Panel estimation of state-dependent adjustment

Empir Econ
DOI 10.1007/s00181-010-0419-y
Panel estimation of state-dependent adjustment
when the target is unobserved
Ulf von Kalckreuth
Received: 30 July 2009 / Accepted: 22 July 2010
© Springer-Verlag 2010
Abstract Understanding adjustment processes has become central in economics.
Empirical analysis is fraught with the problem that the target is usually unobserved.
This article develops and simulates GMM methods for estimating dynamic adjustment
models in a panel data context with partially or entirely unobserved targets and
endogenous, time-varying persistence. In this setup, the standard first differenceGMM
procedure fails. Four estimation strategies are proposed. Two of them are based on
quasi-differencing. The third is characterised by a state-dependent filter, while the last
is an adaptation of the GMM level estimator.
Keywords Dynamic panel data methods · Economic adjustment ·
GMM · Quasi-differencing · Non-linear estimation
JEL Classification C23 · C15 · D21
1 Introduction
New Keynesian economics, with its emphasis on real and financial frictions, has introduced
a focus on microeconomic adjustment dynamics into the empirical literature.
Adjustment dynamics are essential for understanding aggregate behaviour and its sensitivity
towards shocks. Important examples range from price adjustment and its sig-
This article was presented at the 2009 Panel Data Conference at the University of Bonn. It draws on Chap.
3 of the author’s habilitation thesis at the University of Mannheim.
The views expressed in this article do not necessarily reflect those of the Deutsche Bundesbank or its staff.
All the errors and omissions are those of the author.
U. von Kalckreuth (B)
Deutsche Bundesbank Research Centre, Wilhelm Epstein-Str. 14,
60431 Frankfurt am Main, Germany
e-mail: ulf.von-kalckreuth@bundesbank.de
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U. von Kalckreuth
nificance for the New Keynesian Phillips curve (Woodford 2003), over plant level
adjustment and aggregate investment dynamics (Caballero et al. 1995; Caballero
and Engel 1999; Bayer 2006), to aggregate employment dynamics, building from
microeconomic evidence (Caballero et al. 1997). In these studies, as in von Kalckreuth
(2006), the adjustment dynamics itself becomes the principal object of analysis,
instead of being treated as an important, but burdensome obstacle to understanding
equilibrium phenomena.
In a rather general form, economic adjustment can be framed by a ‘gap equation’,
as formalised by Caballero et al. (1995):
yi,t = gi,t , xi,t · gi,t , where
gi,t = yi,t−1 − y∗
i,t
Here, subscripts refer to individual i at time t, and gi,t is the gap between the state
yi,t−1 inherited from the last period and the target y∗
i,t that would be realised if adjustment
costs were zero for one period of time. The speed of adjustment, which is written
as a function of the gap itself and additional state variables xi,t , determines the fraction
of the gap that is removed within one period of time. The adjustment function
will reflect convex or non-convex adjustment costs, irreversibility and indivisibilities,
financing constraints or other restrictions, and the uncertainty of expectations formation.
With quadratic adjustment costs or Calvo-type probabilistic adjustment, will
be a constant.1
Estimating the function is inherently difficult. In general, both y∗
i,t and gi,t will
not be observable. However, some measure of the gap is needed for any estimation,
and if explicitly depends on gi,t , this measure will move to the centre stage. In order
to address this issue, one may try to do the utmost to observe the target as exactly
as possible. The controversy between Cooper and Willis (2004) and Caballero and
Engel (2004) on interpreting the results of gap equation estimates bear testimony to
the problems that may result from imperfect measures of the gap. However, there is
an alternative. In linear dynamic panel estimation, the problem of unobserved targets
can successfully be addressed by positing an error component structure for the measurement
error and eliminating the individual fixed effect by a suitable transformation,
such as first differencing. See Bond et al. (2003) and Bond and Lombardi (2007) for
an error correction model of capital stock adjustment.
In the unrestricted, non-linear case, this approach is not feasible, as a host of
incidental parameters will preclude identification. However, there may be direct qualitative
information on the level of , e.g. from survey data, ratings or market information
services. If one is willing to treat the adjustment process as piecewise linear,
distinguishing regimes of adjustment, then, as will be shown, this information can be
harnessed to eliminate the incidental parameters from the problem completely.
1 Calvo-type adjustment refers to adjustment costs that are infinite with probability 1 − λ and zero with
probability λ. In other words: a randomly drawn share λ of market participants receives the chance to
adjust costlessly. As a modelling device, this assumption is ubiquitous in the monetary Dynamic General
Equilibrium literature. Sometimes this state-independent adjustment is playfully referred to as the working
of the ‘Calvo fairy’.
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Panel estimation of state-dependent adjustment
Linear dynamic panel estimation was pioneered by Anderson and Hsiao (1982),
and it was developed and perfected by Holtz-Eakin et al. (1988); Arellano and Bond
(1991); Arellano and Bover (1995) and Blundell and Bond (1998). This article shows
how classic dynamic panel estimation methodology can be adapted for the analysis of
economic adjustment if the target is unobserved and the nonlinearity takes the form of
discrete regimes. This is not straightforward, as the unknown and time-varying adjustment
coefficient interacts with the equally unknown individual specific measurement
error. However, the reward is substantial: a well-known array of estimation procedures
and tests can be brought to bear on the investigation of economic adjustment.
The estimation methods presented here are geared to short panels that do not allow
a full direct identification of individual targets. The study was motivated by the problem
of characterising the speed of capital stock adjustment as depending on financing
constraints, in an environment where categorical information on the financing situation
is available; see von Kalckreuth (2008a) and von Kalckreuth (2008b).2 The
procedures allow addressing a number of important research questions, including the
state-dependence of pricing behaviour (is there a Calvo fairy?), the adjustment of
the financial structure of companies or banks after shocks, the asymmetry of factor
adjustment (downward rigidities, firing costs), or the implications of irreversibility.
Section 2 of this article characterises the stochastic process to be estimated.
A continuous scalar and a discrete regime vector are evolving jointly, and the
adjustment speed of the continuous-type variable depends on the regime. It is
shown that the standard procedure for estimating linear dynamic panel models
is not applicable. Section 3 assumes predetermined regimes and proposes two
estimators on the basis of quasi-differencing—one of them with the virtue of
great simplicity, the other being more efficient. Both are nonlinear, which may
lead to a small sample bias if in one of the regimes the adjustment speed is
almost zero. A Generalised Methods of Moments (GMM) estimator using statedependent
filtering is suggested, which is immune to this problem. Section 4 works
out sets of moment conditions that can be applied when the regimes are contemporaneously
correlated. Using a level estimator on an amplified equation, the assumption
of predetermined regimes can be dropped at the price of stricter prerequisites regarding
the fixed effect. Under the same conditions, a version involving first differences
is feasible, too. Section 5 compares the moment conditions and discusses their use.
Section 6 tests the proposed routines in a Monte Carlo study. Section 7 concludes.
Appendix A discusses error correction models with state-dependent dynamics, and
Appendix B contains the proofs.
2 A regime-specific adjustment process
A situation where a variable yi,t reverts to some target level y∗
i,twhich is characteristic
of individual i is examined. The speed of adjustment is state-dependent, following the
equation
2 The study successfully applies the estimator QD2, as exposed in Sect. 3 of this article.
123
U. von Kalckreuth
yi,t = − 1 − αi,t−1 yi,t−1 − y∗
i,t + εi,t , (1)
with
αi,t = α
ri,t .
The L-dimensional column vector α holds the state-dependent adjustment coefficients
relevant for each state. The adjustment coefficient αi,t = α
ri,t varies over time and
individuals, depending on the state ri,t, an L-dimensional column vector of regime
indicator variables, with one element taking a value of 1, and all others being zero. The
adjustment speed at date t is given by 1 − αi,t−1 . If the process is stable, it would
eventually settle in the target in the absence of shocks. The target level y∗
i,t is unobservable.
The panel dimension can help identify the adjustment process nonetheless, as it
allows an error component approach for modelling the unobserved target. An assumption
is made of the target to follow an equation that contains an individual-specific
latent term:
y∗
i,t
= x
i,tβ + μi .
The idiosyncratic componentμi in the adjustment equationmay reflect ameasurement
error or unobserved explanatory variables. The vector xi,t may encompass random
explanatory variables, deterministic time trends and also time dummies. In its absence,
the target level is entirely unobservable, but static. A generalized, error correction version
of the adjustment equation is discussed in the Appendix A.
Solving Eq. (1) for yi,t yields:
yi,t = αi,t−1 yi,t−1 + 1 − αi,t−1 x
i,tβ + 1 − αi,t−1 μi + εi,t
la te nt
. (2)
For later purposes, it is useful to state the backward solution to this stochastic difference
equation. For t ≥ 1 and a given starting value yi,0 it is
yi,t = yi,0 − x
i,1β − μi
t−1


