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A note on the nonstationary binary choice logit model
Economics Letters 76 (2002) 267–271 www.elsevier.com / locate / econbase
A note on the nonstationary binary choice logit model Emmanuel Guerre a , Hyungsik Roger Moon b , * a
b
LSTA ( Universite´ Paris 6), Paris, France Department of Economics, KAP 300, University of Southern California, Santa Barbara, Los Angeles, CA 90089, USA Received 15 October 2001; accepted 30 January 2002
Abstract This paper derives the rate and the asymptotic distribution of the MLE of the parameter of a logit model with a nonstationary covariate when the true parameter is zero. The limit distribution of the t-statistic is also given. 2002 Elsevier Science B.V. All rights reserved. Keywords: Binary choice model; Nonstationary covariates JEL classification: C22; C25
1. Introduction Suppose that the observed data s y t , x 9t d9 is generated by y t 5 1h y *t . 0j, y t* 5 b 90 x t 2 ´t , t 5 1, . . . , n.
(1)
The model (1) is the standard binary choice model. When the regressor x t and the error term ´t are stationary, it is well known that under regularity conditions the maximum likelihood estimator (MLE) ] of b0 is Œn-consistent and asymptotically normal. However, if the regressors x t are nonstationary, the MLE of b0 do not have these standard asymptotic properties. Recently, Park and Phillips (2000) considered the model (1) in which the regressors x t are nonstationary, i.e. x t 5 x t21 1 u t and coefficient b0 is not zero, i.e. * Corresponding author. Tel.: 11-213-740-2108; fax: 11-213-740-8543. E-mail addresses: [email protected] (E. Guerre), [email protected] (H.R. Moon). 0165-1765 / 02 / $ – see front matter PII: S0165-1765( 02 )00052-6
2002 Elsevier Science B.V. All rights reserved.
E. Guerre, H.R. Moon / Economics Letters 76 (2002) 267 – 271
268
b0 ± 0. Under regularity conditions, Park and Phillips (2000) obtain dual convergence rates for the MLE. In an orthogonal direction to b0 , the MLE converges to a mixed normal random variable at a rate of n 3 / 4 while in all other directions, the convergence rate is n 1 / 4 . In this note, we derive the asymptotic properties of the MLE of b0 when the true b0 is zero. The model we are considering in this note is quite simple. We assume that (i) x t is univariate, (ii) u t | iids0, 1d, (iii) ´t are iid over t with the logistic distribution Ph´t , xj 5 Lsxd 5 e x / 1 1 e x , (iv) u t and ´s are independent for all t and s. These restrictive assumptions are made to make derivations of asymptotics simple and short. The assumptions in this note can be relaxed at a cost of lengthy calculations and derivations. The main result we find in this paper is that when the true b0 is zero and the regressor of the nonstationary binary choice is nonstationary, the MLE is n-consistent and its limit distribution is similar to the so-called unit root limit distribution. This main result will be derived in Section 2.
2. Results The MLE bˆ maximizes the following log-likelihood function,
O y log Lsbx d 1O s1 2 y d log s1 2 Lsbx dd. n
l ns bd 5
n
t
t
t 51
t
t
t51
As is well known, the log-likelihood function l ns bd is smooth and strictly concave with respect to the parameter b. Thus, at the MLE bˆ , we have ≠l nsbˆ d ]] 5 0. ≠b Define
O
n ≠l ns bd Sns bd 5 ]] 5 x ts y t 2 Ls b x tdd, the score function, ≠b t 51
and
O
n ≠ 2 l ns bd ]]] Jns bd 5 L~ s b x tdx t2 , the hessian function, 2 5 2 ≠b t51
where ex L~ sxd 5 ]]] . s1 1 e xd 2 Now, we find the limit distributions of the score function Sns bd and the hessian function Jns bd at the true parameter b0 5 0. For this, let e t 5 y t 2 Ls0d 5 1h´t , 0j 2 1 / 2. Then, e t | iids0, 1 / 4d. By the conventional functional central limit theorem, we have
E. Guerre, H.R. Moon / Economics Letters 76 (2002) 267 – 271
269
[nr]
O
1 1 ] ] Besrd et Œ] n t51 ⇒ 2 , x [nr] ]] Busrd Œ] n
1 21 2
where Besrd and Busrd are two independent Brownian motions. Using this, it is easy to find that 1
O n
1 1 1 ] Sns0d 5 ] xt et ⇒ ] n n t51 2
E B srd dB srd, u
e
0
and 1
O
E B srd dr.
