bootstrap lm tests for the box cox tobit model
TRANSCRIPT
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Introduction
• This presentation sets out a specification test of the Tobit model against the
alternative of a specification described by the Box Cox transformation.
• An LM test is used to test the null hypothesis of no specification error as this
requires estimates of the restricted (nested) Tobit) model
• The size and power of the test using asymptotic and bootstrap critical values
is estimated by the empirical rejection probabilities for small sample sizes
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1. The Box Cox Tobit Model
• The Tobit model is used to address censoring and corner solution problems.
• When censoring occurs at zero, the model in both applications is written:
where is a `latent’ variable and . The observation rule is:
• In censored data problems, we are usually interested in the features of such
as . For corner solutions however, it is that is of interest.
• Estimation of the parameters , and in (1) is by Maximum Likelihood (ML),
with individual contribution to the log-likelihood given by:
y¤i = x0
i¯+ ²i; i = 1; ::;N (1)
²i »NID¡0; ¾2
¢
yi =
½y¤i if y¤i ¸ 0
0 if y¤i < 0
E [y¤i j xi] E [yi j xi]
y¤i
y¤i
lnLi = di ln
·1¾Á
µyi¡x
0i¯
¾
¶¸+ (1¡ di) ln
·1¡©
µx0i¯)
¾
¶¸
¯ ¾
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1. The Box Cox Tobit Model
• As Moffat (2003) noted however, there are many instances where exhibits
positive skew that cannot be attributed to the asymmetric censoring.
• In the double hurdle model, Moffat takes the following transformation of to
preserve normality:
• The transformation, originally proposed by Box & Cox (1966) for uncensored
data, was designed to ensure that the model for is:
1. Linear in the explanatory variables
2. Has a constant conditional error variance
3. Has a normally distributed error term
• The above properties are essential for the ML-estimators to be consistent
for the true parameters in the Tobit model (1):
yi
yTi =y¸i ¡1¸
0 · ¸ · 1
yi
^̄ p¡! ¯ ¾̂p¡! ¾
yTi
E[²²0 j X] = ¾2IN
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1. The Box Cox Tobit Model
• Applying the Box Cox Transformation (BCT) to the Tobit model therefore, leads
to the following observation rule:
• where is the `transformed’ latent variable with specification:
• This should now satisfy (or approximately) the distributional requirements
for the ML-estimator to be consistent.
• By a change of variables, the contribution to the log-likelihood is:
yTi =
½yT¤i if yT¤i ¸ ¡1=¸
¡1=¸ if yT¤i < ¡1=¸
ith
lnLi = di ln
"y¸¡1i
¾Á
áy¸i ¡ 1
¢=¸¡ x
0i¯
¾
!#+ (1¡ di) ln
"1¡©
Ã1=¸+ x
0i¯
¾
!#(2)
y¤Ti = x0
i¯ + ²i;
yTi ¤
²i »NID¡0; ¾2
¢
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2. LM test of the Tobit specification
• A test of the linearity, homoskedasticity and normality assumptions of the
Tobit specification, is therefore equivalent to a test of:
• against the more general alternative:
• The LM-statistic is the easiest to compute as this requires parameter estimates
under the restrictions imposed by the null .
• Denoting as an vector one 1’s, where
represents the contribution to the unrestricted score
evaluated at the restricted , then the OPG-version of the LM-test is:
H0 : ¸= 1
H1 : ¸ 6= 1
»µ = (
»¯;
»¾; 1)
»gi =
@ lnLi@µ
j»µ
»G= (
»g1; :::;
»gN)
0¿
»µ
LM = ¿0»G(
»G
0»G
0
)¡1»G
0
¿d¡! Â21
ithN £ 1
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2. LM test of the Tobit specification
• In this form, the LM-statistic is simply from artificial regression:
• From (2), the individual elements of are:
• where and . Under the restrictions
imposed by the null, (3) and (4) are the scores of the Tobit model evaluated
at the Tobit MLE’s; (5) can therefore be constructed from these estimates.
»vi1 = yi ¡ (1 + x
0
i
»¯)
»vi2 = 1+ x
0
i
»¯
N £R2u
1=»g0
i¼+ ei»g i
@ lnLi(µ)
@¯j»µ= di
»vi1
»¾2xi + (1¡ di)
¡Á(»vi2=»¾)
1¡©(»vi2=
»¾)
xi»¾
(3)
@ lnLi(µ)
@¾j»µ= di
1»¾
·v2i1»¾2¡ 1
¸+ (1¡ di)
Á(»vi2=
»¾)
1¡©(»vi2=
»¾)
»vi2»¾2
(4)
@ lnLi(µ)
@¸j»µ= di
"lnyi ¡
»vi1»¾2[yi (lnyi ¡ 1) + 1]
#+ (1¡ di)
Á(»vi2=
»¾)
1¡©(»vi2=
»¾)
1»¾
(5)
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3. Bootstrap Critical Values
• The critical value for a test of size- is the solution to where
and is the distribution of the data.
