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Federal Reserve Bank of Chicago
Estimation of a Transformation Model
with Truncation, Interval Observation
and Time-Varying Covariates
Bo E. Honoré and Luojia Hu
WP 2009-16
Estimation of a Transformation Model with Truncation,
Interval Observation and Time–Varying Covariates∗
Bo E. Honor´†
e
Luojia Hu‡
First complete version: December 2007.
This Version: September 2009.
Abstract
Abrevaya (1999b) considered estimation of a transformation model in the presence
of left–truncation. This paper observes that a cross–sectional version of the statistical
model considered in Frederiksen, Honor´, and Hu (2007) is a generalization of the
e
model considered by Abrevaya (1999b) and the generalized model can be estimated
by a pairwise comparison version of one of the estimators in Frederiksen, Honor´, and
e
Hu (2007). Specifically, our generalization will allow for discretized observations of the
dependent variable and for piecewise constant time–varying explanatory variables.
∗
This research was supported by the National Science Foundation and the Gregory C. Chow Econometric
Research Program at Princeton University. We thank seminar participants at Rice University, Universit´
e
Paris 1 Panth´on–Sorbonne and the Federal Reserve Bank of Chicago as well as members of Princeton’s
e
Microeconometric Reading Group for comments. The opinions expressed here are those of the authors and
not necessarily those of the Federal Reserve Bank of Chicago or the Federal Reserve System.
†
Mailing Address: Department of Economics, Princeton University, Princeton, NJ 08544-1021. Email:
honore@Princeton.edu.
‡
Mailing Address: Economic Research Department, Federal Reserve Bank of Chicago, 230 S. La Salle
Street, Chicago, IL 60604. Email: lhu@frbchi.org.
1
1
Introduction
The transformation model
h (Ti∗ ) = g (Xi β) + εi
(1)
is often used to model durations. In models like this it is important to allow for right censoring
and sometimes also for left truncation because the samples used in many applications include
spells that are in progress at the start of the sample period. See Abrevaya (1999b).
It is also sometimes desirable to allow for the dependent variable to be discretized so
that one observes only whether it falls in a particular interval. So the observed duration, Ti ,
would be t if Ti∗ ∈ (t − 1, t]. See Prentice and Gloeckler (1978) or Meyer (1990). Moreover,
in duration models it is often interesting to allow for time–varying covariates, which are not
easily directly incorporated into the transformation model (Flinn and Heckman (1982)).
The contribution of this paper is to specify a statistical model that allows for interval
observations and time–varying covariates but which simplifies to a model with interval observations from (1) when the covariates are time invariant. We then propose an estimator
for the parameters of the model. The estimator can be interpreted as a generalization of the
truncated maximum rank correlation estimator proposed in Abrevaya (1999b).
Consider first the transformation model, (1), with strictly increasing h (·) and g (·) and
with εi independent of Xi and continuously distributed with full support. In this model
P (Ti∗ > t| Xi ) = P (h (Ti∗ ) > h (t)| Xi )
= P (g (Xi β) + εi > h (t)| Xi )
= 1 − F (h (t) − g (Xi β))
where F is the CDF for εi . This gives
P ( Ti∗ > t| Xi , Ti∗ > t − 1) =
1 − F (h (t) − g (Xi β))
1 − F (h (t − 1) − g (Xi β))
where the assumption that εi has full support guarantees that the denominator is not 0.
When 1−F (·) is log–concave (which is implied by the density of εi being log–concave; see
Heckman and Honor´ (1990)), the right–hand side is an increasing function of g (Xi β) and
e
2
hence of Xi β . See the Appendix. This means that one can write the event Ti∗ > t| Xi , Ti∗ >
t − 1 in the form 1 {Xi β > η it } for some (possibly infinite) random variable η it that is independent of Xi and has CDF
1−F (h(t)−·)
.
1−F (h(t−1)−·)
Therefore, if we define
Yit ≡ 1 {Ti∗ ∈ (t − 1, t]} = 1 {Ti = t}
then we can write
Yit = 1 {Xi β − η it ≤ 0} for t such that
Yil = 0
(2)
l≤t−1
In other words, a transformation model with discretized observations of the dependent variable and log–concave errors is a special case of the model
Yit = 1 {Xit β − η it ≤ 0} for t such that
Yil = 0
(3)
l≤t−1
where the difference between (2) and (3) is that the latter allows for time–varying covariates.
Note that this line of argument is valid even if Ti is left truncated.
It is interesting to note that Abrevaya (1999b) also assumes log–concavity of 1 − F (·).1
Note that although log–concavity implies an increasing hazard for h (T ∗ ), it does not impose
such a restriction on T ∗ .2
Equation (3) is a cross–sectional version of the model considered by Frederiksen, Honor´,
e
and Hu (2007). It is well understood that estimators of panel data models can be turned
into estimators of cross–sectional models by considering all pairs of observations as units in
a panel. See, for example, Honor´ and Powell (1994), who apply the idea in Honor´ (1992)
e
e
1
The assumption of log–concavity also appears elsewhere in the literature on truncated regression models.
See, for example, Lee (1993) and (Honor´ 1992).
e
∗
Let λ (·) denote the hazard for ε.
The hazard for h (T ∗ ) is then − ∂ log(P ( h(T )>t|X)) =
∂t
∂ log(1−F (t−g (X β )))
−
= λ (t − g (X β)). When 1 − F (·) is log–concave this is an increasing function of t. On
∂t
2
∗
T
the other hand, the hazard for T ∗ is − ∂ log(P (∂t
>t|X))
∗
= − ∂ log(P ( h(T )>h(t)|X)) = λ (h (t) − g (X β)) h (t).
