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Federal Reserve Bank of Chicago
Estimation of Panel Data Regression
Models with Two-Sided Censoring
or Truncation
Sule Alan, Bo E. Honoré, Luojia Hu, and
Søren Leth–Petersen
WP 2011-08
Estimation of Panel Data Regression Models
with Two-Sided Censoring or Truncation
Sule Alany
Bo E. Honoréz
Luojia Hu
x
Søren Leth–
Petersen
{
November 14, 2011
Abstract
This paper constructs estimators for panel data regression models with individual speci…c
heterogeneity and two–
sided censoring and truncation. Following Powell (1986) the estimation
strategy is based on moment conditions constructed from re–
censored or re–
truncated residuals.
While these moment conditions do not identify the parameter of interest, they can be used to
motivate objective functions that do. We apply one of the estimators to study the e¤ect of a
Danish tax reform on household portfolio choice. The idea behind the estimators can also be
used in a cross sectional setting.
Key Words: Panel Data, Censored Regression, Truncated Regression.
JEL Code: C20, C23, C24.
This research was supported by NSF Grant No. SES-0417895 to Princeton University, the Gregory C. Chow
Econometric Research Program at Princeton University, and the Danish National Research Foundation, through
CAM at the University of Copenhagen (Honoré) and the Danish Social Science Research Council (Leth–
Petersen).
We thank Christian Scheuer and numerous seminar participants for helpful 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.
y
Faculty of Economics, University of Cambridge, Sidgwick Avenue, Cambridge, UK, CB3 9DD. Email:
sule.alan@econ.cam.ac.uk.
z
Department of Economics, Princeton University, Princeton, NJ 08544-1021. Email: honore@Princeton.edu.
x
Economic Research Department, Federal Reserve Bank of Chicago, 230 S. La Salle Street, Chicago, IL 60604.
Email: lhu@frbchi.org.
{
Department of Economics, University of Copenhagen, Øster Farimagsgade 5, Building 26, DK-1353 Copenhagen
K. and SFI, The Danish National Centre for Social Research, Herluf Trolles Gade 11, DK-1052. Email : soren.lethpetersen@econ.ku.dk.
1
1
Introduction
This paper generalizes a class of estimators for truncated and censored regression models to allow
for two–
sided truncation or censoring. The class of estimators is based on pairwise comparisons
and the proposed generalizations therefore apply to both panel data and cross sectional data.
A leading example of when two–
sided censored regression models are useful is when the dependent variable is a fraction. For example, Alan and Leth-Petersen (2006) estimate a portfolio
share equation where the portfolio shares are between 0 and 1, with a signi…cant number of observations on either of the limits. Other recent applications in economics of regression models with
two–
sided censoring include Lafontaine (1993), Petersen and Rajan (1994), Petersen and Rajan
(1995), Houston and Ryngaert (1997), Fehr, Kirchler, Weichbold, and Gachter (1998), Huang and
Hauser (1998), McMillan and Woodru¤ (1999), de Figueriredo and Tiller (2001), Huang and Hauser
(2001), Fenn and Liang (2001), Poterba and Samwick (2002), Nickerson and Silverman (2003), Of…cer (2004), Charness, Frechette, and Kagel (2004), Andrews, Schank, and Simmons (2005) and
Gi¤ord and Bernard (2005). We formulate the two–
sided censored regression model as observations
of (y; x; L; U ) from the model
y = x0 + "
where y is unobserved, but we observe
8
> L
>
>
<
y=
y
>
>
>
: U
and
if
(1)
y <L
if L
if
y
(2)
U
y >U
is the parameter of interest. When y is a share, L and U will typically be 0 and 1, respectively.
We will focus on panel data settings so the observations are indexed by i and t where i = 1; : : : ; n
and t = 1; : : : Ti . This allows for unbalanced panels, but we will maintain the restrictive assumption
that Ti is exogenous in the sense that it satis…es all the assumptions made on the explanatory
variables. In a panel data setting, it is also important to allow for individual speci…c e¤ects in
the errors "it . We will do this implicitly by making assumptions of the type that "it is stationary
conditional on (xi1 : : : xiTi ) or that "it and "is are independent and identically distributed conditional
on some unobserved component
ist .
These will have the textbook speci…cation "it = vi +
it
(f it g
i.i.d.) as a special case. In section 4.2, we discuss how to apply the same ideas to construct
estimators of the cross sectional version of the model.
For the two–
sided truncated regression model, we assume observations of (y; x; L; U ) from the
2
distribution of (y ; x; L; U ) conditional on L
y
U . Two–
sided truncated regression models are
less common than two–
sided censored regression models, but they play a role in duration models.
Suppose, for example, that one wants to study the e¤ect of early life circumstances on longevity by
linking the Social Security Administration’ Death Master File to the 1900–
s
1930 U.S. censuses1 .
This will miss a substantial number of deaths: (1) individuals who died under age 65 (since they
were less likely to be collecting social security bene…ts and thus their deaths were less likely to
be captured); (2) individuals who died before the year 1965 (the beginning of the computerized
Social Security …les); and (3) individuals who died after the last year the data is available. This
is a case of two-sided truncation because we observe an individual only if the dependent variable,
age at death, is greater than 65 years and if the death occurs between 1965 and the last year the
data is available. This is essentially the empirical setting in Ferrie and Rolf (2011), although they
only consider data from the 1900 census, so right–
truncation is unlikely to be an issue in their
application.
Honoré (1992) constructed moment conditions for similar panel data models with one–
sided
truncation or censoring and showed how they can be interpreted as the …rst–
order conditions for
a population minimization problem that uniquely identi…es the parameter vector,
. This paper
generalizes that approach to the case when the truncation or censoring is two–
sided. The main
contribution of the paper is to show that some of those moment conditions can be turned into
a minimization problem that actually uniquely identi…es
when there is two–
sided censoring or
truncation. This is an important step because the moment conditions that we derive do not identify
the parameters of the model. This is a generic problem with constructing estimators based on
moment conditions. For example, Powell (1986) constructed moment conditions for a related cross
sectional truncated and censored regression models based on symmetry of the error distribution.
He also pointed out that while these moment conditions did not identify the parameter of interest,
minimization of an objective function based on them did lead to identi…cation.
The rest of the paper is organized as follows. Section 2 derives the moment conditions and the
associated objective function for models with two–
sided censoring and two–
sided truncation. Section 3 then discusses how these can be used to estimate the parameters of interest. Generalizations
of the two models considered in Section 2 are discussed in Section 4. We present the empirical
application in Section 5 and Section 6 concludes.
1
1930 is the latest year for which this linkage is feasible.
3
2
Identi…cation: Moment Conditions and Objective Functions
The challenge in constructing moment conditions in models with censoring and truncation is that
one typically starts with assumptions on " conditional on x in (1). If one had a random sample
of (y ; x) then these assumptions could be used immediately to construct moment conditions. For
example if E ["x] = 0, then one has the moment conditions E [(y
x0 ) x] = 0. However with
truncation or censoring, y x0 will not have the same properties as ". The idea employed in Powell
(1986), Honoré (1992) and Honoré and Powell (1994) is to apply additional censoring and truncation
to y
x0
in such a way that the the resulting re–
censored or re–
truncated residual satis…es the
conditions assumed on ". For example, Powell (1986) assumed that " is symmetric conditional on
x in a censored regression model with censoring from below at 0. If that is the case, then y
x0 = max f"; x0 g will clearly not be symmetric conditional on x, but the re–
censored residuals,
min fy
x0 ; x0 g will be. This implies moment conditions of the type E [min fy
Unfortunately, this moment condition will not in general identify
to prove that the integral (as a function of b) of E [min fy
b =
x0 ; x0 g x] = 0.
, but Powell (1986) was able
x0 b; x0 bg x] is uniquely minimized at
under appropriate regularity assumptions. Honoré (1992) applied the same insight to a
panel setting where the assumption was that " is stationary conditional on the entire sequence
of explanatory variables. Again, censoring or truncation destroys this stationarity, but it can be
restored for a pair of residuals by additional censoring. Honoré and Powell (1994) then applied the
same idea to any pair of observations in a cross section, and Hu (2002) generalized it to allow for
lagged latent dependent variables as covariates.
The contribution of this paper is to generalize the approach in Honoré (1992) to the case
with two–
sided censoring or truncation. As in Powell (1986), Honoré (1992) and Honoré and
Powell (1994), it is straightforward to construct moment conditions based on some re–
censored
or re–
truncated residuals. However, it is not clear that these moment conditions will identify
the parameters of interest, and we therefore construct (population) objective functions from these
moment conditions, and then explicitly verify that these objective functions are uniquely minimized
at the parameter. It is the construction of the objective functions and verifying that they are
uniquely minimized at the true parameter value that constitute the methodological contribution of
the paper.
