Most of the existing structural-change models presume that the impact of a change is instantaneous and occurs at the same time for all individuals. In this paper, we develop a new structural-change model to measure the lag length between the time when an economic crisis breaks out and the time when the impact is transmitted to various economic sectors. Our model allows different transmission lags for individuals with heterogenous characteristics. Simulation results for the performance of the estimators are reported.

Guoxin Liu

Isabel Kit-Ming Yan

Terence Tai-Leung Chong

English

Research Papers in Economics

The Aggregated Structural-Change Model 
Terence Tai-Leung Chong
The Chinese University of Hong Kong
Isabel Kit-Ming Yan Guoxin Liu
City University of hong Kong York Uninversity
Abstract
Most of the existing structural-change models presume that the impact of a change is
instantaneous and occurs at the same time for all individuals. In this paper, we develop a new
structural-change model to measure the lag length between the time when an economic crisis
breaks out and the time when the impact is transmitted to various economic sectors. Our
model allows different transmission lags for individuals with heterogenous characteristics.
Simulation results for the performance of the estimators are reported.
Citation: Chong, Terence Tai-Leung, Isabel Kit-Ming Yan, and Guoxin Liu, (2007) "The Aggregated Structural-Change
Model." Economics Bulletin, Vol. 3, No. 2 pp. 1-10
Submitted: July 22, 2006.  Accepted: January 10, 2007.
URL: http://economicsbulletin.vanderbilt.edu/2007/volume3/EB-06C20066A.pdf1 Introduction
Existing structural-change models usually presume no lag or identical lag in
the diﬀusion1 of shocks to individuals with heterogeneous economic charac-
teristics. Although this type of abrupt change approach provides a parsi-
monious simpliﬁcation of individual behavior, it is rather unrealistic. For
instance, when there is a stock market crash, people with diﬀerent positions
in the stock market are likely to be aﬀected diﬀerently. Similarly, in an
oil crisis, the surge in the oil price will immediately aﬀect the sectors that
rely heavily on oil, such as the transportation and the plastic manufacturing
sectors, while its impact on other sectors will appear later.
T h eo b j e c t i v eo ft h i sp a p e ri st od e v e l o pam o d e lf o rt h i sk i n do fm u l t i p l e -
order diﬀusion process and estimate the parameters which govern the diﬀu-
sion duration. The model combines the structural-change model (Kurozumi,
2005; Chong, 2001, 2003) and the aggregation model (Theil, 1954; Granger,
1980; Forni and Lippi, 1997; Chong, 2006)2.W ea s s u m et h a tt h eb r e a kd a t e
for each individual is drawn from a distribution with unknown parameters.
By aggregating the individual abrupt-change functions, we get the aggregate
structural-change model.
The remaining of this paper is organized as follows: In Section 2, we
develop the aggregate structural-change model and derive a procedure to
estimate the pre- and post-shift parameters, as well as the parameters of the
underlying distribution that governs the lag length. Section 3 presents the
simulation results of the proposed estimation method. Section 4 concludes
the paper.
1For more details on diﬀusion, one is referred to Jovanovic and MacDonald (1994),
Lippi and Reichlin (1994) and Andolfatto and MacDonald (1998).
2A comprehensive literature review on the problem of aggregations over individuals is
provided in Stoker (1993). More recent works in the area of aggregation include Olsen
(2000) and Aoki (2002).
22 The Model
Suppose that an economic crisis occurs at time t =0in an economy with n
i n d i v i d u a l s .T h ec r i s i si sl i k e l yt ot r i g g e rad o m i n oe ﬀect, as diﬀerent people
may be aﬀected by the crisis diﬀerently. We construct the following simple
structural-break model to capture the heterogeneity in the diﬀusion of shocks
for individual i at time t:
yit =( a1 + b1xit)1{t<k i} +( a2 + b2xit)1{t > ki} + εt + ηit (1)
(i =1 ,2,...,n;t =0 ,1,2,...,T.)
where
ki ∈ {0,1,2...,T} is the lag in the diﬀusion of shocks triggered by a crisis
occurred at time 0. The duration varies across individuals.