∂t
2
The derivative of this with respect to t is λ (h (t) − g (X β)) h (t) + λ (h (t) − g (X β)) h (t), which can be
of either sign.
3
for panel data censored regression models to all pairs of observations in a cross section.
The insights in Frederiksen, Honor´, and Hu (2007) can therefore be used to construct an
e
estimator of β. We pursue this in Section 3 after discussing the model in more detail in
Section 2.
2
The model
Consider a spell with integer–valued duration, Ti , that starts at (integer–valued) time −Vi ≤
0. Following the discussion above, we model the event that the spell lasts s periods conditional on lasting at least s − 1, by the qualitative response model
Yis = 1 {Xis β − η is ≤ 0} ,
(4)
where η is is independent of Xis and the distribution of η is is allowed to change over time.
When there is left truncation, one must distinguish between duration time and calendar
time. We will index the observables, Y and X, by calendar time, and the unobservable η by
duration time. At first sight, this difference is confusing, but it is necessitated by the fact
that the discussion in the previous section implied that one should allow the distribution
of η is to vary by duration time. On the other hand, it seems natural to denote the first
observation for an individual by t = 1.
With this notation, we assume that we observe (Yit , Xit ) starting at t = 1, where
Yit = 1 Xit β − η i,t+Vi ≤ 0 .
(5)
With this notation, ηs with the same time subscript will have the same distribution under
the class of models discussed above. We will let Ti denote the first time that Yit equals 1.
Since Yit is not defined after the end of the spell, and since we want to allow for random
right censoring, we assume that we observe Yit from t = 1 until, and including, Ti or until a
random censoring time Ci − 1 (whichever comes first). In other words, we observe (Yit , Xit )
for t = 1, 2, ..., T i where T i = min {Ti , Ci − 1} So when an observation is censored, Ci will be
4
the first time period in which individual i is not observed. We also assume that we observe
the presample duration, Vi , for each observation.
The statistical assumption on the errors in (5) is that conditional on Vi and on {Yis = 0
for s < t}, η i,t+Vi is independent of Ci , {Xis }s≤t . As explained in Section 1, if the errors
are log–concave and the covariates are time–invariant, this is exactly what is implied by an
underlying transformation model for Ti∗ , where we observe whether a spell that started at
time −Vi and was in progress at time t − 1 is still in progress at time t. Note that when the
covariates are time–varying they are not restricted to be strictly exogenous, and that the
censoring times can be covariate–dependent, as long as they do not depend on the ηs.
In the next section we will apply the insight of Frederiksen, Honor´, and Hu (2007) to
e
construct an estimator for β under these assumptions when the researcher has access to a
random sample of individuals.
3
The estimator
The key insight for the construction of the estimator can be easily illustrated if we ignore
censoring first (so T i = Ti for all i).
Let t1 and t2 be arbitrary. Consider the two events A = {Ti = t1 , Tj > t2 } and B =
{Ti > t1 , Tj = t2 } where t1 + Vi = t2 + Vj . Under the stated assumptions, it then follows
immediately from Lemma 1 of (Frederiksen, Honor´,
e
1
>2
P ( A| A ∪ B, Xit1 , Xjt2 , Vi , Vj )
=1
2
<1
2
and Hu 2007) that
if (Xit1 − Xjt2 ) β > 0,
if (Xit1 − Xjt2 ) β = 0,
if (Xit1 − Xjt2 ) β < 0.
This suggests estimating β by maximizing
i<j
Ti
t1 =1
Tj
t2 =1
1 {t1 + Vi = t2 + Vj }
· 1 {Ti = t1 , Tj > t2 } · 1 (Xit1 − Xjt2 ) b > 0
+ 1 {Ti > t1 , Tj = t2 } · 1 (Xit1 − Xjt2 ) b < 0
5
(6)
(6) is the same as one of the objective functions in (Frederiksen, Honor´, and Hu 2007),
e
except that that paper considers a panel data situation.
It is convenient to rewrite (6) as
i<j
1 {Tj + Vj > Ti + Vi > Vj } · 1
XiTi − Xj,Ti +Vi −Vj b > 0
+1 {Vi < Tj + Vj < Ti + Vi } · 1
(7)
XiTj +Vj −Vi − Xj,Tj b < 0
This has the same structure as the maximum rank correlation estimator developed in Han
(1987).
When there is censoring, (6) can be modified to
i<j
Ti
t1 =1
Tj
t2 =1
1 {t1 + Vi = t2 + Vj , t1 < Ci , t2 < Cj }
(8)
· 1 {Ti = t1 , Tj > t2 } · 1 (Xit1 − Xjt2 ) b > 0
+1 {Ti > t1 , Tj = t2 } · 1 (Xit1 − Xjt2 ) b < 0
.
And equation (7) can be rewritten as
i<j
1 T j + Vj > Ti + Vi > Vj , Ti < Ci · 1
+1 Vi < Tj + Vj < T i + Vi , Tj < Cj · 1
XiTi − Xj,Ti +Vi −Vj b > 0
(9)
XiTj +Vj −Vi − Xj,Tj b < 0 .