n
1 1 1 ]2 Jns0d 5 2 ]2 L~ s0dx t2 ⇒ 2 ] 4 n n t51
u
(2)
2
0
By the mean value theorem,
S
D
Jns b 1d 2 Jns0d Snsbˆ d Sns0d Jns0d ˆ ]]]]] 0 5 ]] 5 ]] 1 ]] n b 1 n bˆ , 2 2 n n n n 1 where b locates between zero and bˆ . Suppose that for some d . 0 we have
sup
1 U]] s J s bd 2 J s0ddU 5 o s1d. n
un 12d bu#1
222d
n
n
(3)
p
Then, by Theorem 10.1 in Wooldridge (1994), n bˆ 5 Ops1d and 1 ] Sns0d n 5 ]]]] 1 o ps1d 1 2 ]2 Jns0d n
n bˆ
⇒2
1E
(4)
21 1
1
2 E
Busrd 2 dr
0
Busrd dBesrd.
0
To prove (3), notice by the mean value theorem that sup
1 U]] s J s bd 2 J s0ddU n 1 sup U]] SO x ( L~ ( b x ) 2 L~ (0))DU 5 n
un 12d bu#1
222d
n
n
n
5
un 12d bu#1
2 t
222d
t51
t
sup un 12d bu#1
U
1 ]] 222d n
SO n
t51
DU
x 3t bL¨ s b 11 x td
,
where L¨ sxd 5 e x /s1 1 e xd 2 2 2e x /s1 1 e xd 3 and b 11 locates between zero and b. Since since u bu , 1 / n 12d and uL¨ sxdu # 3,
E. Guerre, H.R. Moon / Economics Letters 76 (2002) 267 – 271
270
sup un 12d bu#1
U
1 ]] 222d n
SO
DU
n
x bL¨ ( b 11 x t ) 3 t
t51
O
n 1 # 3 sup ]] ux tu 3 . 323d t51 un 12d bu#1 n
Choose 0 , d , 1 / 6. Then, sup un 12dbu#1 1 /n 323d o nt51 ux tu 3 5 o ps1d, and in consequence, sup un 12d bu#1
1 (J ( b ) 2 J (0))U 5 o s1d, U]] n 222d
n
n
p
as required for (3). As a summary, we have the following theorem. Theorem 1. Under the assumptions in the previous section, if b0 5 0, then 21 1
1
n bˆ
⇒2
1E 2 E 1 1E 2 2 Busrd 2 dr
0
Busrd dBesrd
0
21
1
; MN 0, 4
Busrd 2 dr
,
0
where notation ‘ ; ’ signifies equivalence in distribution and MN denotes a mixed normal distribution. Notice that in a univariate nonstationary binary choice model studied by Park and Phillips (2000), when the true parameter b0 ± 0, the MLE is n 1 / 4 -consistent and its limit distribution is mixed normal. When the true parameter b0 5 0, Theorem 1 shows that the MLE has a faster convergence rate n and its limit distribution is similar to the so-called ‘unit root distribution’. In our note, since we assume that u t and ´s are independent, the limit distribution is also mixed normal. The conventional t-statistic in this case is
bˆ t 5 ]]] ]]] . œ 2 Jnsbˆ d By (3) and (2), we have Jns0d Jnsbˆ d 1 2 ]] 2 5 2 ]] 2 1 o ps1d ⇒ ] 4 n n
1
E B srd dr. u
2
0
Thus,
bˆ t 5 ]]] ]]] ⇒ Ns0, 1d. œ 2 Jnsbˆ d Summarizing this, we have the following corollary. Corollary 2. Under the assumptions in the previous section, if b0 5 0, then
E. Guerre, H.R. Moon / Economics Letters 76 (2002) 267 – 271
271
bˆ t 5 ]]] ]]] ⇒ Ns0, 1d. œ 2 Jnsbˆ d References Park, J., Phillips, P.C.B., 2000. Nonstationary binary choice. Econometrica 68, 1249–1280. Wooldridge, J., 1994. Estimation and inference for dependent processes. In: Engle, R., McFadden, D. (Eds.). Handbook of Econometrics, Vol. 4. North Holland, Amsterdam.
