Modeling and Testing Smooth Structural Changes with Endogenous Regressors Bin Chen University of Rochester Abstract: Modeling and detecting parameter stability of econometric models is a long standing problem. Most existing estimation and testing methods are designed for models without endogeneity. Little attention has been paid to models with endogeneous regressors, which may arise in many scenarios in economics. In this paper, we …rst consider a time-varying coe¢ cient time series model with potential time-varying endogeneity. A local linear two stage least squared estimation is developed to estimate coe¢ cient functions. The consistency and asymptotic normality of the estimator are derived. Furthermore, a nonparametric test is proposed to check smooth structural changes as well as abrupt structural breaks with possibly unknown change points in regression models with potential endogeneity. The idea is to compare the …tted values of the unrestricted nonparametric time-varying coe¢ cient model and the restricted constant parameter model. The test has an asymptotic N(0,1) distribution and does not require any prior information about the alternatives. A simulation study highlights the merits of the proposed estimator and test. In an application, we estimate the New Keynesian Phillips Curve for the US nonparametrically and …nd strong evidence against its stability. JEL Classi…cations: C1, C4, E0. Key words: Endogeneity, Instrumental variables, Kernel, Local linear estimation, Model stability, Nonparametric regression, Smooth structural change I am grateful to Pierre Perron and seminar participants at Boston University, the 2013 New York Camp Econometrics and the 2013 Econometric Society China meeting for their useful comments and discussions. I am also grateful to Alastair R. Hall and Otilia Boldea for providing me with their program and to Liquan Huang for excellent research assistance. Any remaining errors are solely mine. Correspondence: Bin Chen, Department of Economics, University of Rochester, Rochester, NY 14627; email: [email protected].

1. INTRODUCTION Modeling and detecting structural changes in economic relationships has been a long standing problem in econometrics. It is particularly relevant for time series over a long time horizon since the underlying economic mechanism is likely to be disturbed by various factors such as preference changes, institutional changes, technological progress. The prevalence of structural instability in macroeconomic and …nancial time series relations has indeed been con…rmed by numerous empirical studies. For example, Stock and Watson (1996) test 76 representative U.S. monthly postwar macroeconomic time series and …nd substantial instability in a signi…cant fraction of the univariate and bivariate autoregressive models. In investigating labor productivity in the U.S. manufacturing/durables sector, Hansen (2001) identi…es strong evidence of a structural break sometime between 1992 and 1996, and weaker evidence of a structural break in the 1960s and the early 1980s. Ang and Kristensen (2012) and Li and Yang (2011) …nd signi…cant time-variation in factor loadings when they test conditional capital asset pricing models. Previous works for estimating and detecting structural changes in time series econometrics mainly focus on abrupt structural breaks. Recently, smooth structural changes have gained increasing attention and some time-varying time series models have appeared as novel tools to capture the evolutionary behavior of economic time series. Among them, a nonparametric timevarying parameter time series model has attracted great interest. It is …rst introduced by Robinson (1989, 1991) and further studied by Orbe, Ferreira and Rodriguez-Poo (2000, 2005, 2006), Cai (2007), Chen and Hong (2012), Kristensen (2012) and Zhang and Wu (2012). One advantage of this nonparametric time-varying parameter model is that little restriction is imposed on the functional forms of both time-varying intercept and slope, except for the condition that they evolve with time smoothly. Despite the success of this class of time-varying parameter model, the maintained assumption all above papers impose is that all explanatory variables are exogenous or predetermined. Those methods are not applicable to macroeconometric models where regressors are correlated with errors. It is well known that endogeneity can arise from many di¤erence sources such as simultaneous equations, measurement errors, omitted variables, etc. All these are relevant in time series regression. For example, the new Keynesian Phillips curve (NKPC) is a forwardlooking model of in‡ation dynamics, where shortrun dynamics in in‡ation is driven by the expected discounted stream of real marginal costs (see, e.g., Fuhrer and Moore 1995, Gali and Gertler 1999, Zhang, Osborn and Kim 2008). In this model, the future/forecasted in‡ation rate and the output gap/marginal cost are endogenous variables, which are correlated with the regression error, namely, the in‡ation surprise. Another related example is the Taylor rule, which stipulates how the fed fund rate responds to in‡ation rate and output. Since the seminal work by Taylor (1993), various versions of the backward-looking or forward-looking Taylor rule for the U.S. monetary policy have been es1

timated. (see, e.g., Clarida, Gali and Gertler 2000, Kim and Nelson 2006, and Orphanides 2004). Similar to the NKPC, endogeneity is unavoidable since the forecast error is correlated with the forecasted in‡ation and the output. A third example is the arbitrage pricing theory, where the expected return of …nancial assets is modelled as a linear function of some macro factors (Chen, Roll and Ross 1986). It has been documented in the literature that macro factors contain large measurement errors (See, e.g., Chen, et al. 1986; Connor and Korajczyk 1986, 1991; Ferson and Harvey 1999), which would lead to the endogeneity problem. This paper develops a two-stage local linear (2SLL) estimator for time-varying models with endogeneous regressors. The …rst step is to estimate a time-varying reduced form of regression model which allows for time-varying correlation between endogenous variables and instruments. The second step is a local linear regression using the projected endogeneous variables as regressors. This estimation can be regarded as a generalization of two-stage least squared (2SLS) from parametric models to nonparametric models. The di¢ culty of this issue lies in the fact that some of the covariates in the second stage are not directly observed but have themselves been estimated in the …rst nonparametric regression. Interestingly, similar to the 2SLS estimator, the 2SLL estimator can be viewed as a LL estimator of the second stage regression, where the projection of the endogeneous variable is replaced by its probability limit. We establish the consistency and asymptotic normality of the 2SLL estimator. As a special case, a time-varying model with a stable reduced form is also studied. The simple LS estimation of the …rst stage has a faster convergence rate and hence has no impact on the second stage LL estimation asymptotically. Furthermore, we develop a Wald-type test for structural changes in models with endogeneous regressors, which can be viewed as a generalization of Hausman’s (1978) test from the parametric framework to the nonparametric framework. Chen and Hong (2012) propose the generalized Hausman’s (1978) test for structural changes of a linear regression model, where the imposed orthogonality condition rules out endogeneous regressors. The current paper …lls the gap. With the …rst stage nonparametric estimation, the theoretical derivation of the test for models with endogeneity is much more involved. And unlike Chen and Hong (2012), we remove the stationarity assumption and thus allow for covariates to be some local stationary processes. Our approach has several attractive features. First, the proposed test can detect smooth structural changes and sudden structural breaks with unknown breakdates or unknown number of breaks in regression models with endogeneous regressors. Second, unlike many tests for structural changes in the literature, which often have nonstandard asymptotic distributions, the proposed test is asymptotically pivotal. Third, our test covers both stable and unstable …rst-stage reduced form models. We need not pretest the stability of the reduced form model. Fourth, no trimming of the boundary region near the end points of the sample period is needed for our test. Fifth, our nonparametric estimation and test are studied in a uni…ed framework. The 2SLL estimators of

2

the time-varying parameters can provide insight into the evolution of the economic relationship. It is worth noting that two recent papers work on testing structural changes in linear regression with endogeneity. Using 2SLS estimation, Hall, Han and Boldea (2012) extend Bai and Perron’s (1998) double maximum tests to allow for endogenous regressors. Their main focus is the model with a stable reduced form. If the reduced form is not stable, Bai and Perron’s (1998) method is …rst applied to identify break dates. Then the whole sample is divided to subsamples accordingly and their test is applied to every subsample. Their testing procedure highly depends on the true data generating process (DGP) of the reduced form. If break dates are identi…ed at the beginning or end of the sample or many break dates are identi…ed, their second step procedure may not be implementable. And sample splitting reduces the e¤ective sample size and hence the power of the test. Perron and Yamamoto (2012) show that under certain assumptions, even in the presence of endogenous regressors, it is still preferable to estimate the break dates and test for structural change using OLS. But if the main interest is to estimate structural parameters, 2SLS is still needed. Moreover, both tests are not suitable for models with smooth structural changes in the reduced form, which is the main focus of this paper. In Section 2, we introduce the framework and develop the two stage estimator. Section 3 derives the consistency and asymptotic normality of the estimator. Section 4 states the hypotheses of interest, the form of the test statistic, and its asymptotic null distribution and power properties. In Section 5, a simulation study is conducted to assess the reliability of the asymptotic theory in …nite samples and an empirical example is used to illustrate the application of our approach. Section 6 provides concluding remarks. All mathematical proofs are collected in the appendix. Throughout the paper, C denotes a generic bounded constant. 2. NONPARAMETRIC ESTIMATION Consider the model Yt = Xt| (t=T ) + "t ; where Yt is a dependent variable, Xt is a d

