This paper deals with the problem of nonstationarity of regressors in binary choice model. The limit distribution of the ML-estimator is mixed normal, but restriction testing shall not be based on standard t-statistic. The results of the conducted Monte Carlo experiment demonstrate that the true size of the restriction test is far from the significance level. Therefore, the t -Student statistic should be modified and this paper proposes its modification. The results of the Monte Carlo investigation point to the superiority of the new statistic.nonstationarity, maximum likelihood, restriction testing

Wojciech Grabowski

English

Research Papers in Economics

Central European Journal of Economic Modelling and Econometrics
Restriction Testing in Binary Choice Model with I(1)
Regressors
Wojciech Grabowski∗
Submitted: 5.10.2009, Accepted: 10.03.2010
Abstract
This paper deals with the problem of nonstationarity of regressors in binary
choice model. The limit distribution of the ML-estimator is mixed normal,
but restriction testing shall not be based on standard t-statistic. The results
of the conducted Monte Carlo experiment demonstrate that the true size of
the restriction test is far from the signiﬁcance level. Therefore, the t-Student
statistic should be modiﬁed and this paper proposes its modiﬁcation. The results
of the Monte Carlo investigation point to the superiority of the new statistic.
Keywords: nonstationarity, maximum likelihood, restriction testing
JEL Classiﬁcation: C12, C25.
∗University of Łódź; email: emfwog@uni.lodz.pl
301 Wojciech Grabowski
CEJEME 1: 301-309 (2009)Wojciech Grabowski
1 Introduction
Though binary choice models have been an important tool of microeconometrics for
many years, there are many settings in which the macroeconomic outcome we seek to
model is a discrete choice between two alternatives, rather than a continuous measure
of some activity. For instance, a central bank decides whether or not to intervene in the
foreign exchange market. Consequently, a currency intervention shall be considered
a binary variable. Many papers treat currency crises and ﬁnancial crises as a binary
variable (crisis, tranquility period). Papers devoted to business cycle modeling distin-
guish between two states (recession, no recession). These macroeconomic time series
often depend on time series integrated of order 1. Park and Phillips (2000) considered
a binary choice model with I(1) regressors and developed asymptotic theory on such
models. The authors proved that the limit distribution of the maximum likelihood
estimator was mixed normal with mixing variates being dependent upon Brownian
local time as well as Brownian motion. In the paper Park and Phillips (2000) it is
shown that the ML estimator converges at a rate T
3
4 along its principal component,
having a slower rate of T
1
4 convergence in all other directions. Guerre and Moon
(2002) and Grabowski (2009) show that if all parameters in a binary choice model
are equal to zero, the maximum likelihood estimator is T-consistent and asymptoti-
cally normal. The problem of nonstationary regressors in a binary choice model was
considered by Grabowski (2007a, 2007b), who included I(2) time series in a binary
choice model. It is well known that in the case of a binary choice model with station-
ary regressors the rate of convergence is T
1
2 and the variance of the ML estimator
is equal to the inverted Fisher Information Matrix. Restriction testing in a binary
choice model with stationary regressors may be based on the traditional t-statistic. If
nonstationary regressors are included in the right-hand side of a binary choice model,
the asymptotic distribution of the ML estimator is normal, but the standard statistic
shall not be used for testing restriction βk = b if b 6= 0. This paper proposes an al-
ternative, appropriate statistic. It shows that the true size of a restriction test diﬀers
from the theoretical size if I(1) variables are included in the binary choice model and
the standard t-student statistic is used.
The paper is organized as follows. Section 2 presents asymptotic theory for binary
choice models having only stationary or I(1) regressors. Section 3 discusses the re-
sults of the Monte Carlo investigation where the true size of the restriction test was
compared with its theoretical size for restriction testing involving the traditional t-
statistic. In Section 4, an alternative statistic is derived and its superiority is proved
by means of another Monte Carlo experiment. The paper concludes with Section 5.
Wojciech Grabowski
CEJEME 1: 301-309 (2009)
302Restriction Testing in Binary Choice Model with I(1) Regressors
2 Asymptotics for the binary choice models
Consider the following binary choice model:
y∗
t = xtβ + ξt, ξt ∼ F t = 1,...,T,
yt = 1{y∗
t ≥0}, (1)
where xt is a row vector of regressors, β is a column vector of parameters and F is
a cumulative distribution function of disturbance ξ. We assume that β 6= 0. If xt
consists only of stationary variables, then the asymptotic distribution of the maximum
likelihood estimator is as follows:
b β
ML ∼ N
￿
β,I(β)
−1￿
, (2)
where I(β) = E
h
−
∂
2 log L(β|X,y)
∂β∂βT
i
. If F(·) is a cumulative distribution function of the
normal distribution (F = Φ), then:
b βML ∼ N