A note on the nonstationary binary choice logit model Emmanuel Guerre a , Hyungsik Roger Moon b , * a
b
LSTA ( Universite´ Paris 6), Paris, France Department of Economics, KAP 300, University of Southern California, Santa Barbara, Los Angeles, CA 90089, USA Received 15 October 2001; accepted 30 January 2002
Abstract This paper derives the rate and the asymptotic distribution of the MLE of the parameter of a logit model with a nonstationary covariate when the true parameter is zero. The limit distribution of the t-statistic is also given. 2002 Elsevier Science B.V. All rights reserved. Keywords: Binary choice model; Nonstationary covariates JEL classification: C22; C25
1. Introduction Suppose that the observed data s y t , x 9t d9 is generated by y t 5 1h y *t . 0j, y t* 5 b 90 x t 2 ´t , t 5 1, . . . , n.
(1)
The model (1) is the standard binary choice model. When the regressor x t and the error term ´t are stationary, it is well known that under regularity conditions the maximum likelihood estimator (MLE) ] of b0 is Œn-consistent and asymptotically normal. However, if the regressors x t are nonstationary, the MLE of b0 do not have these standard asymptotic properties. Recently, Park and Phillips (2000) considered the model (1) in which the regressors x t are nonstationary, i.e. x t 5 x t21 1 u t and coefficient b0 is not zero, i.e. * Corresponding author. Tel.: 11-213-740-2108; fax: 11-213-740-8543. E-mail addresses: [email protected] (E. Guerre), [email protected] (H.R. Moon). 0165-1765 / 02 / $ – see front matter PII: S0165-1765( 02 )00052-6
2002 Elsevier Science B.V. All rights reserved.
E. Guerre, H.R. Moon / Economics Letters 76 (2002) 267 – 271
268
b0 ± 0. Under regularity conditions, Park and Phillips (2000) obtain dual convergence rates for the MLE. In an orthogonal direction to b0 , the MLE converges to a mixed normal random variable at a rate of n 3 / 4 while in all other directions, the convergence rate is n 1 / 4 . In this note, we derive the asymptotic properties of the MLE of b0 when the true b0 is zero. The model we are considering in this note is quite simple. We assume that (i) x t is univariate, (ii) u t | iids0, 1d, (iii) ´t are iid over t with the logistic distribution Ph´t , xj 5 Lsxd 5 e x / 1 1 e x , (iv) u t and ´s are independent for all t and s. These restrictive assumptions are made to make derivations of asymptotics simple and short. The assumptions in this note can be relaxed at a cost of lengthy calculations and derivations. The main result we find in this paper is that when the true b0 is zero and the regressor of the nonstationary binary choice is nonstationary, the MLE is n-consistent and its limit distribution is similar to the so-called unit root limit distribution. This main result will be derived in Section 2.
2. Results The MLE bˆ maximizes the following log-likelihood function,
O y log Lsbx d 1O s1 2 y d log s1 2 Lsbx dd. n
l ns bd 5
n
t
t
t 51
t
t
t51
As is well known, the log-likelihood function l ns bd is smooth and strictly concave with respect to the parameter b. Thus, at the MLE bˆ , we have ≠l nsbˆ d ]] 5 0. ≠b Define
O
n ≠l ns bd Sns bd 5 ]] 5 x ts y t 2 Ls b x tdd, the score function, ≠b t 51
and
O
n ≠ 2 l ns bd ]]] Jns bd 5 L~ s b x tdx t2 , the hessian function, 2 5 2 ≠b t51
where ex L~ sxd 5 ]]] . s1 1 e xd 2 Now, we find the limit distributions of the score function Sns bd and the hessian function Jns bd at the true parameter b0 5 0. For this, let e t 5 y t 2 Ls0d 5 1h´t , 0j 2 1 / 2. Then, e t | iids0, 1 / 4d. By the conventional functional central limit theorem, we have
E. Guerre, H.R. Moon / Economics Letters 76 (2002) 267 – 271
269
[nr]
O
1 1 ] ] Besrd et Œ] n t51 ⇒ 2 , x [nr] ]] Busrd Œ] n
1 21 2
where Besrd and Busrd are two independent Brownian motions. Using this, it is easy to find that 1
O n
1 1 1 ] Sns0d 5 ] xt et ⇒ ] n n t51 2
E B srd dB srd, u
e
0
and 1
O
E B srd dr.