The intuition for the estimator is essentially based on pairwise comparisons. Specifically,
we compare an individual i who was observed to fail at time Ti (and thus had a complete
duration Ti + Vi ) to all other observations j that survived up to the same duration. At
the true parameter value β, if the index for individual i at the time he/she failed, XiTi β, is
larger than the index for the comparable individual j at the time that corresponds to the
same duration, Xj,Ti +Vi −Vj β, then individual j is likely to survive longer than individual i
(that is, T j + Vj > Ti + Vi ). Note the set of comparison observations j includes censored
spells provided the censoring time occurs after Ti + Vi − Vj . The additional inequality in the
indicator function, Ti + Vi > Vj , ensures that the time at which j is being compared to i is
within the sample period (i.e., not truncated).
6
Again this estimator has the same structure as Han’s (1987) maximum rank correlation
estimator and the asymptotic distribution is therefore the one given in Sherman (1993) under
the regularity conditions stated there.
4
Asymptotic properties
Consistency and asymptotic normality can be established as in (Sherman 1993), Abrevaya
(1999b) or Khan and Tamer (2007).
First note that some normalization of the parameter is needed, since the parameter vector
is only identified up to scale. For example, we can normalize the last component of β to be
1.
The two key assumptions for consistency of the estimator are (1) at least one component
of the explanatory variable X is continuously distributed with full support, and (2) the error
η has full support. Without the first assumption, the parameter is not identified, since a
small change in the parameter value could leave the ranking of the index unchanged. The
second assumption on the error guarantees that the set of effective observations that make a
nonzero contribution to the objective function is not empty. Both assumptions are standard
in the semi-parametric estimation literature.
To establish asymptotic normality results, we need some additional notations. Denote
Di = 1 {Ti < Ci }, which is an observable variable indicating a complete (uncensored) spell.
The objective function can be rewritten as
1
n (n − 1)
n
Di · 1 Ti + Vi > Vj , Tj + Vj > Vi , T j + Vj > Ti + Vi
i=1 j=i
·1 XiTi b > Xj,Ti +Vi −Vj b .
Define the function
τ
t, t, d, v, {xs }s≤t , b
≡ E Di · 1 Ti + Vi > v, t + v > Vi , t + v > Ti + Vi · 1 XiTi b > xTi +Vi −v b
7
(10)
+E d · 1 t + v > Vj , Tj + Vj > v, T j + Vj > t + v · 1 xt b > Xj,t+v−Vj b
= E Di · 1 Ti + Vi > v, t + v > Vi , t + v > Ti + Vi · 1 XiTi b > xTi +Vi −v b
+E d · 1 t + v > Vi , Ti + Vi > v, T i + Vi > t + v · 1 xt b > Xi,t+v−Vi b
.
Following Theorem 4 of Sherman (1993), we have
√
∼
n β − β −→ N 0, 4Γ−1 ∆Γ−1
(11)
where
Γ = E
1
Ti , T i , Di , Vi , {Xis } , β
∆ = E
with
2τ
1τ
Ti , T i , Di , Vi , {Xis } , β
and
1τ
Ti , T i , di , Vi , {Xis } , β
denoting the first– and second–derivative operator, respectively.
2
Following Sherman (1993), we can further express the variance-covariance matrix in terms
of “model primitives.” Specifically,
∆ = VX
Xs2 − µs1 Xs2 β
S T, T , D, V, s1 , s2 , Xs2 β gXs1 β Xs2 β
s1 ,s2
and
Γ = EX
Xs2 − µs1 Xs2 β
Xs2 − µs1 Xs2 β
s1 ,s2
S7 T, T , D, V, s1 , s2 , Xs2 β · gXs1 β Xs2 β
where Xs2 is composed of the first K − 1 coordinates of Xs2 (the ones corresponding to the
piece of β that is not normalized to 1), gXs1 β (λ) is the marginal density of Xs1 β,
µs1 (λ) = E Xs1 | Xs1 β = λ
and
S
t, t, d, v, s1 , s2 , λ = E Ai t, t, d, v, s1 , s2 Xis1 β = λ
8
and
Ai t, t, d, v, s1 , s2
= Di · 1 Ti + Vi > v, t + v > Vi , t + v > Ti + Vi , Ti = s1 , Ti + Vi − v = s2
−d · 1 t + v > Vi , Ti + Vi > v, T i + Vi > t + v, t = s2 , t + v − Vi = s1
The asymptotic variance matrix can be estimated by plugging in the estimator β and
calculating sample analogs of Γ and ∆ using numerical derivatives based on a smoothed
version of τ . See Section 6 for more discussion.
5
Relationship to other estimators
The estimator proposed in the previous section is related to a number of existing estimators,
and it coincides with some of them in special situations. For example, when Ci = 2 for all
i, and with no left truncation (so Vi = 0 for all i), (5) is a standard discrete choice model,
and in that case the objective function in (9) becomes
n
1 {Yi > Yj } · 1 (Xi − Xj ) b > 0 + 1 {Yi < Yj } · 1 (Xi − Xj ) b < 0
i<j
which is the objective function for Han’s (1987) maximum rank correlation estimator.
When there is left truncation, and the covariates are time invariant and there is no
censoring, the estimator defined by maximizing (9) is the same as the truncated maximum
rank correlation estimator in (Abrevaya 1999b). This is most easily seen by noting that
without censoring and with time–invariant covariates, (10) becomes
1
n (n − 1)
=
1
n (n − 1)
n
1 {Ti + Vi > Vj , Tj + Vj > Vi , Tj + Vj > Ti + Vi } · 1 Xi b > Xj b
i=1 j=i
n
1 {Ti + Vi > Vj , Tj + Vj > Vi } 1 {Tj + Vj > Ti + Vi } · 1 Xi b > Xj b .
i=1 j=i
Except for the difference in notation and the normalization by n (n − 1), this is exactly
equation (7) in Abrevaya (1999b).