β,
(
T X
t=1
ϕ(xtβ)2
(1 − Φ(xtβ))Φ(xtβ)
xT
t xt
)−1
, (3.a)
and if F(·) is a cumulative distribution function of the logistic distribution (F = Λ),
then:
b βML ∼ N

β,
(
T X
t=1
Λ(xtβ)(1 − Λ(xtβ))xT
t xt
)−1
, (3.b)
where Λ =
exp(xtβ)
1+exp(xtβ). Results (2) and (3) are standard and well-known, likewise
the consequences of asymptotic theory for the maximum likelihood estimator under
satisﬁed regularity conditions. If a researcher wants to test restriction βk = b within a
binary choice model with stationary regressors, they can use the critical values of the
standardized normal distribution for large samples or the critical value of t-student
distribution in smaller samples.
When regressors are nonstationary, asymptotic distributions (3.a) and (3.b) are in-
valid. The asymptotic distribution of the maximum likelihood estimator of a binary
choice model with I(1) regressors was derived by Park and Phillips (2000). Before
presenting the results obtained by Park and Phillips (2000), let us formulate the
following assumptions:
Assumption 1. A data generating process. Process xt is generated in the following
way:
xt = xt−1 + νt, t = 1,...,T
with P(x0 = 0) = 1, where:
νt = Π(L)ηt =
∞ X
i=0
Πiηt−i,
303 Wojciech Grabowski
CEJEME 1: 301-309 (2009)Wojciech Grabowski
with Π(1) 6= 0 and
P∞
i=0 ikΠik < ∞. The innovations ηt are i.i.d. with mean zero
and Ekηtkr < ∞, for some r > 8, have distributions that are absolutely continuous
with respect to the Lebesgue measure and have the characteristic function ϕ, which
satisﬁes limktk→∞ ktkpϕi(t) = 0 for some p > 0.
Assumption 2. Denote by F(·) the cumulative distribution function of εt. Deﬁne
the following functions:
G =
˙ F
F(1 − F)
; K = G · ˙ F = G
2 · F · (1 − F).
F is three times diﬀerentiable so that ˙ F, ¨ F, ˙ G, ¨ G all exist.
As in Park and Phillips (2000), the regressor space is rotated using an orthogonal
matrix H = [h1H2], where h1 =
β √
β0β. We then deﬁne the following processes:
V1 = h
0
1V,
V2 = H0
2V,
where V is a Brownian motion in (Ω,F,P). Deﬁne chronological local time of the
process V1, see Park and Phillips (1999) and (2001), given by:
L1(t,s) = lim
ε→0
1
2ε
Z t
0
1{|V1(r)−s|<ε} dr, (4)
which measures the sojourn time in chronological units that the process spends in the
vicinity of the spatial point s. For a more extended analysis of this process see Revuz
and Yor (1994). After deﬁning the foregoing quantities, we can formulate a theorem,
which gives the asymptotic distribution of the maximum likelihood estimator in a
binary choice model with I(1) regressors.
Theorem 1. Consider the estimation of the parameters of a binary choice model
(1) by maximum likelihood. Let assumptions 1 and 2 hold. Then the asymptotic
distribution of the ML estimator is as follows:
b βML ∼ N
￿
β,T − 1
2Pβq
−1
11.2
￿
, (5)
where: Pβ = β(β0β)−1β0, q11.2 = q11 − q12Q
−1
22 q12 and the elements of matrix
Q =
￿
q11 q12
q21 Q22
￿
are:
q11 = L1(1,0)
R ∞
−∞ s2K(αs)ds,
q12 =
R 1
0 dL1(r,0)V2(r)0 R ∞
−∞ sK(αs)ds,
q21 =
R 1
0 V2(r)dL1(r,0)
R ∞
−∞ sK(αs)ds,
Q22 =
R 1
0 V2(r)V2(r)0dL1(r,0)
