Interval Regression with Sample Selection

Transcription

1 Interval Regresson wth Sample Selecton Géraldne Hennngsen, Arne Hennngsen, Sebastan Petersen May 3, 07 Ths vgnette s largely based on Petersen et al. 07. Model Specfcaton The general specfcaton of an nterval regresson model wth sample selecton s: ε S ε O y S β S x S + ε S { y S 0 f y S 0 otherwse y O β O x O + ε O 3 unknown f y S 0 f < y O and y S y O f < y O 3 and y S. M 0 N 0, [ ρ ρ 4 f M < y O M+ and y S ], 5 where subscrpt ndcates the observaton, y O s a latent outcome varable, y O s a partally observed categorcal varable that ndcates n whch nterval y O les, M s the number of ntervals,,..., M+ are the boundares of the ntervals whereas frequently but not necessarly and M+, y S s a bnary varable that ndcates whether y O s observed, y S s a latent varable that ndcates the tendency that y S s one, x S and x O are column vectors of explanatory varables for the selecton equaton and outcome equaton, respectvely, ε S and ε O are random dsturbance terms that have a jont bvarate normal dstrbuton, and β S and β O are column vectors and ρ and are scalars of unknown model parameters.

2 Log-Lkelhood Functon The probablty that y O s unobserved s: P y S 0 P y S 0 6 P β S x S + ε S 0 7 P ε S β S x S 8 The probablty that y O s observed and ndcates that y O les n the mth nterval s: P y S y O m P y S > 0 m < y O m+ 9 β S x S + ε S > 0 m < β O x O + ε O m+ P P ε S > β S x S m β O x O < ε O m+ β O x O 0 The log-lkelhood contrbuton of the th observaton s: [ ] l y S ln Φ β S x S + y S y O m+ β m ln [Φ O x O, β S x S, ρ m ] Φ m β O x O, β S x S, ρ, where Φ. ndcates the cumulatve dstrbuton functon of the unvarate standard normal dstrbuton and Φ. ndcates the cumulatve dstrbuton functon of the bvarate standard normal dstrbuton. 3 Restrctng coeffcents ρ and The parameter ρ needs to be n the nterval,. In order to restrct ρ to be n ths nterval, we estmate arctanρ nstead of ρ so that the derved parameter ρ tanarctanρ s always n the nterval,. We use the delta method to calculate approxmate standard errors of the derved parameter ρ, whereas the correspondng element of the Jacoban matrx s: tanarctanρ arctanρ ρ arctanρ + ρ 3 The parameter needs to be strctly postve,.e. > 0. In order to restrct to be strctly postve, we estmate log nstead of or so that the derved parameters explog and exp log are always strctly postve. We use the delta

3 method to calculate approxmate standard errors of the derved parameters and, whereas the correspondng elements of the Jacoban matrx are: explog log exp log log explog 4 exp log 5 4 Gradents of the CDF of the bvarate standard normal dstrbuton In order to facltate the calculaton of the gradents of the log-lkelhood functon, we calculate the partal dervatves of the cumulatve dstrbuton functon CDF of the bvarate standard normal dstrbuton: Φ x, x, ρ x x φ a, a, ρ da da, 6 where φ. s the probablty densty functon PDF of the bvarate standard normal dstrbuton: φ x, x, ρ π ρ exp x ρx x + x ρ 7 In the followng, we check equaton 7 by a smple numercal example: > lbrary "mvtnorm" > lbrary "maxlk" > x <- 0.4 > x < > rho < > sgma <- matrx c, rho, rho,, nrow > dens <- dmvnorm c x, x, sgma sgma > prnt dens [] > all.equal dens, * p * sqrt - rho^ ^- * + exp - x^ - * rho * x * x + x^ / * - rho^ [] TRUE 3

4 4. Gradents wth respect to the lmts x and x Φ x, x, ρ x x x x x x φ a, x, ρ da 8 φa x, ρφx da 9 φ a, ρx, ρ φx da 0 a ρx φ ρ φx da ρ φ x ρx ρ Φ a ρx ρ ρ da φx x ρx ρ φa da φx 3 φx, 4 where φ, µ, ndcates the densty functon of a normal dstrbuton wth mean µ and varance. In the followng, we use the same smple numercal example as n the begnnng of secton 4 to check the above dervatons. Frst, we check whether the PDF of the bvarate standard normal dstrbuton,.e. φ x, x, ρ part of equaton 8, s equal to φ x, ρx, ρ φx part of equaton 0 and equal to φ x ρx / ρ ρ φx part of equatons and : > all.equal dens, dnorm x, rho * x, sqrt - rho^ * dnormx [] TRUE > all.equal dens, dnorm x - rho * x / sqrt - rho^ / + sqrt - rho^ * dnormx [] TRUE In the followng, we wll numercally calculate the dervatve of the cumulatve dstrbuton functon of the bvarate normal dstrbuton equaton 6 wth respect to x and check wehther ths partal dervatve s equal to the rght-hand sdes of equatons 8,,, and 4: > funx <- functon a { + prob <- pmvnorm upper c x, a, sgma sgma 4

5 + return prob > grad <- c numercgradent funx, x > prnt grad [] > funx <- functon a { + dens <- rep NA, length a + for n :length a { + dens[] <- dmvnorm c a[], x, sgma sgma + return dens > all.equal grad, ntegrate funx, lower -Inf, upper x $value [] TRUE > funxa <- functon a { + dens <- rep NA, length a + for n :length a { + dens[] <- dnorm a[] - rho * x / sqrt - rho^ / + sqrt-rho^ * dnormx + return dens > all.equal grad, ntegrate funxa, lower -Inf, upper x $value [] TRUE > funxb <- functon a { + dens <- rep NA, length a + for n :length a { + dens[] <- dnorm a[] - rho * x / sqrt - rho^ / + sqrt-rho^ + return dens > all.equal grad, + ntegrate funxb, lower -Inf, upper x $value * dnormx [] TRUE > all.equal grad, + pnorm x - rho * x / sqrt - rho^ * dnorm x [] TRUE 5

6 4. Gradents wth respect to the coeffcent of correlaton ρ Φ x, x, ρ ρ [ x ] x φ a, a, ρ da da ρ 5 6 x x φ a, a, ρ da da 7 ρ x x ρ exp a ρa a + a ρ π ρ da da 8 x x π ρ exp a ρa a + a ρ ρ da da 9 x x ρ π exp a ρa a + a ρ ρ ρ ρ exp a ρa a + a ρ ρ da da ρ 30 x x ρ π a ρa a + a ρ exp a ρa a + a ρ ρ ρ ρ ρ exp a ρa a + a ρ ρ da da 3 6

7 x x π 4ρa ρa a + a ρ a a 4 ρ exp ρ a ρa a + a ρ ρ ρ ρ exp a ρa a + a ρ ρ da da 3 4ρa x x ρa a + a ρ a a 4 ρ ρ ρ ρ π ρ ρ exp a ρa a + a ρ da da x x 4ρa ρa a + a ρ a a ρ + π 4 ρ 5 3 ρ 3 exp a ρa a + a ρ da da x x 4ρa ρa a + a ρ a a ρ + π 4 ρ 5 ρ 3 exp a ρa a + a ρ da da x x ρ ρa ρa a + a a a π ρ 3 ρ 5 exp a ρa a + a ρ da da x x ρ π ρ ρ ρa ρa a + a a a ρ exp a ρa a + a ρ da da

8 π ρ π ρ lm a x x a ρa ρ exp a ρa a + a x ρ a ρx ρ exp a ρa x + x ρ a + ρa ρ a exp ρa a + a da ρ Applyng L Hosptal on the last term leads to x π a ρx ρ ρ exp a ρa x + x ρ 0 π ρ x a ρx ρ exp a ρa x + x ρ da da 40 da 4 π ρ exp a ρa x + x x ρ π ρ exp x ρx x + x ρ 4 43 φ x, x, ρ 44 Ths result s n lne wth Sbuya 960 and Sungur 990. In the followng, we wll numercally calculate the dervatve of the cumulatve dstrbuton functon of the bvarate normal dstrbuton equaton 6 wth respect to ρ and check whether ths partal dervatve s equal to the rght-hand sdes of equaton 44: > # Numercal gradent of the PDF w.r.t. rho > funrho <- functon p { + prob <- dmvnorm x c x, x, + sgma matrx c, p, p,, nrow + return prob > grad <- c numercgradent funrho, rho > prnt grad [] > # Comparson wth analytcal gradent for rho > efun <- exp-x^ - * rho * x * x + x^/* - rho^ 8

9 > all.equal grad, + -*rho*-*rho*x*x+x^+x^ - *x*x*-rho^ * efun/ + *-rho^^3/ + + rho*efun/sqrt-rho^ / + *p*-rho^ [] TRUE > #Eq9 > all.equalgrad, + /*p * + -4*rho*x^-*rho*x*x+x^-*-rho^*-*x*x/4*-rho^^ * + efun * sqrt-rho^/-rho^ - + -rho/sqrt-rho^*efun/-rho^ + [] TRUE > #Eq33 > all.equalgrad, + /*p * + rho/-rho^^3/ - rho*x^-rho*x*x+x^-x*x/ + -rho^^5/ * efun + [] TRUE > #Eq34 > all.equalgrad, + /*p*sqrt-rho^ * + rho/-rho^ - rho*x^-rho*x*x+x^-x*x/ + -rho^^ * efun + [] TRUE > > > # Numercal gradent of the CDF w.r.t. rho > cdfrho <- functon p, xa x, xb x { + prob <- pmvnorm upper c xa, xb, + sgma matrx c, p, p,, nrow + return prob > grad <- c numercgradent cdfrho, rho > prnt grad 9

10 [] > # comparson wth analytcal gradent > all.equal grad, dmvnorm x c x, x, + sgma matrx c, rho, rho,, nrow [] TRUE > # comparsons wth other values > compdervrho <- functon xa, xb, p { + dn <- c numercgradent cdfrho, p, xa xa, xb xb + da <- dmvnorm x c xa, xb, + sgma matrx c, p, p,, nrow + return all.equal dn, da > compdervrho x, x, rho [] TRUE > compdervrho 0.5, x, rho [] TRUE > compdervrho.5, x, rho [] TRUE > compdervrho x, -, rho [] TRUE > compdervrho x, x, 0. [] TRUE > compdervrho x, x, 0.98 [] TRUE 0

11 5 Gradents of the Log-Lkelhood Functon 5. Gradents wth respect to the parameters n the self-selecton decson β S Frst, we use equaton 4, to determne the dervatve of the bvarate standard normal dstrbuton wth respect to the parameter β S as part of the loglkelhood functon: Φ m β O x O, β S x S, ρ Φ m β O x O m β O x O β S Φ ρβ S x S φβ S x S ρ βs x S β S 45 + ρβ S x S φβ S x S ρ x S 46 Usng ths result we can now derve the gradent for β S n the log-lkelhood functon: l β S + [ ] β S y S ln Φ β S x S y S y O m m ln m+ β [Φ O x O m β O x O Φ, β S x S, ρ y S β S + y S y O m ], β S x S, ρ 47 [ ] ln Φ β S x S 48 m m+ β β S ln [Φ O x O ], β S x S, ρ m β O x O Φ, β S x S, ρ β S x S x S 49 Φ β S x S φ y S + y S y O m m Φ m+ β O x O,β S x S, ρ β S Φ m β O x O,β S x S, ρ β S Φ m+ β O x O, β S x S, ρ Φ m β O x O, β S x S, ρ

12 φ β S x S y S x S 50 Φ β S x S + y S y O m m Φ m+ β O x O, β S x S, ρ Φ m β O x O, β S x S, ρ m+ β O x O Φ + ρβ S x S φ β S x S ρ x S Φ + ρβ S x S φ β S x S ρ x S m β O x O φ β S x S y S Φ β S x S + y S y O m m x S 5 Φ m+ β O x O +ρβ S x S ρ Φ m β O x O +ρβ S x S ρ φ β S x S x S Φ m+ β O x O, β S x S, ρ Φ m β O x O, β S x S, ρ