The general approach is to start with a comparison of two observations for a given individual
in a panel. Based on these observations we will construct re–
censored or re–
truncated residuals
4
eits (yit ; xit ; xis ; Lit ; Lis ; Uit ; Uis ; b) and eist (yis ; xis ; xit ; Lit ; Lis ; Uit ; Uis ; b) that have the same properties as "it and "is when b = . This will then imply that if "it and "is are identically distributed
conditional on (xit ; xis ), then
E [ (eits (yit ; xit ; xis ; Lit ; Lis ; Uit ; Uis ; )
eist (yis ; xis ; xit ; Lit ; Lis ; Uit ; Uis ; ))j xit ; xis ] = 0:
(3)
This will form the basis for construction of our estimators. Of course, once it has been established
that eits (yit ; xit ; xis ; Lit ; Lis ; Uit ; Uis ; ) and eist (yis ; xis ; xit ; Lit ; Lis ; Uit ; Uis ; ) are identically distributed then for any function, ( ), we also have the moment condition
E [ ( (eits (yit ; xit ; xis ; Lit ; Lis ; Uit ; Uis ; ))
(eist (yis ; xis ; xit ; Lit ; Lis ; Uit ; Uis ; )))j xit ; xis ] = 0:
(4)
provided that the moment exists.
Moreover, if the errors "it and "is are also independent conditional on (xit ; xis ), then so are
eits (yit ; xit ; xis ; Lit ; Lis ; Uit ; Uis ; ) and eist (yis ; xis ; xit ; Lit ; Lis ; Uit ; Uis ; ). This implies that their
di¤erence is symmetrically distributed around 0, so for any odd function ( ),
E [ ( (eits (yit ; xit ; xis ; Lit ; Lis ; Uit ; Uis ; )
eist (yis ; xis ; xit ; Lit ; Lis ; Uit ; Uis ; )))j xit ; xis ] = 0: (5)
provided that the moment exists.2
In this paper we will focus on (3) and the generalization (5). The reason for this is that in a linear
model without censoring or truncation, (3) will correspond to OLS on the di¤erenced data, whereas
(5) will also accommodate least absolute deviation estimation on the di¤erenced data as a special
case. As already mentioned, the construction of the residuals eits (yit ; xit ; xis ; Lit ; Lis ; Uit ; Uis ; ) is
fairly straightforward, and the challenge is to show that although (3) and (5) may not identify ,
and unconditional version of them can be integrated to yield a population objective function that
is uniquely minimized at b =
. This will involve the integral of
( ), which we will denote by
( ), and in addition to being odd, we will assume that ( ) is also increasing, so
symmetric function. The leading cases are
(d) = jdj and
( ) is a convex
(d) = d2 .
In order to simplify the exposition, we will …rst develop the case when Lit = 0 and Uit = 1. We
will then demonstrate that the result can be adapted to the general case.
2
Of course one could combine the insight in (4) and (5) to get even more general moment conditions. See also the
discussion in Arellano and Honoré (2001) and Honoré and Hu (2004).
5
2.1
Two–
Sided Censoring
Consider …rst the situation with two–
sided censoring. Consider an individual, i, in two time periods,
t and s, and assume that "it and "is are identically distributed. The distribution of yit
be the same as that of "it except that the former is censored from below at
above at 1
x0
it
will
and from
x0 . Figure 1 illustrates this. The dotted line depicts the distribution of "it , while
it
x0 , which typically has point mass at
it
the solid line gives the distribution of yit
1
x0
it
x0
it
(illustrated by the fatter vertical lines). Since x0
it
x0
it
distributions of yit
x0
is
and yis
x0
it
and
will typically di¤er from x0 , the
is
(given (xit ; xis )) will di¤er even if f"it g is stationary
(given (xit ; xis )). However, it is clear that one could obtain identically distributed “residuals” by
arti…cially censoring yit
min f1
x0 ; 1
it
x0 and yis
it
x0 from below at max f x0 ; x0 g and from above at
is
it
is
x0 g. See the dashed lines in Figure 1. One can then form moment conditions
is
from the fact that the di¤erence in these “re–
censored”residuals will be orthogonal to functions of
(xit ; xis ).3 Of course, this construction is only useful if
the supports of yit
x0 and yis
it
1 < x0
it
x0 < 1, because otherwise,
is
x0 will not overlap.
is
In order to proceed, we need explicit expressions for the di¤erence in these “re–
censored”residuals. Consider …rst the case when x0
it
x0 . Then the di¤erence in the arti…cially censored
is
residuals for individual i in periods t and s is
max yit
=
max yit
= max yit
and when x0
it
x0
it
x0
it
x0 ; 1
it
min yit ; 1 + x0
it
= min yit ; 1 + x0
it
3
x0 ; 1
is
min yis
x0
is
x0
is
;0
;0
x0
is
x0
it
(6)
x0
it
min yis ; 1
min yis ; 1
x0
it
x0
is
x0
is
x0
is
;
x0
is
min yit
=
x0 ; x0
it
is
x0
is
x0
is
x0
is
max yis
x0
it
x0 ; x0
is
it
(7)
max yis + x0
it
max yis + x0
it
x0
is
x0
is
;0
x0
it
;0 :
Clearly, one can also use the fact that di¤erences in functions of the re–
censored residuals will be orthogonal to
functions for the explanatory variables. As discussed in Arellano and Honoré (2001), one can also construct moment
conditions based on symmetry under the additional assumption that ("i1 ; :::; "iT ) is exchangeable conditional on
(xi1 ; :::; xiT ). This is the motivation for the approach in Honoré (1992).
6
Figure 1: Illustration of Re–
Censored Residuals when x0 > x0 .
it
is
If we de…ne
8
>
0
>
>
>
>
>
>
1+d
>
>
>
>
>
> min f1 y2 ; y1 g
>
>
<
u (y1 ; y2 ; d) =
y1 y2 d
>
>
>
> max fy
>
1; y2 g
1
>
>
>
>
>
>
d 1
>
>
>
>
:
0
where c1 = min f y2 ; y1
then u (yit ; yis ; x0
it
1g, c2 = max f y2 ; y1
for
d<
1
for
1 < d < c1
for
c1 < d < c2
for
c2 < d < c3
for
c3 < d < c4
for
c4 < d < 1
for
d>1
1g, c3 = min f1
y2 ; y1 g and c4 = max f1
(8)
y2 ; y 1 g
x0 ) will give the di¤erence in the re–
censored residuals discussed above (see
is
Appendix 1). Hence the moment conditions are
E u yit ; yis ; x0
it
x0
is
xit ; xis = 0;
which implies the unconditional moments
E u yit ; yis ; (xit
xis )0
(xit
xis ) = 0:
(9)
Panel A of Figures 2– depict the contribution to the moment condition function u (y1 ; y2 ; d)
4
for pairs of uncensored observations (Figure 2),
7
pairs with one censored and one uncensored
Figure 2: The Functions u (y1 ; y2 ; ) and U (y1 ; y2 ; ). Neither Observation Censored.
observation (Figure 3) and pairs with one observation censored from above and one from below
(Figure 4).
Although the true parameter value will satisfy (9), it is not in general the unique solution to
the moment condition. This is illustrated in Panel A of Figure 5. It considers the case when
yi1
N (0:5; 1) and yi2
N (0:4; 1) and both are censored from below at 0 and from above at 1.
Note that this is the data generation process that one would get with
= 0:1, xi1 = 1 and xi2 = 0
for all i.
It is clear from Figure 5 that the moment condition E [ u (yit ; yis ; x0 b
it
x0 b)j xit ; xis ] = 0 does
is
not identify the parameter, , of the model. The most obvious reason is that, as mentioned, only
observations for which
1 < (xi1
xi2 )0 b < 1 will contribute. In this case xi1
xi2 = 1 for all
observations, so the moment condition is automatically satis…ed when jbj > 1.
Following Powell (1986), we attempt to overcome the non–
identi…cation based on the moment
condition by turning it into the …rst order condition for a minimization problem. It is easy to see
that (9) is (half of minus) the …rst order condition for minimizing
E U yit ; yis ; (xit
8
xis )0 b
;
(10)
Figure 3: The Functions u (y1 ; y2 ; ) and U (y1 ; y2 ; ). One Observation Censored.
Figure 4: The Functions u (y1 ; y2 ; ) and U (y1 ; y2 ; ). Both Observations Censored.
