Using OLS to Estimate and Test for Structural
Changes in Models with Endogenous Regressors
Pierre Perrony
Yohei Yamamotoz
Boston University
Hitotsubashi University
August 22, 2011; Revised November 16, 2012
Abstract
We consider the problem of estimating and testing for multiple breaks in a single
equation framework with regressors that are endogenous, i.e., correlated with the errors.
We show that even in the presence of endogenous regressors, it is still preferable, in
most cases, to simply estimate the break dates and test for structural change using
the usual ordinary least-squares (OLS) framework. Except for some knife-edge cases,
it delivers estimates of the break dates with higher precision and tests with higher
power compared to those obtained using an IV method. Also, the OLS method avoids
potential weak identi…cation problems caused by weak instruments. To illustrate the
relevance of our theoretical results, we consider the stability of the New Keynesian
hybrid Phillips curve. IV-based methods only provide weak evidence of instability. On
the other hand, OLS-based ones strongly indicate a change in 1991:1 and that after
this date the model loses all explanatory power.
JEL Classi…cation Number: C22.
Keywords: structural change, instrument variables, parameter variations, Phillips
Curve.
This is a revised version of parts of a paper previously circulated under the title “Estimating and Testing
Multiple Structural Changes in Models with Endogenous Regressors”. Perron acknowledges …nancial support
for this work from the National Science Foundation under Grant SES-0649350. We are grateful to Zhongjun
Qu and two referees for useful comments.
y
Department of Economics, Boston University, 270 Bay State Rd., Boston, MA, 02215 (perron@bu.edu).
Ph: 617-353-3026, Fax: 617-353-4449.
z
Department of Economics, Hitotsubashi University, 2-1, Naka, Kunitachi, Tokyo, 186-8601, Japan
(yohei.yamamoto@econ.hit-u.ac.jp)
1
Introduction
Both the statistics and econometrics literature contain a vast amount of work on issues
related to structural changes with unknown break dates (see, Perron, 2006, for a detailed
review). For the problem of multiple structural changes, Bai and Perron (1998, 2003) provided a comprehensive treatment of various issues in the context of multiple structural change
models: consistency of estimates of the break dates, tests for structural changes, con…dence
intervals for the break dates, methods to select the number of breaks and e¢cient algorithms
to compute the estimates. Perron and Qu (2006) extended the analysis to the case where arbitrary linear restrictions are imposed on the coe¢cients of the model. In doing so, they also
considerably relaxed the assumptions used in Bai and Perron (1998). Bai, Lumsdaine and
Stock (1998) considered asymptotically valid inference for the estimate of a single break date
in multivariate time series allowing stationary or integrated regressors as well as trends with
estimation carried using a quasi maximum likelihood (QML) procedure. Also, Bai (2000)
considered the consistency, rate of convergence and limiting distribution of estimated break
dates in a segmented stationary VAR model estimated again by QML when the break can
occur in the parameters of the conditional mean, the variance of the error term or both. Qu
and Perron (2007) considered a multivariate system estimated by quasi maximum likelihood
which provides methods to estimate models with structural changes in both the regression
coe¢cients and the covariance matrix of the errors. Kejriwal and Perron (2009, 2010) provide
a comprehensive treatment of issue related to testing and inference with multiple structural
changes in a single equation cointegrated model. Zhou and Perron (2007) considered the
problems of testing jointly for changes in regression coe¢cients and variance of the errors.
More recent work considered the case of a single equation with regressors that are endogenous, i.e., correlated with the errors. Perron and Yamamoto (2012) provide a very simple
proof of the consistency and limit distributions of the estimates of the break dates by showing that using generated regressors, the projection of the regressors of the space spanned
by the instruments, to account for potential endogeneity implies that all the assumptions of
Perron and Qu (2006) (or those of Bai and Perron, 1998) obtained with original regressors
contemporaneously uncorrelated with the errors, are satis…ed. Hence, the results of Bai and
Perron (1998) carry through, though care must be exercised when the reduced form contains
breaks not common to those of the structural form. For an earlier, more elaborate though
less comprehensive treatment, see also Hall et al. (2012) and Boldea et al. (2012).
In this paper, we show that even in the presence of endogenous regressors, in general it
1
is still preferable to simply estimate the break dates and test for structural change using the
usual ordinary least-squares (OLS) framework. The idea is simple yet compelling. First,
except for a knife-edge case, changes in the true parameters of the model imply a change
in the probability limits of the OLS parameter estimates, which is equivalent in the leading
case of regressors and errors that have a homogenous distribution across segments. Second,
one can reformulate the model with those probability limits as the basic parameters in
a way that the regressors and errors are contemporaneously uncorrelated. We are then
back to the framework of Bai and Perron (1998) or Perron and Qu (2006) and we can use
their results directly to obtain the relevant limit distributions. More importantly, the OLS
framework involves the original regressors while the IV framework involves as regressors the
projection of these original regressors on the space spanned by the instruments. This implies
that the generated regressors in the IV procedure have less quadratic variation than the
original regressors. Hence, in most cases, a given change in the true parameters will cause
a larger change in the conditional mean of the dependent variable in the OLS framework
compared to the corresponding change in an IV framework. Accordingly, using OLS delivers
consistent estimates of the break fractions and tests with the usual limit distributions and
also improves on the e¢ciency of the estimates and the power of the tests in most cases.
This is shown theoretically and also via simulations. Also, the OLS method avoids potential
weak identi…cation problems inherent when using IV methods. Some care must, however, be
exercised. Upon a rejection, one should verify that the change in the probability limit of the
OLS parameter estimates is not due to a change in the bias term and carefully assess the
values of the changes in the structural parameters and the bias terms. In most applications,
there will be no change in the bias but given the possibility that it can occur one should
indeed be careful to assess the source of the rejection. This is easily done since after obtaining
the OLS-based estimates of the break dates one would estimate the structural model based
on such estimates. The relevant quantities needed to compute the change in bias across
segments are then directly available.
To illustrate the relevance of our theoretical results, we consider the stability of the New
Keynesian hybrid Phillips curve as put forward by Gali and Gertler (1999). The results show
that IV-based methods only provide weak evidence of structural instability. On the other
hand, an OLS-based sup-Wald test for one break is highly signi…cant, with 1991:1 as the
estimate of the break date (with a very tight con…dence interval (1990:4 to 1991:2)). The
higher discriminatory power of OLS-based methods over IV-based ones occurs despite the
fact that the instruments are highly correlated with future in‡ation. The estimates for the
2
period 1960:1-1991:1 are close to the full sample estimates reported earlier and support Gali
and Gertler’s (1999) conclusion. On the other hand, the estimates for the period 1991:21997:4 are all very small and insigni…cantly di¤erent from zero. Hence, the hybrid New
Keynesian Phillips curve speci…cation has lost any explanatory power for in‡ation. This
forecast breakdown of the New Keynesian Phillips curve for in‡ation is interesting and can
be traced back to the change in the behavior of in‡ation (e.g., Stock and Watson, 2007).
The structure of the paper is as follows. Section 2 introduces the model and summarizes
the relevant result about the limit distribution of the estimates of the break dates using
IV procedures. Section 3 shows how using the OLS framework is not only valid but also
preferable in that it delivers estimates of the break dates with higher precision and tests
with higher power. Section 4 substantiates our theoretical results via simulations and shows
their practical importance. Section 5 presents our empirical illustration related to the New
Keynesian hybrid Phillips curve. Section 6 provides brief concluding remarks.
2
The Model and Results about the IV Procedure
Consider a multiple linear regression model with mx breaks occurring at fT1x ; :::; Tmx x g,
y = X + u;
(1)
where y = (y1 ; :::; yT )0 is the dependent variable and X = diag(X1 ; :::; Xmx +1 ) is a T by
(mx + 1)p matrix with Xi = (xTix 1 +1 ; :::; xTix )0 for i = 1; :::; mx + 1 with the convention
that T0 = 0 and Tmx +1 = T . We denote the true break dates with a 0 superscript, i.e.,
T1x0 ; :::; Tmx0x
with the associated diagonal partition of X by X 0 and the true value of the
coe¢cient vector 0 . Note that this framework includes the case of a partial structural
change model by introducing linear restrictions R = r with R a k by (mx + 1)p matrix with
rank k and r a k dimensional vector of constant. It will be useful also to de…ne the break
x
x
fractions ( x0 ; :::; x0 ) = (T x0 =T; :::; T x0 =T ) with corresponding estimates ( ^ ; :::; ^ ) =
1
mx
1
mx
1
mx
(T^1x =T; :::; T^mx x =T ).
We allow for some regressors to be correlated with the errors and assume that there
exists a set of q variables zt that can serve as instruments, and we de…ne the T by q matrix
Z = (z1 ; :::; zT )0 . We consider a reduced form linking Z and X that itself exhibits mz changes,
so that
X = Z0
0
+ v;
(2)
0
with Z 0 = diag(Z10 ; :::; Zm
), the diagonal partition of Z at the break dates (T1z0 ; :::; Tmz0z )
z +1
and
0
= ( 01 ; :::;
0
mz +1 ).
Also, v = (v1 ; :::; vT )0 is a T by q matrix, which can be cor3
related with ut but not with zt . Given estimates (T^1z ; :::; T^mz z ) obtained in the usual way
using the method of Bai and Perron (1998, 2003), one can construct the diagonal partition Z^ = diag(Z^1 ; :::; Z^mz+1 ), a T by (mz + 1)q matrix with Z^l = (zT^z +1 ; :::; zT^z )0 for
l 1
l
^
l = 1; :::; mz + 1. Let ^ be the OLS estimate of the parameter in a regression of X on Z.