1{·} is an indicator function which equals one if the condition inside the
bracket is true, and equals zero otherwise.
The error components εt and η1t,η 2t,...,ηnt are pair-wise independent,
with zero means and with variances var(εt)=σ2
ε and var(ηit)=σ2
η for all i.
It is also assumed that εt are serially independent.
Given the structural-break functions of all the individuals in the economy,
we can derive the aggregate structural-change model by performing a linear
aggregation. Our analysis is based on the per capita series, which has the
same time series properties as the aggregate series. Let
3yt =
1
n
n X
i=1
yit
= a1
1
n
n X
i=1
1{t<k i} + b1
1
n
n X
i=1
xit1{t<k i}
+a2
1
n
n X
i=1
1{t ≥ ki} + b2
1
n
n X
i=1
xit1{t > ki} + εt +
1
n
n X
i=1
ηit . (2)
Let
xt =
1
n
n X
i=1
xit.
We assume that ki are independent across i and that the regressor xit is
independent of ki for all i and t.
As n →∞ ,w eh a v e
1
n
n X
i=1
1{t<k i} → Pr(ki >t ),
1
n
n X
i=1
1{t > ki} → Pr(ki 6 t),
1
n
n X
i=1
xit1{t<k i} → xt Pr(ki >t ),
1
n
n X
i=1
xit1{t > ki} → xt Pr(ki 6 t),
1
n
n X
i=1
ηit → 0,
and
4yt =( a1 + b1xt)Pr(ki >t )+( a2 + b2xt)Pr(ki ≤ t)+εt. (3)
A salient feature of this model is that it is derived from the aggregation of
individual structural change functions that allow for heterogeneous individual
speciﬁc characteristics.
2.1 Estimation in Finite Sample
For ﬁnite value of T, we use a Poisson distribution3 (with mean m0) to model
the arrival time of the impact. Thus,
Pr(ki = j)=
m
j
0 exp(−m0)
j!
,j =0 ,1,2,... (4)
where 0 ≤ m0 ≤ T is assumed. Hence,
Pr(ki 6 t)=
t X
x=0
mx
0 exp(−m0)
x!
=
Γ(t,m0)
Γ(t)
, (5)
where
Γ(t,m)=
t X
x=0
mx exp(−m)
x!
Γ(t) (6)
and
Γ(t)=( t − 1)!. (7)
Thus, we have
3Aoki (2002) studies the aggregation of individual binary decision rules, he shows that
under certain conditions, one can obtain a Poisson distribution.
5yt =( a1 + b1xt)
µ
1 −
Γ(t,m)
Γ(t)
¶
+( a2 + b2xt)
Γ(t,m)
Γ(t)
+ εt. (8)
We estimate the model by using a two-step nonlinear LS estimation pro-
cedure. For any given m,w eﬁnd the LS estimators b a1T (m), b a2T (m),
b b1T (m) and b b2T (m) to minimize the total sum of squared residuals RSS (m)
deﬁned as
T X
t=0
µ
yt −
³
b a1T (m)+b b1T (m)xt
´µ
1 −
Γ(t,m)
Γ(t)
¶
−
³
b a2T (m)+b b2T (m)xt
´ Γ(t,m)
Γ(t)
¶2
.
(9)
We then search for a value of m which minimizes RSS (m). The estimator
for m is deﬁned as
b m = Arg min
m∈(0,T)
RSS (m). (10)
After obtaining the b m,t h eﬁnal structural estimators are
b aiT (b m) and b biT (b m) for i =1 ,2.( 1 1 )
2.2 Estimation in Large Sample
In order to allow for asymptotic analysis, we need to convert the time scale
from {0,1,2,3...,T} into the [0,1] interval. The Poisson distribution does
not have a continuous counterpart in the zero-one interval. Thus, we need to
employ a continuous distribution in this interval. Since the time is re-scaled,
the model should be rewritten as
yt =( a1 + b1xt)Pr(li >h )+( a2 + b2xt)Pr(li ≤ h)+εt . (12)
6where
h =
t
T
∈ [0,1];
li =
mi
T
∈ [0,1] is the diﬀusion duration of a crisis for individual i, i.e.,
the lag length in time before individual i is impacted by the crisis.
For simplicity, we employ the Beta distribution, which is in [0,1]. For the
Beta distribution, we have
Pr(li ≤ h)=
Z h
0
f(τ;α,β)dτ, (13)
where
f(τ;α,β)=
(
Γ(α+β)τα−1(1−τ)β−1
Γ(α)Γ(β) if 0 ≤ τ ≤ 1,
0 otherwise.
(14)
When h =0 , we have Pr(li ≤ 0) = 0; when h =1 , we have Pr(li ≤ 1) = 1.
For any given positive α and β, the residual sum of squares is equal to
RSS(v)=
T X
t=0
[yt − (a1 + b1xt)Pr(li >h ) − (a2 + b2xt)Pr(li ≤ h)]
2 , (15)
where
v =( a1,a 2,b 1,b 2,α,β). (16)
The nonlinear LS estimators are
b v =( b a1,b a2,b b1,b b2,b α,b β)=a r gm i n
v∈R4×R2
+
RSS(v). (17)
73 Simulations
In this section, we study the performance of our estimators for the following
mean-shift model:
yit = a11{t<k i} + a21{t > ki} + εt + ηit.
The pre- and post-shift parameters are a1 = −1 and a2 =1respectively.
All the error terms are drawn independently from a standard normal distri-
bution.
Experiment 1:
The objective of this experiment is to demonstrate the performance of the
estimators of the pre- and post-shift regression coeﬃcients and the Poisson
mean coeﬃcient m0. The sample sizes of the simulations are T =5 0 ,100
and 200. The true values of m0 are set to 0.5T and 0.3T.W ec a r r yo u tt h e