9
Khan and Tamer (2007) consider a model with left censoring as well as right censoring,
whereas we allow for left truncation as well as right censoring. When there is neither left
censoring nor left truncation, and when the covariates are time invariant, the estimator
defined by maximizing (9) coincides with the estimator proposed by Khan and Tamer (2007),
except that ours applies to discretized durations and theirs to exactly measured durations,
and we allow for time–varying covariates. Whether right censoring or right truncation is
more interesting depends on the specific application. Left truncation will, for example, be
relevant if as in Frederiksen, Honor´, and Hu (2007), the duration of interest is the length of
e
employment on a given job, and one has information on a sample of workers observed between
two fixed points in time. In this case the durations are left truncated, because spells that
ended before the start of the sampling will not appear in the data, and it is crucial for our
approach that one observes the duration of employment in the current job at the start of the
sampling. On the other hand, models with both left and right censoring are, for example,
useful for estimation of models where the dependent variable is a fraction, which is restricted
to be between zero and one, and where both zeros and ones are likely to be observed. See,
for example Alan and Leth-Petersen (2006). Both Khan and Tamer (2007) and we allow the
censoring points at the right to be observed only when the observation equals a censoring
point. Khan and Tamer (2007) also allow the left–censoring point to be unobserved when an
observation is not left–censored, whereas we assume that the truncation point is observed for
everybody who is not truncated, but not for truncated durations. Both papers assume that
one observes the actual duration, and not just the duration from the censoring/truncation
point. In the duration contexts we have in mind, this is the most severe assumption.3
The framework here is also closely related to standard statistical duration models with
discretized observations. The proportional hazard model can be written as
Z (T ) = −x β + ε,
3
See also Heckman and Singer (1986) for a discussion of the effect of different sample schemes on the
analysis of duration data.
10
where Z is the log integrated baseline hazard and ε has an extreme value distribution. Prentice and Gloeckler (1978), Meyer (1990) and (Hausman and Woutersen 2005) study a version
of this model with interval observations. Meyer (1990) and Hausman and Woutersen (2005)
also allow for time–varying explanatory variables and for ε to be a sum of an extreme value
distributed random variable and a random variable that captures unobserved heterogeneity.
While the estimation in Meyer (1990) is likelihood–based and hence fundamentally different
from ours, the structure of the estimator proposed in Hausman and Woutersen (2005) shares
many of the features of the estimator proposed here. The main difference is that theirs is
based on a comparison of the integrated hazards rather than just the current index, X β.
As a result, the approach does not seem to generalize to models with left truncation. On
the other hand, log–concavity plays no role in Hausman and Woutersen (2005).
6
Monte Carlo experiment
In this section, we conduct a small scale Monte Carlo study to illustrate the proposed estimation method and investigate its finite sample performance. We also demonstrate how
to conduct inference and examine how good an approximation the asymptotic distribution
provides for finite samples.
The designs are based on the following:
• All of the designs have two explanatory variables.
• β = (β 1 , β 2 ) = (1, 2) . The parameter of interest is θ = β 2 /β 1 . The fact that this is
one dimensional greatly simplifies the computations.
• Time–varying intercept β 0 = −4 + (s/10)1.2 . This introduces duration dependence
beyond the duration dependence introduced by the shape of F and by the choice of
h (·).
11
• The time between the start of a spell and the first period of observation is uniformly
distributed on the integers between 1 and 5.
• The censoring time is generated as the minimum of ten periods from the start of the
spell and Q periods from the start of the sample, where Q is uniformly distributed on
the integers between 1 and 8.
We consider a number of designs within this framework.
Design 1: Dynamic Probit.
The two explanatory variables are generated by i.i.d. draws from a bivariate normal
distribution with zero means, variances equal to 2 and 1, respectively, and covariance equal
to 1. In this design, η i,t is i.i.d N (0, 4).
Design 2: Transformation Model Hazard.
This design is set up as a generalization of a transformation model. Specifically, using
the notation of Section 1, we assume that h (u) = log (u), g (u) = u, and ε ∼ N (0, 1). Using
the derivation in Section 1, this yields
P (Ti = t) = P Ti∗ < t| {Xis }s≤t , Ti∗ > t − 1 = 1 −
1 − Φ (log (t) − Xit β)
.
1 − Φ (log (t − 1) − Xit β)
As in Design 1, the two explanatory variables are generated by i.i.d. draws from a bivariate
normal distribution with zero means, variances equal to 2 and 1, respectively, and covariance
equal to 1..
Design 3: Feedback.
Recall that our model does not require the explanatory variables to be strictly exogenous. In this design we therefore allow for feedback from the error η to future values of X.
Specifically, we follow Design 1 except that the explanatory variables are defined by
• X2t = η t−1 for t > 1 and standard normal for t = 1.
• X1t = X2t + N (0, 1).
12
Design 4: Covariate–Dependent Censoring and Truncation.
Our model allows censoring and truncation to be correlated with explanatory variables.
In this design, we follow the basic structure of Design 1 but let censoring be defined by the
outcome of a probit with explanatory variable X1t .
Design 5: Dynamic Probit 2.
This design is like Design 1 except that
• X2s ∼ N (0.5 − 0.2s, 1).