R ∞
−∞ K(αs)ds,
α =
√
β0β
Wojciech Grabowski
CEJEME 1: 301-309 (2009)
304Restriction Testing in Binary Choice Model with I(1) Regressors
This theorem was formulated and proved by Park and Phillips (2000).
It can be noticed that, if the binary choice model’s regressors are integrated of order
1, then the asymptotic distribution of the ML estimator is normal too. Obviously,
restriction testing in smaller samples can use the t-Student distribution. But with
b 6= 0 the t-statistic for a model with stationary regressors is diﬀerent from the t-
statistic for a model with I(1) regressors. To derive the appropriate statistic, we
present a corollary of Park and Phillips (2000) that results from Theorem 1.
Corollary 1. Let assume that 1 and 2 hold. If
R ∞
−∞ sK(s)ds = 0 and β 6= 0 then
T
1
4
￿
b βML − β
￿
d −→
β
kβk
￿
L1(1,0)
Z ∞
−∞
s2K (skβk) ds
￿− 1
2
W(1), (6)
where kβk =
p
βTβ and W are a univariate standard Brownian motion independent
of V .
This corollary was formulated and proved by Park and Phillips (2000).
3 Restriction testing based on a traditionally
calculated t-Student statistic
Consider a binary choice model and the following hypothesis:
H0 : βk = b,
H1 : βk 6= b , (7)
where b 6= 0. According to asymptotic distributions (3.a), (3.b) of the maximum
likelihood estimator of a binary choice model with stationary regressors, the testing
of hypothesis (7) can utilize t-statistic given by the formula:
tb βk =
b βML
k − b
dkk
, (8)
where dkk is the k-th diagonal element of matrix ￿
PT
t=1
ϕ(xtb β
ML)
2
(1−Φ(xtb βML))Φ(xtb βML)xT
t xt
￿−1
for a probit model and the k-th diago-
nal element of matrix
nPT
t=1 Λ(xtb βML)
￿
1 − Λ
￿
xtb βML
￿￿
xT
t xt
o−1
in the case of a
logit model. For regressors integrated of order 1 and b = 0, testing is also based on
statistic (8).
However, the latter is inappropriate if the binary choice model (1) has nonstationary
regressors. To make it clear, let us run a Monte Carlo experiment for the logit model.
In our experiment, explanatory variables are generated according to the following
formula:
305 Wojciech Grabowski
CEJEME 1: 301-309 (2009)Wojciech Grabowski
xi
jt = xi
jt−1 + εi
jt, t = 1,...,T, xi
j0 = 0, εi
jt ∼ N(0,1), j = 1,2 (9)
where i = 1,2,...,I denotes the number of replication. An unobservable dependent
variable is generated by formula:
yi∗
t = 1 + 0.001xi
1t + 0.001xi
2t + ξi
t, t = 1,...,T, ξi
t ∼ Λ. (10)
The observable values of variable yt are obtained as follows:
y
i
t = 1{yi∗
t >0}, t = 1,...,T. (11)
In the next step of our experiment, parameters β0, β1, β2 of the binary choice model:
yi∗
t = β0 + β1xi
1t + β2xi
2t + ξi
t, t = 1,...,T, ξi
t ∼ Λ (12)
are estimated for each replication. Consider now the following statistic:
tb βi
1 =
b βi
1 − 0.001
b σb βi
1
, (13)
where b βi
1 is the maximum likelihood estimate of parameter β1 for the i-th iteration.
b σb βi
1 is the estimate of the standard deviation of the maximum likelihood estimator
for the i-th iteration. This estimate is a square root of the second diagonal element
of the following matrix:
b D2
￿
b βi
￿
=
 