simulation for a sample with 100 individuals (n =1 0 0 )a n das a m p l ew i t h
a larger number of individuals (n =1 0 0 0 ). The number of replications is
R =2 0 0 .
The Poisson random variable with mean m0 is simulated by using the
following inverse transformation method:
(1) Generate a uniform(0,1) random variable U.
(2) i =0 , p =e x p( −m0), F = p.
(3) If U<F ,s e tX = i and stop.
(4) p =
λp
i +1
, F = F + p, i = i +1 .
(5) Go to step (3) until all the simulated observations are obtained.
Table 1 reports the average estimates across all replications. Note that
the estimates get closer to the true values as T increases. When we com-
8pare the results with diﬀerent number of individuals (n=100 and 1000),t h e
performance of the estimators are similar.
Table 1: The Mean of the Estimators in the Poisson Case
T 50 50 100 100 200 200
m0 25 15 50 30 100 60
(n =1 0 0 ,R=2 0 0 )
b a1 (true value = -1) −1.0108 −1.0260 −1.0060 −0.9943 −0.9998 −1.0090
b a2 (true value = 1) 1.0281 1.0304 1.0124 1.0209 1.0060 1.0016
b m 24.87 15.16 49.95 30.28 99.88 59.80
(n = 1000,R=2 0 0 )
b a1 (true value = -1) −1.0131 −1.0034 −1.0166 −1.0143 −1.0090 −0.9969
b a2 (true value = 1) 1.0479 1.0188 0.9973 1.0060 0.9942 0.9968
b m 24.94 15.21 50.07 29.88 99.67 60.17
Experiment 2:
This experiment evaluates the performance of the estimators in the Beta
case. The sample sizes are T =5 0 ,100 and 150. The numbers of individuals
are set to n =1 5 0 ,300 and 1000. The number of replications is set to R =
500. The parameters of Beta distribution are α and β.L e tb a1, b a2, b α and b β be
the average of the estimates from the 500 replications. There are four cases
of interest, namely, (i) α<1 and β<1, (ii) α>1 and β<1, (iii) α<1
and β>1, and (iv) α>1 and β>1. We exclude the cases where α or β
equals 1. The results are reported in Table 2.
9T a b l e2 :T h eM e a no ft h eE s t i m a t o r si nt h eB e t aC a s e
T 50 50 50 100 100 100 150 150 150
n 150 300 1000 150 300 1000 150 300 1000
Case 1: α =0 .4,β=0 .7
b a1 0.983 0.986 0.990 0.981 0.988 0.982 0.979 0.986 0.988
b a2 −1.007 −1.060 −1.011 −1.017 −1.011 −1.016 −1.014 −1.012 −1.012
b α 0.397 0.379 0.394 0.384 0.387 0.386 0.390 0.384 0.392
b β 0.685 0.696 0.679 0.680 0.693 0.689 0.688 0.691 0.683
Case 2: α =1 .3,β=0 .5
b a1 0.989 0.993 0.991 0.990 0.988 0.992 0.987 0.991 0.997
b a2 −1.015 −1.011 −1.012 −1.010 −1.014 −1.007 −1.011 −1.012 −1.002
b α 1.297 1.294 1.289 1.293 1.296 1.292 1.288 1.291 1.297
b β 0.495 0.487 0.489 0.494 0.490 0.493 0.498 0.494 0.491
Case 3: α =0 .5,β=1 .6
b a1 0.981 0.984 0.989 0.993 0.984 0.994 0.985 0.987 0.995
b a2 −1.020 −1.017 −1.012 −1.011 −1.015 −1.008 −1.011 −1.019 −1.017
b α 0.487 0.490 0.492 0.484 0.492 0.489 0.488 0.485 0.486
b β 1.587 1.589 1.592 1.584 1.587 1.586 1.595 1.592 1.597
Case 4: α =1 .2,β=2
b a1 0.989 0.986 0.987 0.983 0.995 0.982 0.987 0.994 0.992
b a2 −1.017 −1.015 −1.013 −1.003 −1.011 −1.012 −1.012 −1.012 −1.010
b α 1.189 1.192 1.190 1.192 1.194 1.192 1.189 1.190 1.189
b β 1.989 1.991 1.996 1.982 1.985 1.987 1.982 1.990 1.992
10Table 2 shows that the estimation method provides good estimates of the
pre- and post-shift regression coeﬃcients and the distribution parameters α
and β without the need of using a sample with very large number of individu-
als. Forall speciﬁed values of T,the estimators perform well for n =1 5 0 , 300
and 1000. Similarly, for all speciﬁed values of n, the estimators also perform
well for T =5 0 , 100 and 150.
4C o n c l u d i n g R e m a r k s
This paper contributes to the existing literature by developing a model that
combines the features of an aggregation model and a structural-change model.
The model is constructed by aggregating the individual abrupt-change mod-
els, with the transmission lag for each individual drawn from a distribution
with an unknown mean. A salient feature of the model is that it allows for
individual heterogeneity in the diﬀusion lag of economic crisis. The model
can be estimated via the nonlinear LS method. Simulation results suggest
that our estimators perform quite well in ﬁnite samples.
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Habit Formation: Deep and Uncertain

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