τ=0
αi,τ + x
i,tβ + μi + Ai,t , (3)
with
Ai,t =
t−1

l=1
εi,l − x
i,l+1β
t−1


τ=l
αi,τ + εi,t . (4)
The solution has three components. The first term captures the influence of the initial
deviation. The second term is the target level at time t, x
i,tβ+μi . The third term, Ai,t ,
represents the effect of shocks and target changes, past and present. In the long run,
when the influence of the initial conditions has died out, Ai,t is equal to the deviation
from the target.
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Panel estimation of state-dependent adjustment
In Eq. (2), both the individual effect and xi,t interact with a time-varying and endogenous
variable. This precludes the classical strategy for estimating linear dynamic
panel equations with fixed effects, namely to transform the equation by taking first
differences and use moment conditions involving higher lags of the dependent and
explanatory variables to accommodate for the fact that the transformed residual will
be correlated with the lagged endogenous variable. First differencing the Eq. (2) yields
yi,t = α

ri,t−1 yi,t−1 + (1 − α)

ri,t−1x
i,t β − α

ri,t−1μi + εi,t . (5)
Unlike the case of linear adjustment, the expression containing the unobserved μi is
not differenced out, and we have to deal with a time-varying error component that
is correlated with the explanatory variables. The following sections are devoted to
finding moment restrictions that make estimation feasible. The last set of restrictions
that will be discussed actually involves an amplified version of Eq. (5).
3 Predetermined regimes
In most applications, it will not be possible to treat ri,t as fully exogenous. If, for
example, εi,t is the error term in a capital accumulation equation and ri,t is a regime
indicating the degree of financing constraints, then the two variables should be correlated.
This section examines the case when the regime indicator, ri,t−1, can at least
be considered as predetermined with respect to the contemporaneous error term, εi,t .
Let us start by assuming the error term to be a martingale difference sequence:
E εi,t i,t−1 = 0, with
i,t−1 =
ri,t−1, ri,t−2, . . . , xi,t−1, xi,t−2, . . . , εi,t−1, εi,t−2, . . . , μi , y0i . (6)
Accommodation of the more general assumption
E εi,t

∗
i,t−k = 0, k ≥ 1, with

∗
i,t−k
=
ri,t−1, ri,t−2, . . . , xi,t−k , xi,t−k−1, . . . , εi,t−k, εi,t−k, . . . , μi , y0i , (7)
is straightforward. Note that
∗
i,t−k in assumption (7) is not simply a lagged version
of i,t−1, as the generalisation maintains the assumption of a predetermined ri,t−1.
The case of contemporaneously correlated regime indicators will be treated in Sect. 4.
3.1 Two moment conditions based on quasi-differencing
This subsection discusses two nonlinear transformations of the adjustment equation
that serve to eliminate the unobserved heterogeneity. Holtz-Eakin et al. (1988)
proposed quasi-differencing as a strategy in a case where fixed effects are subject
to time-varying shocks that arecommonacross individuals.3 It is nowexplored whether
3 See also Chamberlain (1983), pp. 1263–1264.
123
U. von Kalckreuth
this method can be generalised to themore complicated case at hand, where adjustment
coefficients are endogenous and vary over time and individuals.
Applied to the problem at hand, the quasi-differencing procedure as proposed by
Holtz-Eakin et al. (1988) would involve lagging Eq. (2), multiplying both sides by
1 − αi,t−1 / 1 − αi,t−2 and subtracting the result from Eq. (2). After reordering
coefficients, this gives
yi,t−1 − αi,t−1
1 − αi,t−2
αi,t−2 yi,t−1 − 1 − αi,t−1 x
i,tβ=εi,t − 1 − αi,t−1
1 − αi,t−2
εi,t−1.
(8)
The unobserved heterogeneity has duly been eliminated, but the error structure is difficult
to deal with, because αi,t−1 will in general be correlated with εi,t−1 and αi,t−2.
The underlying idea nonetheless leads to useful moment conditions, actually in two
different ways. First, dividing Eq. (8) by 1 − αi,t−1 gives
1
1 − αi,t−1
yi,t − αi,t−2
1 − αi,t−2
yi,t−1 − x
i,tβ = ψi,t ,
with ψi,t = εi,t
1 − αi,t−1
− εi,t−1
1 − αi,t−2
. (9)
This transformation—which shall be referred to as ‘QD1’—corresponds to solving
Eq. (1) for the deviation from the target, yi,t−1 − x
i,tβ − μi , and then solving the
lagged version of (1) for the past deviation from the target, yi,t−2 −x
i,t−1β −μi , and
finally differencing μi out. On the basis of Eq. (9), moment conditions for parameter
estimation can be formulated.
Second, we may multiply Eq. (9) by 1 − αi,t−2, to obtain
1 − αi,t−2
1 − αi,t−1
yi,t − αi,t−2 yi,t−1 − 1 − αi,t−2 x
i,tβ = ξi,t , (10)
with ξi,t = 1 − αi,t−2
1 − αi,t−1
εi,t − εi,t−1. (11)
This transformation shall be labelled ‘QD2’. It corresponds to multiplying Eq. (1) by
1 − αi,t−2 / 1 − αi,t−1 and subtracting the lag of the original adjustment equation.
Proposition 1 Under assumption (6) assuming the absence of serial correlation in
the error term, the levels yi,t−p, p ≥ 2, are instruments in Eqs. (9) and (10):
E yi,t−pψi,t = 0, (12)
E yi,t−pξi,t = 0. (13)
Proof See Appendix B.
Likewise, it can be shown that xi,t−p and the regime indicators ri,t−p, p ≥ 2,
are instruments in the Eqs. (9) and (10). If assumption (6) of no serial correlation is
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Panel estimation of state-dependent adjustment
replaced by (7), then the set of instruments is pushed backwards in time accordingly:
The lags yi,t−k−p and xi,t−k−p, p ≥ 1 are instruments in the Eqs. (9) and (10). Note
that the regime indicator ri,t−1 is still assumed to be predetermined with respect to
εi,t ; thus, all lags ri,t−p, p ≥ 2 are instruments irrespective of k.
To discuss estimation on the basis of the two sets of moment conditions, it is useful,
however, to restate the transformations (9) and (10). Equation (9) has the convenient
feature that x
i,tβ enters additively. Collecting terms, one obtains
ψi,t = yi,t−1 + 1
1 − αi,t−1
yi,t − 1
1 − αi,t−2
yi,t−1
− x
i,tβ
= yi,t−1 + γ
ri,t−1 yi,t − x
i,tβ
= yi,t−1 + γ

ri,t−1 yi,t − x
i,tβ (14)
with γ
= 1
1−α1
. . . 1
1−αL
. (15)
Equation (14) is linear in the coefficient vectors γ and β, and can be estimated by
linear GMM using the moment conditions (12) of Proposition 1. The structural coefficients
α are related to the elements of γ by the nonlinear one-to-one transformation
(15). Inverting this transformation, therefore, gives a nonlinear GMM estimator of α.
Standard deviations and co-variances can be assessed using the delta method.
Making use of QD2 for GMM estimation is trickier. Let d ri,t−2, ri,t−1 be an
L2 × 1 indicator vector, where each element is a dummy variable indicating one of
the possible combinations of ri,t−2 and ri,t−1. Let λ be the vector of coefficients
1 − αi,t−2 / 1 − αi,t−1 corresponding to the elements of d (·):
λ
= 1 1−α1
1−α2
1−α1
1−α3
· · · 1−αL
1−αL−2
1−αL
1−αL−1
1 .
Let furthermore δ be a vector of products of the adjustment coefficients and β:
δ = (1 − α) ⊗ β =
⎛
⎜⎜⎜⎝
(1 − α1) β
(1 − α2) β
...
(1 − αL ) β
⎞
⎟⎟⎟⎠
.
Finally, let
h (α, β) =
⎛
⎝
λ
−α
−δ
⎞
⎠
(16)
123
U. von Kalckreuth
be an L (L + 1 + K) × 1 vector of reduced form coefficients, of which L (L + K)
are unknown. This results in
ξi,t = λ
d ri,t−2, ri,t−1 yi,t − α
ri,t−2 yi,t−1 − δ
ri,t−2 xi,t
= d ri,t−2, ri,t−1