9
Figure 5: The Functions E [u (y1 ; y2 ; )] and E [U (y1 ; y2 ; )].
where
U (y1 ; y2 ; d) =
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
1 + 2c1 + c2
1
2d
2c3 c1 + 2c3 c2 + (y1
d2 + 2c1 + c2
1
2c3 c1 + 2c3 c2 + (y1
2c3 d + 2c3 c2 + (y1
(y1
y2
1 + c2
4
2c4
2c4
c2 )2
for
2
c2 )
d<
1
y2
2c2 c4 + 2c2 c3 + (y1
2c2 c4 + 2c2 c3 + (y1
y2
2
y2
c3 )
2
c3 )
c1 < d < c2
c2 < d < c3 :
for
c3 )2
1 < d < c1
for
c2 )
2
for
for
y2
y2
2
d)
2c2 d + 2c2 c3 + (y1
d2 + 2d + c2
4
y2
c3 < d < c4
for
c4 < d < 1
for
d>1
Panel B of Figures 2– depict the contribution to the objective function U (y1 ; y2 ; d) for pairs
4
of uncensored observations (Figure 2), pairs with one censored and one uncensored observation
(Figure 3) and pairs with one observation censored from above and one from below (Figure 4). Like
the estimator for the panel data one–
sided censored regression model developed in Honoré (1992),
the objective function is piecewise quadratic or linear. However, surprisingly, going from one–
sided
to two–
sided censoring ruins the convexity of the objective function, and the shape of the function
is more similar to the objective function in Powell (1986) although that estimator was developed
for a cross section model with symmetrically distributed errors.
It is clear from the discussion above that the true
10
will solve the …rst order condition for
minimizing the population objective function in (10). However, since U is constant, linear, quadratic
and convex, and quadratic and concave over di¤erent regions, it is not at all obvious that
will be
the unique solution to these …rst order conditions. The key step for establishing identi…cation of
is therefore to establish that the function in (10) is minimized at . We establish this in Appendix
1 (Section 7.1), and the result is illustrated in Panel B of Figure 5.
As mentioned, it is also possible to construct moment conditions based on (5). Let
and symmetric, and let
() =
0(
be convex
) (when it exists). When "it and "is are independent and
identically distributed, we also have the moment conditions
E
u yit ; yis ; x0
it
x0
is
xit ; xis = 0
which imply the unconditional moments
E
where
(u (y1 ; y2 ; d)) =
u yit ; yis ; (xit
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
0
xis )0
(xit
for
xis ) = 0
d<
(11)
1
(1 + d)
for
1 < d < min f y2 ; y1
1g = c1
(c3 )
for
c1 < d < max f y2 ; y1
1g = c2
y2
d) for
c2 < d < min f1
y2 ; y1 g = c3
(c2 )
for
c3 < d < max f1
y2 ; y1 g = c4
(y1
(d
1)
for
for
0
c4 < d < 1
d>1
Except for a multiplicative constant, (11) is the …rst order condition for minimizing
E U
yit ; yis ; (xit
xis )0 b
(12)
where U is found by integrating (u (y1 ; y2 ; d)) over each of the regions and insisting on continuity
at the boundaries between the regions:
8
>
(0)
(1 + c1 )
>
>
>
>
>
> (1 + d)
(1 + c1 )
>
>
>
>
>
>
>
(c3 ) d +
>
<
U (y1 ; y2 ; d) =
>
>
>
>
>
(c2 ) d +
>
>
>
>
> (d 1)
>
(c4 1)
>
>
>
>
:
(0)
(c4 1)
(c3 ) c1 + (c3 ) c2 +
(c3 ) c1 + (c3 ) c2 +
(c3 ) c2 +
(y1
(y1
y2
(y1
y2
y2
y2
(c2 ) c4 + (c2 ) c3 +
(c2 ) c4 + (c2 ) c3 +
d<
1
c2 ) for
1 < d < c1
for
c1 < d < c2
c2 < d < c3 :
for
c3 < d < c4
c3 ) for
c4 < d < 1
c2 )
c3 )
(y1
(y1
y2
y2
The appendix establishes that (12) is uniquely minimized at b = .
11
for
for
y2
(y1
c2 )
d)
(c2 ) c3 +
(y1
c3 )
for
d>1
2.2
Two–
sided Truncation
Mimicing the argument for the model with two–
sided censoring, it is clear that if "it and "is are
independent and identically distributed conditional on (xit ; xis ), then the observed errors, yit x0 ,
it
will be i.i.d. except that the sampling scheme will have truncated them at di¤erent points,
and
x0
is
x0
it
from below and 1
x0
is
and 1
from above. We can then construct identically
x0 at
it
distributed residuals by arti…cially truncating yit
above (and similarly for yis
E r yit ; yis ; (xit
x0
it
x0 from below and at 1
is
x0 from
is
x0 ). This yields many moment conditions, including
is
xis )0
E
yit
yis
x0
it
1
xis )0
(xit
1
x0
is
yis
x0
it
1
x0
is
x0
it
yit
(xit
1
xis )
x0
is
(13)
= 0
It is an easy exercise to see that except for a multiplicative constant (13) is the …rst order
condition for minimizing
xis )0 b
E R yit ; yis ; (xit
where R (y1 ; y2 ; d) is de…ned by
8
> 1 (y1 y2 max fy1
> 2
>
<
>
>
>
:
1
2
1
2
(y1
(y1
y2
y2
1; y2 g)2 if
d)2
min fy1 ; 1
y2 g)2
(14)
max fy1
1; y2 g > d
if max fy1
1; y2 g
d
min fy1 ; 1
if
d > min fy1 ; 1
y2 g
y2 g
Figure 6 depicts the function R and its derivative, whereas Figure 7 shows their expectation when
y1
N (0:5; 1) and y2
N (0:4; 1) and both are truncated from below at 0 and from above at 1.
More generally, again let
be convex and symmetric, and let
() =
0(
) (when it exists).
Then
xis )0
yit
yis
(xit
1
x0
is
yit
x0
it
1
1
E
x0
it
x0
is
yis
x0
is
1
x0
it
(xit
xis ) = 0
1 (xit
xis ) = 0
or
E
yit
yis
(xit
xis )0
1 0
yit
xis )0
(xit
1 0
12
1
yis + (xit
xis )0
Figure 6: The Functions r (y1 ; y2 ; ) and R (y1 ; y2 ; ).
Figure 7: The Functions E [r (y1 ; y2 ; )] and E [R (y1 ; y2 ; )].
13
or with
= (xit
xis )0 ,
0 = E [ (yit
yis
) 1 f0
yit
1g 1 f0
= E [ (yit
yis
) 1 f0
yit
= E [ (yit
yis
) 1f
yit g 1 fyit
= E [ (yit
yis
) 1 fmax fyit
yis +
g 1 fyit
1g (xit
1g 1 f0
1
yis + g 1 fyis +
g 1 f yis
1; yis g
xis )]
g 1f
min fyit ; 1
1
yis gg (xit
1g (xit
yis g (xit
xis )]
xis )]
xis )]
This is minus the derivative of E R yit ; yis ; (xit xis )0 b evaluated at b = , where
8
> (max fy1 1; y2 g) for
>
d < max fy1 1; y2 g
>
<
R (y1 ; y2 ; d) =
(y1 y2 d)
for max fy1 1; y2 g d min fy1 ; 1
>
>
>
:
(min fy1 ; 1 y2 g) for
d > min fy1 ; 1 y2 g
y2 g
As was the case for the censored model, the argument above only establishes that the true
will solve the …rst order condition for minimizing E R
yit ; yis ; (xit
hand, it is clear that without additional strong assumptions,
xis )0 b . On the other
will not be the unique minimizer.
The reason is that we know that, in general, the truncated regression model will not be identi…ed
with exponentially distributed errors. As a result, assumptions must be added that rule out the
exponential distribution. In Appendix 1 (section 7.2), we show that
E R yit ; yis ; (xit
xis )0 b , and more generally of E R
yit ; yis ; (xit
is the unique minimizer of
xis )0 b , provided that the
errors have a log–
concave probability distribution.
3
Estimation
The arguments leading to identi…cation of the parameters of interest,
comparing two observations for the same individual and we showed that
the unique minimizer of an expectation of the form E Q yit ; yis ; (xit
Q. This suggests estimating
, above were based on
could be expressed as
xis )0 b
for some function
by minimizing a sample analog of this such as
n
X Ti
b = arg min 1
b n
2
1
i=1
X
Q yit ; yis ; (xit
xis )0 b
1 s<t Ti
In this aggregation, observations get di¤erent weight depending on the number of observations for
a given individual. Alternatively, one could also use objective functions of the type
n
arg min
b
1X
n
X
wist Q yit ; yis ; (xit
i=1 1 s<t Ti
14
xis )0 b
where the wist ’ are exogenous weights. In particular, with unbalanced panels, one might want wist
s
to depend on Ti , the number of time periods for individual i. For example, one can think of the
usual …xed e¤ects estimator in a linear regression model as minimizing
n
X
X
i=1 1 s<t Ti
1
yis
Ti
so a simple natural choice for wist could be
yit
(xis
xit )0 b
2
;
1
Ti .
For two–
sided censoring, the resulting estimator is4
b = arg min
b
or more generally
b = arg min
b
n
X
X
wist U yit ; yis ; (xit
xis )0 b
X
wist U
xis )0 b
i=1 1 s<t Ti
n
X
yit ; yis ; (xit
(15)
i=1 1 s<t Ti
where the functions U and U are de…ned in Section 2.1. Standard arguments5 yield
Theorem 1 Consider a random sample of size n from Ti; fyit ; xit gTi : If
t=1
1.
yit =
2. ("i1 ; "i2 ;
8
>
>
>
<
>
>
>
:
0
if
x0 + "it if 0
it
1
if
x0 + "it < 0
it
x0 + "it
it
1 ;
x0 + "it > 1
it
; "iTi ) is continuously distributed conditional on Ti; fxit gTi
t=1 with a density that
is continuous and positive everywhere,
3. the sequence "i1 ; "i2 ;
; "iTi is stationary conditional on Ti; fxit gTi , and for any s; t
t=1
there exists a random variable,
st ,
i
such that "is and "it are independent conditional on
Ti
st ,
i
4. the matrix
E (xis
xit ) (xis
xit )0
1 < (xis
xit )0
<1
has full rank
4
A Stata-program for calculating this estimator can be found at www.princeton.edu/~honore/stata.
5
Consistency follows from Theorem 4.1.1 of Amemiya (1985) and asymptotoc normality from, for example, The-
orem 3.3 of Pakes and Pollard (1989).
15
then
p
where b is de…ned in (15) and
=
dE
d
n b
1
! N 0;
P
s<t wi;t s
u yit ; yis ; (xit
db0
1
V
xis )0 b
(xit
xis )
b=
and
0
V = E vi vi
with
vi =
X
wi;t
s
xis )0
u yit ; yis ; (xit
(xit
xis ) :
s<t
These assumptions are consistent with a “…xed e¤ects” model in which "it =
unrestricted and the sequence
feit gTi
" t=1
i
+ eit with
"
i
independent and identically distributed. The assumptions
also allow for some correlation in the eit ’ For example if (eis ; eit ) is bivariate normal with the
" s.
" "
same variance, then they can be written as eis = Zis + Qi and eit = Zit + Qi where Zit , Zis and
"
"
Qi are independent normals. So eis and eit are independent conditional on Qi . The assumption
"
"
that ("i1 ; "i2 ;
; "iTi ) is continuously distributed is necessary if one wants to allow
di¤erentiable(i.e.,
(d) = jdj). Without it, the derivative in the expression for
to be non–
might not exist.
(d) = d2 , condition 3 can be reduced to assuming that the sequence "i1 ; "i2 ;
When
stationary conditional on
Ti; fxit gTi
t=1
; "iTi is
, and condition 2 is not necessary. In that case the terms
in the asymptotic variance reduce to
"
X
=E
wi;t s 1
1 < (xis xit )0
<1
s<t
1
1 < (xis
1
yit < (xis
xit )0
< yis
0
xit )
1
1 0 < (xis
<0 +1 1
yit < (xis
xit )0
< yis
0
xit )
<1
(xis
xit )
and
0
V = E vi vi
with
vi =
X
wi;t
su
yis ; (xis
s<t
16
xit )0
(xis
xit ) (xis
0
xit )
#
Following standard arguments, these are consistently estimated by
"
n
n
o
1X X
b=
wi;t s 1
1 < (xis xit )0 b < 1
n
i=1 s<t
n
o
n
o
1
1 < (xis xit )0 b < yis 1
1 0 < (xis xit )0 b < yis
1
n
o
n
xit ) b < 0 + 1 1
0
yit < (xis
and
yit < (xis
xit ) b < 1
0
o
(xis
xit ) (xis
0
xit )
#
n
1X 0
b
vi vi
bb
V =
n
i=1
with
vi =
b
X
wi;t
s
u yis ; (xis
s<t
xit )0 b (xis
xit )
For two–
sided truncation the resulting estimator is
b = arg min
b
or more generally
b = arg min
b
n
X
X
wist R yit ; yis ; (xit
xis )0 b
(16)
X
wist R
xis )0 b
(17)
i=1 1 s<t Ti
n
X
yit ; yis ; (xit
i=1 1 s<t Ti
where the functions R and R are de…ned in Section 2.2.
We have
Theorem 2 Consider a random sample of size n from Ti; fyit ; xit gTi : If
t=1
1. yit is drawn from the distribution of x0 + "it conditional on 0
it
2. ("i1 ; "i2 ;
x0 + "it
it
1
; "iTi ) is continuously distributed conditional on Ti; fxit gTi
t=1 with a density that
is continuous and positive everywhere
; "iTi is stationary conditional on Ti; fxit gTi , and for any s; t
t=1
3. the sequence "i1 ; "i2 ;
there exists a random variable,
density conditional on
st ,
i
such that "is and "it are independent and have log–
concave
st ,
i
4. the matrix
E (xis
Ti
xit ) (xis
xit )0
has full rank
17
1 < (xis
xit )0
<1
then
p
where b is de…ned in (17) and
=
P
dE
d
n b
s<t wi;t s
1
! N 0;
r yit ; yis ; (xit
db0
1
V
xis )0 b
(xit
xis )
b=
and
0
V = E vi vi
with
vi =
X
wi;t
s
r yit ; yis ; (xit
xis )0
(xit
xis )
s<t
4
Extensions
4.1
Mixed Censored/Truncation
Having considered models with two–
sided censoring or truncation, it is natural to also consider a
regression model with censoring from one side and truncation from the other:
yit = x0 + "it
it
(yit ; xit ) = (min fyit ; Uit g ; xit ) conditional on Lit
yit
(18)
To simplify the notation, we again focus on the case where Lit = 0 and Uit = 1. In this case
the moment condition based on the same logic as above is
0 = E 1 yit
min yit
x0 >
it
x0
is
x0 ; 1
it
= E 1 yit > (xit
xis )0
min yit ; 1 + (xit
= E t yit ; yis (xit
x0
is
xis )0
xis )0
1 yis
x0 >
is
min yis
1 yis >
x0
it
x0 ; 1
is
(xit
min yis ; 1
x0
it
xit ; xis
xis )0
(xit
xis )0
(xit
xis )0
xit ; xis
xit ; xis :
where we have assumed that "it and "is are independent and identically distributed conditional on
(xit ; xis ).
This implies the unconditional moment condition
E t yit ; yis (xit
xis )0
18
(xit
xis ) = 0
where
t (y1 ; y2 ; d) =
where e1 =
c
y2 , c2 = max fy1
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
0
1
(y1
d < e1
c
for
for e1 < d < c2
c
y2
y2
y1
d) for c2 < d < c3
for c3 < d < e4
c
1
0
e4 < d
c
for
y2 ; y1 g and e4 = y1 . Note that c2 and c3
c
1; y2 g, c3 = min f1
are de…ned as before, but e1 and e4 di¤er from c1 and c4 .
c
c
Let
T (y1 ; y2 ; d) =
8
>
>
>
>
>
>
>
>
>
<
y2 ) e1 + 2 (1
c
2 (1
2 (1
y2 ) d + 2 (1
(y1
>
>
>
>
>
>
>
>
>
:
y2 ) c2 + (y1
y2
c2 )2 for
y2 ) c2 + (y1
y2
c2 )2
d)2
y2
for e1 < d < c2
c
for c2 < d < c3
2 (y1
1) d + 2 (y1
1) c3 + (y1
y2
c3 )2
2 (y1
1) e4 + 2 (y1
c
1) c3 + (y1
y2
c3 )2 for
We then de…ne the estimator of
for c3 < d < e4
c
by minimizing
XX
i
d < e1
c
wits T yit ; yis ; (xit
e4 < d
c
xis )0 b
t<s
The function T and its derivative are depicted in Figures 8 and 9 for a pair of uncensored
observations and for a pair with one censored and one uncensored observation, respectively.
Figure 10 shows the moment condition and the expected value of the objective function when
y1
N (0:5; 1) and y2
N (0:4; 1) and both are truncated from below at 0 and censored from
above at 1.
As before, we also have
E
where
t yit ; yis (xit
is convex and symmetric, and ( ) =
Let
T (y1 ; y2 ; d) =
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
(1
(1
y2 ) e1 + (1
c
y2 ) d + (1
(y1
xis )0
0(
(xit
xis ) = 0
) (when it exists)
y2 ) c2 +
(y1
y2
c2 ) for
y2 ) c2 +
(y1
y2
c2 )
y2
d)
for e1 < d < c2
c
for c2 < d < c3
for c3 < d < e4
c
(y1
1) d + (y1
1) c3 +
(y1
y2
c3 )
(y1
1) e4 + (y1
c
1) c3 +
(y1
y2
c3 ) for
19
d < e1
c
e4 < d
c
Figure 8: The Functions t (y1 ; y2 ; ) and T (y1 ; y2 ; ). Neither Observation Censored.
Figure 9: The Functions t (y1 ; y2 ; ) and T (y1 ; y2 ; ). One observation censored.
20
Figure 10: The Functions E [t (y1 ; y2 ; )] and E [T (y1 ; y2 ; )].
We then de…ne the estimator of
by minimizing
XX
i
4.2
wits T
yit ; yis ; (xit
xis )0 b :
t<s
Pairwise Di¤erence Versions
If we can estimate a panel data with 2 observations per unit, then we can apply the same idea to
any two observations in a cross section, treating the constant in the cross sectional model as an
individual–
speci…c e¤ect. This idea was explicitly used in Honoré and Powell (1994) to construct
estimators for the parameters of cross sectional (one–
sided) censored and truncated regression models based on the panel data estimator in Honoré (1992). Among others, this also characterizes the
relationship between the estimators in Manski (1987) and Han (1987) and between the estimators in
Kyriazidou (1997) and Powell (1987). The same idea can be applied to the models with two–
sided
censoring and truncation considered here, and the asymptotic properties follow from the arguments
used in Honoré and Powell (1994):
21
4.3
General Censoring Points.
It is easy to generalize the results above to the case where the truncation and censoring points are
not all 0 and 1. For example, consider the model
8
>
>
Li
>
<
yit =
x0 + "it
> it
>
>
:
Ui
then
8
>
>
>
<
yit
Ui
and we could estimate
by
Li
=
>
Li
>
>
:
b = arg min
b
XX
i
with two–
sided censoring
if
yit < Li
if Li
if
xit
Ui Li
if
"it Li
Ui Li
+
1
t<s
yit < Li
if Li
if
yit
Ui
wits U
Ui
yit > Ui
0
0
yit
Li yis
;
Li Ui
yit
Ui
yit > Ui
Li
;
Li
xit
Ui
xis
Li
0
b
(19)
This simple approach does not work when the censoring points are time–
varying, because then
"it Lit
Uit Lit
is not stationary.
In order to proceed, we need explicit expressions for the di¤erence in these “re–
censored”residuals. We …rst note that only pairs for which the support of the re–
censored residuals overlap can play
a role in the moment conditions leading to the objective function. These pairs are characterized by
Lit
Uis < x0
it
x0 < Uit
is
Lis
and for such pairs, the di¤erence in the arti…cially censored residuals for individual i in periods t
and s is
mami Lis
x0 ; yit
is
= mami Lis ; yit
x0
it
x0 ; Uis
it
x0
is
x0
is
mami Lit
x0 ; yis
it
mami Lit ; yis + x0
it
; Uis
x0 ; Uit
is
x0
is
x0
it
; Uit + x0
it
x0
is
where we use the notation mami fa; x; bg = max fa; min fx; bgg.
If we de…ne
k (L; U; y; d) =
and
u (y1 ; y2 ; d; L1 ; L2 ; U1 ; U2 ) = 1 fL1
8
>
>
>
<
>
>
>
:
U
y
for
d
L
d<y
y
for
U2 < d < U1
22
U
U <d<y
d>y
L
L
L2 g (k (L2 ; U2 ; y1 ; d)
k (L1 ; U1 ; y2 ; d) + d)
then
E u yit ; yis ; x0
it
x0
is
; Lit ; Lis ; Uit ; Uis
(xit ; xis ) = 0
and hence
E u yit ; yis ; x0
it
x0
is
; Lit ; Lis ; Uit ; Uis (xit
xis ) = 0:
These will be the moment conditions that lead to the estimator in this case.
Also de…ne
K (L; U; y; d) =
8
> 2yU
>
>
<
>
>
>
: 2yL
U 2 for
2dU
d)2
(y
y
L2
2dL
d<y
for
U
U <d<y
d>y
L
L
S (y1 ; y2 ; d; L1 ; L2 ; U1 ; U2 ) = K (L2 ; U2 ; y1 ; d) + K (L1 ; U1 ; y2 ; d)
d2
and
V (y1 ; y2 ; d; L1 ; L2 ; U1 ; U2 ) =
and the estimator for
8
> S (y1 ; y2 ; L1
>
>
<
U2 ; L1 ; L2 ; U1 ; U2 ) for
S (y1 ; y2 ; d; L1 ; L2 ; U1 ; U2 )
for L1
>
>
>
: S (y ; y ; U
L2 ; L1 ; L2 ; U1 ; U2 ) for
1 2
1
d < L1
U2
U2 < d < U1
d > U1
L2
L2
is then de…ned by
arg min
b
n
XX
wits V yit ; yis ; (xit
xis )0 b; Lit ; Lis ; Uit ; Uis
i=1 t<s
A version of this can be developed for a general loss function.
All of these extensions assume that the censoring and truncation points are exogenous in the
sense that one must make assumptions on the error terms conditional on them. In a recent paper,
Khan, Ponomareva, and Tamer (2011) consider a (one–
sided) censored regression model with endogenous censoring. Their approach only leads to partial identi…cation, but it would be interesting
to generalize it to more general versions of the models considered here.
5
Empirical Application
In this section we apply the estimator in Section 2.1 to analyze the portfolio-reshu- ing e¤ect of a
tax reform that increased the after-tax capital income on bonds relative to stocks in Denmark in
1987. We use a panel data set constructed from administrative records covering two years before
and after the reform to estimate a portfolio share equation for bonds as a function of marginal tax
23
rates on capital income. The analysis presented here follows the literature on taxation and portfolio
structure, e.g., Feldstein (1976), Hubbard (1985), King and Leape (1998), Samwick (2000), Poterba
and Samwick (2002), Poterba (2002) and Alan, Crossley, Atalay, and Jeon (2010). These papers
analyze (repeated) cross sections of households.6 Here the analysis is extended by using panel
data and controlling for time–
invariant correlated heterogeneity, i.e., …xed e¤ects. Controlling for
correlated unobserved …xed factors is likely to be important in this context, since the portfolio
composition of a household is likely to be in‡
uenced by time–
invariant factors such as risk aversion
and time discounting.
In the next subsection we give a brief overview over the tax reform. After this, we introduce
the data and present the results.
5.1
The Tax Reform
The tax reform, announced in 1985 and implemented in 1987, broke the link between the marginal
tax rates on earned income and capital income. Before the reform, all income was taxed at the
same marginal tax rate. With the reform the tax rate on positive capital income for high-income
households was decreased from 73 percent to 56 percent. The reform thereby increased the aftertax return on interest-bearing assets and therefore encouraged households to shift their portfolios
toward such assets. The reform also changed the tax value of interest deductions from 73 to about
50 percent, and this substantially increased the cost of debt, primarily mortgages, for leveraged
high-income households. For such households the reform e¤ectively brought a negative wealth
shock, giving them a strong incentive to lower their debt burden.7
The exact changes, however, di¤ered across municipalities. The Danish income tax system is
built around a proportional local government tax and a progressive tax collected by the central
government. While the progressive schedule is the same for everybody in Denmark, the local
6
Bakija (2000) uses the limited panel module of the American Survey of Consumer Finances (SCF) to study
portfolio changes around the 1988 tax reform. However, his data set is very small (984 households) and unrepresentative due to the well-known attrition problem in the SCF panel module; see Kennickell and Woodburn (1997).
More important in this context, the estimators applied do not exploit the full potential of the panel data in handling
unobserved heterogeneity. Ioannides (1992) also employs the 1983-1986 SCF panel module but does not control for
unobserved heterogeneity.
7
Alan and Leth-Petersen (2006) document that the reduced value of the interest deduction led households to
liquidate …nancial assets to lower their mortgage debt. This was possible because pre-payment of mortgage debt is
not restricted in Denmark.
24
Figure 11: Marginal Tax Rate for High–
Tax Municipality.
government tax rates vary across municipalities. A tax ceiling, however, insured that the marginal
tax rate could be at the maximum 73 percent. After the reform the tax ceiling on earned income
was reduced to 68 percent in the highest bracket8 and 56 percent in the middle bracket. Capital
income was now taxed at the same rate independently of the level of earned income. The marginal
tax rates across tax brackets before and after the reform are summarized in Table 1 (see Appendix
2)
The application of a tax ceiling together with the heterogeneous local government tax rates
implies that the reform had di¤erential e¤ects on people living in di¤erent municipalities. Figures
11 and 12 illustrate the changes in marginal tax rates due to the reform for a high-tax and a low-tax
municipality, respectively.
For a high-income person living in the municipality with the high local government tax, the
marginal tax rate on positive net capital income falls by 14.5 percentage points and the marginal
tax rate on negative net capital income falls by 20.5 percentage points. For a similar person living
in the municipality with the low local government tax rate, the marginal tax rate on positive
8
Approximately 20 percent of the population belong in the top bracket.
25
Figure 12: Marginal Tax Rate for Low–
Tax Municipality.
net capital income falls by 16.1 percentage points and negative net capital income falls by 22.1
percentage points. It is these di¤erences in changes of marginal tax rates that we will exploit for
identifying the e¤ect of changes in marginal tax rates on the portfolio allocation when using the
…xed e¤ects estimator.
The marginal tax rates on capital income refer to income received in the form of dividends on
stocks and interest payments from interest bearing accounts and bonds. Both before and after the
reform, realized capital gains/losses associated with trading assets were generally not taxed. The
exemption from this rule is capital gains from corporate stocks held for less than three years. Such
capital gains are taxed as earnings. Dividend payments were low relative to interest received from
bonds.9 This suggests that lowering the marginal tax rate on positive capital income a¤ected bonds
and stocks di¤erentially, favoring mainly income from bonds. In the empirical analysis we therefore
focus on reshu- ing between bonds and stocks.
9
The median household in the sample holding stocks received dividends corresponding to 2 percent of the value of
the stocks. The median household in the sample holding bonds received interest payments from these corresponding
to 10 percent of the value of the bonds.
26
5.2
Data
The data set is drawn from a random sample of 10 percent of the Danish population observed in
the years 1984 to 1988. Information on portfolio allocations, income, wealth and demographics is
collected and merged from di¤erent public administrative registers for all adult members of the
household that the sampled person belongs to. Portfolio and income information is obtained from
the income tax register. The portfolio information exists because Denmark had a wealth tax that
required all wealth holdings to be reported to the tax authorities. This information allows us
to break the wealth of each household into holdings of stocks and bonds. “Stocks” includes all
holdings of publicly and privately traded stocks, and “bonds” includes government and corporate
bonds. The holdings of stocks and bonds are self–
reported through the tax return and then audited
by the tax authority.
5.2.1
Sample selection
For our analysis we exclude observations if one of the household members is self-employed, since
register data are not likely to contain a good measure of own business wealth and because taxable
income is quite volatile for those individuals. Sampled individuals younger than 18 or older than
60 are dropped as are students and individuals living together with his/her parents or living in
a common household, i.e., a household with more than one family. To keep the focus on the
importance of tax incentives, we include only stable couples, i.e., couples where the partner is the
same in 1984 through 1988. On the same grounds we also exclude couples moving in the sample
period. For the purpose of the analysis we require that households entering the sample be observed
in all years in the period 1984-1988 so that we have a balanced panel.
Our objective is to investigate whether households reshu- e their portfolios in response to a
change in tax incentives. As in most industrialized countries many Danish households have fairly
undiversi…ed portfolios. Since the decrease in the value of interest deductions generated a large
negative wealth shock, clearly, these households are not likely to engage in portfolio reshu- ing and
hence cannot give us a clean answer regarding portfolio readjustments. We therefore construct a
sub-sample of households holding positive amounts of stocks or bonds of at least 5,000 DKK in
1984. We also require households to hold a positive amount of either stocks or bonds throughout the
rest of the observation period. This selection is introduced because we want to focus on households
with a potential to reshu- e between stocks and bonds. Also renters are deselected because there
27
are few renters with diversi…ed portfolios.10 The …nal subsample includes 8,577 households.11
5.3
Results
In this section we investigate if households reshu- ed their portfolio of bonds and stocks as a response
to the changes in relative after-tax returns on assets brought about by the 1987 tax reform. To do,
this we employ the estimator presented above and estimate a portfolio share equation where the
fraction of bonds in …nancial wealth, de…ned as the sum of bonds and stocks, is regressed on the
marginal tax rate on positive capital income and some control variables.
The distinguishing feature of our data set is the panel dimension. This facilitates estimating
portfolio share equations allowing for correlated unobserved time–
invariant heterogeneity. This is
important because we believe that unobserved time–
invariant factors, such as risk aversion and
time preferences, are correlated with wealth. High risk aversion may, for example, lead to a higher
portfolio share of safe assets, such as bonds, for a given level of wealth.
Before the reform, capital income and earnings were lumped together and taxed according to a
progressive tax scheme. This implies that households choose their tax bracket when choosing their
portfolios and that the marginal tax rate on capital income is likely to be an endogenous regressor.
We address this by calculating the marginal tax rate on capital income based on the household’
s
income in 1984, the year before the reform was announced, but using current year rules. In this way
the individual level tax bracket is allocated based on information that was predetermined relative
to the portfolio response to the reform.
We regress portfolio shares on the marginal tax rate on positive capital income, the log of total
…nancial assets, i.e., assets held in stocks and bonds, and a set of year dummies. Tax rate changes
vary across municipalities, but most of the change in tax rates is common across municipalities.
Year dummies control for the e¤ect of this common part, thereby also removing the major part
of the wealth e¤ect brought about by the reform. E¤ectively, by introducing year dummies, the
coe¢ cients on marginal tax rates are identi…ed by di¤erences in changes of marginal tax rates.
Year dummies may also pick up common e¤ects related to ‡
uctuations in assets. Financial assets
10
For assessing portfolio reshu- ing renters could have been included. We have chosen to leave them out of this
analysis because there are only a few renters (898) with positive …nancial wealth of at least 5000 DKK in 1984.
Moreover, renters generally do not provide a good comparison group for homeowners, since di¤erent preference
parameters may govern their behavior.
11
See Alan and Leth-Petersen (2006) for a more detailed analysis.
28
control for any remaining wealth e¤ect that might be present.12
Table 2 of Appendix 2 presents the parameter estimates from estimating random e¤ects Tobit
and …xed e¤ects censored regression models for the portfolio share of bonds in …nancial wealth.
The estimated parameter on the marginal tax rate is negative.13 If year dummies and …nancial
assets pick up the wealth e¤ect related to the reform, in particular the e¤ect of the reduction in
the value of interest deduction that led households to liquidate …nancial assets, then this is exactly
what economic theory predicts. Households should substitute from stocks toward bonds, whose
relative after-tax return increased, and this is what the results indicate.
Considering the corresponding random e¤ects estimates, we can see that the parameter estimates on …nancial assets and on year dummies are quite di¤erent, and the test of equality of all
the parameters in the random e¤ects and …xed e¤ects speci…cations rejects.
6
Concluding Remarks
This paper constructs estimators for panel data regression models with individual speci…c heterogeneity and two–
sided censoring and truncation. Following Powell (1986) the estimation strategy is
based on moment conditions constructed from re–
censored or re–
truncated residuals. While these
moment conditions do not identify the parameter of interest, they can be used to motivate objective
functions that do. This part is the main methodological contribution of the paper. We apply one
of the estimators to study the e¤ect of a Danish tax reform on household portfolio choice. We …nd
that a random e¤ects speci…cation can be rejected in favor of the “…xed”e¤ects speci…cation studied here, although both models yield the same sign of the key parameter that one would anticipate
from economic theory. The estimators are fairly easy to implement and a link to a program that
calculates the leading estimator is provided at http://www.princeton.edu/~honore/stata.
12
An alternative identi…cation strategy could be based on comparing the behavior of households in di¤erent tax
brackets. Households in the lowest tax bracket faced only very small changes in marginal tax rates on capital income,
and households in the middle tax bracket faced di¤erent changes in marginal tax rates than households in the highest
tax bracket. In our case this is not a natural approach to follow. High–and low–
income people are di¤erent in terms
of wealth levels and portfolio composition and possibly di¤erent with respect to preference parameters such as the
discount rate and the level of risk aversion. Households in lower tax brackets therefore do not represent a natural
control group for high–
income households.
13
As explained in Honoré (2008), the parameter estimates for both the random e¤ects and the …xed e¤ects models
can be converted to marginal e¤ects by multiplying them by the fraction of observations that are not censored.