1 vector,

(1)

( ) : [0; 1] ! Rd is some unknown smooth

function, "t is an unobservable regression disturbance. We allow some regressors to be correlated with the errors. If Xt is exogenous, Robinson’s (1989, 1991) kernel estimation and Cai’s (2007) local linear estimation can be used. However, their methods would yield inconsistency due to the endogeneity problem. Suppose there exists an instrument vector Zt such that E("t jZs ; s

t) = 0 and E(Zt Xt| ) =

Mzx (t=T ); where Mzx ( ) : [0; 1] ! Rm d : Specify the reduced form for Xt as Xt =

|

(t=T )Zt + vt ;

3

(2)

where Zt is an m 1 instrument, ( ) = [ Rm and

rank(Zt Xt| )

1

( ) ; :::;

d

( )] is of dimension m d; each

j

( ) : [0; 1] !

= d: We allow for time-varying parameters in the reduced form, which is

in line with the rational expectation hypothesis. Economic agents adapt their optimal behavior to policy changes, which leads to reduced-form equations that exhibit time-varying parameters. However, parameters of interests that describe preferences and technology might remain constant and our test proposed in Section 4 aims at checking the stability of those parameters. The speci…cation that parameters ( ) and ( ) are some functions of ratio t=T rather than time t only is a common scaling scheme in the literature (see, e.g., Phillips and Hansen 1990, Robinson 1991 and Cai 2007). The reason for this speci…cation is that nonparametric estimators for

t

and

t

will not be consistent unless the amount of data on which they depend increases,

and merely increasing the sample size will not necessarily improve estimation of

t

and

t

at some

…xed point t; even if some smoothness conditions are imposed: The amount of local information must increase suitably if the variance and bias of nonparametric estimators of

t

and

t

are to

decrease suitably. We apply 2SLL to estimate the model (1) and (2). The …rst stage local linear parameter estimator is obtained by minimizing the local sum of squared residuals (SSRs): t+bT h1 c

X

min2m

j 2R

=

| 0;j Zs

kst Xs;j

| 1;j

s=t bT h1 c

t+bT h1 c

X

s

2

t

(3)

Zs

T

| 2 j Lst ) ;

kst (Xs;j

s=t bT h1 c

where kst =

1 k h1

s t T h1

is a kernel function with k ( ) : [ 1; 1] ! R+ , h1 is the bandwidth in the

…rst stage with h1 ! 0 and T h1 ! 1 as T ! 1; j = 1; :::; d,

vector, and

i;j

is an m

j

( sT t )i Zs ;

1 coe¢ cient vector for

= (

| 0;j ;

| | 1;j )

is a 2m

i = 0; 1; Lst = (1;

s t | ) T

1 vector with

Zs is a 2m

1

is the Kronecker product. Examples of k( ) include the uniform, Epanechniov and

quartic kernels. By solving (3), we obtain the solution 2

^ t;j = 4

t+bT h1 c

X

s=t bT h1 c

The local linear estimator for

t

3

kst Lst L|st 5

1

t+bT h1 c

X

kst Lst Xs;j :

s=t bT h1 c

is the intercept estimator: ^ t = (e|1

Im )^ t :

We note that if E(Xt Zt| ) is not time varying, the conventional least squared estimation will be 4

su¢ cient in the …rst stage. Let ^ t = ^ |t Zt : X The second stage local linear parameter estimator is obtained by minimizing the local SSRs as t+bT h2 c

min

2R2d

X

lst Ys

| 1

s=t bT h2 c

t+bT h2 c

X

=

| ^ 0 Xs

|

lst (Ys

s

t T

2

^s X

(4)

P^st )2 ; ;

s=t bT h2 c

where lst =

1 l h2

s t T h2

is a kernel function, which need not be the same as kst ; h2 is the bandwidth ^s in the second stage with h2 ! 0 and h1 =h2 ! 0; = ( |0 ; |1 )| is a 2d 1 vector, P^st = (1; s t )| X T

is a 2d

1 vector.