yi,t r
i,t−1 yi,t−1 r
i,t−1 xi,t h (α, β) . (17)
In this case, there is no convenient one-to-one transformation from the elements
of h (α, β) to the underlying structural parameters. The nonlinearity of the problem
therefore has to be treated explicitly. Consider the simplest case, with two states and
no explanatory variables xi,t . Then λ and α have two elements each and one can write
π
= h (α)
= 1 1−α1
1−α2
1−α2
1−α1
1 −α1 −α2 .
Though nonlinear in the parameters, this equation is linear in the transformed variables.
This makes it easy to apply the Gauss–Newton method for solving the optimisation
problem inherent in GMM estimation. The Gauss–Newton method iterates
on a linearised moment function, sequentially improving the estimation. Calculating
pseudo-observations for each step, the estimation problem can be solved using routines
for the estimation of linear econometric models.4 As initial values for the iteration,
one can use the results from QD1 estimation exposed earlier in this section.
The transformations QD1 and QD2 are nonlinear, and the stochastic properties of
the transformed residuals depend on the adjustment parameters. Consider the transformed
residuals ψi,t = εi,t /(1 − αi,t−1) − εi,t−1/(1 − αi,t−2) on the one hand and
ξi,t = (1 − αi,t−2)/(1 − αi,t−1)εi,t − εi,t−1 on the other. The variance of ψi,t , will
become large if one or both alpha-coefficients are in the neighbourhood of 1, creating
problems in small samples. An adjustment coefficient approaching 1 will affect
the transformed error term of QD2, ξi,t , to a lesser degree. First, only one of the two
components of the difference is affected. Second, the effect is mitigated by the denominator,
1 − αi,t−2. Indeed, if the alpha coefficients in different regimes are of similar
size, the random factor will stay in the neighbourhood of 1. Therefore, when the alpha
coefficients are high (i.e. adjustment speed is low), considerable efficiency gains can
be expected from using QD2. This will be investigated in a simulation study in Sect. 6.
3.2 Generalised Differencing
As has been exposed above, the nonlinear transformations QD1 and QD2 may lead
to poor results if in one or more of the regimes the adjustment speed is very low.
The transformations cannot be used at all if one of the regimes is characterised by an
adjustment speed of exactly zero. This is a case of considerable theoretical interest,
4 The Gauss–Newton method has originally been developed for nonlinear least squares problems. See
Davidson and MacKinnon (1993) on the use of Gauss–Newton in nonlinear least squares and instrumental
variables estimation, Hayashi (2000), on GMM estimation, and Judge et al. (1985) on numerical methods
in maximisation. An unpublished appendix on the use of Gauss–Newton in the current context is available
from the author upon request.
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Panel estimation of state-dependent adjustment
as the presence of fixed adjustment costs or irreversibility leads to bands around the
target where no adjustment takes place—the solution to the stochastic control problem
triggers adjustment when some threshold level is surpassed. Threshold behaviour
should be expected for decisions on single projects, not for firms or sectors, where
many such projects are aggregated. However, for small units it is certainly useful to
explicitly consider regimes of no adjustment, as have done Caballero et al. (1995) in
the context of plant level investment.
Therefore, it is worth asking whether there is a transformation that eliminates the
fixed effect in the target equation without affecting the size of the idiosyncratic errors.
It turns out that there is such a transformation, provided that the regime indicator has
limited memory with respect to εi,t . Consider again the first-differenced adjustment
Eq. (5) above:
yi,t = α

ri,t−1 yi,t−1 + (1 − α)

ri,t−1x
i,t β − α

ri,t−1μi + εi,t .
Whenever ri,t−1 = ri,t−2, this simplifies to
yi,t = α
ri,t−1 yi,t−1 + (1 − α)
ri,t−1 x
i,tβ + εi,t .
This expression looks very much like the first difference in the linear case, although
there is more than one adjustment coefficient to estimate. It is only taking first differences
of observations that belong to different regimes which leads to a latent term
−α

ri,t−1μi that will be correlated with the lagged dependent variable under a variety
of circumstances.
As it is this term that precludes the use of the standard technique, the following strategy
comes to mind: Differences are only formed for observations with ri,t−2 = ri,t−1.
The first element of α1 is estimated on the basis of cases where two consecutive observations
belong to the first regime, and using differences of observations that both
belong to the second regime leads to inference on the second adjustment coefficient,
etc. In this straight fashion, however, the idea will not work. If ri,t−1 and εi,t−1 are
correlated and groups of observations are formed according to regimes, then the transformed
residual εi,t will have a (conditional) expectation different from zero in those
groups. This will lead to biased estimators.
Under certain additional assumptions, however, a straightforwardmodification will
yield useful moment conditions:
1. Let q be the maximum τ for which there is a correlation between ri,t and εi,t−τ ,
e.g. as a consequence of a moving average structure of the state variable driving
the regime indicator. Then the observation is to be transformed subtracting past
observations of the same regime with a lag of at least = q + 2.
2. If an observation is not matched by a 2 + q-lag in the same regime, then it may
be transformed using a higher lag > q + 2.
The first part of the rule proposes a dynamic filter, which varies according to regimes.
The second avoids the loss of many observations in cases where regimes in t and t +q
do not match.
123
U. von Kalckreuth
The th difference is
yi,t − yi,t− = α
ri,t−1 yi,t−1 − ri,t− −1 yi,t− −1
+(1 − α)
ri,t−1x
i,t
− ri,t− −1x
i,t− β
−α
ri,t−1 − ri,t− −1 μi + εi,t − εi,t− ,
which simplifies to
yi,t − yi,t− = α
ri,t−1 yi,t−1 − yi,t− −1 + (1 − α)
ri,t−1 x
i,t
− x
i,t− β
+εi,t − εi,t− , (18)
if the two observations are characterised by the same regime, such that ri,t−1 =
ri,t− −1. When does the expectation of the residual term, εi,t − εi,t− , conditional
on ri,t−1 and the equality ri,t−1 = ri,t− −1, become zero? It is sufficient that εi,t and
εi,t− are both uncorrelated with the two conditioning variables ri,t−1 and ri,t− −1.
According to assumption (6), εi,t is uncorrelated with ri,t−1 and ri,t− −1. Then the
same is true with respect to εi,t− and ri,t− −1. Therefore, by choosing , it only
remains to make sure that εi,t− and ri,t−1 are uncorrelated. With = 1, this will
not be the case if εi,t and ri,t are contemporaneously correlated. However, if ri,t is
uncorrelated with all lags of εi,t , then = 2 will ensure that
E εi,t − εi,t− ri,t−1, ri,t−1 = ri,t− −1 = 0. (19)
More generally, if there is correlation between ri,t and εi,t−τ up to lag τ = q, the
difference that guarantees the above equation to hold will have to be at least of order
= q + 2. However, one is not restricted to using only differences of the order that
is ‘just right’, i.e. q + 2. Any other difference of order ≥ q + 2 will fulfil Eq. (19)
just as well. It is straightforward to construct a difference using the most proximate
observation of the same regime with lag ≥ q + 2. With respect to admissibility of
instruments, the rules of the classic first-difference approach apply: the instruments
need to be uncorrelated with the earlier of the two observations that make up the difference.
In the following, this procedure is called the Generalised Difference estimator.
For the moment conditions to hold, it is necessary to strengthen assumption (6). In
addition to the variables in the conditioning set i,t−1, εi,t must also be uncorrelated
with the future regimes ri,t+q+1, ri,t+q+2, . . ..
Proposition 2 Let the conditional expectation of εi,t satisfy
E εi,t i,t−1, ri,t+q+1, ri,t+q+2, . . . = 0, (20)
with i,t−1 defined as in assumption (6). Then the lagged levels yi,t− −p, p ≥ 1 are
instruments in Eq. (18), the adjustment equation transformed by taking the th difference,
with ≥ q + 2, conditional on the regimes being the same in each pair of
observations:
E εi,t − εi,t− yi,t− −p ri,t−1, ri,t−1 = ri,t− −1 = 0, with ≥ q + 2.
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Panel estimation of state-dependent adjustment
Proof See Appendix B.
Likewise, it can be shown that xi,t− −p and the regime indicators ri,t− −p are
instruments in Eq. (18), given ri,t−1 = ri,t− −1. As in Proposition 1 above, if i,t−1
in (20) is replaced by
∗
i,t−k , as defined in assumption (7), with k being the minimum
τ such that εi,t does not vary with εi,t−τ and xi,t−τ , the set of instruments is pushed
backwards in time: The lags yi,t− −k−p and xi,t− −k−p, p ≥ 1 are instruments. As
the regime indicator ri,t−1 is still assumed to be predetermined all lags ri,t−p, p ≥ 2
are instruments irrespective of k.
It is an identifying assumption for the process that drives the regime indicator to
have finite memory with respect to innovations εi,t . This is a limitation. If ri,t are correlated
with all past values of εi,t , then the conditional expectation of the transformed
error term resulting from a difference of two observations from the same regime will
not disappear. The resulting bias can be expected to wane if the minimum lag length is
chosen to be large. However, doing so would result in the loss of many observations,
exacerbating another weakness of the estimation strategy. In principle, assuming a
finite memory of the regime indicator with respect to εi,t is rather similar in kind to
the assumption of a finite memory of εi,t with respect to earlier shocks,which is needed
to use lagged endogenous variables as instruments in the standard approach. Whether
the condition (20) can be expected to hold or not will depend on the estimation problem
at hand. In the context of estimating the microeconomic adjustment of the capital
stock under financing constraints, it may be realistic to assume that, after the shock to
capital demand, the financing structure of a firm will be restored in finite time.5
3.3 Testing finite memory and deciding on the length of memory
In order to use Generalised Differencing, it is necessary to test the condition (20) and
decide on the length of the memory of the process driving the regime with respect
to εi,t . There are two simple solutions. The first is to use the test of overidentifying
restrictions associated with Sargan (1958) and Hansen (1982) to check the validity of
the moment conditions. The drawback is that this test is generally used as an omnibus
test of the specification, including the choice of the instruments. It is preferable to
have a more specific test concerning the appropriate lag length.
Such a specific test can be based on the fact that the expected value of the residual
will not disappear if the lag length chosen is too short. In that case, the choice
of observations according to regime will select positive or negative outcomes of εi,t ,
because of the correlation between the regime variable and the error component εi,t .
If regime dummies are added to the adjustment equation, then their coefficients will
be estimated as positive or negative quantities according to the direction of selectivity,
although they should be zero according to the basic specification. Furthermore,
it is known how these estimates for regime constants are distributed under the null
of a correct specification. Using a GMM estimator, they are asymptotically normal,
with mean zero, and their standard deviation is given by the standard deviation of the
coefficient. Therefore, the t-value on these coefficients is a valid test statistic.
5 For a theoretical model that makes this prediction, see von Kalckreuth (2004, 2008b, Chap. 1).
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U. von Kalckreuth
It may be argued that this test ignores the possibility that the regime-specific constants
truly belong into the equation. Consider a trend in the term in the brackets of
Eq. (1) that makes the target level of yi,t change over time:
yi,t = − 1 − αi,t−1 yi,t−1 − κt − μi + εi,t .
Solving for yi,t yields
yi,t = αi,t−1 yi,t−1 + 1 − αi,t−1 κt + 1 − αi,t−1 μi + εi,t .
After transforming the equation by subtracting an observation belonging into the same
regime, lagged periods, one obtains
yi,t − yi,t− = αi,t−1 yi,t−1 − yi,t− −1 + 1 − αi,t−1 κ + εi,t − εi,t− .
Regime-specific constants may thus be the result of a trending target variable. Actually,
this is a case of misspecification: the time trend should have figured in xi,t. The
regime constants should be proportional to each other, with a factor of proportionality
given by the adjustment speeds.6 More generally, they should not be of different sign,
as it will be the case if the coefficient on the regime dummy collects the residuals
selected for their high or low value.
4 Moment restrictions for contemporaneously correlated regimes
All moment restrictions discussed in the previous section require the regime indicator
to be predetermined with respect to the current shock term. This may hold in many
applications, specifically if there are long planning and gestation lags as in the case of
fixed investment. In other circumstances, the error term in the adjustment equation and
the threshold variable governing the adjustment regime may be contemporaneously
correlated. Let us investigate an approach that can be brought to bear in this case.
For greater clarity, the adjustment equation shall be rewritten with a modified dating,
to highlight the possibility of a contemporaneous correlation between the speed of
adjustment and εi,t :
yi,t = − 1 − αi,t yi,t−1 − x
i,tβ − μi + εi,t , (21)
or
yi,t = αi,t yi,t−1 + 1 − αi,t x
i,tβ + μi + εi,t . (22)
It will now be shown that the requirement of predetermined regimes can be dropped at
the cost of additional assumptions regarding the fixed effect. Under these assumptions,
6 Let z1 and z2 be two regime dummy coefficients, with α1 and α2 the corresponding adjustment coefficients.
If the regime dummies result from a trending target as above, then the nonlinear restriction between
coefficients is z1/z2 = (1 − α1)/(1 − α2). It is rather straightforward to test this restriction after estimation.
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Panel estimation of state-dependent adjustment
it is possible to leave the fixed effect in an equation amplified by regime dummies and
use first differences as instruments. Under the same conditions, first differences will
also serve as instruments for a modified version of the first differenced Eq. (5).
Level estimation was introduced by Arellano and Bover (1995) and Blundell and
Bond (1998) as a response to a specific problem arising in the standard autoregressive
model with fixed effects. If the coefficient of the lagged dependent variable is in the
neighbourhood of one, then the level behaves like a random walk, and it will be a
weak instrument in the differenced equation. These authors use the following moment
condition for estimation in the estimation of the standard autoregressive model:
E yi,t−p μi + εi,t = 0,
with p ≥ 1. If εi,t is serially uncorrelated, then it is sufficient that yi,t is mean
stationary and displays a constant correlation with μi for the moment equation to
hold. This implies a requirement on the initial conditions: the deviation of the starting
value from the stationary level needs to be uncorrelated with the stationary level itself.
The latent term of Eq. (22) is given by 1 − αi,t μi + εi,t. In the attempt to use
first differences as instruments for levels, let us first take a look at
E yi,t−p 1 − αi,t μi + εi,t .
This expectation will be zero if, first, E yi,t−p = 0, and second, yi,t−p is uncorrelated
with both 1 − αi,t−1 μi and εi,t . The first condition requires the process to
be mean stationary, as in the derivation of Blundell/Bond and Arellano/Bover. The
second condition is hard to fulfil. To see the reason, one may adjust the backward
solution in (3) and (4) to the modified dating:
yi,t = yi,0 − x
i,1β − μi
t


τ=1
αi,τ + x
i,tβ + μi + Ai,t ,
where
Ai,t =
t

l=2
εi,l−1 − x
i,lβ
t


τ=l
αi,τ + εi,t .
Plugging this back into (21) yields the expression:
yi,t = − 1 − αi,t

yi,0 − x
i,1β − μi
t−1


τ=1
αi,τ + Ai,t−1 − x
i,tβ

+ εi,t .
(23)
The difference yi,t−p is a function of all εi,τ, xi,τ and αi,τ , τ ≤ t − p, as well as of
the initial condition, the deviation yi,0 − x
i,1β − μi . One of the requirements for the
covariance of yi,t−p and 1 − αi,t μi to disappear is therefore a limited memory of
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U. von Kalckreuth
αi,t = α
ri,t with respect to its own past. Fixed effects in ri,t are thus excluded. This
would be hard to defend in many applications, given the presence of a fixed effect in
the law of motion governing yi,t .
In order to weaken the requirements, one may decompose the individual target
level, μi , into its expectation over all individuals, μe, and the individual deviation
from this expectation, μ
∗
i . Let, therefore,
μi = μe + μ
∗
i , with μe = Ei (μi ).
By definition, E μ
∗
i = 0. Rewriting the adjustment equation in (22) gives
yi,t = α
ri,t yi,t−1 + (1 − α)
ri,tx
i,tβ + μe (1 − α)
ri,t + μ
∗
i (1 − α)
ri,t + εi,t
laten t term
.
(24)
Written this way, the equation contains a regime-specific shift term μe (1 − α) ri,t .
In estimation, this term can be taken into account by introducing the regime vector
ri,t as a regressor into the equation.
Proposition 3 Consider the conditions
E εi,t εi,t−k, εi,t−k−1, . . . , xi,t−k ,
xi,t−k−1, . . . , ri,t−k , ri,t−k−1, . . . , yi,0 − x
i,1β − μi = 0, (25)
E μ
∗
i
εi,t ,
ri,t ,
xi,t , yi,0 − x
i,1β − μi = 0, (26)
with k ≥ 1, where a term in curly brackets denotes an entire time series. Jointly, these
conditions are sufficient for the following moment restrictions to hold in Eq. (24):
E yi,t−p εi,t + 1 − αi,t μ
∗
i = 0 with p ≥ k, (27)
Proof See Appendix B.
It follows immediately from the condition (25) that appropriately lagged values
xi,t−p and ri,t−p can also be used as instruments. Some comments are in order.
It is natural that one has to impose conditions on μ
∗
i , now that μi is not differenced out
of the error term. The invariance of expected μ
∗
i with respect to the time path
εi,t
is rather unproblematic. It agrees well with the basic structure of the error component
model. The irrelevance of the regime process is less innocuous. It is well conceivable
that a real-world data generating process for ri,t may contain a fixed effect that is
correlated with μ
∗
i . Similar reservations apply with respect to the required irrelevance
of
xi,t . Finally, the necessity of having an expected value of μi that is independent
of the initial deviation was also found by Blundell and Bond (1998) when investigating
the use of moment equations for levels in a linear context. The condition is not
innocuous either: it excludes an initial condition such as yi,0 = 0. It can be replaced
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Panel estimation of state-dependent adjustment
by the requirement that the process has been running for a ‘very long’ time, as the first
term inside the bracket of Eq. (23) will disappear asymptotically.7
As a corollary to Proposition 3, it follows that lags of yi,t can also be used as
instruments in a differenced version of the augmented adjustment Eq. (24):
yi,t = α

ri,t yi,t−1 + (1 − α)