n
1 1 1 ]2 Jns0d 5 2 ]2 L~ s0dx t2 ⇒ 2 ] 4 n n t51
u
(2)
2
0
By the mean value theorem,
S
D
Jns b 1d 2 Jns0d Snsbˆ d Sns0d Jns0d ˆ ]]]]] 0 5 ]] 5 ]] 1 ]] n b 1 n bˆ , 2 2 n n n n 1 where b locates between zero and bˆ . Suppose that for some d . 0 we have
sup
1 U]] s J s bd 2 J s0ddU 5 o s1d. n
un 12d bu#1
222d
n
n
(3)
p
Then, by Theorem 10.1 in Wooldridge (1994), n bˆ 5 Ops1d and 1 ] Sns0d n 5 ]]]] 1 o ps1d 1 2 ]2 Jns0d n
n bˆ
⇒2
1E
(4)
21 1
1
2 E
Busrd 2 dr
0
Busrd dBesrd.
0
To prove (3), notice by the mean value theorem that sup
1 U]] s J s bd 2 J s0ddU n 1 sup U]] SO x ( L~ ( b x ) 2 L~ (0))DU 5 n
un 12d bu#1
222d
n
n
n
5
un 12d bu#1
2 t
222d
t51
t
sup un 12d bu#1
U
1 ]] 222d n
SO n
t51
DU
x 3t bL¨ s b 11 x td
,
where L¨ sxd 5 e x /s1 1 e xd 2 2 2e x /s1 1 e xd 3 and b 11 locates between zero and b. Since since u bu , 1 / n 12d and uL¨ sxdu # 3,
E. Guerre, H.R. Moon / Economics Letters 76 (2002) 267 – 271
270
sup un 12d bu#1
U
1 ]] 222d n
SO
DU
n
x bL¨ ( b 11 x t ) 3 t
t51
O
n 1 # 3 sup ]] ux tu 3 . 323d t51 un 12d bu#1 n
Choose 0 , d , 1 / 6. Then, sup un 12dbu#1 1 /n 323d o nt51 ux tu 3 5 o ps1d, and in consequence, sup un 12d bu#1
1 (J ( b ) 2 J (0))U 5 o s1d, U]] n 222d
n
n
p
as required for (3). As a summary, we have the following theorem. Theorem 1. Under the assumptions in the previous section, if b0 5 0, then 21 1
1
n bˆ
⇒2
1E 2 E 1 1E 2 2 Busrd 2 dr
0
Busrd dBesrd
0
21
1
; MN 0, 4
Busrd 2 dr
,
0
where notation ‘ ; ’ signifies equivalence in distribution and MN denotes a mixed normal distribution. Notice that in a univariate nonstationary binary choice model studied by Park and Phillips (2000), when the true parameter b0 ± 0, the MLE is n 1 / 4 -consistent and its limit distribution is mixed normal. When the true parameter b0 5 0, Theorem 1 shows that the MLE has a faster convergence rate n and its limit distribution is similar to the so-called ‘unit root distribution’. In our note, since we assume that u t and ´s are independent, the limit distribution is also mixed normal. The conventional t-statistic in this case is
bˆ t 5 ]]] ]]] . œ 2 Jnsbˆ d By (3) and (2), we have Jns0d Jnsbˆ d 1 2 ]] 2 5 2 ]] 2 1 o ps1d ⇒ ] 4 n n
1
E B srd dr. u
2
0
Thus,
bˆ t 5 ]]] ]]] ⇒ Ns0, 1d. œ 2 Jnsbˆ d Summarizing this, we have the following corollary. Corollary 2. Under the assumptions in the previous section, if b0 5 0, then
E. Guerre, H.R. Moon / Economics Letters 76 (2002) 267 – 271
271
bˆ t 5 ]]] ]]] ⇒ Ns0, 1d. œ 2 Jnsbˆ d References Park, J., Phillips, P.C.B., 2000. Nonstationary binary choice. Econometrica 68, 1249–1280. Wooldridge, J., 1994. Estimation and inference for dependent processes. In: Engle, R., McFadden, D. (Eds.). Handbook of Econometrics, Vol. 4. North Holland, Amsterdam.