• Unless is known, cannot be obtained and we use critical values from
the limiting distribution under , i.e.: .
• The size of the test using is which can be determined through
the asymptotic expansion . This error can be large
• An alternative approach is to obtain critical values from the bootstrap null
distribution which replaces with a consistent estimator . Then:
• which has a smaller error of order . The critical value solving
be found by Monte Carlo simulation as the
quantile of the B ordered bootstrap statistics
Gn(c;F0) = Pr(LM · c)
H0
F0 = F(xi; yi; µ0)
® Gn(cn;®;F0) = 1¡®
F0 cn;®G1(c1;®) = Pr(Â21 · c1;®) = 1¡®
Gn(c;F0) =G1(c) +O(n¡1)®+O(n¡1)c1;®
Gn(c;Fn) F0 Fn
Gn(c;F0) =Gn(c;Fn) +O(n¡3=2) (6)
O(n¡3=2)
Gn(cyn;®;Fn) = 1¡® 1¡®
cyn;®
LMy1 ; :::; LM
yB
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4. The Parametric Bootstrap Algorithm
• The null is rejected if .
• In the - simulations, each bootstrap sample is generated by re-sampling
from the EDF, while generating from . The algorithm is:
H0 : ¸= 1
F(yi; j xi;»µ)
xiyi
1. Estimate the Tobit model parameters: , .This imposes the constraint
2. Draw a random sample of size from the EDF of and denote these
3. Generate errors from and denote these
4. Use the values in steps 2 and 3 to generate a bootstrap sample of size
and compute
5. Estimate the Tobit model using the bootstrap sample and compute the
contributions to the scores
6. Estimate the artificial regression and compute
7. Repeat steps 2 – 6 a total of - times and compute the critical value as
the percentile of the ordered bootstrap LM-test statistics.
¸= 1
N
xyi ; :::; x
ynN
LM > cyn;®
N(0; ¾̂2) ²y1; :::; ²
yn
y¤yi = x
y0i^̄ + ²
yi
yyi =max(0; y
¤yi )
N
»gyi ; :::;
»gyN
1 =»gy0
i ± + ui LMyb =N £R2
u
B
1¡®
cyn;®B
B
^̄ ¾̂
xi
1010
5. Monte-Carlo Design
• The size and power of the LM-test using bootstrap and first-order asymptotic critical values can be estimated from the empirical rejection probabilities.
• The data for the Monte-Carlo experiments is generated from the DGP:
The experiments consist of the following steps:
.
• As , then . Thus forand ,
yTi =
½yT¤i if yT¤i ¸ ¡1=¸
¡1=¸ if yT¤i < ¡1=¸
yi =¡¸yT¤i + 1
¢1=¸
1. Generate values for and from a specified DGP and compute
2. Estimate the LM statistic for testing as detailed earlier
3. Compute the bootstrap critical value at the -level for testing
4. Repeat steps 1-3, - times and count the rejections . The empirical rejection
probability , is an estimate of the true rejection probability .
²i xi
y¤Ti = x0
i¯ + ²i;
y¤Ti , yTi yi
® H0 : ¸= 1
T R
R=T
H0 : ¸= 1
N
R» B(T; p)pT (R=T ¡ p)
d¡!N [0; p(1¡ p)] p= 0:05
Pr(0:04 · R=T · 0:06 j p= 0:05) ¼ 0:95
p
T = 2000
1111
• Under , the empirical rejection probability is an estimate of the size
of the LM-test using bootstrap & asymptotic critical values .
• For these experiments , , , ,
and where: , and
. The size estimates are:
• Using bootstrap critical values there is no size distortion. This is not the
case using asymptotic critical values which result in large size distortions
5.1 Size Estimates
¯0 = 1N = 25 B = 499 T = 2000
H0 : ¸= 1
®= 0:05
¯1 2 f¡:5;¡:55;¡:6;¡:65;¡:7;¡:75;¡:8;¡:85;¡:9;¡:95g
²i »NID(0;1)
x0
i¯ = ¯0 + ¯1xi1 lnxi »N(1;0:5)
.05
.1.1
5.2
.25
.3
Em
pir
ica
l Reje
ctio
n P
rob
ab
ility
-1-.9-.8-.7-.6-.5 b1
Asymptotic Test Bootstrap Test
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5.2 Power Estimates (1)
• Under , the empirical rejection probability is an estimate of the
power of the LM-test against the alternative.