T X
t=1
Λ(xi
tb βi)
￿
1 − Λ
￿
xi
tb βi
￿￿￿
xi
t
￿T
xi
t
!−1
. (14)
It is clear that if the size of the restriction test based on statistic (13) is equal to the
signiﬁcance level, then:
lim
I→∞
1
I
I X
i=1
1￿
|tb βi|>t
1− λ
2 ;T−1
￿ = λ, (15)
where t1− λ
2 ;T−1 denotes an appropriate percentile of t-student distribution with T −1
degrees of freedom. We run 100 000 replications for each case and calculate - for the
diﬀerent numbers of observations and the diﬀerent signiﬁcance levels - the following
quantity:
S(T,λ) =
1
100 000
100 000 X
i=1
1￿
|tb βi|>t
1− λ
2 ;T−1
￿. (16)
Quantity (16) is interpreted as the true size of the signiﬁcance test (with statistic
(8)), when the regressors in a binary choice model are integrated of order 1. Table
Wojciech Grabowski
CEJEME 1: 301-309 (2009)
306Restriction Testing in Binary Choice Model with I(1) Regressors
11 presents the values taken by this quantity. As most economic time series have
numbers of observations varying between 20 and 300, the samples range from 20
to 300. The values are reported for the most frequently used statistical inference
signiﬁcance levels, namely (α = 0.1;0.05;0.02;0.01).
Table 1: Simulated quantity (16) for logit model regressors integrated of order 1 and
a traditional t-statistic
λ
0.1 0.05 0.02 0.01
T
20 0.071 0.019 0.000 0.000
50 0.086 0.037 0.006 0.003
100 0.081 0.031 0.009 0.003
200 0.084 0.030 0.008 0.003
300 0.086 0.032 0.010 0.004
The results show that the diﬀerences between the true size and the signiﬁcance level
are large, so traditional t-Student (8) shall not be used for inferring about signiﬁcance
in the binary choice model with I(1) regressors.
4 A new statistic for restriction testing involving
binary choice model with I(1) regressors
The results of the Monte Carlo experiment have shown that the traditional t-student
statistic shall not be used for verifying restriction in the binary choice models with
I(1) regressors. This section of the paper proposes an alternative solution, which is
based on corollary 1. According to the corollary, if we set hypothesis (7), we shall use
the following statistic:
tb βk =
b βk − b
r￿PT
t=1 K
￿
xtb βML
￿
x2
kt
￿−1
q
c L1(1,0)
b wk
, (17)
where:
b wk =
b βk qPK
i=1 b β2
i
, (18)
c L1(1,0) = lim
ε→0
1
2ε
Z 1
0
1{|c V1(r)|<ε} dr (19)
and
307 Wojciech Grabowski
CEJEME 1: 301-309 (2009)Wojciech Grabowski
∀r∈[0,1] V1(r) =
PK
i=1 b βi qPK
i=1 b β2
i
V (r). (20)
K(s) =
ϕ(s)
2
Φ(s)(1−Φ(s)) in the case of a probit model and K(s) = 1
Λ(s)(1−Λ(s)) in the logit
case. Because statistic (17) diﬀers from statistic (8), hence with a non-zero restriction
imposed on parameter βk we shall not use statistic (8), because the obtained results
would be inappropriate then.
In order to demonstrate the superiority of statistic (17) when restriction (7) is tested in
a binary choice model with I(1) regressors, we shall conduct a Monte Carlo experiment
to compare the true size with its signiﬁcance level, when restriction (7) is tested and
statistic (17) is used. Following the setup of the ﬁrst Monte Carlo investigation, the
samples range from 20 to 300, the data are generated according to formulas (9)-(12)
and quantity (16) is computed.
Table 2: Simulated quantity (16) values for logit model regressors integrated of order
1 and statistic (17)
λ
0.1 0.05 0.02 0.01
T
20 0.102 0.050 0.020 0.011
50 0.099 0.050 0.020 0.010
100 0.100 0.050 0.020 0.010
