.. Daw a value η, denoted as η fom a tunated standad nomal tunated at V n i. 4 3 L i is (J- x (J-. 4 Refe to Tain (003 on how to obtain a daw fom a tunated standad nomal distibution. It is impotant to note that in dawing fom a tunated standad nomal, U(0, daws ae used. 4

5 ( V + η ( V + η 3. Calulate P( η < η = η = Φ(. 4. Daw a value η, denoted as η fom a tunated standad nomal tunated at ( Vn i + η. 5. Calulate ( Vn3i + 3η + 3η ( Vn3i + 3η + 3η P( η3 < η = η η = η = Φ( Repeat steps -5 fo all altenatives exept i. 7. The simulated pobability fo the th daw of η, η J is given as ( Vn i ( V + η ( Vn3i + 3η + 3η P = Φ( Φ( Φ( L Repeat steps -7 fo =,, R whee R is the numbe of epliations fo the GHK simulato. ( R ( 9. Finally, the simulated hoie pobability is P = P. R = This algoithm is implemented in Shmidheiny s MATLAB toolbox. 3 Halton Sequene This setion povides a non-tehal intodution to Halton Sequenes. Fo a moe tehal teatment, onsult the efeenes ited in Sando and Andas (004. Conside a one-dimensional Halton sequene in base. Divide the ut line into equal patitions oesponding to the base hosen (i.e. ut in half. The fist element of the sequene is -. Add - to 0 and 0.5 to get the seond and thid elements of the sequene espetively. Ou sequene now is {0.5, 0.5, 0.75}. Add -3 to zeo and eah element of the sequene in pope ode. Ou sequene is now {0.5, 0.5, 0.75, 0.5, 0.65, 0.375, 0.875}. Repeat the poess until one has the desied numbe of elements. The main ationale fo hoosing a Halton sequene ove a simple pseudo-andom geneato is that we want ou daws to mimi a ufom distibution as muh as possible. A possible way to do this is to intodue some negative oelation between suessive daws so that eah side of the distibution is epesented almost equipopotionally. This is the same woking idea fo using antitheti Monte Calo tehques. With the Halton sequene, eah suessive daw jumps fom one side of the distibution to the othe. One measue of pefomane fo pseudo-andom o detemisti samples that appoximate the ontinuous ufom distibution is Disepany. Mathematially, Disepany is defined as n D( x,..., x = sup Sn ( G nxl x X I m X m 5

6 whee I m is an m-dimensional hypeube, G X is a subspae of I m, and S n (G X is a funtion that ounts the numbe of sample points ontained in eah subspae. Lowe disepany means that ou sample points ae moe dispesed in I m whih is ou desied haateisti. Conside a ompaison of the pseudo-andom geneato of MATLAB (and( and a (andomized Halton sequene in a two-dimensional spae: Figue a: Pseudo-Random Geneato Figue b: (Randomized Halton Sequene By visual inspetion, Figue a has highe disepany ompaed to Figue b. In othe wods, the Halton sequene is moe dispesed aoss the egion ompaed to the simple pseudo-andom geneato and theefoe we expet that the fome will pefom bette than the latte in estimation via simulation. 4 Data and Results Shmidheiny s MATLAB toolbox ontains the data set and applies the MNP outine using the data set as a demo. The data set oiginally omes fom Geene and Henshe (997 and is losely studied in 5 The two-dimensional andomized Halton sequene is of base and 3. See the appendix fo the MATLAB ode. 6

7 Geene (003. The data set ontains 840 obsevations on 4 tavel modes fo 0 individuals. The 4 modes onsist of Ai, Tain, Bus o Ca. The hoie attibutes in the model ae g (genealized ost measue, ttime (teminal waiting time, hin (household inome, and hinai (household inome inteated with ai dummy. Estimation is pefomed using maximum simulated likelihood. The MSLE esults ae given below: Table : MSLE Results A B C D E Constants Ai ( ( ( ( ( Tain *** *** ***.6474***.57948*** ( (0.363 ( ( ( Bus 0.906*** *** *** *** *** (0.988 (0.775 ( (0.538 ( Slope Coeffiients g *** *** *** *** *** ( ( ( ( ( ttime ** ** ** *** *** ( ( ( ( ( hinai ** * ** * ( ( ( ( ( # of Obsevations (Individuals Type of daws and( and( Halton and( Halton # of GHK Repliations Substitution Patten Unestited, Nomalized Unestited, Nomalized Unestited, Nomalized Unestited, Nomalized Unestited, Nomalized Computational Time (in minutes Sigfiane Levels: % (***, 5% (**, 0% (* Note: Nomalization algotihm fo unestited substitution patten is based on Tain (003 and is implemented in Shmidheiny s toolbox 7