• X1s = 1 {X2s + N (0, 1) > 0}
• η i,s ∼ N (0, 0.1 + (0.15s))
The summary statistics for the five designs are reported in Table 1. For each design,
100,000 observations are drawn from the underlying data–generating process. We then compute the fraction of the sample that is censored, the fraction that is truncated, and the mean
and standard deviation of the underlying duration.
Below, we report Monte Carlo results for the point estimates of β as well as for the
performance of tests statistics based on the asymptotic distribution in Section 4. To do this,
we estimate the components of the variance of the estimators by sample analogs of smoothed
versions of the components.4
Recall that
Γ=E
4
2τ
Ti , T i , Di , Vi , {Xis } , β
A recent paper by Subbotin (2007) has shown that the nonparametric bootstrap can be used to estimate
the quantiles and variance of various maximum rank correlation estimators. The structure of our estimator
is essentially the same as that of the maximum rank correlation estimators he considers. We therefore
conjecture that the bootstrap could have been used to estimate the variance in our case as well, although
this would increase the computational burden.
13
and
∆=E
1τ
Ti , T i , Di , Vi , {Xis } , β
1τ
Ti , T i , Di , Vi , {Xis } , β
where
τ
t, t, d, v, {xs }s≤t , b
= E Di · 1 Ti + Vi > v, t + v > Vi , t + v > Ti + Vi · 1 XiTi b > xTi +Vi −v b
+E d · 1 t + v > Vi , Ti + Vi > v, T i + Vi > t + v · 1 xt b > Xi,t+v−Vi b
.
We then estimate τ by the smoothed version
τ
=
t, t, d, v, {xs }s≤t , b
1
n
n
+
Di · 1 Ti + Vi > v, t + v > Vi , t + v > Ti + Vi · Φ
i=1
n
1
n
d · 1 t + v > Vi , Ti + Vi > v, T i + Vi > t + v · Φ
i=1
XiTi b − xTi +Vi −v b
h
xt b − Xi,t+v−Vi b
h
Then
1τ
1
nh
=
t, t, d, v, {xs }s≤t , b
n
Di · 1 Ti + Vi > v, t + v > Vi , t + v > Ti + Vi
i=1
XiTi b − xTi +Vi −v b
XiTi − xTi +Vi −v
h
+d · 1 t + v > Vi , Ti + Vi > v, T i + Vi > t + v
xt b − Xi,t+v−Vi b
φ
xt − Xi,t+v−Vi ,
h
φ
and
2τ
=
−1
nh3
t, t, d, v, {xs }s≤t , b
n
Di · 1 Ti + Vi > v, t + v > Vi , t + v > Ti + Vi
i=1
XiTi b − xTi +Vi −v b φ
XiTi − xTi +Vi −v
14
XiTi b − xTi +Vi −v b
h
XiTi − xTi +Vi −v
.
+d · 1 t + v > Vi , Ti + Vi > v, T i + Vi > t + v
xt b − Xi,t+v−Vi b
XiTi b − xTi +Vi −v b φ
h
xt − Xi,t+v−Vi
And therefore
Γ=
and
1
∆=
n
1
n
xt − Xi,t+v−Vi
.
n
2τ
Ti , T i , Di , Vi , {Xis } , β
i=1
n
1τ
Ti , T i , Di , Vi , {Xis } , β
1τ
Ti , T i , Di , Vi , {Xis } , β
.
i=1
As mentioned earlier, β is only identified up to scale. One possibility is to normalize one
of the coefficients to 1, and hence essentially focus on β 2 /β 1 or β 1 /β 2 . Unfortunately, this
normalization will lead to different MAE and RMSE depending on which of the coefficients
is normalized. So if one were to compare different estimators, one might reach different
conclusions depending on a seemingly innocent normalization. This is unsatisfactory in
models where there is only one parameter. For this reason, it is likely to be better to
consider θ = log (β 2 /β 1 ) = log (β 2 ) − log (β 1 ) the parameter of interest. This means that the
true parameter is log (2) ≈ 0.693 for all of the designs. We estimate θ by a grid search over
the interval between − log (6) and log (6) with equal grids of size
1
.
200
Since the parameter of
interest is one dimensional, the line search is feasible despite the fact that the calculation of
the objective function requires O (n2 ) operations. When it is of higher dimension, it would
be beneficial to use the method described in Abrevaya (1999a) to calculate the objective
function in O (n · log (n)) operations. We calculate the variance of θ by applying the so–
called δ–method to (11).
For each design, the Monte Carlo experiment is conducted with 5,000 replications for
each of the 5 sample sizes: 100, 200, 400, 800 and 1,600. The results are reported in Tables
2–6. Overall, the results across the designs are broadly consistent with predictions from the
asymptotic theory. Some additional remarks are in order.
First, the results illustrate the consistency of the estimator, since both the median ab15
solute error (MAE) and root mean squared error (RMSE) decrease as sample size increases.
Moreover, the estimator is close to median unbiased even for small sample sizes.
Second, the theory predicts that the estimator converges to the true parameter value
√
at the rate n . This is borne out in the simulation as the MAE and RMSE decrease
√
toward zero at a rate of approximately 2 when the sample size is doubled. For example, a
regression of the log of the median absolute error on the log of the sample size (and design
dummies) yields a coefficient of −0.543 with a standard error of 0.006.
Third, to examine the normality prediction from the asymptotic theory, we estimate the
density for θ − θ and plot the kernel estimate in Figures 1-5. The left–hand side of each
figure gives the estimated density of the estimator of (β 2 /β 1 ) centered at the true value.
They show severe asymmetry in the distribution of the estimator: it tends to be skewed to
the left, especially in small samples. As mentioned, this is expected because of the somewhat
unnatural normalization. The right–hand side of the figures shows the estimated density of
the estimator of log (β 2 /β 1 ), again centered at the true value. There one can see that the
distribution becomes more symmetric and closer to normal as sample size increases.
Finally, the asymptotic theory suggests that we can conduct inference using t–tests. Under the null, the test statistic should follow a standard normal distribution. In the simulation,
we compute a t–statistic for each of the 5,000 estimates θ and calculate the fraction of times
the null is rejected at the 20% level. We focus on tests with (nominal) size of 20% rather
than the conventional 5% because the results for the latter are likely to be more erratic for
a finite number of simulations. The results are reported for various bandwidths used in the
estimation of the asymptotic variance-covariance matrix of the estimator.
In general, the rejection rate is closer to the nominal size of the test when the bandwidth is
smaller and the sample size is larger. For example, for the bandwidth (0.05, 0.20) and sample
size 1,600, the reject rate is 21.8%, 22.7%, 20.8% and 21.6% for Designs 1 through 4.5 These
5
Different bandwidths are used in estimating the matrix ∆ and Γ. The latter is based on a second
derivative, and one would therefore expect it to require a larger bandwidth than the former.
16
are close to being statistically indistinguishable from the nominal size. The performance of
the test under some other combinations of bandwidth and sample size is less encouraging.
The test also performs less well under Design 5. We speculate that this is because of the
discreteness of xi1 .
The last row reports the reject rates computed using the average of the variance–covariance
matrix estimated over all the bandwidth choices. Overall, the t-test tends to over–reject the
null.
It is interesting to compare our results to a standard logit or probit estimation of (5)
where one uses x1 , x2 and time dummies as explanatory variables. Since we expect them
to perform comparably, we focus on the logit maximum likelihood estimator.6 Designs 1, 3
and 4 are all correctly specified probit models, so one would expect the logit estimator to do
well for this design. This is confirmed in panels one, three and four of Table 7. The bias is
small and the MAE and RMSE fall at a rate close to root–n. It is less clear what to expect
for Designs 2 and 5. Panel 2 of Table 7 shows that the logit estimator does well for Design
2. It appears to be close to unbiased and its MAE and RMSE fall at a rate close to root–n.
One potential explanation for this is that misspecified maximum likelihood estimators often
do well when the explanatory variables are jointly normally distributed. See, for example,
Ruud (1983). Design 5 shows a situation where the logit estimator does relatively poorly.
The bias is quite high, and as a result, the MAE and RMSE do not fall rapidly as the sample
size increases.
7
Conclusion
In this paper we propose a generalization of the transformation model that is appropriate for
studying duration outcome with truncation, censoring, interval observations of the dependent
variable, and time-varying covariates. We develop an estimator for this model, discuss its
6
Since our estimator of θ was calculated by a grid search over the interval between − log (6) and log (6),
we censored the logit maximum likelihood estimator of β 2 /β 1 to be in the interval between
17
1
6
and 6.
asymptotic properties and investigate its finite sample performance via a Monte Carlo study.
Overall, the results suggest that the estimator performs well in finite samples, and the
asymptotic theory provides a reasonably good approximation to the distribution. We also
investigate test–statistics for the estimator. Those require estimation of the asymptotic
variance of the estimator. This is somewhat sensitive to different choices of bandwidth.
Investigating the optimal bandwidth choice in this case could be an interesting future research
topic.
References
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19
8
Appendix
Assume that H (·) is a log–concave function and let
f (w) =
H (a2 − w)
H (a1 − w)
where a2 > a1 . Let w1 < w2 and
∆a = a2 − a1 ,
∆w = w2 − w1
and
λ=
∆a
∆a + ∆w
then
a2 − w2 = λ (a2 − w1 ) + (1 − λ) (a1 − w2 )
so by concavity of ln (H (·)),
ln (H (a2 − w2 )) > λ ln (H (a2 − w1 )) + (1 − λ) ln (H (a1 − w2 )) .
(12)
Also
a1 − w1 = (1 − λ) (a2 − w1 ) + λ (a1 − w2 )
so
ln (H (a1 − w1 )) > (1 − λ) ln (H (a2 − w1 )) + λ ln (H (a1 − w2 ))
Adding (12) and (13) yields
ln (H (a2 − w2 )) + ln (H (a1 − w1 )) > ln (H (a2 − w1 )) + ln (H (a1 − w2 ))
and
ln (f (w2 )) − ln (f (w1 )) = (ln (H (a2 − w2 )) − ln (H (a1 − w2 )))
− (ln (H (a2 − w1 )) − ln (H (a1 − w1 )))
> 0
Hence f is an increasing function.
20
(13)
Table 1: Summary Statistics for the Designs
Design 1 Design 2 Design 3 Design 4
Fraction Truncated
0.260
0.349
0.316
0.260
Fraction Censored
0.297
0.100
0.190
0.425
Mean Duration
5.317
3.592
4.369
5.317
Standard Deviation of Duration
3.422
2.023
2.918
3.422
Median
MAE
Mean
RMSE
0.05, 0.20
0.05, 0.40
0.10, 0.20
0.10, 0.40
0.20, 0.20
0.20, 0.40
0.40, 0.20
0.40, 0.40
average
Table 2: Results for Design 1
Performance of Estimator
n = 100 n = 200 n = 400 n = 800 n = 1600
0.721
0.695
0.702
0.693
0.693
0.331
0.215
0.155
0.105
0.070
0.727
0.706
0.702
0.698
0.694
0.488
0.331
0.227
0.155
0.106
Significance when testing at 20% level
n = 100 n = 200 n = 400 n = 800 n = 1600
0.332
0.240
0.212
0.207
0.218
0.163
0.152
0.187
0.241
0.271
0.421
0.323
0.280
0.256
0.246
0.243
0.232
0.261
0.296
0.302
0.494
0.382
0.323
0.285
0.260
0.324
0.301
0.305
0.326
0.319
0.536
0.423
0.350
0.303
0.269
0.390
0.347
0.337
0.343
0.329
0.289
0.259
0.261
0.274
0.273
21
Design 5
0.183
0.543
7.270
3.937
Median
MAE
Mean
RMSE
0.05, 0.20
0.05, 0.40
0.10, 0.20
0.10, 0.40
0.20, 0.20
0.20, 0.40
0.40, 0.20
0.40, 0.40
average
Median
MAE
Mean
RMSE
0.05, 0.20
0.05, 0.40
0.10, 0.20
0.10, 0.40
0.20, 0.20
0.20, 0.40
0.40, 0.20
0.40, 0.40
average
Table 3: Results for Design 2
Performance of Estimator
n = 100 n = 200 n = 400 n = 800 n = 1600
0.718
0.713
0.698
0.693
0.692
0.540
0.370
0.255
0.175
0.125
0.656
0.712
0.705
0.697
0.695
0.752
0.553
0.392
0.274
0.186
Significance when testing at 20% level
n = 100 n = 200 n = 400 n = 800 n = 1600
0.423
0.350
0.268
0.236
0.227
0.164
0.175
0.180
0.224
0.265
0.515
0.422
0.335
0.280
0.253
0.269
0.252
0.245
0.273
0.296
0.578
0.477
0.372
0.302
0.269
0.359
0.317
0.293
0.303
0.313
0.620
0.514
0.394
0.317
0.279
0.426
0.363
0.322
0.319
0.323
0.333
0.299
0.274
0.273
0.275
Table 4: Results for Design 3
Performance of Estimator
n = 100 n = 200 n = 400 n = 800 n = 1600
0.683
0.699
0.693
0.696
0.688
0.400
0.263
0.174
0.120
0.085
0.691
0.716
0.707
0.700
0.694
0.569
0.405
0.271
0.185
0.124
Significance when testing at 20% level
n = 100 n = 200 n = 400 n = 800 n = 1600
0.389
0.289
0.214
0.201
0.208
0.216
0.150
0.143
0.202
0.255
0.468
0.372
0.288
0.264
0.247
0.280
0.226
0.225
0.273
0.302
0.551
0.455
0.345
0.308
0.273
0.368
0.307
0.287
0.316
0.327
0.608
0.503
0.384
0.330
0.287
0.438
0.370
0.325
0.344
0.342
0.326
0.268
0.247
0.266
0.273
22
Median
MAE
Mean
RMSE
0.05, 0.20
0.05, 0.40
0.10, 0.20
0.10, 0.40
0.20, 0.20
0.20, 0.40
0.40, 0.20
0.40, 0.40
average
Median
MAE
Mean
RMSE
0.05, 0.20
0.05, 0.40
0.10, 0.20
0.10, 0.40
0.20, 0.20
0.20, 0.40
0.40, 0.20
0.40, 0.40
average
Table 5: Results for Design 4
Performance of Estimator
n = 100 n = 200 n = 400 n = 800 n = 1600
0.706
0.693
0.703
0.698
0.693
0.385
0.260
0.178
0.118
0.085
0.704
0.703
0.704
0.701
0.694
0.557
0.397
0.273
0.182
0.124
Significance when testing at 20% level
n = 100 n = 200 n = 400 n = 800 n = 1600
0.355
0.280
0.235
0.195
0.216
0.176
0.161
0.186
0.219
0.270
0.437
0.369
0.307
0.250
0.249
0.250
0.235
0.255
0.269
0.300
0.515
0.428
0.356
0.280
0.268
0.335
0.305
0.307
0.305
0.320
0.568
0.472
0.383
0.296
0.280
0.402
0.353
0.340
0.324
0.333
0.300
0.272
0.270
0.259
0.276
Table 6: Results for Design 5
Performance of Estimator
n = 100 n = 200 n = 400 n = 800 n = 1600
0.711
0.704
0.682
0.698
0.693
0.567
0.409
0.285
0.195
0.135
0.646
0.710
0.720
0.718
0.703
0.789
0.612
0.438
0.305
0.207
Significance when testing at 20% level
n = 100 n = 200 n = 400 n = 800 n = 1600
0.518
0.414
0.365
0.355
0.372
0.506
0.464
0.567
0.637
0.661
0.548
0.485
0.429
0.399
0.401
0.528
0.557
0.638
0.666
0.677
0.607
0.541
0.469
0.425
0.411
0.584
0.618
0.672
0.681
0.683
0.631
0.566
0.480
0.431
0.412
0.627
0.648
0.682
0.684
0.683
0.511
0.503
0.505
0.508
0.513
23
Median
MAE
Mean
RMSE
Median
MAE
Mean
RMSE
Median
MAE
Mean
RMSE
Median
MAE
Mean
RMSE
Median
MAE
Mean
RMSE
Table 7: Performance of the Logit MLE
Design 1
n = 100 n = 200 n = 400 n = 800
0.699
0.699
0.692
0.694
0.240
0.165
0.114
0.084
0.715
0.702
0.696
0.695
0.369
0.252
0.173
0.123
Design 2
n = 100 n = 200 n = 400 n = 800
0.696
0.703
0.692
0.689
0.413
0.282
0.199
0.139
0.691
0.720
0.701
0.694
0.639
0.443
0.303
0.214