13 5. Gradents wth respect to the parameters n the outcome decson β O Analogous to β S and by usng equaton 4 we derve the gradent of β O : Φ m β O x O, β S x S, ρ β O Φ βs x S + ρ m βo x O ρ φ m β O x O xo Usng ths result we derve the gradent for the outcome parameter β O for the loglkelhood functon: l β O [ ] β O y S ln Φ β S x S + y S y O m m ln Φ m β O x O y S y O m m β O Φ m β O x O y S y O m m y S y O m m Φ m+ β [Φ O x O ], β S x S, ρ ln βs x S + ρ m+ β O x O ρ Φ m+ β [Φ O x O ], β S x S, ρ Φ m+ β O x O,β S x S, ρ β O, β S x S, ρ, β S x S, ρ Φ m β O x O,β S x S, ρ β O Φ m+ β O x O, β S x S, ρ Φ m β O x O, β S x S, ρ βs x S + ρ m βo x O ρ Φ m+ β O x O, β S x S, ρ Φ m β O x O, β S x S, ρ φ φ m+ β O x O m β O x O xo xo 56 3

14 y S y O m m Φ βs x S + ρ m+ β O x O ρ Φ βs x S + ρ m βo x O ρ Φ m+ β O x O, β S x S, ρ Φ m β O x O, β S x S, ρ φ φ m+ β O x O m β O x O xo Gradents wth respect to the coeffcent of correlaton ρ Gven the result that the dervatve of the CDF wth respect to ρ s equal to the PDF see equaton 44, we can also derve the gradent of the correlaton parameter ρ: l ρ ρ + y S y O m y S ln m ln [ ] Φ β S x S m+ β [Φ O x O ], β S x S, ρ m β O x O Φ, β S x S, ρ y S y O m m+ β ln [Φ O x O, β S x S, ρ ρ ] m Φ m β O x O y S y O m m, β S x S, ρ φ m β O x O, β S x S, ρ φ m+ β O x O, β S x S, ρ Φ m+ β O x O, β S x S, ρ Φ m β O x O, β S x S, ρ 60 l arctanhρ l ρ ρ arctanhρ l ρ ρ Gradents wth respect to the standard devaton used for normalsaton Fnally, we derve the gradent for n the same way as we dd for β S and β O : 4

15 Φ m β O x O, β S x S, ρ Φ βs x S + ρ m βo x O ρ φ m β O x O βo x O m 6 Φ m β O x O, β S x S, ρ lm m lm Φ Φ Φ m Φ βs x S + ρ m βo x O ρ m β O x O βs x S + ρ m βo x O φ ρ lm φ m β O x O m βs x S + ρ m βo x O ρ βs x S + ρ m βo x O ρ lm m β O x O m φ m β O x O βo x O m 63 βo x O m Φ Φ lm m φ m β O x O βs x S + ρ m βo x O ρ βs x S + ρ m βo x O ρ lm m lm m m βo x O φ m β O x O φ m β O x O φ m β O x O m β O x O m β O x O Smlarly: Φ m β O x O, β S x S, ρ lm m

16 l [ ] y S ln Φ β S x S + y S y O m+ β m ln [Φ O x O m ], β S x S, ρ m β O x O Φ, β S x S, ρ y S y O m m+ β ln [Φ O x O, β S x S, ρ ] m Φ m β O x O, β S x S, ρ y S y O m Φ m+ β O x O, β S x S, ρ Φ m β O x O, β S x S, ρ m Φ m+ β O x O,β S x S, ρ m β Φ O x O,β S x S, ρ y S yo m 74 m Φ m+ β O x O, β S x S, ρ Φ m β O x O, β S x S, ρ Φ βs x S + ρ m+ β O x O m+ β O x O φ βo x O m+ ρ Φ βs x S + ρ m βo x O ρ φ m β O x O βo x O m References l log l log l 75 Petersen, Sebastan, Géraldne Hennngsen, and Arne Hennngsen, Whch Households Invest n Energy-Savng Home Improvements? Evdence From a Dansh Polcy Interventon, 07. Unpublshed Manuscrpt. Department of Management Engneerng, Techncal Unversty of Denmark. Sbuya, Masaak, Bvarate Extreme Statstcs, I, Annals of the Insttute of Statstcal Mathematcs, 960,,

17 Sungur, Engn, Dependence Informaton n Parameterzed Copulas, Communcatons n Statstcs - Smulaton and Computaton, 990, 9 4,