0
^ = Z^ ^ = diag(X
^ 10 ; :::; X
^m
^ l = Z^l (Z^ 0 Z^l ) 1 Z^ 0 X
e
The instruments are then X
)0 where X
l
l l with
z +1
0
el = (x ^z
X
Tl 1 +1 ; :::; xT^lz ) , so that its value in regime l is obtained using only data from that
regime. The relevant IV regression is then
y=X
+u
e;
(3)
^ 1 ; :::; X
^ mx +1 ), a T by (mx + 1)p matrix
subject to the restrictions R = r, where X = diag(X
^ j = (^
with X
xTjx 1 +1 ; :::; x^Tjx )0 for j = 1; :::; mx + 1. Also, u
e = (e
u1 ; :::; u
eT )0 with u
et = ut + t
where t = (x0t x^0t ) j for Tjx01 + 1 t Tjx0 . The estimates of the break dates are then
(T^1x ; :::; T^mx x ) = arg min SSRTR (T1 ; :::; Tmx );
(4)
T1 ;:::;Tmx
where SSRTR (T1 ; :::; Tmx ) is the sum of square residuals from the restricted OLS regression (3)
evaluated at the partition fT1 ; :::; Tmx g. We also de…ne the break fractions ( 1x0 ; :::;
x
x
(T1x0 =T; :::; Tmx0x =T ) with corresponding estimates ( ^ 1 ; :::; ^ mx ) = (T^1x =T; :::; T^mx x =T ).
x0
mx )
=
Let the union of the break dates in the structural and reduced form be denoted by
(T10 ; :::; Tm0 )
with the corresponding regime indexed by i for i = 1; :::; m. Note that m need
not equal mx + mz if some break dates are common. It will be convenient to state the set of
assumptions used, which will permit us to de…ne the notation to be used throughout. These
are the same used in Perron and Yamamoto (2012) following Perron and Qu (2006).
Assumption A1: Let wt = (x0t ; zt0 )0 . For each i = 1; :::; m + 1, let
then
Ti0 1 +[ Ti0 s]
X
1= Ti0
t=Ti0 1 +1
wt wt0 !p Qi (s)
2
4
QiXX (s)
QiXZ (s)
QiZX (s)
QiZZ (s)
0
Ti0 = (Ti+1
Ti0 ),
3
5;
a non-random positive de…nite matrix uniformly in s 2 [0; 1].
Assumption A2: There exists an n0 > 0 such that for all n > n0 ; the minimum
PTi0 +n
PTi0
0
eigenvalues of (1=n) t=T
z z 0 are bounded away from zero
0 +1 zt zt and of (1=n)
t=T 0 n t t
i
i
(i = 1; :::; m).
Assumption A3: rank[E(xt zt0 )] = p and the matrix
T for some > 0.
4
Pk1
t=k2
zt zt0 is invertible for k1
k2
Assumption A4: Let the Lr -norm of a random matrix A be de…ned by kAkr = (
P P
i
j
E jAij jr )1=r
for r
1: (Note that kAk is the usual matrix norm or the Euclidean norm of a vector.)
With fFt : t = 1; 2; ::g a sequence of increasing -…elds, we assume that fzt ut ; Ft g forms a
Lr -mixingale sequence with r = 2 + " for some " > 0. That is, there exist nonnegative constants fct : t
1g and f
j
:j
0g such that
we have: (a) kE(zt ut jFt j )kr
ct
P1 1+k
maxt ct
K < 1; (d) j=0 j
some K; M; N > 0.
j;
j
j
(b) kzt ut
# 0 as j ! 1 and for all t
E(zt ut jFt+j )kr
ct
j+1 :
1 and j
0;
Also assume (c)
< 1; (e) kzt k2r < M < 1 and kut k2r < N < 1 for
Assumption A5: fzt vt ; Ft g also satis…es Assumption A4 and supt jjzt zz0 jj = Op (log1=2 T ).
Assumption A6: Ti0 = [T
0
i]
0
1
, where 0 <
< ::: <
0
m
< 1.
Assumption A7: The minimization problem de…ned by (4) is taken over all possible
partitions such that Ti
Ti
Assumption A8: Let
T for some > 0.
1
T;i
=
0
i+1
0
i.
Assume
T;i
= vT
i,
for some
i
independent of
T where vT > 0 is a scalar satisfying either a) vT is …xed, or b) vT ! 0 and T 1=2
for some h 2 (0; 1=2).
Assumption A9: Let Ti0 = Ti0
Ti0 ! 1, uniformly in s 2 [0; 1]:
1.( Ti0 )
1
2. ( Ti0 )
3. ( Ti0 )
4. ( Ti0 )
P
Ti0 1 +[s Ti0 ]
t=Ti0 1 +1
wt wt0 !p sQi
PTi0
h
vT ! 1
Ti0 1 , for i = 1; :::; m, and wt = (x0t ; zt0 )0 . Then, as
2
s4
QiXX QiXZ
QiZX
QiZZ
3
5;
0
1 +[s Ti ]
u
e2t !p se2i ;
t=Ti0 1 +1
PTi0 1 +[s Ti0 ] PTi0 1 +[s Ti0 ]
1
(zr zt0 u
er u
et )0 !p s iZ Ue ;
r=Ti0 1 +1
t=Ti0 1 +1
PTi0 1 +[s Ti0 ]
1=2
zt u
et ) BZi Ue (s) with BZi Ue (s) a multivariate Gaussian
t=Ti0 1 +1
mean zero and covariance E[BZi Ue (s)BZi Ue (r)0 ] = minfs; rg iZ Ue :
1
process
on [0; 1] with
Perron and Yamamoto (2012) provide a short proof about the consistency, rate of convergence and limit distribution of the estimates of the break dates. These are summarized
in the following Proposition.
Proposition 1 a) Under Assumptions A1-A8: for every > 0, there exists a C < 1, such
0
for every j = 1; :::; mx . b) Let j = 1; :::; mx
that for all large T , P (jT vT2 ( ^ j
j )j > C) <
index the mx break dates of the structural form and let i denote the position of the j th break in
the structural form amongst the m total break dates in both the structural and reduced forms
5
(note that i can range from 1 to m). Also suppose that the ith overall regime corresponds to
regime l in the reduced form and the (i + 1)th overall regime corresponds to regime l in the
reduced form. Under Assumptions A1-A9, we have:
(
0 i;l
j QHH
0 i;l
j HU
e
j)
2
vT (T^jx
j
(i;l)
Tjx0 ) !d arg max VH (s);
s
(i)
(i;l)
where VH (s) = W1 ( s)
(i)
with W1
0 i;l
i QHH
jsj =2 if s 0 and
q
(i)
(i;l)
i;2
i;1
i
VH (s) =
H ( H = H )W2 (s)
i
H
jsj =2 if s > 0;
(i)
and W2
independent Wiener processes de…ned on [0; 1], (
i;2 2
i
0 (i+1);l
0 (i+1);l
0 (i+1);l
i=
i, ( H ) =
i = i QHH
i and H =
i HU
i QHH
e
i;1 2
H) =
0 i;l
i QHH
i;l
e
HU
0
i
i=
i.
They also show that all results pertaining to the problem of testing the null hypothesis
of no structural change using equation (3) remain the same as in Bai and Perron (1998),
although we do not restate these explicitly.
3
Estimating and testing using OLS estimates
In this section, we show that, in most cases, it is preferable to estimate the break dates using
the standard OLS method rather than an IV procedure even in the presence of endogenous
regressors. The reasons are very simple. Assume for simplicity that the break dates are
known and let QiXX be de…ned in Assumption A1, p limT !1 E(Xi u) = i for i = 1; :::; m + 1,
then the probability limit of the OLS estimate ^ from (1) is given by, under A9,
=
0
=
0
+ p lim (X00 X0 ) 1 X00 u =
+
0
1
+ diag(Q1XX ; :::; Qm+1
XX ) ( 1 ; :::;
T !1
1
0
[(Q1XX ) 1 1 ; :::; (Qm+1
m+1 )] :
XX )
Any change in the parameter
0
m+1 )
0
will imply a change in the limit value of the OLS estimate
,
except for a knife-edge case such that the change in the bias term (QXX ) 1 exactly o¤sets
the change in 0 . Hence, one can still identify parameter changes using OLS estimates. The
second feature is the well known inequality jjPZ Xjj
jjXjj so that using an IV procedure
leads to regressors that have less quadratic variation than when using OLS. These facts imply
that the estimates of the break dates will, in general, be less precisely estimated using an
IV procedure. The main cause for this is the fact that a change in the parameter 0 will, in
general, cause a larger change in the conditional mean of the dependent variable in the OLS
framework compared to the corresponding change in an IV regression.
6
To make the above arguments more precise, consider writing the DGP (1) as
0
y = X0
= X0 [
PX0 )u and
where u = (I
+ PX0 u + (I
0
T
PX0 )u
+ (X00 X0 ) 1 X00 u] + (I
PX0 )u = X0
= [ 0 + (X00 X0 ) 1 X00 u] for which
T
+u ;
T
!p
. So we can consider
a regression in terms of the population value of the parameters, viz., y = X0
+ u . It is
clear that in this framework X0 is uncorrelated with u so that the OLS estimate, say ^ ,
will be consistent for . This suggests estimating the break dates by minimizing the sum of
squared residuals from the following regression
y=X
+u :
(5)
Then, the estimates of the break dates are (T^1 ; :::; T^m ) = arg minT1 ;:::;Tm SSRT (T1 ; :::; Tm ),
X )0 (y
where SSRT (T1 ; :::; Tm ) = (y
X ). We then obtain directly from Perron and
Qu (2006) that, under standard conditions, the estimates of the break fractions are consistent and have the same convergence rate as in the usual OLS framework with regressors
contemporaneously uncorrelated with the errors. The limit distribution of the estimates of
the break dates is given in the following proposition.
Proposition 2 Under Assumptions A1–A4 and A6-A9 with A2-A4 and A9 stated with xt
instead of zt or wt , A4 and A9 stated with ut instead of ut and u
et (using the notations
i
instead of i ;
and BXU
0
0
instead of i = i+1
i and
i
e;
ZU
i
XU
(
0
i
i
QiXX
0
(i)
where V (i) (s) = W1 ( s)
i
XU
2
i;1
=
i
0
i
XU
i=
i)
2
vT (T^i
i
0
p
QiXX
2
i;2
=
i+1
i
s
0 and
(i)
i ( i;2 = i;1 )W2 (s)
i,
i
Ti0 ) !d arg max V (i) (s)
i
jsj =2 if s
V (i) (s) =
with
i
and BZi Ue ), with A8 stated in term of
6 0, we have, for i = 1; :::; m:
=
i,
=
i
0
i+1
XU
i
i=
jsj =2
i
0
Qi+1
XX
if s > 0;
i,
i
=
i
0
Qi+1
XX
i=
i
0
QiXX
The proof of the above result is a simple consequences of the model (5) and Perron and
Qu (2006), hence is omitted. Note that the limit distribution depends on i = i+1 i . But
since the OLS estimates are consistent for
this quantity can still be consistently estimated.
One can then compare the limit distributions of the estimates of the break dates using OLS
7
i.
or the IV procedure. For simplicity, consider the case where the instruments, regressors
and errors have a constant distribution throughout the sample so the moment matrices do
not change across regimes. Also, suppose that the errors are uncorrelated. Then, the limit
distribution of the break fractions based on the IV procedure, denoted ^ IV , is given by
0
T;i QHH
e2i
(i)
(i)
T;i
T ( ^ i;IV
(i)
0
i)
!d arg max VH (s);
s
(i)
0, VH (s) = (
where VH (s) = W1 ( s) jsj =2 if s
i;2 2
i;1 2
0 i
0
with ( H ) = i H Ue i = i QHH i , ( H ) = 0i i+1
e i=
HU
fractions based on the OLS procedure, denoted ^ OLS , is
0
T;i QXX
2
2
where
1
= p limT !1 T
T;i
T ( ^ i;OLS
0
i)
(i)
i;1
i;2
jsj =2 if s > 0
H = H )W2 (s)
0
i QHH i , and that of the break
!d arg max[W (s)
jsj=2];
s
u^ 0 u^ = p limT !1 T
1
X ^ )0 (y
(y
X ^ ). Note that, even in
i;1
this case, the limit distribution of the IV estimate will not be symmetric since i;2
H 6= H
even with homogenous distributions for the errors and regressors. This makes di¢cult a
precise comparison of the relative e¢ciency of the two estimates. Note, however, that the
component that contributes most to the variability of each estimate is the scaling factor on
the left-hand side. Hence, to gain some insights about the relative e¢ciency of the OLS
and IV estimates, we shall analyze the conditions for which QXX = 2 QHH =e2i is positive
in the scalar case with one regressor. We then have yt = xt
QHH =
2 2
z,
QXX =
= E(ut vt ) such that
2 2
z
2
QXX e2i
D
= (
2 2
z
= 2
2 2
z
QHH
2
2
v )(
+
i
e2i
+ 2v ,
2 2
v.
+
2
2
=
+2
2
v
2
that D
2
=
2
2
i
2 2 2 2
z i v
+
2 2
i v)
+
2 2
v
2 2
z
and
e2i
2 2
2
z(
2
i v
+2
2
(
2 2
z
+
2 1
v) )
2 4
i v
+
+
2 2
z
. First, when
(
2 2 2 2 2
( z
z
2 2
z
2 2 2
z i
i
+
+
i
= 0, QHH
+
0. Consider …rst the case where
2
=
=
2
+
2 1
v)
and
2
+2
i
+
2
i
+
QXX and e2i =
0 and OLS dominates IV. Also, when
8
2 1
v) :
2
v
= 0. Then
hold, hence OLS and IV are equivalent
> 0, we can, without loss of generality, normalize
D=2
with
+ ut and xt = zt + vt with
;
We seek su¢cient conditions such that D
(D = 0). When
i
2
Finally,
+2
= 0 so that QHH = QXX =
+
i
2 2
i v,
2
v
= 1 so that
2 2 2 2 2
( z
z
+ 1) 1 ;
2
2 2
2 1
so
z + v)
2 2
2
QXX =
z + v and
= 0, QHH =
2
=
2 2
z
2
2
(
e2i =
2
+
2 2
i v
2
=
2
so that OLS again dominates IV. When
i
6= 0, de…ne a =
2 2
z
and xi = = i so that D= 2i = 2axi + a + ( 2 = 2i ) + 2xi + 1 + (a=(a + 1))x2i . Using the fact
2
that 2
a su¢cient condition for D 0 is that D
0 where
D = 2axi + a + x2i + 2xi + 1 + (a=(a + 1))x2i :
After some algebra we get
a+1
(a + 1)2
(a + 1)2
a+1
[xi + a + 1]
D = x2i + 2
xi +
= xi +
2a + 1
2a + 1
2a + 1
2a + 1
Hence, D
0 when xi
(a + 1) or xi
(a + 1=(2a + 1)). Using R2 =
these conditions can be expressed as, without the normalization 2v = 1,
R
2
max 1 +
2
i v
; 1
2
i v
2 2
2 2
z =(
z
+
:
2
v)
(6)
Accordingly, OLS dominates IV except for a narrow cone centered around i 2v = = 1.
Though this case can possibly occur, it is unlikely in practice. For example, if 2v = 1, it
requires that the correlation between the errors and regressors ( ) be exactly the negative
of the value of the parameter in the …rst regime ( i ). We nevertheless consider the …nite
sample properties of the estimates in that case in Section 4.1.
In summary, it is, in most cases, preferable to estimate the break dates using the simple
OLS-based method. As we shall see below, the loss in e¢ciency can be especially pronounced
when the instruments are weak as is often the case in applications. Of course, the ultimate
goal is not to get estimates of the break dates per se but of the parameters within each
regime, one should then use an IV regression but conditioning on the estimates of the break
dates obtained using the OLS-based procedure. Their limit distributions will, as usual, be
the same as if the break dates were known since the estimates of the break fractions converge
at a fast enough rate. Using the same logic, we can expect tests for structural change to be
more powerful when based on the OLS regression rather than the IV regression. This will
be investigated through simulations in the next section.
4
Simulation evidence
In this section, we provide simulation evidence to assess the advantages of estimating the
break dates and constructing tests using an OLS regression compared to using an IV regression. Three scenarios are possible for a change between regimes i and i
9
1, say: 1)
i
6=
i 1
and
0
i
6=
0
i 1;
2)
i
6=
i 1
and
0
i
=
0
i 1;
3)
i
=
i 1
and
0
i
6=
0
i 1.
The …rst is the
most interesting since one can test and form con…dence intervals using either the OLS or IV
method. We discuss this scenario in Section 4.1 with the leading case of regressors that have
a homogenous distribution across segments and with the correlation between regressors and
errors also invariant. In Section 4.2, we consider the case for which either can be changing,
a case not covered by our theory. In Section 4.3, we consider the second scenario. The
third possibility is a knife-edge case that would require a change in the bias term to exactly
o¤set the change in the structural parameters. Even though in practice this case is a remote
possibility, we nevertheless discuss it in Section 4.4.
4.1
Homogenous distributions across segments
The data are generated by yt = xt
t
+ ut where yt , xt and ut are scalars. We consider, for
simplicity, the case of a single break in the parameter
t
=
t
occurring at mid-sample, so that
c for t 2 [1; T =2] ;
c for t 2 [T =2 + 1; T ] :
(7)
First, de…ne the following random variables: ut and vt are i:i:d: N (0;
2
u)
and i:i:d: N (0;
2
v)
with E(ut vt ) = ; also t
i:i:d: N (0; 2 ) and "t
i:i:d: N (0; 2 ), mutually uncorrelated
and also uncorrelated with ut and vt . The regressor xt is kept the same throughout the
speci…cations and is generated by xt =
+
+ vt and, again for simplicity, we consider the
p
p
"t with 0
2. Note that ut
case of single instrument zt generated by zt =
t+ 2
t
and vt are correlated so the regressors xt are correlated with the errors ut . The variable t is
a component common to xt and zt and the parameter controls the extent of the correlation
between the regressors and the instruments. The component "t is added in the generation of
zt to keep the variability of the instrument constant when varying . The data-generating
p
p
process can also be expressed as yt = xt + ut with xt = + t + vt and zt =
"t ,
t+ 2
so that the reduced form can be expressed as:
p
p
(2
)= "t );
xt =
+ (1= )zt + (vt
p
P
P
p
=
+ [(1= )
(2
)= p lim ( Tt=1 zt "t )( Tt=1 zt2 ) 1 ]zt
T !1
{z
}
|
=
+[vt
|
p
(2
+ zt + et :
)= "t +
p
(2
)= p lim (
{z T !1
et
10
PT
t=1 zt "t )(
PT
2 1
t=1 zt ) zt ];
}
The error et is then not correlated with zt by construction and it is a valid reduced form
representation. The limit of the …rst stage R2 is R2 ! V ar( zt )=V ar(xt ), where
r
1
2
2
2
Cov(zt ; "t )V ar(zt ) 1 +
Cov(zt ; "t )2 V ar(zt ) 2 V ar(zt );
V ar( zt ) =
p
and V ar(xt ) = V ar( t ) + V ar(vt ). Using Cov(zt ; "t ) =
p
2
2
; V ar(zt ) = 2 2 , V ar(vt ) =
2
v,
and V ar(xt ) = 2 + 2v , then the limit of the R2 can be simpli…ed to R2 ! 2 =[2( 2 + 2v )].
Throughout the simulations is adjusted according to the speci…ed value of R2 . Since is
bounded above by 2, the highest possible R2 is
so that the maximum R2 is 0:83. Here, we set
2
2
u
=(
2
= 1;
2
v ).
+
2
v
Speci…cally, we set
= 1;
= 1 and
2
=5
2
v
= :5. We also
2
consider a stronger endogeneity parameter = 1:0. We report results for R = :8, :5, :3 and
:1, and c = :25; :5; and 1:0. The sample size is T = 100 and the number of replications is
3000. The results are presented in Figure 1 for the cumulative distribution functions.
The OLS-based estimates are always more precise than the IV-based estimates. The
extent to which the IV-based estimates perform badly increases as R2 decreases, and even
for a value as high as 0:5, which is larger than what can be obtained in most applications, the
IV-based estimates show much higher variability than the OLS-based estimates. In the case
of weak instruments (R2 = :1), the IV-based estimates are not informative. Also, when the
magnitude of change increases, the precision of the OLS-based estimate increases noticeably.
The increase in the precision of the IV-based estimate is, however, not as substantial as c
increases. Hence, even with large breaks, the OLS-based estimates are far superior. The
results are robust when we consider stronger endogeneity with
= 1:0. The simulations
show that it is indeed highly preferable to estimate the break dates using the standard OLS
framework even when the regressors are contemporaneously correlated with the errors.
We also simulated the power of the sup-Wald test for a single break of Quandt (1958,
1960) as developed by Andrews (1993), given by sup W = sup[ T ]<Tb <[(1
)T ]
[SSRTr
SSR(Tb )]=[SSR(Tb )=(T 1)], where SSRTr is the restricted sum of squared residuals from a
regression assuming no break, and SSR(Tb ) is the sum of squared residuals when allowing
for a break at date Tb . The trimming parameter is set to
= :15 and we report results for
tests with a 5% nominal size. We also considered the UDmax test of Bai and Perron (1998),
for which there is no need to pre-specify the number of breaks. The results were qualitatively
similar and are not reported. To get the power functions, we varied c between 0 and :5. The
results, presented in Figure 2, show that the OLS-based test has the highest power. The
power of the IV-based test is noticeably inferior and decreases as the instrument gets more
weakly correlated with the regressor. The results are again robust to the stronger endogeneity
11
case. Hence, testing should also be performed using the usual OLS-based methods.
4.1.1
The case when IV dominates OLS
As argued in Section 3, there is a region of the parameter space for which IV may dominate
OLS in large samples. It is given by (6) and depicted in Figure 3 and corresponds to a
2
v=
narrow cone around
=
1. It remains to be seen whether using OLS in such cases
leads to noticeably inferior estimates in …nite samples. To that e¤ect, we report simulations
setting yt = xt i + ut , xt = zt + vt then choose parameter values i 2v = = 1 to maximize
the chances that IV is superior. Again, we consider a single break and we use the following
parameter values:
1
= 1,
= (R2 =(1
R2 ))1=2 ,
z
=
=
v
= 1 and
=
1 so that
2
1 v=
= 1 is satis…ed. The sample size is T = 100, and the parameter has one time
break at mid-sample such that 1 = 1 for t
50 and 2 = 0:5 afterwards. Again 3,000
replications are used to construct the cdf of the break date estimates for four di¤erent values
of the …rst stage R2 . These are reported in Figure 4. They show that indeed IV can
dominate OLS if the …rst stage R2 is high (R2 = 0:5). The di¤erences are, however, small.
When the R2 is smaller, OLS dominates IV and with weak instruments the IV estimate is
uninformative, as expected. Note also the pronounced skewness of the distribution of the
IV estimate. We tried di¤erent parameter values within the set de…ned by Figure 3 and
the results were qualitatively similar. In particular, slight deviations from values satisfying
2
1 lead to having OLS-based estimates be more precise than IV-based ones. This
v= =
is important for two reasons. First, it shows the relevance and adequacy of our large sample
approximation to obtain su¢cient conditions for IV to be superior to OLS. Second, it shows
how restricted, indeed unlikely, is the set of parameters for which IV dominates OLS. Hence,
these results suggest that the cost in …nite samples of using OLS when IV is superior to OLS
using large sample arguments is small, especially compared to the substantial gains of using
OLS in most of the parameter space, as shown above.
4.2
Non-homogenous segments : Scenario 1
We now consider the case for which the distribution of the regressors and the correlation
between the regressors and errors can vary across segments so that the change in the pseudo
limit values
0
i+1
0
i.
i
=
i+1
i
need not equal the change in the true population values
i
=
Given the nature of the limit distributions stated in Propositions 1-2, it is di¢cult
to provide theoretical results; hence, we resort to simulations. The data generating process
is similar to the one used in the previous section except that we allow some key parameters
12
to change across regimes. More speci…cally, we now have xt =
2
tv ),
t
+
t
+ vt , vt
i:i:d:
2
t
N (0;
i:i:d: N (0; ) and E(ut vt ) = t . We consider three cases: 1) a change in
t
QXX induced by a change in the variance of the regressor such that 2tv = 21v for t 2 [1; T =2]
2
tv
and
=
2
2v
for t 2 [T =2 + 1; T ]; 2) a change in the correlation between the regressor and
the errors such that
t
=
1
for t 2 [1; T =2] and
t
=
2
for t 2 [T =2 + 1; T ]; 3) a change
in QXX induced by a change in the mean of the regressor such that t = 1 for t 2 [1; T =2]
and t = 2 for t 2 [T =2 + 1; T ]. The other speci…cations are the same. Throughout the
simulations, we again set
2
t
= 5
2
tv
so that R2 is …xed for the whole sample. Note that
= (12=5)R2 is also …xed. Again, in all cases, the sample size is T = 100, the number of
replications is 3000 and we report results for R2 = :8, :5, :3 and :1. The results reported
are the cumulative distribution functions of the IV and OLS-based estimates. In all cases,
we consider changes such that the value of the relevant parameter is smaller or greater in
the …rst regime compared to the second, since the distributions are not symmetric for equal
changes occurring in the two segments. The change in the parameter is given by (7) with
c = 0:25. We use 2u = 1; 2v = 1; 2 = 5 2v ; = 0:5, and = 1 unless otherwise stated.
Consider …rst case (1) with a smaller variance of the regressor in the …rst regime so that
2
2v
= 1 and
2
1v
= a with a = 0:8, 0:5 and 0:25. The results are presented in Figure 5 (left
column). Here 1 is greater in absolute value than 1 , so we can expect an increase in
the e¢ciency of the OLS-based estimate of the break date. When comparing with the base
case in Figure 1, this is indeed the case and more so as a decreases. The results in Figure
5 (right column) pertain to the case of a smaller variance in the second regime so that
2
1v
1
2
2v
is smaller in absolute value than 1 . The speci…cations used are
= 1 and
= b with
b = 0:8, 0:5 and 0:25. One can see a reduction in the e¢ciency of the OLS-based estimate.
It nevertheless remains more e¢cient than the IV-based estimate and again more so as the
extent of the correlation between the regressor and instrument decreases.
Consider now a change in the correlation between the errors and the regressors across
regimes. The …rst case is one for which the correlation is smaller in the …rst regime given
by
1.
2
= 1 and
1
= a = 0:5, 0:25 and 0:1, for which
1
is smaller in absolute value than
As depicted in Figure 6 (left column), this translates into a reduction in the precision
of the OLS-based estimate of the break date. We have some instances for which the IVbased estimate is marginally more precise than the OLS-based one. This, however, requires
a very high correlation between the regressor and instrument and a very large change in the
correlation between the regressor and the errors. The results with a correlation higher in
the …rst regime are presented in Figure 6 (right column) for the speci…cations
13
1
= 1 and
2
= b = 0:5, 0:25 and 0:1. Here
1
is greater in absolute value than
1
and the results
show that in all cases the OLS-based estimate is more precise than the IV-based one.
The last case considered has the mean of the regressor changing across regimes. Figure
7 (left column) presents the results when the mean is higher in the …rst regime, speci…ed
by
= 1 and
2
1
= a = 1:5, 2:0 and 5:0. In this case,
1
is smaller in absolute value
than 1 . The OLS-based estimate nevertheless remains more e¢cient unless the change in
mean is very large (a = 5) and the correlation between the regressor and instrument is high,
even then the di¤erences are minor. When the mean of the regressor is higher in the second
regime
1
is greater in absolute value than
1.
The results in Figure 7 (right column) for
the speci…cations 1 = 1 and 2 = b = 1:5, 2:0 and 5:0 show that the OLS-based estimate is
more e¢cient than the IV-based one and more so as the change in mean increases.
Overall, the simulations presented showed that, in general, the OLS-based estimate is
more precise than the IV-based one even when the distribution of the regressors or the
correlation between the regressors and the errors change across regimes. There are cases
for which this is not true, though with small di¤erences, but they occur for unrealistically
high values of the correlation between the regressors and instruments and very large changes
in either the correlation between the regressors and errors or the means of the regressors.
Hence, for all practical purposes, it is more bene…cial to use an OLS-based method.
4.3
Scenario 2
In the second scenario
i
6=
i 1
and
0
i
=
0
i 1,
so that there are no changes in the structural
parameters but there is a change in either the marginal distribution of the regressors xt or
in the correlation between these regressors and the errors ut so that the limit value of the
OLS estimates i changes. Since there are no changes in the structural parameters, it is
uninformative to compare the relative e¢ciency of the estimate of the break dates. Hence,
we concentrate on the rejection frequencies of the tests for structural change using the IV
and OLS methods. The presumption is that the IV method would correctly identify no
change, while the OLS method would incorrectly identify a change. We use simulations to
address this issue and discuss afterwards the practical implications.
Consider …rst the case in which the correlation between the regressors and the errors
changes across segments. Note …rst that the limit distribution of the sup-Wald OLS-based
test does not have the stated limit distribution, while the limit distribution of the IV based
test is not a¤ected. Hence, we can expect size-distortions using the OLS-based method.
Though these are minor for the sup-Wald test, we nevertheless also report the rejection
14
frequencies using critical values obtained using the …xed regressors bootstrap method of
Hansen (2000). We use the same data-generating process as in Section 4.2, except that
1 and 1. The results presented in
t = 1 throughout. We set 2 = 0:5 and vary 1 between
Table 1 show the IV-tests to retain an exact size close to 5%, as expected, and the OLS-based
test can have high rejection frequency if the induced change in bias is large. This is also the
case when the bootstrap tests are used.
Consider now the case of a change in the marginal distribution of the regressors, either a
change in the variance and a change in the mean of xt . Consider …rst a change in the variance
of xt . Since the variance change in xt and that in zt (hence, in x^t ) occur, none of the IV or
OLS-based sup-Wald tests have the same limit distributions. From results in Hansen (2000)
(see also Kejriwal and Perron, 2010, in a related context) the size distortions of the sup-Wald
test are minor in the presence of a change in the mean of the regressors but substantial when
the variance of the errors change. In the IV case, the variance of the errors in the second
stage regression t = (xt x^t ) also changes with a change in the marginal distribution of the
regressors xt . Hence, we also present results using the …xed regressors bootstrap methods
of Hansen (2000) using the heteroskedastic version. To assess the …nite sample rejection
frequencies, we use the same data-generating process as in Section 4.2, except that
2
2v
t
=1
2
1v
and we set
= 1 while varying
between 10 to 0:25. The results are presented in Table
2. They show the OLS-based test to exhibit liberal size distortions when the bias change is
large. The IV-test has better size provided the heteroskedastic bootstrap method is used,
though even then some size-distortions occur with a large change.
The last case is a change in the mean of the regressors xt . Here, again none of the
IV or OLS-based sup-Wald tests have the stated limit distribution that applies when the
distribution of the regressors is homogeneous. This is so for the IV-based method since a
change in the marginal distribution of xt implies a change in the marginal distribution of
x^t . If the reduced form is stable, a change in the mean of xt must be associated with a
change in the mean of the instruments and accordingly in the mean of the …tted values
x^t . If the reduced form is unstable but the breaks are properly accounted for, the mean of
x^t varies by construction. Again, in the IV case, the variance of the errors in the second
stage regression change. Hence, we also present results using the …xed regressors bootstrap
methods of Hansen (2000) using the heteroskedastic version. Given that the reduced form is
unstable, for the IV-based tests we proceed as follows. We …rst estimate the break date in
the reduced form and test whether the change is signi…cant using a sup-Wald test. If it is, the
break is accounted for when estimating the structural model using the full sample method of
15
Perron and Yamamoto (2012). Table 3 reports the rejection frequencies of the tests using the
same DGP as in section 4.2, except again that t = 1 throughout and 2 = 1 and 1 varying
between 0:1 and 10 (to obtain a range for the bias similar to the previous experiments).
Here, things are quite di¤erent. As expected, the OLS-based test does have high power
if the change in mean is large. The IV-based test using the standard asymptotic critical
values has the correct size if the instrument is very strongly correlated with the regressor
and the change is small. As the instrument gets more weakly correlated, the size of the test
increases rapidly. The rejection frequencies of the IV-based test exceed those of the OLSbased test when the R2 of the …rst stage regression is small. Intuitively, as the instruments
get weaker, x^t gets closer to a white noise process so that the change in x^t cannot fully
account for the change in the mean of yt induced by the change in xt without a concurrent
change in either the regression coe¢cient, the intercept or both, thereby inducing a change
in the parameters. With tests constructed using the heteroskedastic bootstrap, the size
distortions are considerably reduced for the IV-based test, but can remain important when
the correlation between the regressor and the errors is weak. As expected, the OLS-based
tests have high power even when the heteroskedastic bootstrap version is used.
4.4
Scenario 3
The third scenario pertains to a knife-edge case of 0i 6= 0i 1 but i = i 1 so that the change
in the structural parameters is exactly o¤set by the bias change induced by a change in the
moments of the regressors or in the correlation between the regressors and the errors. This
is arguably a case of lesser practical relevance but we still include a discussion about it for
completeness. We again consider three cases; 1) a change in the variance of the regressor,
2) a change in the correlation between the regressor and the errors, and 3) a change in the
mean of the regressor. We use
2
u
= 1,
2
=5
2
v,
= 1,
2
v
= 1 and
= 1 unless otherwise
stated. The changes occur at T =2. We present the distributions of the break date estimates
at i = i 1 (but 0i 6= 0i 1 ) in the left column of Figure 8. We also provide power functions
of the sup-Wald test with a 5% nominal size using the asymptotic critical values with c
ranging from 0 to 1 in the right column of Figure 8.
For the …rst case, we set the change from
2
1v
= 10 to
2
2v
= 1 (the largest change
considered in Section 4.3). However, even with such a large variance change, the e¤ective
break in the structural parameter is small (c = 0:063). Figure 8 (the top panel in the left
column) shows that the OLS-based estimate is, as expected, not informative. However, the
cumulative distribution functions of the IV-based estimates are also quite ‡at. What is
16
more instructive is to look at the relative performance of both methods in a neighborhood
of the knife-edge case. To that e¤ect, we computed the power functions for this case with c
ranging from 0 to 1. The results are in the top right panel of Figure 8. Both the OLS and
IV-based tests have important size distortions at c = 0, consistent with the results presented
in Section 4.3 (though these could be reduced using the bootstrap method). The power of
the OLS-based test decreases up to c = 0:063 (when i = i 1 ), but it increases rapidly
afterwards. The OLS method can dominate the IV-based one when the instrument is weak.
We next consider the second case with
1
=
1 and
2
= 1. This represents an (unreal-
istically) large change that requires the correlations to vary from perfect negative correlation
to perfect positive correlation at mid-sample. Such a large change is needed to have at least
a modest bias change with magnitude 0:286. The distribution of the OLS estimate of the
break date with c = 0:143 (corresponding to
i
=
i 1)
shows that it is uninformative. The
IV estimates are more precise if the correlations between the regressors and the instruments
are strong, however, it becomes less informative as the instruments get weaker. Looking at
the neighborhood around the knife-edge case, the power function of the OLS-based test has
again a U-shape around c = 0:143 but the power quickly increases as c increases.
For the third case, we again need to consider a large change from
1
= 5 to
2
= 1 along
with a maximum correlation between the regressors and the errors. These yield a bias change
of 0:133 so that the knife-edge case occurs at c = 0:067. The results presented in the bottom
left panel of Figure 8 show that the OLS method is, as expected, uninformative but also
that the IV-based estimate is barely superior irrespective of the strength of the instruments.
Looking at the power function around the knife-edge case, the power of the OLS-based test
dominates the IV-based test regardless of the strength of the instruments (power is above
size when c = 0:067 because of distortions arising from the very large mean shift).
4.5
Practical implications
Our analyses showed the OLS-based procedure to outweigh, by possible large margins, the
IV-based procedure when a change in the structural form occurs (more precise estimate
and tests with higher power). There are exceptions but they all apply to so-called knifeedge cases that arguably are not likely to occur in practice, and even in such cases IV is
only marginally better than OLS. We also discussed how OLS-based methods can spuriously
detect a change in the structural form when none occurs due to possible changes in the
marginal distributions of the regressors or the correlation between the errors and regressors.
These are of clear relevance in practical applications. Should the fact that the OLS-based
17
tests reject the null hypothesis of no change when
i
6=
i 1
and
0
i
=
0
i 1
be of a concern
and a reason to abandon it in favor of the IV-based test? The answer is clearly no but with
the caveat that care must be exercised in practice. Upon a rejection, one should verify that
it is not due to a change in the bias term and carefully assess the values of
terms (QiXX )
1
i.
0
i
and the bias
In most applications, there will be no change in the bias but given the
possibility that it can occur one should indeed be careful to assess the source of the rejection.
The price of discarding the OLS-based method in favor of the IV-based one is too high.
First, in most cases the power of the IV-based method is lower and much more so as the
instruments gets weaker, a common feature in applications. Second, as illustrated by the
case with a change in the mean of the regressors the IV-based test can su¤er from serious
size distortions, which are, however, reduced to a great extent using Hansen’s (2000) heteroskedastic bootstrap test. Since these are …nite sample problems, one would not be inclined
to question the source of the rejection and would reach the wrong conclusion. The fact that
we could have i 6= i 1 and 0i = 0i 1 does not invalidate the OLS-based method, it simply calls for a more careful analysis. This is easily done since after obtaining the estimates
of the break dates one would estimate the structural model based on such estimates. The
relevant quantities needed to compute the change in bias across segments are then directly
available. Given that the residuals from the IV regression are given by u
et = ut + (x0t x^0t ) j
for Ti0 1 + 1 t Ti0 , an estimate of the change in bias from regime (i 1) to regime i, can
be obtained as
T^i
P
(
t=T^i
(
xt x0t )
1 +1
T^P
i 1
t=T^i
T^i
P
1
t=T^i
xt x0t ) 1
2 +1
1 +1
T^P
i 1
t=T^i
b
(xt u
et
2 +1
x^0t )^i )
xt (x0t
b
(xt u
et
xt (x0t
(8)
x^0t )^i 1 )
b
where T^i are the estimates of the break dates obtained using the OLS-based method, u
et and
^i are the estimated residuals and coe¢cient estimates, respectively, from the estimated IV
regression.
5
Empirical example
To illustrate the relevance of our results, we consider the stability of the New Keynesian
hybrid Phillips curve as put forward by Gali and Gertler (1999). The basic speci…cation is
t
=
+
t 1
+ x t + Et
18
t+1
+ ut
(9)
where
t
is the in‡ation rate and Et is the expectation operator conditional on information
available at time t. The variable xt is a real determinant of in‡ation usually taken as a
measure of real economic activity such as the output gap, though Gali and Gertler (1999)
argue that using a measure of real marginal cost is preferable. The key ingredient is the
component Et
t+1
which makes the Phillips-curve forward looking. The one-period lag of
in‡ation is usually introduced on the ground that it improves the …t of the model but can be
rationalized by supposing that a proportion of …rms use backward-looking rules to set prices.
In this speci…cation, one expects
than
and
to be positive and that
so that the forward looking component dominates. Also,
is substantially larger
is expected to be positive
so that an increase in real activity is associated with an increase in in‡ation. As argued by
Gali and Gertler (1999), the estimate of is usually negative when using the output gap but
is positive, though quite small, when using real marginal cost.
We follow the approach taken by Gali and Gertler (1999) and use basically the same
speci…cations and data. The data was obtained from Andre Kurmann’s website (and used
in Kurmann, 2007). It is for the U.S.A. and quarterly for the period 1960:1-1997:4 as in Gali
and Gertler (1999). The in‡ation rate
t
is the quarterly change in the GDP de‡ator and xt
is either 1) labor income share as a proxy of real unit labor cost (nonfarm business unit labor
cost de‡ated by the GDP de‡ator), or 2) detrended real GDP assuming a quadratic linear
trend as a measure of the output gap. The only di¤erence is that we use as instruments
only one lag of the following variables: in‡ation, labor income share, the output gap, the
interest rate spread (10 years US treasury bill rate minus the 3 years bill rate), wage in‡ation
(quarterly change in the nonfarm business nominal wage rate) and the commodity price
in‡ation (quarterly change in the spot market price index). Using four lags as in Gali and
Gertler (1999) would imply too many instruments so that testing for changes in the reduced
form, needed for the IV-based methods, becomes impossible with the tests having no power.
Gali and Gertler (1999) estimated the model using a non-linear IV procedure given that
their model implies restrictions across the coe¢cients which are functions of some basic
parameters. We use a linear IV method of estimation. The results are, however, in line with
those of Gali and Gertler (1999) for the full sample speci…cation and doing so does not a¤ect
our conclusions. The parameter estimates for the full sample are presented in Table 4 for
both cases in which xt is speci…ed as the real marginal labor cost or as the output gap. The
results are in close agreement with those of Gali and Gertler (1999). The coe¢cient
on
future expected in‡ation is large and signi…cant and about twice as large as the coe¢cient
on lagged in‡ation, indicating that the forward-looking behavior is dominant. The estimate
19
of
is positive when using the real marginal cost of labor and negative when using the
output gap, though the values are small and insigni…cant. Note also that the instruments
are quite strongly correlated with future in‡ation, the R2 of the …rst-stage regression being
0:73. The results are qualitatively similar using four lags of each variables as instruments.
In order to proceed with the IV-based methods, we need to consider the stability of the
reduced form. This is not needed for the OLS-based method as long as tests are constructed
using the …xed regressors bootstrap method of Hansen (2000). The results presented in Table
5 points to the presence of 1 or 2 breaks. Applying the Bai and Perron (1998) sequential
method, the test for one break is signi…cant at the 1% level using the asymptotic critical
values and at the 10% level using the critical values from Hansen’s (2000) bootstrap method.
The test for 2 versus 1 break is again signi…cant at the 1% level using the asymptotic critical
values but not using the boostrap method, a di¤erence possibly due to the large number of
regressors. In the following, we consider both cases with 1 and 2 breaks.
We now turn to the issue of the stability of this hybrid New-Keynesian Phillips curve.
Consider …rst the OLS-based method. When using the labor income share, the test for one
break is signi…cant at the 1% level using the asymptotic critical values and at the 10% level
using the bootstrap ones. The estimate of the break date is at 1991:1 with a very tight
con…dence interval (1990:3 to 1991:3). There is no evidence for the presence of an additional
break. When using the output gap, the test for one break is again signi…cant at the 1%
level using the asymptotic critical values but not signi…cant using the bootstrap ones. The
estimate of the break date is again 1991:1 with an even tighter con…dence interval (1990:4
to 1991:2). There is again no evidence for the presence of a second break.
We now consider the results obtained using IV-based methods. We start with the subsample approach advocated by Hall et al. (2012), which looks at evidence for changes in the
structural form using the sub-samples induced by the estimates of the break dates in the
reduced form. The results in Table 6 show that one cannot …nd any statistically signi…cant
evidence for the presence of a break. This is true whether one considers a single or two
breaks in the reduced form. Things are somewhat di¤erent when using the full-sample
method advocated by Perron and Yamamoto (2012). Consider …rst the case with one break
in the reduced form at date 1980:4. The results are the same using either the labor income
share or the output gap. The test for one break is signi…cant at the 5% level using the
asymptotic critical values but not when using the bootstrap ones. There is no evidence for
the presence of a second break and the estimate of the break date is 1967:2, very di¤erent
from that obtained using the OLS-based method. Overall, the evidence for a break in the
20
structural form is weaker. Consider now the case with two breaks in the reduced form at
dates 1973:1 and 1980:4. When using the labor income share the results are similar to the
case of a single break in the reduced form. The test for one break is barely signi…cant at
the 10% level using the asymptotic critical values and not using the bootstrap ones. The
estimate of the break date is again 1967:2. Things are di¤erent when using the output gap
with results identical to those obtained using the OLS-based method. The test for one break
is signi…cant at the 1% level using the asymptotic critical values but not signi…cant using
the bootstrap ones. The estimate of the break date is again 1991:1 with a tight con…dence
interval (1990:3 to 1991:3). There is again no evidence for the presence of a second break.
It remains to be seen whether this change in 1991:1 is economically signi…cant. To that
e¤ect we estimated the parameters of the model by IV using the same instruments allowing
for a change in all parameters. The estimates are presented in Table 7. They are indeed very
interesting. The estimates for the period 1960:1-1991:1 are close to the full sample estimates
reported earlier and support Gali and Gertler’s (1999) conclusion. On the other hand, the
estimates for the period 1991:2-1997:4 are all very small and insigni…cantly di¤erent from
zero in both cases. Hence, both of these hybrid New Keynesian Phillips curve speci…cations
have lost any explanatory power for in‡ation. Again, the results are qualitatively similar
using four lags of each variables in the instrument list.
As discussed above, the full-sample IV-based method pointed to a break in 1967:2 when
considering one break in the reduced form with either the labor income share or the output
gap and also when considering two breaks in the reduced form when using the labor income
share. To see which of 1967:2 or 1991:1 is the more appropriate break date, we present in
Table 8 the parameter estimates of the structural form with a break in 1967:2. The estimates
show no signi…cant di¤erences between the samples 1960:1-1967:2 and 1967:3-1997:4. The
parameter
shows a slight increase that is not signi…cant, while
shows an increase (from
negative to positive) when using the labor income share. With one break in the reduced
form, however, it shows an increase when using the output gap. So the results are sensitive
to the exact speci…cation and also not signi…cant as the standard errors are large in both subsamples. The changes when considering a break in 1991:1 are economically more important
and statistically signi…cant than those obtained with a break in 1967:2. This reinforces the
conclusion that the OLS-based method provides more reliable results.