200 0.100 0.050 0.020 0.010
300 0.100 0.050 0.020 0.010
The second Monte Carlo experiment shows that testing restriction (7) in a binary
choice model with I(1) regressors and using β 6= 0 and statistic (17) we obtain more
credible results than for the traditional statistic. The true size and the signiﬁcance
level are nearly equal in most cases, proving the superiority of statistic (17).
As mentioned above, statistic (8) is appropriate when restriction βk = 0 is veriﬁed.
The same observation was made by Guerre and Moon (2002) and Grabowski (2009).
5 Conclusions
Because of the problem of non-stationaryregressorsappearing sometimes in the binary
choice models, restriction βk = b has to be dealt with by means of an appropriate
statistic. For b 6= 0, the ML estimator of the parameters of a binary choice model
with I(1) regressors has a mixed normal limit distribution. Limit distribution (6)
shows, however, that in a binary choice model with I(1) regressors the ML estimator
converges to the parameter more slowly than in a model with stationary regressors.
Therefore, a larger sample is needed.
The results of the Monte Carlo investigation show that the true size of a restriction
Wojciech Grabowski
CEJEME 1: 301-309 (2009)
308Restriction Testing in Binary Choice Model with I(1) Regressors
test diﬀers from the signiﬁcance level when a traditional t-statistic is used in a binary
choice model with I(1) regressors. Accordingly, this paper proposes an alternative
statistic whose superiority has been proven by the second Monte Carlo investigation.
Acknowledgements
The useful comments from the anonymous referees are gratefully acknowledged.
References
[1] Grabowski W., (2009), A note on the nonstationary probit model, Unpublished
Manuscript, University of Lodz.
[2] Grabowski W., (2007a), Integrated Time Series in Binary Choice Models, [in:]
33rd International Conference MACROMODELS’06, [ed.:] A.Welfe, W.Welfe,
215-233.
[3] Grabowski W., (2007b), Asymptotics for Integrated Time Series in Binary Choice
Models, presented at 62-nd Econometric Society European Meeting, Budapest,
27-31 August.
[4] Guerre E., Moon H.R., (2002), A note on the nonstationary binary choice logit
model, Economics Letters 76, 267-271.
[5] Park J.Y., Phillips P.C.B., (1999), Asymptotics for nonlinear transformations of
integrated time series, Econometric Theory 15, 269-298.
[6] Park J. Y., Phillips P.C.B., (2000), Nonstationary binary choice, Econometrica
68, 1249-1280.
[7] Park, J.Y. and Phillips P.C.B., (2001), Nonlinear regressions with integrated
time series, Econometrica 69, 117-161.
[8] Revuz, D., Yor M., (1994), Continuous Martingale and Brownian Motion, 2 ed.,
Springer-Verlag, New York.
309 Wojciech Grabowski
CEJEME 1: 301-309 (2009)

A note on the nonstationary binary choice logit model,

A note on the nonstationary probit model,

Asymptotics for Integrated Time Series in Binary Choice Models, presented at 62-nd Econometric Society European Meeting,

Asymptotics for nonlinear transformations of integrated time series,

Nonlinear regressions with integrated time series,

Nonstationary binary choice,

Restriction Testing in Binary Choice Model with I(1) Regressors

http://cejeme.org/publishedarticles/2010-38-27-634105175172500000-4821.pdf

Restriction Testing in Binary Choice Model with I(1) Regressors

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics