Model Process
Resume:
Dynamic censored regression and the Open Market
Desk reaction function
Robert de Jong Ana Mar Herrera
a
March 9, 2009
Abstract
The censored regression model and the Tobit model are standard tools in economet-
rics. This paper provides a formal asymptotic theory for dynamic time series censored
regression when lags of the dependent variable have been included among the regres-
sors. The central analytical challenge is to prove that the dynamic censored regression
model satis es stationarity and weak dependence properties if a condition on the lag
polynomial holds. We show the formal asymptotic correctness of conditional maximum
likelihood estimation of the dynamic Tobit model, and the correctness of Powell s least
absolute deviations procedure for the estimation of the dynamic censored regression
model. The paper is concluded with an application of the dynamic censored regression
methodology to temporary purchases of the Open Market Desk.
1 Introduction
The censored regression model and the Tobit model are standard tools in econometrics. In
a time series framework, censored variables arise when the dynamic optimization behavior
Department of Economics, Ohio State University, 429 Arps Hall, Columbus, OH 43210, email
abo8zg@r.postjobfree.com
Department of Economics, Michigan State University, 215 Marshall Hall, East Lansing, MI 48824, email
abo8zg@r.postjobfree.com. We thank Selva Demiralp and Oscar Jord` for making their data available, and
a
Bruce Hansen for the use of his code for kernel density estimation.
1
of a rm or individual leads to a corner response for a signi cant proportion of time. In
addition, right-censoring may rise due to truncation choices made by the analysts in the
process of collecting the data (i.e., top coding). Censored regression models apply to vari-
ables that are left-censored at zero, such as the level of open market operations or foreign
exchange intervention carried out by a central bank, and in the presence of an intercept in
the speci cation they also apply to time series that are censored at a non-zero point, such
as the clearing price in commodity markets where the government imposes price oors, the
quantity of imports and exports of goods subject to quotas, and numerous other series.
The asymptotic theory for the Tobit model in cross-section situations has long been
understood; see for example the treatment in Amemiya (1973). In recent years, asymp-
totic theory for the dynamic Tobit model in a panel data setting has been established using
large-N asymptotics; see Arellano and Honor (1998) and Honor and Hu (2004). However,
e e
there is no result in the literature that shows stationarity properties of the dynamic cen-
sored regression model, leaving the application of cross-section techniques for estimating the
dynamic censored regression model in a time series setting formally unjusti ed. This paper
seeks to ll this gap. After all, a justication of standard inference in dynamic nonlinear
models requires laws of large numbers and a central limit theorem to hold. Such results
require weak dependence and stationarity properties.
While in the case of linear AR models it is well-known that we need the roots of the
lag polynomial to lie outside the unit circle in order to have stationarity, no such result
is known for nonlinear dynamic models in general and the dynamic regression model in
particular. The primary analytical issue addressed in this paper is to show that under some
conditions, the dynamic censored regression model as de ned below satis es stationarity and
weak dependence properties. This proof is therefore an analogue to well-known proofs of
stationarity of ARMA models under conditions on the roots of the AR lag polynomial. The
dynamic censored regression model under consideration is
p
i yt i + xt + t ),
yt = max(0, (1)
i=1
where xt denotes the regressor, t is a regression error, we assume that Rq, and we
de ne 2 = E t . One feature of the treatment of the censored regression model in this
2
paper is that t is itself allowed to be a linear process (i.e., an MA process driven by
an i.i.d. vector of disturbances), which means it displays weak dependence and is possibly
correlated. While stationarity results for general nonlinear models have been derived in e.g.
Meyn and Tweedie (1994), there appear to be no results for the case where innovations are
not i.i.d. (i.e. weakly dependent or heterogeneously distributed). The reason for this is that
the derivation of results such as those of Meyn and Tweedie (1994) depends on a Markov
2
chain argument, and this line of reasoning appears to break down when the i.i.d. assumption
is dropped. This means that in the current setting, Markov chain techniques cannot be used
for the derivation of stationarity properties, which complicates our analysis substantially,
but also puts our analysis on a similar level of generality as can be achieved for the linear
model.
A second feature is that no assumption is made on the lag polynomial other than that
max (z ) = 1 p=1 max(0, i )z i has its roots outside the unit circle. Therefore, in terms
i
of the conditions on max (z ) and the dependence allowed for t, the aim of this paper is to
analyze the dynamic Tobit model on a level of generality that is comparable to the level of
generality under which results for the linear model AR(p) model can be derived. Note that
intuitively, negative values for j can never be problematic when considering the stationarity
properties of yt, since they pull yt back to zero . This intuition is formalized by the fact
that only max(0, j ) shows up in our stationarity requirement.
An alternative formulation for the dynamic censored regression model could be
yt = yt I (yt > 0) where (B )yt = xt + t,
(2)
where B denotes the backward operator. This model will not be considered in this paper,
and its fading memory properties are straightforward to derive. The formulation considered
in this paper appears the appropriate one if the 0 values in the dynamic Tobit are not
caused by a measurement issue, but have a genuine interpretation. In the case of a model for
the di erence between the price of an agricultural commodity and its government-instituted
price oor, we may expect economic agents to react to the actually observed price in the
previous period rather than the latent market clearing price, and the model considered in this
paper appears more appropriate. However, if our aim is to predict tomorrow s temperature
from today s temperature as measured by a lemonade- lled thermometer that freezes at zero
degrees Celsius, we should expect that the alternative formulation of the dynamic censored
regression model of Equation (2) is more appropriate.
The literature on the dynamic Tobit model appears to mainly consist of (i) theoretical
results and applications in panel data settings, and (ii) applications of the dynamic Tobit
model in a time series setting without providing a formal asymptotic theory. Three notewor-
thy contributions to the literature on dynamic Tobit models are Honor and Hu (2004), Lee
e
(1999), and Wei (1999). Honor and Hu (2004) considers dynamic Tobit models and deals
e
with the problem of the endogeneity of lagged values of the dependent variable in panel data
setting, where the errors are i.i.d., T is xed and large-N asymptotics are considered. In fact,
the asymptotic justi cation for panel data Tobit models is always through a large-N type
argument, which distinguishes this work from the treatment of this paper. For a treatment
of the dynamic Tobit model in a panel setting, the reader is referred to Arellano and Honor e
(1998, section 8.2).
3
Lee (1999) and Wei (1999) deal with dynamic Tobit models where lags of the latent
variable are included as regressors. Lee (1999) considers likelihood simulation for dynamic
Tobit models with ARCH disturbances in a time series setting. The central issue in this paper
is the simulation of the log likelihood in the case where lags of the latent variable (in contrast
to the observed lags of the dependent variable) have been included. Wei (1999) considers
dynamic Tobit models in a Bayesian framework. The main contribution of this paper is the
development of a sampling scheme for the conditional posterior distributions of the censored
data, so as to enable estimation using the Gibbs sampler with a data augmentation algorithm.
In related work, de Jong and Woutersen (2003) consider the dynamic time series binary
choice model and derive the weak dependence properties of this model. This paper also con-
siders a formal large-T asymptotic theory when lags of the dependent variable are included as
regressors. Both this paper and de Jong and Woutersen (2003) allow the error distribution to
be weakly dependent. The proof in de Jong and Woutersen (2003) establishes a contraction
mapping type result for the dynamic binary choice model; however, the proof in this paper
is completely di erent, since other analytical issues arise in the censored regression context.
As we mentioned above, a signi cant body of literature on the dynamic Tobit model con-
sists of applications in a time series setting without providing a formal asymptotic theory.
Inference in these papers is either conducted in a classical framework, by assuming the max-
imum likelihood estimates are asymptotically normal, or by employing Bayesian inference.
Papers that estimate censored regression models in time series cover diverse topics. In the -
nancial literature, prices subject to price limits imposed in stock markets, commodity future
exchanges, and foreign exchange futures markets have been treated as censored variables.
Kodres (1988, 1993) uses a censored regression model to test the unbiasedness hypothesis
in the foreign exchange futures markets. Wei (2002) proposes a censored-GARCH model
to study the return process of assets with price limits, and applies the proposed Bayesian
estimation technique to Treasury bill futures.
Censored data are also common in commodity markets where the government has histor-
ically intervened to support prices or to impose quotas. An example is provided by Chavas
and Kim (2006) who use a dynamic Tobit model to analyze the determinants of U.S. butter
prices with particular attention to the e ects of market liberalization via reductions in oor
prices. Zangari and Tsurumi (1996), and Wei (1999) use a Bayesian approach to analyze the
demand for Japanese exports of passenger cars to the U.S., which were subject to quotas
negotiated between the U.S. and Japan after the oil crisis of the 1970 s.
Applications in time series macroeconomics comprise open market operations and foreign
exchange intervention. Dynamic Tobit models have been used by Demiralp and Jord` (2002) a
to study the determinants of the daily transactions conducted by the Open Market Desk,
and Kim and Sheen (2002) and Frenkel, Pierdzioch and Stadtmann (2003) to estimate the
intervention reaction function for the Reserve Bank of Australia and the Bank of Japan,
4
respectively.
The structure of this paper is as follows. Section 2 present our weak dependence results
for (yt, xt ) in the censored regression model. In Section 3, we show the asymptotic validity of
the dynamic Tobit procedure. Powell s LAD estimation procedure for the censored regres-
sion model, which does not assume normality of errors, is considered in Section 4. Section
5 studies the determinants of temporary purchases of the Open Market Desk. Section 6
concludes. The Appendix contains all proofs of our results.
2 Main results
We will prove that yt as de ned by the dynamic censored regression model satis es a weak
dependence concept called Lr -near epoch dependence. Near epoch dependence of random
variables yt on a base process of random variables t is de ned as follows:
De nition 1 Random variables yt are called Lr -near epoch dependent on t if
sup E yt E (yt t M, t M +1, . . ., t+M ) r = (M )r 0 as M . (3)
t Z
The base process t needs to satisfy a condition such as strong or uniform mixing or
independence in order for the near epoch dependence concept to be useful. For the de nitions
of strong and uniform mixing see e.g. Gallant and White (1988, p. 23) or P tscher
o
and Prucha (1997, p. 46). The near epoch dependence condition then functions as a device
that allows approximation of yt by a function of nitely many mixing or independent random
variables t .
For studying the weak dependence properties of the dynamic censored regression model,
assume that yt is generated as
p
yt = max(0, i yt i + t ). (4)
i=1
Later, we will set t = xt + t in order to obtain weak dependence results for the general
dynamic censored regression model that contains regressors.
When postulating the above model, we need to resolve the question as to whether there
exists a strictly stationary solution to it and whether that solution is unique in some sense.
See for example Bougerol and Picard (1992) for such an analysis in a linear multivariate set-
ting. In the linear model yt = yt 1 + t, these issues correspond to showing that j t j
j =0
is a strictly stationary solution to the model that is unique in the sense that no other function
of ( t, t 1, . . .) will form a strictly stationary solution to the model.
5
An alternative way of proceeding to justify inference could be by considering arbitrary
initial values (y1, . . ., yp ) for the process instead of starting values drawn from the stationary
distribution, but such an approach will be substantially more complicated.
The idea of the strict stationarity proof of this paper is to show that by writing the
dynamic censored regression model as a function of the lagged yt that are su ciently remote
in the past, we obtain an arbitrarily accurate approximation of yt . Let B denote the backward
operator, and de ne the lag polynomial max (B ) = 1 p=1 max(0, i )B i . The central result
i
of this paper, the formal result showing the existence of a unique backward looking strictly
stationary solution that satis es a weak dependence property for the dynamic censored
regression model is now the following:
Theorem 1 If the linear process t satis es t = ai ut i, where a0 > 0, ut is a sequence
i=0
of i.i.d. random variables with density fu, E ut r
fu (y + a) fu (y ) dy M a
1/(1+r )
for some constant M whenever a for some > 0, t=0 Gt 0 (5)
for some function F, then (i) there exists a solution yt to the model of Equation (4) such
that (yt, t ) is strictly stationary; (ii) if zt = f ( t, t 1, . . .) is a solution to the model, then
yt = zt a.s.; and (iii) yt is L2 -near epoch dependent on t . If in addition, ai c1 exp( c2 i)
for positive constants c1 and c2, then the near epoch dependence sequence (M ) satis es
(M ) c1 exp( c2 M 1/3 ) for positive constants c1 and c2 .
Our proof is based on the probability of yt reaching 0 given the last p values of t always
being positive. This property is the key towards our proof and is established using the linear
process assumption in combination with the condition of Equation (5). Note that by the
results of Davidson (1994, p. 219), our assumption on t implies that t is also strong mixing
1/(1+r )
with (m) = O ( m+1 Gt ). Also note that for the dynamic Tobit model where errors
t=
are i.i.d. normal and regressors are absent, the condition of the above theorem simpli es to
the assumption that max (z ) has all its roots outside the unit circle.
One interesting aspect of the condition on max (z ) is that negative i are not a ecting
the strict stationarity of the model. The intuition is that because yt 0 a.s., negative i
can only pull yt back to zero and because the model has the trivial lower bound of 0 for
yt, unlike the linear model, this model does not have the potential for yt to tend to minus
in nity.
6
3 The dynamic Tobit model
De ne =, where = ( 1, . . ., p ), and de ne b = (r, c, s) where r is a (p 1)
vector and c is a (q 1) vector. The scaled Tobit loglikelihood function conditional on
y1, yp under the assumption of normality of the errors equals
T
1
LT (b) = LT (c, r, s) = (T p) lt (b), (6)
t=p+1
where
p
lt (b) = I (yt > 0) log(s 1 ((yt ri yt i c xt )/s))
i=1
p
ri yt i c xt )/s)).
+I (yt = 0) log( (( (7)
i=1
In order for the loglikelihood function to be maximized at the true parameter, it ap-
pears hard to achieve more generality than to assume that t is distributed normally given
yt 1, . . ., yt p, xt . This assumption is close to assuming that t given xt and all lagged yt
is normally distributed, which would then imply that t is i.i.d. and normally distributed.
Therefore in the analysis of the dynamic Tobit model below, we will not attempt to consider
a situation that is more general than the case of i.i.d. normal errors. Alternatively to the
result below, we could also nd conditions under which T converges to a pseudo-true value
. Such a result can be established under general linear process assumptions on (x t, t ), by
the use of Theorem 1. It should be noted that even under the assumption of i.i.d. errors, no
results regarding stationarity of the dynamic Tobit model have been derived in the literature
thus far.
Let T denote a maximizer of LT (b) over b B . De ne wt = (yt 1, . . ., yt p, x t, 1) . The
1 at the end of the de nition of wt allows us to write b wt . For showing consistency, we
need the following two assumptions. Below, let denote the usual matrix norm de ned as
M = (tr(M M ))1/2, and let X r = (E X r )1/r .
Assumption 1 The linear process zt = (x t, t ) satis es zt = j vt j, where the vt are
j =0
i.i.d. (k 1) vectors, vt r 0.
3. B, where B is a compact subset of Rp+q+1, and B = R where inf > 0.
p
i yt i + xt > ) is positive de nite for some positive .
4. Ewt wt I ( i=1
p
Theorem 2 Under Assumption 1 and 2, T .
For asymptotic normality, we need the following additional assumption.
Assumption 3
1. is in the interior of B .
2. I = E ( / b)lt ( / b )lt = E ( / b)( / b )lt is invertible.
d
Theorem 3 Under Assumptions 1, 2, and 3, T 1/2 ( T ) N (0, I 1 ).
4 Powell s LAD for dynamic censored regression
For this section, de ne =, where = ( 1, . . ., p ), de ne b = (r, c ) where r is
a (p 1) vector and c is a (q 1) vector, and wt = (yt 1, . . ., yt p, x t ) . This rede nes the
b and vectors such as to not include s and respectively; this is because Powell s LAD
estimator does not provide a rst-round estimate for 2 . Powell s LAD estimator T of the
dynamic censored regression model is de ned as a minimizer of
T
1
ST (b) = ST (c, r, s) = (T p) s(yt 1, . . ., yt p, xt, t, b)
t=p+1
p
T
1
ri yt i + c xt )
= (T p) yt max(0, (9)
t=p+1 i=1
p +q
over a compact set subset B of R . We can prove consistency of Powell s LAD estimator
of the dynamic time series censored regression model under the following assumption.
8
Assumption 4
1. B, where B is a compact subset of Rp+q .
2. The conditional distribution F ( t wt ) satis es F (0 wt) = 1/2, and f ( wt) = F ( w )
is continuous in on a neighborhood of 0 and satis es c2 f (0 wt) c1 > 0 for con-
stants c1, c2 > 0.
p
3. E xt 3 ) is nonsingular for some positive .
i=1
p
Theorem 4 Under Assumptions 1 and 4, T .
For asymptotic normality, we need the following additional assumption. Below, let
(wt, t, b) = I (b wt > 0)(1/2 I ( t + ( b) wt > 0))wt . (10) can be viewed as a heuristic derivative of s with respect to b.
Assumption 5
1. is in the interior of B .
2. De ning G(z, b, r ) = EI ( wtbwt z ) wt r, we have for z near 0, for r = 0, 1, 2,
sup G(z, b, r ) K1 z. (11)
b 0)wt wt is invertible.
4. For some r 2, E xt 2r 0)[ 2 I (yt
ull.dvi
Contact this candidate