Overall, the results point to the following conclusions. With the OLS-based method, the
evidence for a break in the structural form is strong and points to the same break date,
1991:1, whether using the labor income share or the output gap. When using the IV-based
21
methods, things are not as precise. First, the sub-sample approach yields no evidence for
a break in the structural equation. The more powerful full-sample approach suggested by
Perron and Yamamoto (2012) can deliver strong results in line with the OLS-based ones,
as obtained when considering the output gap. But it can also be less powerful as in the
case for which the labor income share is used. Note …nally that the higher discriminatory
power of OLS-based methods over IV-based ones occurs despite the fact that the instruments
are highly correlated with future in‡ation. The OLS-based method is also much easier to
implement without the need to worry about changes in the reduced form.
Finally, we need to assess that the rejection is not due to change in the bias term. To that
e¤ect we estimated the change in the bias from (8) with a break in the srtuctural equation in
1991:1. For the case with one break in the reduced form in 1980:4, the estimated value was
0:0388 with the labor income share and 0:0513 with the output gap. For the case with two
breaks in the reduced form in 1973:1 and 1980:4, the estimated value was 0:0107 with the
labor income share and 0:0212 with the output gap. These compare to a change in coe¢cient
of 1:011 with the labor income share and 1:052 with the output gap. Hence, the change in
the bias is negligible compared to the change in the parameters so that a rejection using the
OLS-based sup-Wald test genuinely re‡ects a change in the structural parameters.
This forecast breakdown of the New Keynesian Phillips curve for in‡ation is interesting
and can be traced back to the change in the behavior of in‡ation. As argued by Stock and
Watson (2007) even though in‡ation has become more stable after the mid-80’s it also has
become more di¢cult to forecast (see also Rossi and Sekhposyan, 2010). Our evidence is
consistent with theirs, though stronger perhaps because of the di¤erent break date identi…ed.
6
Conclusions
In this paper, we considered the problem of multiple structural changes in a single equation
framework with regressors that are endogenous, i.e., correlated with the errors. We showed
that even in the presence of endogenous regressors, it is still preferable to simply estimate
the break dates and test for structural change using the usual ordinary least-squares framework. The reasons are simple. First, changes in the true parameters of the model imply a
corresponding change in the probability limits of the OLS parameter estimates, except for
a possible knife-edge case. Second, one can reformulate the model with those probability
limits as the basic parameters in a way that the regressors and errors are contemporaneously
uncorrelated. We are then simply back to the framework of Bai and Perron (1998) or Perron
and Qu (2006) and we can use their results directly to obtain the relevant limit distributions.
22
Since the OLS framework involves the original regressors, while the IV framework involves
as regressors the projection of these original regressors on the space spanned by the instruments, this implies that the generated regressors in the IV procedure have less quadratic
variation than the original regressors. Accordingly, using OLS not only delivers consistent
estimates of the break fractions and tests with the usual limit distributions, it also improves
on the e¢ciency of the estimates and the power of the tests in the vast majority of cases.
Lastly, it is important to note that the OLS procedure avoid potential weak identi…cation
problems when estimating and testing for structural changes.
23
References
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change point. Econometrica 61: 821-856.
Bai J. 2000. Vector autoregressive models with structural changes in regression coe¢cients
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Bai J, Lumsdaine RL, Stock JH. 1998. Testing for and dating breaks in multivariate time
series. Review of Economic Studies 65: 395-432.
Bai J, Perron P. 1998. Estimating and testing linear models with multiple structural changes.
Econometrica 66: 47-78.
Bai J, Perron P. 2003. Computation and analysis of multiple structural change models.
Journal of Applied Econometrics 18: 1-22.
Boldea O, Hall AR, Han S. 2012. Asymptotic distribution theory for break point estimators
in models estimated via 2SLS. Econometric Reviews 31: 1-33.
Gali J, Gertler M. 1999. In‡ation dynamics: a structural econometric analysis. Journal of
Monetary Economics 44: 195-222.
Hall AR, Han S, Boldea O. 20121. Inference regarding multiple structural changes in linear
models with endogenous regressors. Journal of Econometrics 170: 281-302.
Hansen BE. 2000. Testing for structural change in conditional models. Journal of Econometrics 97: 93-115.
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models with multiple structural changes. Journal of Econometrics 146: 59-73.
Kejriwal M, Perron P. 2010. Testing for multiple structural changes in cointegrated regression
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Kurmann A. 2007. VAR-based estimation of Euler equations with an application to New
Keynesian pricing. Journal of Economic Dynamics and Control 31: 767-796.
Perron P. 2006. Dealing with structural breaks. In Palgrave Handbook of Econometrics, Vol.
1: Econometric Theory, K. Patterson and T.C. Mills (eds.), Palgrave Macmillan, 278-352.
Perron P, Qu Z. 2006. Estimating restricted structural change models. Journal of Econometrics 134: 373-399.
24
Perron P, Yamamoto Y. 2012. A note on estimating and testing for multiple structural
changes in models with endogenous regressors via 2SLS. Forthcoming in Econometric Theory.
Qu Z, Perron P. 2007. Estimating and testing multiple structural changes in multivariate
regressions. Econometrica 75: 459-502.
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two separate regimes. Journal of the American Statistical Association 53: 873-880.
Quandt, RE. 1960. Tests of the hypothesis that a linear regression system obeys two separate
regimes. Journal of the American Statistical Association 55: 324-330.
Rossi B, Sekhposyan T. 2010. Has models’ forecasting performance for US output growth and
in‡ation changed over time, and when? International Journal of Forecasting 26: 808-835.
Stock JH, Watson MW. 2007. Why has U.S. in‡ation become harder to forecast? Journal of
Money, Credit and Banking 39: 3-33.
Zhou J, Perron P. 2007. Testing jointly for structural changes in the error variance and
coe¢cients of a linear regression model. Unpublished Manuscript, Department of Economics,
Boston University.
25
F e
1:
Cu ie d r
f
f te e e
s
a
f te re k de
c=0.25 (Ф=0.5)
c=0.25 (Ф=1.0)
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
OLS
R2=0.8
0.3
0.3
R2=0.5
0.2
R2=0.3
R2=0.1
0.1
0.2
0.1
0.0
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
-0.3
0.3
-0.2
-0.1
c=0.5 (Ф=0.5)
0.0
0.1
0.2
0.3
c=0.5 (Ф=1.0)
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
-0.3
0.3
-0.2
-0.1
c=1.0 (Ф=0.5)
0.0
0.1
0.2
0.3
c=1.0 (Ф=1.0)
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
2:
o
P w
r functions of the sup-Wald structural change test;
Ф=0.5
1.0
OLS
0.9
R2=0.8
R2=0.5
0.8
R2=0.3
R2=0.1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
(c)
Ф=1.0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
(c)
Figure 3 : Set of parameter values with which IV estimates can be more e cient than OLS
estimates;
R2
1.0
0.8
0.6
0.4
0.2
0.0
-5
-4
-3
-2
-1
0
1
2
3
4
Figure 4 : Finite sample distributions of break date estimates under the condition
5
2
v
=
1;
R2 = 0.3
R2 = 0.5
1.0
1.0
0.8
0.8
0.6
0.6
0.4
OLS
0.4
0.2
IV
0.2
0.0
0.0
-0.35
0
0.35
-0.35
R2 = 0.1
0
0.35
R2 = 0.001
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
-0.35
0
0.35
-0.35
0
0.35
Figure 5 : Change in the variance of the regressor;
(Left : 2v = 1 and 1v = a, Right : 1v = 1 and 2v = b)