29
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7
Appendix 1: Proofs and Derivations
7.1
Two–
Sided Censoring
This section provides justi…cation for the statements about the estimators for the model with two–
sided censoring. We …rst verify that u (yit ; yis ; x0
it
x0 ) in equation (8) does indeed yield the
is
di¤erence in the re-censored residuals de…ned in (6) and (7). Write d = x0
it
x0 , and consider
is
…rst the case where d > 0. In this case, the di¤erence in the re-censored residuals is
max fyit
d; 0g
min fyis ; 1
dg
which is most easily analyzed by considering a number of cases for d between 0 and 1. As mentioned
earlier, the contribution to the moment condition should be 0 when d is outside the interval between
1 and 1,which is consistent with the de…nition of u:
There are four cases based on combinations of whether yit
Case 1 (yit
(yit
d)
d
(1
Case 2 (yit
0
(1
d) = yit
d
d) = d
Case 3 (yit
(yit
d)
0 and yis
1
d:14
1
d): In this case, max fyit
d; 0g
min fyis ; 1
dg =
1
d): In this case, max fyit
d; 0g
min fyis ; 1
dg =
1
d): In this case, max fyit
d; 0g
min fyis ; 1
dg =
1.
d
d
0 and yis
1.
0 and yis
0 and yis
yis = yit
Case 4 (yit
d
yis
0 and yis
d.
1 d): In this case, max fyit
d; 0g min fyis ; 1
dg = 0 yis =
yis .
Case 3 corresponds to values of d close to (or at) 0. Speci…cally, the region for Case 3 is
(0; min fyit ; 1
yit
yis
yis g) = (0; c3 ). Noting that c2
0, it is clear that max fyit
d; 0g min fyis ; 1
d is consistent with the de…nition of u in equation (8).
Case 2 corresponds to values of d close to (or at ) 1, speci…cally the region (max fyit ; 1
(c4 ; 1), and it is again clear that u delivers max fyit
d; 0g
The region that de…nes the other two cases is (min fyit ; 1
min fyis ; 1
yis g ; 1) =
dg.
yis g ; max fyit ; 1
Cases 1 and 4 give di¤erent expressions depending on whether yit or 1
14
dg =
yis g) = (c3 ; c4 ).
yis is larger, but these
Since both the di¤erence in the re-censored residuals and u are continuous in d, it is not necessary to distinguish
between closed and open intervals in the following discussion.
34
expressions correspond exactly to the two cases for the max in the de…nition of u in (8) over the
interval (c3 ; c4 ).
The case d < 0 is dealt with in exactly the same manner.
We now turn to the question of why
xis )0 b
E U yit ; yis ; (xit
is uniquely minimized at the true . This is the key for consistency of the proposed estimator.
The result follows from the following lemma
Lemma 3 Suppose
yi1 = mami f0; + "i1 ; 1g
and
yi2 = mami f0; "i2 ; 1g
where "i1 and "i2 are identically distributed random variables with support on the whole real line.
Then
arg max E [U (yi1 ; yi2 ; d)] =
d2[ 1;1]
Proof: For 1
d
0
E [u (yi1 ; yi2 ; d)] = E [max fyi1
d; 0g
8
>
>
>
<
1 if
>
>
>
: 1
min fyi2 ; 1
= E [max fmami f0; + "i1 ; 1g
= E [max fmami f0
= E [mami f0;
d;
if
1<
if
1
d; 0g
dg
<1
dg]
d + "i1 ; 1
d + "i1 ; 1
1
min fmami f0; "i2 ; 1g ; 1
dg]
dg ; 0g
dg]
mami f0; "i2 ; 1
mami f0; "i2 ; 1
dg]
If "i1 (and "i2 ) have full support, then this is negative for d > and positive for d < .
For
1
d
0
E [u (yi1 ; yi2 ; d)] = E [min fyi1 ; 1 + dg
max fyi2 + d; 0g]
= E [min fmami f0; + "i1 ; 1g ; 1 + dg
max fmami f0; "i2 ; 1g + d; 0g]
= E [mami f0; + "i1 ; 1 + dg
max fmami fd; "i2 + d; 1 + dg ; 0g]
= E [mami f0; + "i1 ; 1 + dg
mami f0; "i2 + d; 1 + dg]
35
If "i1 (and "i2 ) have full support, then this is negative for d > and positive for d < .
Since
E U 0 (yi1 ; yi2 ; d) = E [u (yi1 ; yi2 ; d)]
the argument above shows that
arg max E [U (yi1 ; yi2 ; d)] =
d2[ 1;1]
Corollary 4 Consider the model
8
>
>
>
<
1 if
1
if
>
>
>
: 1
1<
if
<1
1
yit = x0 + "it
it
yit
8
> 0 if
>
yit < 0
>
<
=
y
if 0 yit 1
> it
>
>
: 1 if
yit > 1
for t = 1; 2. If "it is stationary conditional on (xi1 ; xi2 ) with support on the whole real line, then
the set of solutions to
max E U yi1 ; yi2 ; mami
b
xi2 )0 b; 1
1; (xi1
is
b : P mami
1; (xi1
xi2 )0 b; 1 = mami
1; (xi1
xi2 )0 ; 1
=1
The Corollary above requires that the errors are stationary conditional on the regressors. This is
much more general than the usual assumption that the individual— speci…c e¤ect and the contemporaneous errors interact additively. To see that E U
yit ; yis ; (xit
xis )0 b
in (12) is uniquely
minimized, it is convenient to assume that "it and "is are independent and identically distributed conditional on some individual— speci…c e¤ect, vi . With this assumption, the argument for
why E U
yit ; yis ; (xit
Speci…cally, when 1
d
xis )0 b
is uniquely minimized follows essentially the same logic as above.
0
(u (yit ; yis ; d)) = (mami f0;
Let
= (xit
d + "it ; 1
xis )0 .
36
dg
mami f0; "is ; 1
dg)
If "it and "is are independent and identically distributed conditional on vi , then
E [ (u (yit ; yis ; ))j xit ; xis ] = E [ E [ (u (yit ; yis ; ))j
= E [ E [ (mami f0; "it ; 1
because mami f0; "it ; 1
g
g
mami f0; "is ; 1
i ; xit ; xis ]j xit ; xis ]
mami f0; "is ; 1
g)j
i ; xit ; xis ]j xit ; xis ]
=0
g is symmetrically distributed conditional on
i,
and ( ) is an odd function.
For d >
mami f0;
d + "it ; 1
dg
mami f0; "is ; 1
dg
mami f0; "it ; 1
with probability 1 (conditional on
E [ (mami f0;
d + "it ; 1
dg
i ),
and since
mami f0; "is ; 1
dg
is increasing
mami f0; "is ; 1
E [ (mami f0; "it ; 1
The line of argument is the same when d <
dg
dg)j
dg
i ; xit ; xis ]
mami f0; "is ; 1
dg)j
i ; xit ; xis ]
=0
. Strict inequalities follow from a full support
assumption on "i1 (and "i2 ).
We therefore have that E [ U (yit ; yis ; d)j xit ; xis ] is decreasing to the left of (xit
increasing to the right. Hence it is minimized at d = (xit
this implies that E U
7.2
yit ; yis ; (xit
xis )0
and
xis )0 . Subject to a rank condition,
xis )0 b xit ; xis is minimized at b = :
Two–
Sided Truncation
The following Lemma (combined with the obvious rank–
condition) establishes that minimization
of E R
yi1 ; yi2 ; (xi1
x2i )0 b
will identify
if the distribution of " is log–
concave. This is the
assumption that was made in a number of other papers (including Honoré (1992), Honoré and
Powell (1994) and Abrevaya (1999); see also the discussion in Chen (forthcoming)). In Section
2.2, we only consider the case with (two–
sided) truncation at 0 and 1. It is just as easy to prove
identi…cation for general individual– and time–
speci…c truncation points. In the following we
therefore denote the truncation points by L and U .
Lemma 5 Let (L; U ) be a vector of random variables such that L < U with probability 1. Assume
that " is independent of (L; U ) and has a continuous, log–
concave distribution with support on the
whole real line. Let yit =
it +"it
for some real number,
37
it ,
and consider two draws (yi1 ; Li1 ; Ui1 ) and
(yi2 ; Li2 ; Ui2 ) from the distribution of (y; L; U ) conditional on L < y < U . then E [R (yi1 ; yi2 ; d)]
is uniquely minimized at d =
7.3
i2 .
i1
Proof of Lemma 5.
Let E denote expectation conditional on truncation and E in population.