8 Models A, B and D ae based on the pseudo-andom geneato and( and Model A is used as the benhmak model. Models C and E utilize daws fom andomized Halton sequenes of k- x dimension whee k is the numbe of altenatives (oesponds to the numbe of bases used and is the numbe of epliations (oesponds to the numbe of elements in eah base. With a few mino exeptions, all of the models have simila estimation esults whih an mean two things. Fist, the (odinay GHK simulato is obust enough to handle extemely small epliations and exhibits the same pefomane as a Halton sequene-based GHK simulato. Seond, ou expeiment is not entiely onlusive fom a theoetial point of view sine thee seems to be some divegene in tems of ou objetive and the measue of pefomane used in the analysis and ompaison of the two samples. Speifially, ou objetive is to see how well the diffeent samples atually estimate the hoie pobabilities but the yadstik of pefomane is based on estimated oeffiients fo the hoie attibutes. We ae atually looking at how well these samples yield an estimate via MSLE whih is affeted by the numbe of obsevations used in the study and not simply on the numbe of epliations. With a lage enough sampled population, the estimate will be athe peise even if the numbe of epliations is small (Tain, 003. Nonetheless fom a patial point of view, if the eseahe has a suffiiently lage numbe of obsevations and the only issue is the numbe of epliations to be used, then an odinay GHK simulato pefoms as well as a Halton-sequene based GHK simulato 6, as mentioned peviously. 5 Conlusion As we have seen in the pevious setion, a GHK simulato pefoms as well as a Halton sequenebased GHK simulato, assuming the eseahe has enough obsevations. Though a andomized Halton sequene is easy to inopoate in any pogamming outine fo MNP models (see the Appendix, a onsideable amount of omputing time is expended duing estimation and thus may not be woth it (again, given that we have a suffiiently lage numbe of obsevations. Nevetheless, othe quasi-monte Calo samples and samples based on othogonal aays an be inopoated. How these samples atually pefom in patial appliations still emains to be studied. 6 Given that the numbe of obsevations is suffiiently lage, using a Halton sequene-based GHK simulato has only maginal impovements with espet to the estimates but takes onsideably moe time to pefom the estimation. Theefoe, on these gounds and as explained above, an odinay GHK simulato would be most likely pefeed in patial appliations. 8

9 Appendix The andomized Halton sequene geneato an be easily inopoated into the GHK simulato by eplaing the funtion that alls and( with a funtion that gives us a andomized Halton sequene of size k- x. 7 The MATLAB ode fo geneating andomized Halton sequenes is as follows: funtion = RandHaltVe(m,n % Retuns a matix of andom Halton daws of size m x n (n daws fom m bases = []; vp = GeneatePime(m; fo i = :m (i,: = RandomHalton(n,vP(i; end funtion = RandomHalton(m,b % Randomizes Halton daws with base b and m elements Halton_nonR = GetHalton(m,b; u = and(,; % The following ensues that ou andomized Halton element is within % (0,: fo i = : m if Halton_nonR(i + u < ; t(i = Halton_nonR(i + u; else t(i = Halton_nonR(i + u - ; end end =t'; whee GetHalton(m,b is a funtion that povides a Halton sequene in base b and with size m x (Bandimate, 00. The funtion RandomHalton(m,bis based on a andomizing poedue suggested by Kenneth Tain in his online ouse in UC Bekeley 8. 7 This ous when we daw a sample point fom a tunated standad nomal

10 Refeenes Bosh-Supan, A. and V. Hajivassiliou (993, Smooth Unbiased Multivaiate Pobability Simulation fo Maximum Likelihood Estimation of Limited Dependent Vaiable Models, Jounal of Eonometis 58, Bandimate, P. (00, Numeial Methods in Finane: A MATLAB -Based Intodution, Wiley & Sons, NY. Geweke, J. (989, Bayesian infeene in eonometi models using Monte Calo integation, Eonometia 57, Geweke, J. (99, Effiient simulation fom the multivaiate nomal and Student-t distibutions subjet to linea onstaints, in E. M. Keamidas, ed., Compute Siene and Statistis: Poeedings of the Twenty-Thid Symposium on the Intefae, pp Intefae Foundation of Noth Ameia In., Faifax. Geweke, J., M. Keane, and D. Runkle (994, Altenative Computational Appoahes to Infeene in the Multinomial Pobit Model, Review of Eonomis and Statistis 76, Gouieoux, C. and A. Monfot (996, Simulation-Based Eonometi Methods, Oxfod Uvesity Pess, NY. Geene, W (003, Eonometi Analysis, 5 th Ed., Pentie Hall, NJ. Geene, W. and D. Henshe (997, Multinomial Logit and Disete Choie Models, in Geene, W., LIMDEP Vesion 7.0 Use s Manual, Revised, Eonometi Softwae In., NY. Hajivassiliou, V. and D. MFadden (998, The method of simulated soes fo the estimation of LDV models, Eonometia 66, Hajivassiliou, V., D. MFadden and P. Ruud (996, Simulation of Multivaiate Nomal Retangle Pobabilities and Thei Deivatives: Theoetial and Computational Results, Jounal of Eonometis 7,

11 Sando, Z. and P. Andas (004, Altenative Sampling Methods fo Estimating Multivaiate Nomal Pobabilities, Jounal of Eonometis 0, Sando, Z. and K. Tain (00, Quasi-Random Simulation of Disete Choie Models, mimeo. Shmidheiny, K. (003, Inome Segegation and Loal Pogessive Taxation: Empiial Evidene fom Switzeland, Uvesity of Ben, mimeo. Tain, K. (003, Disete Choie Methods with Simulation, Cambidge Uvesity Pess, NY. Tuffin, B. (996, On the Use of Low-Disepany Sequenes in Monte Calo Methods, Monte Calo Methods and Appliations, Wang, X. and F. J. Hikenell (000, Randomized Halton Sequenes, Mathematial and Compute Modelling 3,