a = 0.8
1.0
b = 0.8
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
OLS
0.4
0.4
R2=0.8
0.3
R2=0.5
R2=0.3
0.2
0.3
0.2
R2=0.1
0.1
0.1
0.0
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
-0.3
0.3
-0.2
-0.1
a = 0.5
0.0
0.1
0.2
0.3
0.1
0.2
0.3
0.1
0.2
0.3
b = 0.5
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
a = 0.25
1.0
-0.3
-0.2
-0.1
b = 0.25
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
-0.3
-0.2
-0.1
0.0
Figure 6 : Change in the correlation between the regressor and errors;
(Left : 2 = 1 and 1 = a, Right : 1 = 1 and 2 = b)
a = 0.5
1.0
b = 0.5
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
OLS
0.4
0.4
R2=0.8
0.3
R2=0.5
R2=0.3
0.2
R2=0.1
0.1
0.3
0.2
0.1
0.0
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
a = 0.25
1.0
-0.3
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
-0.2
-0.1
0.0
0.1
0.2
0.3
a = 0.1
1.0
-0.1
0.0
0.1
0.2
0.3
0.1
0.2
0.3
0.1
0.2
0.3
b = 0.25
1.0
0.9
-0.3
-0.2
0.0
-0.3
-0.2
-0.1
b = 0.1
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
-0.3
-0.2
-0.1
0.0
Figure 7 : Change in the mean of the regressor;
(Left : 2 = 1 and 1 = a, Right : 1 = 1 and 2 = b)
a = 1.5
1.0
b = 1.5
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
OLS
0.4
R2=0.8
0.3
R2=0.5
R2=0.3
0.2
R2=0.1
0.1
0.4
0.3
0.2
0.1
0.0
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.1
0.2
0.3
0.1
0.2
0.3
b = 2.0
a = 2.0
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
a = 5.0
1.0
-0.3
0.3
-0.2
-0.1
b = 5.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
-0.3
-0.2
-0.1
0.0
Figure 8 : The cases of i = i 1 and 0i 6= 0i 1 ;
(Left : distributions of the break date estimate,
Right : power functions of the sup-Wald test)
Change in the variance of the regressor
(Break Date)
(Power Function)
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
OLS
R2=0.8
R2=0.5
R2=0.3
R2=0.1
0.3
0.2
0.4
0.2
0.1
0.1
0.0
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
OLS
R2=0.8
R2=0.5
R2=0.2
R2=0.1
0.3
c
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Change in the correlation between the regressor and the errors
(Break Date)
(Power Function)
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
c
0
0.3
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Change in the mean of the regressor
(Break Date)
(Power Function)
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
c
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Table 1. Rejection frequencies of the sup-Wald tests
Change in the correlation between the regressor and errors
asymptotic test
φ1
bias change
OLS
IV(R2 = 0.8)
IV(R2 = 0.5)
IV(R2 = 0.3)
IV(R2 = 0.1)
1.0
0.07
0.137
0.069
0.064
0.059
0.063
0.1
−0.06
0.113
0.067
0.072
0.065
0.057
−0.5
−0.14
0.338
0.070
0.068
0.067
0.069
bootstrap test
−1.0
−0.21
0.687
0.112
0.081
0.075
0.062
1.0
0.07
0.088
0.032
0.013
0.015
0.031
0.1
−0.06
0.119
0.023
0.024
0.020
0.014
−0.5
−0.14
0.394
0.105
0.166
0.128
0.050
−1.0
−0.21
0.651
0.080
0.094
0.124
0.145
Table 2. Rejection frequencies of the sup-Wald tests
Change in the variance of regressor
asymptotic test
σ 21v
bias change
OLS
IV(R2 = 0.8)
IV(R2 = 0.5)
IV(R2 = 0.3)
IV(R2 = 0.1)
10.0
−0.06
0.619
0.239
0.202
0.167
0.138
5.0
−0.06
0.360
0.138
0.123
0.114
0.092
2.0
−0.03
0.116
0.083
0.073
0.077
0.073
bootstrap test
0.25
0.13
0.455
0.109
0.094
0.082
0.068
10.0
−0.06
0.318
0.052
0.057
0.118
0.219
5.0
−0.06
0.094
0.065
0.114
0.115
0.115
2.0
−0.03
0.092
0.084
0053
0.020
0.027
0.25
0.13
0.269
0.016
0.011
0.017
0.090
Table 3. Rejection frequencies of the sup-Wald tests
Change in the mean of the regressor
asymptotic test
μ1
bias change
OLS
IV(R2 = 0.8)
IV(R2 = 0.5)
IV(R2 = 0.3)
IV(R2 = 0.1)
0.1
0.01
0.063
0.036
0.031
0.048
0.156
1.5
−0.01
0.066
0.024
0.023
0.035
0.101
5.0
−0.06
0.337
0.180
0.249
0.368
0.685
bootstrap test
10.0
−0.07
0.798
0.600
0.737
0.869
0.967
0.1
0.01
0.076
0.093
0.080
0.075
0.222
1.5
−0.01
0.057
0.066
0.012
0.016
0.009
5.0
−0.06
0.035
0.033
0.036
0.061
0.101
10.0
−0.07
0.176
0.012
0.028
0.060
0.158
Table 4. IV full sample coefficient estimates for the hybrid Philips curve
xt
μ (const)
0.000
(0.001)
0.000
(0.000)
labor income share
GDP gap
γ (πt−1 )
0.367
(0.103)
0.260
(0.162)
κ (xt )
0.003
(0.008)
−0.010
0.009
β (Et πt+1 )
0.619
(0.116)
0.745
(0.190)
Table 5. Tests for structural changes in the reduced form equation
number of breaks
1
2
3
test
∗∗∗[∗]
SupF=43.7
F(2|1)= 52.0∗∗∗
F(3|2)= 21.0
break dates
1980:4
1973:1, 1980:4
1973:1, 1980:4, 1990:1
Table 6. Tests for structural changes in the hybrid Philips curve equation
1. One break in the reduced form (1980:4)
(a) xt is labor income share
OLS
IV(full sample)
IV(segment 1)
IV(segment 2)
SupF
F(2|1)
30.6∗∗∗[∗]
18.2∗∗
5.93
5.96
11.4
14.5∗
−
−
(b) xt is the GDP gap
SupF
F(2|1)
break date
1991:1
1967:2
1976:3
1991:1
95%
1990:3
1964:4
1976:2
1990:3
C.I.
1991:3
1969:4
1976:4
1991:3
break date
95%
1990:4
1965:3
1965:3
1990:3
C.I.
1991:2
1969:1
1969:1
1991:3
break date
1991:1
1967:2
1967:2
1974:1
1991:1
95%
1990:3
1967:1
1967:1
1969:3
1990:3
C.I.
1991:3
1967:3
1967:3
1978:3
1991:3
break date
1991:1
1991:1
1968:3
1977:2
1991:1
95%
1990:4
1990:3
1968:2
1974:3
1990:3
C.I.
1991:2
1991:3
1968:4
1980:1
1991:3
22.2∗∗∗
12.6
1991:1
21.0∗∗∗
12.6
1967:2
10.8
−
1967:2
8.7
−
1991:1
2. Two breaks in the reduced form (1973:1, 1980:4)
(a) xt is labor income share
OLS
IV(full sample)
IV(segment 1)
IV(segment 2)
SupF
OLS
IV(full sample)
IV(segment 1)
IV(segment 2)
IV(segment 3)
∗∗∗[∗]
30.6
16.9∗
9.3
3.5
6.0
(b) xt is the GDP gap
OLS
2SLS(full sample)
IV(segment 1)
IV(segment 2)
IV(segment 3)
F(2|1)
11.4
13.1
−
−
−
SupF
F(2|1)
22.2∗∗∗
24.8∗∗∗
7.1
2.7
8.7
12.6
14.3
−
−
−
Table 7. IV sub-sample estimates for the hybrid Phillips curve equation (break in 1991:1)
1. One break in the reduced form (1980:4)
(a) xt is labor income share
μ (const) γ (πt−1 ) κ (xt )
1960:1-1991:1
0.000
0.351
0.000
(0.002)
(0.111) (0.008)
1991:2-1997:4
0.001
0.130
0.042
(0.005)
(0.157) (0.036)
(b) xt is the GDP gap
μ (const) γ (πt−1 ) κ (xt )
1960:1-1991:1
0.000
0.310
−0.006
(0.000)
(0.129) (0.008)
1991:2-1997:4
0.007
0.047
−0.042
(0.002)
(0.170) (0.018)
β (Et πt+1 )
0.624
(0.107)
−0.328
(0.385)
β (Et πt+1 )
0.674
(0.132)
−0.340
(0.363)
2. Two breaks in the reduced form (1973:1 and 1980:4)
(a) xt is labor income share
μ (const) γ (πt−1 ) κ (xt ) β (Et πt+1 )
1960:1-1991:1
0.000
0.302
0.001
0.683
(0.001)
(0.070) (0.007)
(0.067)
1991:2-1997:4
0.001
0.130
0.042
−0.328
(0.005)
(0.157) (0.036)
(0.385)
(b) xt is the GDP gap
μ (const) γ (πt−1 ) κ (xt ) β (Et πt+1 )
1960:1-1991:1
0.000
0.279
−0.007
0.712
(0.000)
(0.076) (0.007)
(0.078)
1991:2-1997:4
0.007
0.047
−0.042
−0.340
(0.002)
(0.170) (0.018)
(0.363)
Table 8. IV sub-sample estimates with a break in 1967:2
1. One break in the reduced form (1980:4)
(a) xt is labor income share
μ (const) γ (πt−1 ) κ (xt )
1960:1-1967:2
0.000
−0.028
0.006
(0.004)
(0.374) (0.019)
1967:3-1997:4
−0.003
0.167
0.023
(0.002)
(0.136) (0.013)
(b) xt is the GDP gap
μ (const) γ (πt−1 ) κ (xt )
1960:1-1967:2
0.004
0.218
0.055
(0.001)
(0.327) (0.026)
1967:3-1997:4
0.000
0.158
−0.008
(0.001)
(0.162) (0.009)
β (Et πt+1 )
0.554
(0.306)
0.807
(0.129)
β (Et πt+1 )
0.646
(0.360)
0.831
(0.158)
2. Two breaks in the reduced form (1973:1 and 1980:4)
(a) xt is labor income share
μ (const) γ (πt−1 ) κ (xt ) β (Et πt+1 )
1960:1-1967:2
−0.002
−0.358
0.014
0.970
(0.003)
(0.248) (0.013)
(0.230)
1967:3-1997:4
−0.003
0.243
0.024
0.723
(0.002)
(0.081) (0.011)
(0.075)
(b) xt is the GDP gap
μ (const) γ (πt−1 ) κ (xt ) β (Et πt+1 )
1960:1-1967:2
0.001
−0.215
0.004
0.716
(0.002)
(0.302) (0.022)
(0.394)
1967:3-1997:4
0.000
0.251
−0.007
0.729
(0.000)
(0.083) (0.007)
(0.078)
Notes for Tables 4-8:
1. In parentheses below the coefficient estimates are the heteroskedasticity robust estimates of the standard errors.
2. The Sup-F test uses a 10% trimming. *, ** and *** indicate significance at the 10%, 5% and 1% level,
respectively, using asymptotic critical values. In brackets are the significance level using the fixed regressors
bootstrap method of Hansen (2000).
3. The confidence intervals of the estimates of the break date are based on the two-sided 95% nominal level
symmetric method described in Bai (1997).