The moment condition can then be written as
E [ (4yi
4di ) 1 fLi2
= E [ (4yi
di2
yi1
4di ) 1 fLi2 + 4di
= E[ (4"i + 4
1 fLi1 + (4
where 4ai = ai1
4di )
i
i ) 1 fLi2
= E [ (4"i +
yi1
P (Li1
yi1
i
4di )
i
i
i
"i1 +
yi2
4di
Ui2
yi2
di2
Ui1
(4
i
+
i
Ui1
di1 g]
4di g]
4di )g
4di )g]
i
Ui2
i2
di1
4di )
"i1
i
i g 1 fLi1
i2
Ui2
i2
i1
i g 1 fLi1
i1
+
"i2
"i2
i
Ui1
i1
Ui1
+
i1
+
Ui1 g 1 fLi2 yi2 Ui2 g]
1
:
Ui1 ; Li2 yi2 Ui2 )
It su¢ ces to show that this is nonpositive for
left of 0, 0 for
di2 g 1 fLi1
Ui2 + 4di g 1 fLi1
"i1
i
i2
Ui2
Ui1 + (4
i1
= (4
i2
i ) 1 fLi2
1 fLi1
(4
"i2 +
ai2 . Letting
= E[ (4"i +
yi1
4di ) 1 fLi2
i
di1
= 0; nonnegative for
i
i
< 0, strictly negative in a neighborhood to the
> 0, and strictly positive in a neighborhood to the right
of 0. Now consider the term
E [ (4"i +
1 fLi1
i1
i ) 1 fLi2
"i1
De…ning wi1 =
1
2
Ui1
("i1
i2
i
"i1
bj
i1 g 1 fLi2
"i2 ) and wi2 =
i2
1
2
i g 1 fLi1
i2
"i2
bj
i1
+
i
"i2
Ui1
i1
38
ig
i2 g]:
("i1 + "i2 ) (so "i1 = (wi1 + wi2 ) and "i2 = wi2
and 4"i = 2wi1 )
+
wi1
i g]
ig
E [ (4"i +
1 fLi1
i ) 1 fLi2
"i1
i1
= E [ (2wi1 +
i2
Ui1
Ui2
i1 g 1 fLi2
i ) 1 fLi2
+
"i1
i
i2
wi2
1 fLi1
i1
1 fLi1
i1
wi1 + wi2
Ui1
1 fLi2
i2
wi2
Ui2
= E [ (2wi1 +
1 Li2
"i2
i
Ui1
i1
+
ig
i2 g]
ig
i2
1
Ui1 +
1 Li1
1
= E [ (2wi1 +
1
2
min Ui2
= E
= E
i2
1
2
+ wi2
1
2
i
wi1 +
1
2
1
2
i
wi1 +
i
i
+ wi2 +
1
2
1
2
wi1 +
i
1
2
wi1 +
i
Ui2
Li1 +
i
Ui1
i1
i2
i
1
2
+ wi2
wi2 +
i1
Li2 +
i
1
2
wi2
i2
1
2
i
1
2
i
i
+ wi2 +
]
i)
wi2
i2
1
i ; Ui1 +
2
i1
1
i ; Li1
2
+ wi2
wi2 +
i1
1
i ; Ui2 +
2
i2
1
2
i
i
g]
+ wi2 +
1
2 i
+ wi2 +
i
i2
(2wi1 +
E
1
2
wi2 +
i1
Ui2 +
1fmax Li2
wi1 +
i1
ig
i)
wi2
i2
i1
+
Ui2
+
i2 g]
wi1
Ui1
Ui2
i1
i1 g
i
wi1
"i2
i2
wi1 + wi2
i
i g 1 fLi1
i2
wi2
i) 1
(2wi1 +
where ci = min Ui2
i2
1
i ; Li1 +
2
1
ci wi1 +
2
i) 1
wi2
ci
i1
1
i ; Ui1
2
i1
wi2 +
1
i ; Li2 +
2
i2
+ wi2 +
1
2
ci
i
wi1 +
1
2 i;
+ wi2
1
2
Li1 +
ci
i
i1
+ wi2
wi2 ; Li1 ; Ui1 ; Li2 ; Ui2
1
2 i ; Ui1
i1
wi2 +
1
2 i;
Li2 +
i2
Strict log–
concavity of "it , implies that wi1 is strictly unimodal and symmetric conditional on wi2 .
39
.
It therefore follows that
E
E
(2wi1 +
i) 1
ci
wi1 +
1
2
ci
i
8
>
0
>
>
>
>
>
>
> <0
>
<
=0
>
>
>
>
>
0
>
>
>
>
: >0
wi2 ; Li1 ; Ui1 ; Li2 ; Ui2
if
i
<0
if
i
< 0,
if
i
=0
if
if
P
ci
wi1 +
1
2 i
ci > 0
i
>0
i
> 0, P
ci
wi1 +
1
2 i
ci > 0:
Since the "’ (and hence the wi ’ are continuous, the condition that P
s
s)
ci
wi1 +
1
2 i
ci > 0
will be satis…ed if P (ci > 0) > 0.
We will next show that for
1 fci > 0g = 1
1
= 1
1
= 1
i
2 (0; k), P (ci > 0) > 0 for some k > 0. Note that
1
1 Ui1
wi2 +
i >0
i1
2
1
Li2 + i2 + wi2 +
1 Ui2
wi2
i >0
i2
2
1
1
wi2 > Li1
1 Ui1
i1 +
i
i1 +
i > wi2
2
2
1
1
wi2 > Li2
1 Ui2
i2
i
i2
i > wi2
2
2
1
1
Ui1
1 Ui2
i1 +
i > wi2 > Li1
i1 +
i
2
2
Li1 +
i1
+ wi2
1
2
i
1
2
i2
>0
i
>0
1
2
i
> wi2 > Li2
i2
This will have positive probability provided that
P
Li1
i1
+
1
2
i
< Ui2
i2
1
2
>0
which follows from Lemma 6.
Lemma 6 If (U1 ; V1 ) and (U2 ; V2 ) are two independent draws of a random vector (U; V ) with
P (U < V ) = 1, then there exists a k > 0 such that for 0 <
Proof.
1
2.
Since V
) > 0.
U > 0 with probability 1, there exists an m > 0 such that P (V
Now consider the space f(u; v) : v
U > m)
u > mg. This can be divided into a countable number
of regions, Ak , such that for (u1 ; v1 ) ; (u2 ; v2 ) 2 Ak , ju1
P (V
< k, P (U1 < V2
u2 j <
m
2
and jv1
v2 j <
m
2.
Since
U > m) = 1 , at least one of these regions, Ak , must have positive probability. Hence there
2
is positive probability that (U1 ; V1 ) and (U2 ; V2 ) in the statement of the lemma both belong to this
Ak , in which case U1 < V1
m < V2
m
2.
40
1
2
i
:
8
Appendix 2: Empirical Results
Table 1: Marginal tax rates before and after
implementation of the 1987 tax reform
Before Reform
After Reform
Tax bracket
Earnings + Cap inc
0-113
M + 19:75
0
113-186
M + 34:15
130
186-
M + 44:95
Tax ceiling
73
Earnings
inc < 0
inc: > 0(1)
130
M + 22:00
M + 22:00
M + 22:00
200
M + 28:00
M + 22:00
M + 28:00
M + 40:00
M + 22:00
M + 28:00
Tax bracket
200
Tax ceiling
68:00=56:00(2)
56:00
Note: M is the local government tax rate. Threshold values for the tax brackets are given
in 1000 DKK. Thresholds are adjusted yearly. Threshold values used in the table are for 1986
(before the reform) and 1987 (after the reform). The marginal tax rates refer to personal income
(as opposed to household income).
(1) The tax brackets for positive net capital income refer to the sum of earnings and positive
net capital income.
After the reform positive capital income is taxed progressively up to the …rst threshold, 130,000
DKK. For a married couple the progression threshold is 260,000 based on the sum of their joint
positive net capital income and earnings.
(2) The 68 percent tax ceiling applies only to the top bracket.
41
Table 2:. Random and Fixed E¤ects Censored Regression Estimates
of the Portfolio Share of Bonds in Financial Wealth.
Fixed E¤ects
Random E¤ects
0:130
0:206
(0:071)
(0:045)
0:177
0:085
(0:008)
(0:003)
0:214
0:157
(0:006)
(0:006)
0:314
0:248
(0:009)
(0:006)
0:318
0:285
(0:013)
(0:008)
0:383
0:331
(0:013)
(0:008)
MTR capital income
Ln(Financial Assets)
D85
D86
D87
D88
Constant
—
0:047
(0:046)
# households/observations
8,577 / 42,885
# left/right censored obs
7,529 / 15,655
Test of Parameter Equality (d.f.)
167 (6)
Standard errors in parenthesis
42
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5
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