ri,tx
i,t β − μeα

ri,t + εi,t − μ
∗
i α

ri,t
laten t term
.
Under conditions (25) and (26), the following restriction will hold8:
E yi,t−p−1 εi,t − αi,tμ
∗
i = 0 with p ≥ k. (28)
Note that the moment restrictions for differences in (28) do not use all the information
contained in the moment restriction for levels: the first are implied by the latter but not
vice versa. Furthermore, because the residuals in (28) are first differenced, one observation
is lost, and the instruments have to be removed one period in time. However,
the moment condition is not necessarily useless: estimators based on condition (28)
may be more robust against violations of assumption (26) regarding the fixed effect,
especially when regime changes are relatively infrequent, as μ
∗
i is differenced out of
(28) whenever ri,t = ri,t−1.
5 A synopsis
At this point, it is interesting to compare the conditions for Propositions 1, 2 and 3.
All of them require the expected value of εi,t to be invariant with respect to past
values εi,t−k, εi,t−k−1, . . ., the levels or first differences of xi,t−k , xi,t−k−1, . . . as well
as to μi and/or the initial deviation. Propositions 1 and 2 also need εi,t to be uncorrelated
with ri,t−1, the regime indicator figuring in the current date adjustment equation,
whereas for Proposition 3, invariance of εi,t with respect to lag k and earlier of the
regime indicator is sufficient. As an additional identifying assumption for the Generalised
Differencing approach, the memory of ri,t needs to be finite with respect to lags
of εi,t . This excludes, for example, an autoregressive process for the state variable
underlying the adjustment indicator, with the innovation contemporaneously correlated
to εi,t . The level estimator, for its part, needs the expected value of the individual
effect μi to be unrelated to the process governing the idiosyncratic error, changes
in the forcing term xi,t , the regimes and the initial deviation. Both these restrictions
may impose considerable limitations. However, estimators based on Propositions 2
and 3 are able to fulfil special tasks. The Generalised Difference estimator will be
unbiased even if some of the alpha coefficients are large—in fact, it still works if
7 Such a process may also be observed by means of a ‘short’ panel—what matters is not the length of
the panel, but whether or not the process has been running long enough to bring the effect of the initial
condition in Eq. (23) into the neighbourhood of zero.
8 This follows directly from E yi,t−p−1 εi,t + 1 − αi,t μ
∗
i = 0 and E( yi,t−p−1(εi,t +
(1 − αi,t )μ
∗
i )) = 0.
123
U. von Kalckreuth
one of them is exactly equal to 1 or even greater. Like the standard first-difference
estimator in the linear case, the Generalised Difference estimator can be supposed to
deliver imprecise results if all the adjustment coefficients are in the neighbourhood
of 1, as then the level instruments are weak. In this case, the level estimator will perform
better. Perhaps even more importantly, this latter estimator is also capable of
dealing with regime indicators that are contemporaneously correlated with the error
term.
6 Implementing and simulating the estimators
This section compares the four sets of moment conditions exposed in the Propositions
1, 2 and 3, using them separately for estimation on simulated panel data sets.
6.1 Setting up the simulation
For the regime indicator, a threshold process is specified. The kth element of ri,t is
given by
r(k)i,t = Ind ¯sk−1 ≤ si,t ≤ ¯sk .
The numbers ¯s0, . . . , ¯sL are thresholds, with the first and the last element being equal
to−∞and∞, respectively. As an example for a threshold process with infinite memory
with respect to the error term, an AR(1) is used as a process for the latent state
si,t :
si,t = asi,t−1 + υi,t ,
where the current shock υi,t is contemporaneously correlated with the error term εi,t .
Alternatively, as an example of a process with finite memory, it is assumed that the
threshold process is driven by an MA(q):
si,t = b +
q

j=0
c jηi,t−j , with c0 = 1.
The elements of the moving average conform to
E ηi,t = 0, E ηi,tηi,t−p = 0∀p > 0, E ηi,t εi,t = 0, E ηi,t εi,t−p = 0∀p > 0.
Concretely, the two interrelated processes
ri,t , yi,t are simulated as follows:
Regime-dependent error correction process: εi,t is standard normal, μi is distributed
N (1, 1) , εi,t and μi are independent.
Regime indicator process: Regarding the number of regimes, let L = 2. If the
threshold process is driven by an AR(1), then let E υ2
i,t
= 1, E υi,t εi,t = 0.8, υi,t
being calculated as a weighted sum of εi,t and an independent Gaussian process. The
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Panel estimation of state-dependent adjustment
AR-parameter a is 0.8. Likewise, for the MA(q), the stochastic structure is chosen as
E η2
i,t
= 1, E ηi,t εi,t = 0.8, with ηi,t being calculated as a weighted sum of εi,t
and an independent Gaussian process. The threshold level is set equal to zero, resulting
in an equal number of observations in each regime on average. Let us experiment
with a MA(0) (uncorrelated regimes states) and a MA(1) with c1 = 0.8. Note that the
assumed contemporaneous correlation between the shocks in the regime equation and
the error term is very high.
Panel structure: The panel is unbalanced, with individuals carrying either 8, 9 or
10 observations, 1,000 individuals of each type, that is, 3,000 individuals in total. For
each individual, the process is simulated for 50 periods, and only the last 8, 9 or 10
observations are used for estimation.
All the estimators are implemented by first calculating the transformed observations
and the instruments and then adapting and using the routines supplied with the
DPD module for Ox proposed by Doornik et al. (2002) to perform GMM estimates
and tests.9 Details on the estimation routines are given below and in the notes to the
tables.
6.1.1 Quasi-difference estimations QD1 and QD2
Let us assume an AR(1) as a process driving the threshold variable that constitutes the
regime. The estimation equations are transformed in the way described in Sect. 3. The
first quasi-differencing approach, QD1, is implemented by estimating the transformed
equation using a standard linear GMM estimator and then calculating the structural
parameters by inverting Eq. (15). The more complicated QD2 estimation is performed
by treating the moment as a nonlinear function of the structural parameters, using the
iterative Gauss–Newton method.
Estimates on the basis of the QD1 transformation are used as initial values. As
instruments, levels lagged twice are used. It turns out that the instruments are more
informative (the estimates being more precise) if they are separated out in regimes,
which means: For purposes of instrumentation, the lags of yi,t−2 are interacted with
regime dummies, ri,t−2.
6.1.2 Generalised Difference estimation
The transformation described in Proposition 2 consists in taking the th difference,
with chosen such that regimes ri,t−1 and ri,t− −1 match, subject to some minimum
order of difference. Available instruments are levels lagged + 1, + 2, . . .. As the
appropriate depends on the regime process, so does the set of instruments. By taking
the earlier of the two observations as a point of reference yi,t and assigning to it
the nearest lead yi,t+ of the same regime with ≥ 2 + q, the definition of suitable
instruments is straightforward. One can uniformly use lags yi,t−1, yi,t−2 and earlier as
instruments. As in Quasi-Difference estimation, let us interact the lagged levels yi,t−1
9 Ox is an object-oriented matrix programming language. For a complete description of Ox, see Doornik
(2001).
123
U. von Kalckreuth
with regime indicators ri,t−1. In order to test the validity of the transformation, regime
dummies are included as additional RHS variables. They also enter the instrument
set.
6.1.3 Level estimation
As described in Sect. 4, the level estimator is implemented by specifying an auxiliary
equation that contains a set of regime dummies as an additional RHS variable. Instruments
are first differences of lagged endogenous variables, interacted with regime
indicators, ri,t−1 yi,t−1 ri,t−2 yi,t−2 (four variables!) plus differenced indicators
for regime 1 taken from ri,t−1, ri,t−2. Simulations are performed both for the case
where a predetermined regime regime ri,t−1 enters the adjustment equation, and for
the case of a contemporaneously correlated regime ri,t governing the adjustment.
6.2 Simulation results
Tables 1 and 2 show estimates on the basis of quasi-difference transformations QD1
and QD2 (1,000 runs). The theoretical discussion has shown that the finite sample
properties of the estimators may depend on the size of the regime-specific coefficients,
notably on their difference from 1. Therefore, estimations for a whole range
of parameters are shown. The true value for α1 is set as 0.3, whereas the value for α2
ranges from 0.3 to 0.9. Larger ranges and finer steps are plotted in Figs. 1 and 2.
Table 1 and Fig. 1 display results for the simpler QD1 transformation. Although
for smaller coefficient values, the estimator performs well and yields correct estimates
with a good precision, it is less reliable if one of the regime-specific coefficients is
large. For α1 = α2 = 0.3, the mean bias is only of the order of −0.004 for both
parameters. It will be 0.0133 for ˆα2 when α2 is raised to 0.7, and for α2 = 0.9, the
finite sample bias of ˆα2 becomes a non-negligible −0.0414.10 The estimates ˆα1 also
deteriorate, although less markedly. The table also gives t-values and Sargan statistics.
The bias leads the t-tests reject the true value too often when one of the coefficients is
too high: In the extreme case of α2 = 0.9, the true value is rejected 77.9% of the times.
The same is true for the Sargan test of instrument validity: with large regime-specific
coefficients, it rejects the instruments 81.6% of the times when α2 = 0.9. One can
conclude that slow speeds of adjustment (high persistence) create a problem for QD1
estimation.
Table 2 and Fig. 2 give results for the QD2 transformation. As is expected, for large
values of regime specific adjustment coefficients the estimator performs better than
its counterpart based on QD1. In the extreme cases of α1 = 0.3 and α2 = 0.9, the bias
is still only 0.0152 and −0.0209, respectively. For smaller values of regime-specific
coefficients, there is hardly any bias at all. Sargan statistics and t-values are reliable,
except for very high values of α2.
10 Whether one considers the bias as large will also depend on the way one looks at the parameter. The
state-dependent speed of adjustment is given by 1 − αi,t−1. A bias of −0.0415 when the true value of α2
is 0.9 will, therefore, overestimate the adjustment speed by 41.5%.
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Panel estimation of state-dependent adjustment
Table 1 Quasi-differences, QD1 transformation, 1,000 runs
Simulation # (1) (2) (3) (4)
Specification state variable underlying regimes AR(1)
True α1 0.3 0.3 0.3 0.3
True α2 0.3 0.5 0.7 0.9
α1
Mean parameter estimate 0.2930 0.2955 0.2939 0.2687
Mean bias –0.0041 –0.0045 –0.0061 –0.0313
Mean estimated std. deviation 0.0220 0.0236 0.0276 0.0351
Std. dev. parameter estimate 0.0218 0.0247 0.0298 0.0533
RMSE 0.0222 0.0251 0.0304 0.0618
Freq. rejections of true value on 5% conf. level 4.6% 6.8% 5.9% 25.7%
α2
Mean parameter estimate 0.2957 0.4938 0.6868 0.8586
Mean bias –0.0043 –0.0062 –0.0133 –0.0414
Mean estimated std. deviation 0.0194 0.0189 0.0177 0.0139
Std. dev. parameter estimate 0.0197 0.0190 0.0188 0.0203
RMSE 0.0202 0.0200 0.0230 0.0262
Freq. rejections of true value on 5% conf. level 6.0% 5.4% 12.3% 77.9%
Freq. rejection by Sargan–Hansen on 5% conf. level 8.1% 9.4% 16.4% 81.6%
Valid obs. in estimation 21,000 21,000 21,000 21,000
Notes: the table shows GMM estimates of α1 and α2 on the basis of the transformation QD1, see Proposition
1. Columns vary by parameters α1 and α2 used for generating panels according to Eq.(2). Each column
represents 1,000 repetitions of two-stage GMM estimates using an unbalanced panel of 3,000 individuals
with 10, 9 and 8 observations (1,000 individuals each). The number of valid observations is reduced
by the need to transform variables. Instruments are the levels of ri,t−2 yi,t−2 (i.e. two interaction terms)
and a constant. Estimated standard deviations are derived from reduced form estimates using the delta
method. Sargan-Hansen test is the test of overidentifying restrictions associated with Sargan (1958) and
Hansen (1982). Estimation is executed using DPD package version 1.2 on Ox version 3.30 and additional,
user-written routines
The theoretical discussion in Sect. 3 has shown that the precision of the QD2 estimator
should depend on the ratio of adjustment speeds. If both of them are high, but
of similar size, then the ratio 1 − αi,t−2 / 1 − αi,t−1 in the definition of the transformed
error term ξi,t cancels out in Eq. (11). The error term in QD1, in contrast,
depends on the absolute distance of the regime-specific coefficients from unity. To
study this issue, the simulations of QD1 and QD2 estimation are performed using a
value of α1 = 0.8 as a platform and varying over α2.The result is shown in Figs. 3 (QD1
estimation) and 4 (QD2 estimation). Here, the QD1 estimates are biased throughout
the range. The bias of ˆα2 switches from positive to negative, whereas the bias of ˆα2 is
negative throughout. In contrast, with QD2, the bias practically disappears when both
parameters are large, to be noticeable only when α1 is small.
Table 3 and Figs. 5 and 6 give results using GMM on observations transformed by
Generalised Differences. InColumns 1 and 2, the estimator is correctly used. Thememory
of the regime process is restricted—Column (1) assumes uncorrelated regimes,
and Column (2) assumes a threshold process driven by anMA(1). The minimum leads
used in transformation are 2 and 3, respectively. In both cases, the Generalised
Difference estimator performs well. The estimates are unbiased. The standard deviations
are similar to what can be obtained from the quasi-difference estimates for the
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U. von Kalckreuth
Table 2 Quasi-differences, QD2 transformation, 1,000 runs
Simulation # (1) (2) (3) (4)
Specification state variable underlying regimes AR(1)
True α1 0.3 0.3 0.3 0.3
True α2 0.3 0.5 0.7 0.9
α1
Mean parameter estimate 0.2998 0.3006 0.3021 0.3152
Mean bias −0.0002 0.0006 0.0021 0.0152
Mean estimated std. deviation 0.0221 0.0229 0.0261 0.0418
Std. dev. parameter estimate 0.0217 0.0235 0.0270 0.0463
RMSE 0.0217 0.0235 0.0271 0.0487
Freq. rejections of true value on 5% conf. level 4.7% 5.8% 5.8% 9.5%
α2
Mean parameter estimate 0.2985 0.4982 0.6943 0.8791
Mean bias −0.0014 −0.0018 −0.0057 −0.0209
Mean estimated std. deviation 0.0195 0.0194 0.0187 0.0174
Std. dev. parameter estimate 0.0195 0.0192 0.0188 0.0170
RMSE 0.0196 0.0193 0.0197 0.0269
Freq. rejections of true value on 5% conf. level 5.9% 4.5% 5.9% 23.0%
Freq. rejection by Sargan–Hansen on 5% conf. level 5.2% 6.0% 6.0% 22.9%
Valid obs. in estimation 21,000 21,000 21,000 21,000
Notes: the table shows GMM estimates of α1 and α2 on the basis of the transformation QD2, see Proposition
1. Columns vary by parameters α1 and α2 used for generating panels according to Eq. (2). Each
column represents 1,000 repetitions of a two-stage GMM procedure iterating on pseudoregressors, using
an unbalanced panel of 3,000 individuals with 10, 9 and 8 observations (1,000 individuals each). As an
initial value, an estimate on the basis of QD1 was used. The number of valid observations is reduced by
the need to transform variables. Instruments are the levels of ri,t−2 yi,t−2 (i.e. two interaction terms) and
a constant. Estimated standard deviations are calculated as a by-product from the final Gauss–Newton iteration
step. Sargan-Hansen test is the test of overidentifying restrictions associated with Sargan (1958) and
Hansen (1982). Estimation is executed using DPD package version 1.2 on Ox version 3.30 and additional,
user-written routines
smaller of the two coefficients and actually somewhat lower for the higher coefficient.
In the case of an MA(1) regime process, standard deviations are higher, as less
observations can be used. Column (1), with a minimum lead of 2, yields an average of
15,058 valid observations per estimation. This number decreases to 11,277 in Column
(2), when a minimum lead of 3 is imposed. On the same set of simulated data, the
estimates based on quasi-differencing can use 21,000 observations each run. Figure 5
shows that the average deviation of the Generalised Difference estimator from the true
parameter value is very small when the conditions for its use are met and does not
depend systematically on the size of the adjustment coefficients. Even regime-specific
coefficients equal to or larger than 1 can be accommodated, as long as the overall
process remains stable. Columns (3) and (4) do ‘the wrong thing’. For Column (3), a
minimum lead of 2 is used on data generated with a regime process generated by an
MA(1), where a lead of ≥ 3 is warranted. Column (4) assumes an AR(1) process
driving the threshold variable: this process has infinite memory. Unsurprisingly, in
both cases, the estimator turns out to be biased. However, in spite of a strong correlation
between the shock in the regime variable and the error term, the bias is moderate.
In Column (3), only the estimates ˆα2 are biased, to a degree that is similar to the
performance of the QD2 estimator under the same (unfavourable) parameter values.
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Panel estimation of state-dependent adjustment
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-0.040
-0.035
-0.030
-0.025
-0.020
-0.015
-0.010
-0.005
Quasi-Differences 1: Bias as a function of alpha2
bias alpha1 × alpha2 bias alpha2 × alpha2
Fig. 1 Mean bias for estimates on the basis of QD1, with α1 = 0.3 and α2 varying
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.010
0.015
Quasi-Differences 2: Bias as a function of alpha2
bias alpha1 × alpha2 bias alpha2 × alpha2
Fig. 2 Mean bias for estimates on the basis of QD2, with α1 = 0.3 and α2 varying
When, as assumed in Column (4), the regime process is driven by a process with
infinite memory, the resulting bias is larger, similar in size to the weak performance
of the QD1 estimator when one of the coefficients is large. Figure 6 shows how in this
latter case the bias depends on the alpha-parameters.
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U. von Kalckreuth
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
Quasi-Differences 1: Bias as a function of alpha2
bias alpha1 × alpha2 bias alpha2 × alpha2
Fig. 3 Mean bias for estimates on the basis of QD1, with α1 = 0.8 and α2 varying
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-0.0100
-0.0075
-0.0050
-0.0025
0.0000
0.0025
0.0050
0.0075
0.0100 Quasi-Differences 2: Bias as a function of alpha2
bias alpha1 × alpha2 bias alpha2 × alpha2
Fig. 4 Mean bias for estimates on the basis of QD2, with α1 = 0.8 and α2 varying
The specification tests do not fail to detect the erroneous assumption regarding
the warranted order of differentiation. In both cases, the regime constant test rejects
the specification in 100% of the cases. As the estimated coefficients are of opposite
sign, they cannot be caused by trending target values. The regime dummies
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Panel estimation of state-dependent adjustment
Table 3 Generalised Differences estimation, (α1, α2) = (0.3, 0.8), 1,000 runs
... using appropriate leads ... using inappropriate leads
Specification state variable (1) (2) (3) (4)
underlying regimes MA(0) MA(1) MA(1) AR(1)
lead = 2 lead = 3 lead = 2 lead = 2
α1
Mean estimate (true value 0.3) 0.2990 0.2994 0.2950 0.2767
Mean est. std. dev. 0.0215 0.0286 0.0243 0.0223
Mean bias −0.0010 −0.0006 −0.0050 −0.0232
RMSE 0.0118 0.0276 0.0239 0.0322
Freq. rejections of true value 6.4% 3.8% 5.1% 18.0%
on 5% conf. level
α2
Mean estimate (true value 0.8) 0.7978 0.7995 0.7736 0.7568
Mean est. std. dev. 0.0261 0.0304 0.0298 0.0276
Mean bias −0.0021 −0.0005 −0.0264 −0.0432
RMSE 0.0113 0.0296 0.0399 0.0518
Freq. rejections of true value 3.7% 4.5% 14.2% 34.6%
on 5% conf. level
Specification tests
G1
Mean estimate −0.0001 0.0000 −0.0765 −0.0977
Mean est. std. dev. 0.0116 0.0145 0.0114 0.0102
Freq. rejections of zero value 5.8% 5.4% 100% 100%
on 5% conf. level
G2
Mean estimate −0.0007 −0.0010 0.0822 0.0977
Mean est. std. dev. 0.0114 0.0141 0.0114 0.0102
Freq. rejection of zero value 4.9% 4.7% 100% 100%
on 5% conf. level
Freq. rejection by Sargan–Hansen 5.1% 4.8% 91.9% 23.2%
on 5% conf. level
Av. no. of valid observations 15,058 11,277 14,107 15,117
Notes: the table shows GMM estimates of α1 and α2 on the basis of Generalised Differencing, see Proposition
2. Columns vary by the stochastic specification of the regime indicator when generating the panels
according to Eq. (2) and by the lead used for transformation. Columns (1), (2), and (3) specify processes
where the memory of the regime variable is limited over time, and the state variable that underlies the regime
indicator follows an MA process. In column (4), the regime process is supposed to have infinite memory.
In all columns, α1 = 0.3 and α2 = 0.8. Each column represents 1,000 repetitions of two-stage GMM
estimates using an unbalanced panel of 3,000 individuals with 10, 9 and 8 observations (1,000 individuals
each). The number of valid observations is reduced by the need to transform variables. Instruments are the
levels of ri,t−1 yi,t−1 (i.e. two interaction terms) and a constant. G1 and G2 are regime dummy coefficients
introduced as a specification test for the correct lag length, see Subsection 3.3. Sargan-Hansen test is the test
of overidentifying restrictions associated with Sargan (1958) and Hansen (1982). Estimation is executed
using DPD package version 1.2 on Ox version 3.30 and additional, user-written routines
have ‘captured’ the regime-specific non-zero expectations of the differenced residuals
E εi,t − εi,t−2 ri,t−1,ri,t−1 = ri,t−3 for the two values that ri,t−1 can take.
The Sargan test is sensitive for the misspecification in Column (3) where the wrong
lead is used, rejecting 91.9% of the estimates. Detecting an infinite memory of the
regime variable is harder for the Sargan test: only 23.2% of estimates in Column (4)
are rejected.
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U. von Kalckreuth
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
-0.0030
-0.0025
-0.0020
-0.0015
-0.0010
-0.0005
0.0000
0.0005 GD estimation: bias as a function of alpha2
bias alpha1 × alpha2 bias alpha2 × alpha2
Fig. 5 Mean bias for Generalised Differences estimates, with α1 = 0.8 and α2 varying. Here regime
process uncorrelated over time, correct lead of 2
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-0.055
-0.050
-0.045
-0.040
-0.035
-0.030
-0.025
-0.020
-0.015
GD estimation: bias as a function of alpha2
bias alpha1 × alpha2 bias alpha2 × alpha2
Fig. 6 Mean bias for Generalised Differences estimates, with α1 = 0.8 and α2 varying. Here regime
process unlimited memory AR(1), misspecified lead of 2
Table 4, together with Figs. 7 and 8, show simulation results for the level estimator,
both for the case of a predetermined regime and a contemporaneous regime.
In both cases, a regime process with infinite memory is assumed. In the table and
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Panel estimation of state-dependent adjustment
Table 4 Level estimation, 1,000 runs
Simulation # (1) (2) (3) (4)
Regime indicator Predetermined Contemporaneous
State variable underlying regimes AR(1)
True α1 0.3 0.3 0.3 0.3
True α2 0.8 1.1 0.8 1.1
α1
Mean parameter estimate 0.3031 0.3004 0.3006 0.2951
Mean bias 0.0031 0.0004 0.0006 −0.0049
Mean estimated std. deviation 0.0197 0.0074 0.0255 0.0073
Std. dev. parameter estimate 0.0187 0.0078 0.0252 0.0079
RMSE 0.0190 0.0078 0.0252 0.0094
Freq. rejections of true value on 5% conf. level 4.3% 4.2% 4.7% 10.5%
α2
Mean parameter estimate 0.7987 1.1001 0.7891 1.1004
Mean bias −0.0013 0.0001 −0.0109 0.0004
Mean estimated std. deviation 0.0188 0.0013 0.0283 0.0017
Std. dev. parameter estimate 0.0191 0.0013 0.0285 0.0017
RMSE 0.0192 0.0013 0.0305 0.0017
Freq. rejections of true value on 5% conf. level 5.7% 5.2% 7.0% 5.6%
Auxiliary regime constants
G1
Mean estimate 0.6985 0.698 0.6795 0.6922
Theoretically expected 0.7 0.7 0.7 0.7
G2
Mean estimate 0.2030 −0.0984 0.2434 −0.0852
Theoretically expected 0.2 −0.1 0.2 −0.1
Freq. rejection by Sargan–Hansen on 5% conf. level 4.1% 3.0% 5.2% 4.9%
Valid obs. in estimation 24,000 24,000 24,000 24,000
Notes: the table shows GMM estimates of α1 and α2 on the basis of level estimation (see Proposition 3).
Columns vary by parameters α2 and by the stochastic specification of the regime indicator used for generating
the panels according to Eq. (2). In all cases, the regime process is supposed to have infinite memory,
following an AR(1) process. Columns (1) and (2) relate to processes where the regime variable is predetermined
in the adjustment equation, and Columns (3) and (4) relate to results for regime variables that are
contemporaneously correlated with the error term. In all columns, α1 = 0.3. While Columns (1) and (3)
specify α2 = 0.8, columns (2) and (4) show results for α2 = 1.1. Each column represents 1,000 repetitions
of two-stage GMMestimates using an unbalanced panel of 3,000 individuals with 10, 9 and 8 observations
(1,000 individuals each). Instruments are first differences of lagged endogenous variables, interacted with
the regime indicators, ri,t−1 yi,t−1 ri,t−2 yi,t−2 (i.e. four variables) plus dummies for the first regime
from ri,t−1 and ri,t−2. G1 and G2are coefficients of regime dummies introduced into the equation to capture
the regime-specific shift term in Eq. (24). Sargan–Hansen test is the test of overidentifying restrictions
associated with Sargan (1958) and Hansen (1982). Estimation is executed using DPD package version 1.2
on Ox version 3.30 and additional, user-written routines
the figures α2 varies, with a fixed value of α1 = 0.3. In the predetermined case,
there is little bias over the whole range of parameters, with the possible exception
of α2 = 1, where the bias of ˆα1 assumes a moderate value of 0.0117 (not shown in
the table). Standard deviations are similar to those that were obtained with the other
estimators. If α2 assumes a value larger than 1, then the estimates become extremely
exact.
Columns (3) and (4), as well as Fig. 8, show that the level estimator indeed successfully
copes with contemporaneous regime variables, a problem that cannot be
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U. von Kalckreuth
solved by any of the other approaches. There is a moderate bias that peaks at 0.012
for ˆα2 when α2 = 0.9 (not shown in the table), and the standard deviations are higher
than with a predetermined regime for α2 < 1. Again, for α2 > 1, the level estimates
become very exact. In all the columns, the regime dummy is very near the theoretical
value of E 1 − αi,t μe, a term that is introduced into Eq. (24) by splitting up the firm
fixed effect into its expectation and a deviation uncorrelated with the shocks in the
other processes.
7 Conclusion and outlook
Four different ways of estimating an adjustment equation with time-varying persistence
have been presented, all within a GMM framework, albeit with a different set
of moment conditions.
Two estimation techniques rely on transforming the original equation using quasidifferences.
Both quasi-differences estimators are very precise when all the coefficients
are small. When both coefficients are large and of similar size (high persistence
throughout the regimes), the results of QD1 estimation have been shown to be unusable
in simulation, whereas the QD2 approach continues to deliver correct results. In
von Kalckreuth (2008a), the QD2 estimator is successfully employed for estimating
differential adjustment speeds for the capital stock. The most difficult parameterisation
is observed when coefficients are widely different, while one of them is large.
While affected by small sample problems, the QD2 estimator performs clearly better
in this situation. In direct comparison, the major virtue of the QD1 estimator lies in
its surprising simplicity.
The third method involves transformation using Generalised Differences, with a
lead that is long enough to overcome the memory of the εi,t -shocks in the process
driving the regime indicator. This method is applicable only when the memory of the
regime process is limited.We have seen above how to test this requirement. Although
a limited memory may be a good approximation in a number of circumstances, such
as investment under financing constraints, the requirement will not always be fulfilled.
If the conditions are met, then this method leads to a linear estimator which remains
unbiased also if some of the coefficients are in the neighbourhood of 1 or larger. The
fourth method leaves the equation untransformed, and past differences are used as
instruments. Regime dummies are employed to capture and neutralise the time-varying
non-zero expected value of the residual process. Thememory of the regime process
is irrelevant for this technique. However, one needs to assume the individual-specific
deterministic equilibrium as being unrelated to the process governing the idiosyncratic
error, changes in the forcing term xi,t , the initial deviation and the regimes. The level
estimator is very precise with regard to larger coefficients. This is not really surprising:
the use of level equations has originally been proposed to overcome the problem
of weak instruments in cases where the autoregressive parameter approaches unity.
More important is another virtue of the fourth method: the level estimator is the sole
procedure that can be used when the regime indicator is contemporaneous to the error
term in the adjustment equation.
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Panel estimation of state-dependent adjustment
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
-0.0025
0.0000
0.0025
0.0050
0.0075
0.0100
Level estimation: bias as a function of alpha2
bias alpha1 × alpha2 bias alpha2 × alpha2
Fig. 7 Mean bias for level estimation with predetermined regimes, with α1 = 0.3 and α2 varying
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
-0.012
-0.010
-0.008
-0.006
-0.004
-0.002
0.000
Level estimation: bias as a function of alpha2
bias alpha1 × alpha2 bias alpha2 × alpha2
Fig. 8 Mean bias for level estimation with contemporaneous regimes, with α1 = 0.3 and α2 varying
In dealing with a practical estimation problem, one first of all needs to decide
whether the assumption of contemporaneous regimes is warranted in the given situation.
If this is the case, then it is the quasi-differencing methods that impose the least
stringent conditions. They can be used and interpreted like first-difference estimations
in the standard model. Owing to the fact that the nonlinear transformation will affect
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U. von Kalckreuth
the latent terms in QD1 stronger than in QD2, the latter is to be preferred, although
the former may be used as a starting point for specification search and as a convenient
way to generate initial values for QD2 estimation. If adjustment speeds are low and
dissimilar, then even QD2 can lead to non-trivial small sample biases. In this case, the
Generalised Differences method may be preferable, subject to a test on the requirement
of finite memory, in spite of losing many observations.
If adjustment regimes are likely to be contemporaneously correlated, then estimation
using the level approach is possible if the fixed effect is ‘benevolent’, as explained
above. In that case, the level approach will also be a useful device if the speed of adjustment
is slow. Under the same conditions, one may also use the lagged first differences
as instruments in a differenced equation. This, however, is less efficient, as observations
are lost and not all the information in the moment restriction is used.
In deciding upon the use of the moment conditions, one may ask whether they
can be meaningfully combined in estimation. In general, this will not be the case.
If regimes are contemporaneously correlated with the idiosyncratic error term, then
the moment conditions from Propositions 1 and 2 should not be used at all. If the
regime can be considered as predetermined, then one will want to avoid imposing the
additional conditions needed for the level estimator. They are more restrictive than
in the standard autoregressive case. With predetermined regimes, the moment conditions
from Proposition 1 and 2 are to be regarded as alternatives. For higher speeds
of adjustment, they are very similar, and a possible gain from augmenting QD2 by
Generalised Differences will not be worth the risk of erroneously imposing additional
constraints, whereas, for very low and dissimilar speeds, even QD2 breaks down and
Generalised Differencing should be used on its own.
Appendix A: A state-dependent ECM
Formally, the state-dependent adjustment equation considered in this article involves a
lagged dependent variable and a forcing term xi,t . However, in addition, higher-order
adjustment processes can be accommodated, by redefining states appropriately.
Consider a linear autoregressive process with distributed lags in a forcing term xi,t
and an individual specific constant μi :
A (L) yi,t = B (L) xi,t + μi + εi,t .
where A (L) and B (L) are lag polynomials. As is well known, the process can always
be written in the error correction format. If, for example, A (L) and B (L) are of order
2, then this leads to
yi,t = −φ yi,t−1 − β
xi,t−1 − μ
∗
i + γ 0
xi,t + γ 1
xi,t−1 + ω yi,t−1 + εi,t .
In the first line, the term in brackets is the deviation from the static equilibrium, where
β may be interpreted as a cumulative long-run effect of a shock in xi,t . The transformed
constant μ
∗
i is equal to [A (L)]−1 μi. The termφ is the speed of adjustment. If
the process is stable, then |φ| < 1. The second line depicts the transitional dynamics,
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Panel estimation of state-dependent adjustment
which is not directly related to the deviation from equilibrium. With A (L) or B (L)
of order higher than 2, the transitional dynamics in the error correction format would
involve higher-order lags of differences xi,t and yi,t .
A generalisation of the adjustment process considered hitherto makes φ, γ 0, γ 1,
and ω state-dependent, while leaving the transformed constant μ
∗
i and the long-run
effect β time invariant. The latter imposes a constraint on the time-varying coefficients:
yi,t = −φi,t−1 yi,t−1 − β
xi,t−1 − μ
∗
i + γ 0
i,t−1