Design 3
n = 100 n = 200 n = 400 n = 800
0.691
0.692
0.695
0.692
0.302
0.202
0.137
0.098
0.711
0.710
0.701
0.696
0.454
0.307
0.209
0.144
Design 4
n = 100 n = 200 n = 400 n = 800
0.687
0.695
0.696
0.696
0.288
0.201
0.138
0.097
0.706
0.702
0.701
0.699
0.438
0.302
0.209
0.146
Design 5
n = 100 n = 200 n = 400 n = 800
0.884
1.008
1.046
1.055
0.521
0.406
0.361
0.362
0.761
1.012
1.080
1.082
0.948
0.685
0.540
0.476
24
n = 1600
0.693
0.059
0.693
0.085
n = 1600
0.695
0.100
0.695
0.149
n = 1600
0.694
0.069
0.695
0.103
n = 1600
0.692
0.067
0.693
0.100
n = 1600
1.063
0.370
1.073
0.428
Figure 1: Density of Estimation Error for Estimator and Its Log for Design 1
Figure 2: Density of Estimation Error for Estimator and Its Log for Design 2
25
Figure 3: Density of Estimation Error for Estimator and Its Log for Design 3
Figure 4: Density of Estimation Error for Estimator and Its Log for Design 4
26
Figure 5: Density of Estimation Error for Estimator and Its Log for Design 5
27
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Sumit Agarwal and Brent W. Ambrose
WP-08-09
The Choice between Arm’s-Length and Relationship Debt: Evidence from eLoans
Sumit Agarwal and Robert Hauswald
WP-08-10
Consumer Choice and Merchant Acceptance of Payment Media
Wilko Bolt and Sujit Chakravorti
WP-08-11
Investment Shocks and Business Cycles
Alejandro Justiniano, Giorgio E. Primiceri, and Andrea Tambalotti
WP-08-12
New Vehicle Characteristics and the Cost of the
Corporate Average Fuel Economy Standard
Thomas Klier and Joshua Linn
WP-08-13
Realized Volatility
Torben G. Andersen and Luca Benzoni
WP-08-14
Revenue Bubbles and Structural Deficits: What’s a state to do?
Richard Mattoon and Leslie McGranahan
WP-08-15
The role of lenders in the home price boom
Richard J. Rosen
WP-08-16
Bank Crises and Investor Confidence
Una Okonkwo Osili and Anna Paulson
WP-08-17
Life Expectancy and Old Age Savings
Mariacristina De Nardi, Eric French, and John Bailey Jones
WP-08-18
Remittance Behavior among New U.S. Immigrants
Katherine Meckel
WP-08-19
Birth Cohort and the Black-White Achievement Gap:
The Roles of Access and Health Soon After Birth
Kenneth Y. Chay, Jonathan Guryan, and Bhashkar Mazumder
WP-08-20
Public Investment and Budget Rules for State vs. Local Governments
Marco Bassetto
WP-08-21
Why Has Home Ownership Fallen Among the Young?
Jonas D.M. Fisher and Martin Gervais
WP-09-01
Why do the Elderly Save? The Role of Medical Expenses
Mariacristina De Nardi, Eric French, and John Bailey Jones
WP-09-02
Using Stock Returns to Identify Government Spending Shocks
Jonas D.M. Fisher and Ryan Peters
WP-09-03
5
Working Paper Series (continued)
Stochastic Volatility
Torben G. Andersen and Luca Benzoni
WP-09-04
The Effect of Disability Insurance Receipt on Labor Supply
Eric French and Jae Song
WP-09-05
CEO Overconfidence and Dividend Policy
Sanjay Deshmukh, Anand M. Goel, and Keith M. Howe
WP-09-06
Do Financial Counseling Mandates Improve Mortgage Choice and Performance?
Evidence from a Legislative Experiment
Sumit Agarwal,Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff
WP-09-07
Perverse Incentives at the Banks? Evidence from a Natural Experiment
Sumit Agarwal and Faye H. Wang
WP-09-08
Pay for Percentile
Gadi Barlevy and Derek Neal
WP-09-09
The Life and Times of Nicolas Dutot
François R. Velde
WP-09-10
Regulating Two-Sided Markets: An Empirical Investigation
Santiago Carbó Valverde, Sujit Chakravorti, and Francisco Rodriguez Fernandez
WP-09-11
The Case of the Undying Debt
François R. Velde
WP-09-12
Paying for Performance: The Education Impacts of a Community College Scholarship
Program for Low-income Adults
Lisa Barrow, Lashawn Richburg-Hayes, Cecilia Elena Rouse, and Thomas Brock
Establishments Dynamics, Vacancies and Unemployment: A Neoclassical Synthesis
Marcelo Veracierto
WP-09-13
WP-09-14
The Price of Gasoline and the Demand for Fuel Economy:
Evidence from Monthly New Vehicles Sales Data
Thomas Klier and Joshua Linn
WP-09-15
Estimation of a Transformation Model with Truncation,
Interval Observation and Time-Varying Covariates
Bo E. Honoré and Luojia Hu
WP-09-16
6
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