• For these experiments, , , , ,
and where , , and
. The power estimates are:
• With the exception of , the LM-test using bootstrap critical values
at the 5% level of significance seems reasonably powerful for
H1 : ¸= ¸1
¯1 =¡:5¸= ¸1 2 f:1; :15; :2; :::;1:3g
N = 25 ®= 0:05 B = 499 T = 2000 ²i »NID(0;1)
x0
i¯ = ¯0 + ¯1xi1 lnxi »N(1;0:5) ¯0 = 1
0.2
.4.6
.81
Em
pir
ica
l Reje
ctio
n P
rob
ab
ility
.2 .4 .6 .8 1 1.2lambda
Asymptotic test Bootstrap Test
¸= 0:5
N = 25
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5.3 Power Estimates (2)
• Whilst the LM-test exhibits reasonable power for , it is worth examining
the power against DGP’s where a would necessary for consistency
• For these experiments, , , , , and the data are
generated using similar DGP’s to those used by Drukker(2002):
• The are generated from, , , and , distributions and the function
for homoskedastic and for hetroskedastic errors.
• The following table sets out the power estimates:
¸ 6= 1
¸ 6= 1
y¤i = 1+xi1 +xi2 + xi3 + ²iph(z
0i®);
xi1 »N(0;1) xi2 = :3x1i +ui2;
xi3 = :3x1i +ui3;
ui2 »N(0;1)
ui3 »N(0;1)
Distribution h(z0
i®) = 1
N(0; 1)
t4
Â25
N = 100 ®= 0:05 B = 499 T = 2000
²i N(0; 1) t4 Â25
h(z0
i®) = 1 h(z0
i®) = e2xi1
h(z0
i®) = e2xi1
0.8720:140
0:795
N=A 0:734
0:085
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6. Description of `bctobit’ Program
bctobit [, Fixed Nodots bfile(string) reps(integer 499)]
Description
• bctobit computes the LM-statistic for testing against in the Box Cox Tobit model. This is equivalent to testing the linearity, normality and homoskedasticity assumptions of the Tobit specification.
• The regressors are assumed to be random, and critical values are obtained from the bootstrap null distribution of the LM test statistic by repeated sampling from the (parametric) bootstrap DGP.
Options
• Fixed - specifies that the regressors are fixed in the bootstrap null distribution
• Nodots – suppresses the replication dots
• bfile(name) – the name of the saved file which contains the LM-statistics computed from the bootstrap samples
• reps(#) - the number of samples to be drawn from the bootstrap DGP to estimate the percentiles of the bootstrap null distribution. Default is 499
H0 : ¸= 1 H1 : ¸ 6= 1
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6. Description of `bctobit’ Program
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7. Further Research....
• A natural extension would be to consider the alternative of a Box Cox
transformation with an error term that is hetroskedastic
• where is an unknown function , with , and
• A test of the joint hypothesis: against the alternaitve of
is equivalent to testing the validity of the Tobit specification.
• The LM statistic would now be based on the additional components of the
score vector, evaluated at the restrictions given by the null. These are:
• As such . The size and power using bootstrap critical
values can be estimated from empirical rejection probabilities as before.
H1 : ¸= 1; ´ = 0
yT¤i = x0
i¯ + ²iph(z
0i®);
H1 : ¸ 6= 1; ´ 6= 0
@ lnLi(µ)
@®j»µ= di
1
2
"»v2
i1
¾2¡ 1
#·zi + (1¡ di)
¡Á(»vi2=¾)1¡©(
»vi2=¾̂)
·zi
2»¾
h h0(:) 6= 0 h(0) = 1 h0(0) = ·
LMd¡! Â2
1+dim(z)
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8. References
• Box, G. E. P. and D. R. Cox (1964) “An Analysis of Transformations”, Journal
of the Royal Statistical Society, 26, 211-243.
• Drukker, D. M. (2002) “Bootstrapping a conditional moments test for
normality after tobit estimation”, The Stata Journal, 2, 125-139
• Moffatt, P. G. (2003) “Hurdle models of loan default” , School of Economic
and Social Studies, University of East Anglia, Norwich, UK