xi,t
+γ 1
i,t−1

xi,t−1 + ωi,t−1 yi,t−1 + εi,t .
Now let again ri,t be an indicator variable characterising the speed of adjustment. As
the adjustment process is parameterised over two lags, it is straightforward to model
the time-varying parameters as a function involving the state variables in two periods,
t −1 and t −2. Finally, let di,t−1 be an indicator vector of dummies for all the possible
values ri,t−1, ri,t−2 can take. Then we can write
φi,t−1 = ϕ
di,t−1, ωi,t−1 = ω
di,t−1, γ 0
i,t−1
=
0di,t−1, γ 1
i,t−1
=
1di,t−1,
with ϕ, ω,
0 and
1 vectors and matrices of state-dependent adjustment coefficients
remaining to be estimated. Written this way, the problem is fully equivalent to the one
that has been treated in this article, with di,t−1 taking the place of ri,t−1 with respect
to the adjustment speed, φi,t−1, and using appropriate interaction terms for all the
other state-dependent coefficients.With the help of quasi-differencing or Generalised
Differencing, one can eliminate the fixed effect from the adjustment equation. With
contemporaneous adjustment coefficients, one may use the level estimator. It has to
be noted though that—compared to a first-order adjustment process—the Generalised
Difference estimator will be difficult to use, as there are L2 states to be considered
here, and only pairs of observations belonging to the same regime with a given minimum
time distance can be used. The other two estimation principles are not affected
by this profusion of states, except for the fact that the number of coefficients is higher.
Appendix B: Proofs
Proof of proposition 1 If E εi,t
i,t−1 = 0, then any function f i,t−1 will be
orthogonal to εi,t , because
E f i,t εi,t = E i,t E f i,t−1 εi,t
i,t−1
= E i,t f i,t−1 E εi,t i,t−1 = 0. (29)
Consider first E yi,t−pψi,t , with p ≥ 2. Equation (3) and (4) show that yi,t−p is a
function of ri,t−p−1, ri,t−p−2, . . . , xi,t−p, xi,t−p−1, . . . , εi,t−p, εi,t−p−1, . . . μi , yi,0 .
The expressions 1/(1 − αi,t−1) and 1/(1 − αi,t−2) are functions of ri,t−1 and ri,t−2.
Applying (29) to the products yi,t−p/(1 − αi,t−1)εi,t and yi,t−p/(1 − αi,t−2)εi,t−1
yields E yi,t−pψi,t = 0. The same argument holds for E yi,t−pξi,t , with p ≥ 2.


123
U. von Kalckreuth
Proof of proposition 2 The proposition follows from the law of iterated expectations:
E yi,t− −p εi,t − εi,t− ri,t−1, ri,t− −1
= Eyi,t− −p E yi,t− −p εi,t − εi,t− ri,t−1, ri,t− −1, yi,t− −p
= Eyi,t− −p yi,t− −p · E εi,t − εi,t− ri,t−1, ri,t− −1, yi,t− −p = 0,
because the conditional expectation within the brackets is zero for ≥ 2 + q. The
backward solution (3) decomposes yi,t into the initial deviation, yi,0 − x
i,1β − μi ,
and the history of xi,t , ri,t and εi,t . The assumption (20) ensures that conditioning
on ri,t−1, ri,t− −1 and yi,t− −p, p ≥ 1 will preserve a zero expectation of εi,t and
εi,t− −1.

Proof of Proposition 3 The restriction (27) holds for p = k if, first,
E yi,t−kεi,t = E yi,t−k · E εi,t
yi,t−k = 0, (30)
and second,
E yi,t−k 1 − αi,t μ
∗
i = 0. (31)
Given the backward solution (23), condition (25) is sufficient for the expectation
in the bracket of (30) to be identically zero, as yi,t−k is a function of
εi,t−k, εi,t−k−1, . . . , xi,t−k, xi,t−k−1, . . . , ri,t−k , ri,t−k−1, . . . , yi,0−x
i,1β − μ .
Similarly, one has
E yi,t−k 1 − αi,t μ
∗
i = E yi,t−k 1 − αi,t · E μ
∗
i yi,t−k 1 − αi,t .
If the expectation of μ
∗
i is zero conditional on all random variables that constitute
yi,t−k according to its reduced form in (23), then the expectation in (31) vanishes.


Acknowledgments The author thanks Jörg Breitung for important discussions, encouragement and
patience. Olympia Bover made a vital comment that gave the article a new turn. Vassilis Hajivassiliou
and Sarah Rupprecht discussed earlier conference versions. Two anonymous referees made extremely helpful,
detailed and constructive comments. This article has been presented in part or fully at the 2009 Panel
Data Conference in Bonn, the 2008 Econometric Society European Meeting in Mailand, the 2007 Deutsche
Bundesbank and Banque de France Spring Conference on Microdata Analysis and Macroeconomic
Implications in Eltville, the 2007 Annual Meeting of the Verein für Socialpolitik in Munich and the 2007
CES-Ifo Conference on Survey Data in Economics—Methodology and Applications, in Munich.
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