The solution to (4) is 2

^t = 4

t+bT h2 c

X

s=t bT h2 c

and the local linear estimator for

t

3

1

lst P^st P^st| 5

t+bT h2 c

X

lst P^st Ys

s=t bT h2 c

is the intercept estimator ^ = (e| t 1

Id )^t :

Similar to Cai (2007), the 2SLL estimator ^ t and the …rst stage local linear estimator ^ t could be viewed as the OLS estimators of the transformed models 1=2 1=2 ^ | lst Ys = lst X s

1=2

0

and kst Xs = kst Zs| 1=2

1=2

1=2

0

+ kst

+ lst

s

s

t T

t T

Zs|

1

^| X s

1

+ "s ;

+ vs ;

s = 1; :::; T;

respectively. Hence the estimation can be implemented by any standard econometric software. Note that our framework can accommodate the partially time-varying coe¢ cient model: | Yt = X1;t

where

1;x

1;x

| + X2;t

is time-invariant.

5

2;x

(t=T ) + "t ;

Following Zhang and Wu (2012), we can estimate ^ = 1

Z

1

1;x

by

w (t) ^ 1 (t) dt;

0

where ^ 1 (t) is the nonparametric estimator corresponding to X1;t and w(t) is some weighting p function. Similar to Zhang and Wu (2012), it can be shown that T ( ^ 1 1;x ) = OP (1): But for simplicity, we focus on (1) for the rest of the paper. 3. ASYMPTOTIC DISTRIBUTION To derive the asymptotic distribution of ^ t ; we impose the following regularity conditions. Assumption A.1: (i) Rt = fZt| ; vt| ; "t g| is a -mixing process with hmixing coe¢ cients f (j)g 1 2 j=1 j

satisfying (ii) sup1

t T

(j) 1+ < C for some 0 <

< 1; where

E kRt k8+ < 1.

(j) = supk E supA2Fk+j P AjF k 1 1

Assumption A.2: (i) f"t g is a martingale di¤erence sequence (m.d.s.) such that E ("t jFt 1 ) =

0; where Ft It

1

1

= Zt| ; Zt| 1 ; :::; "t 1 ; "t 2 ; ::: : (ii) fvt g is a m.d.s. such that E (vt jIt 1 ) = 0; where

= Zt| ; Zt| 1 ; :::; vt 1 ; vt 2 ; ::: :

Assumption A.3: (i) Mzz (t=T ) = E (Zt Zt| ) and Mz" (t=T ) = E (Zt Zt| "2t ), where Mzz ( ) and

Mz" ( ) are smooth functions such that their second order derivatives are continuous in [0; 1]: (ii) (t=T ) is a smooth function such that its second order derivative is continuous in [0; 1]: (iii)

t

=

t

= (t=T ) is a smooth function such that its second order derivative is continuous in [0; 1]: Assumption A.4: (i) k : [ 1; 1] ! R+ is a symmetric bounded probability density function.

(ii) l : [ 1; 1] ! R+ is a symmetric bounded probability density function.

The -mixing condition in Assumption A.1 imposes a restriction on the temporal dependence

in fRt g: A similar condition has been used in Juhl and Xiao (2012) and Kristensen (2012). Unlike

Cai (2007), Chen and Hong (2012) and Robinson (1989, 1991), we do not impose stationarity assumption and hence regressors and instruments could be nonstationary. In macroeconomics, lagged values of Yt and Xt are often included in the empirical models and hence allowing potentially nonstationary regressors and instruments is useful. Assumption A.2 allows dynamic regression models with conditional heteroscedasticity of unknown form. Assumption A.2 requires the linear regression model to be correctly speci…ed under H0 and the violation of correct model speci…cation may lead to spurious rejection of model stability. Assumption A.3 imposes moment conditions on R1 R1 Rt : We allow for time-varying moments. Assumption A.4 implies 1 r(u)du = 1; 1 ur(u)du = 0 R1 and 1 u2 r(u)du < 1; where r ( ) = k ( ) or l ( ) : All examples noted in Section 2 satisfy this

assumption.

We now state the asymptotic properties of ^ ( ):

6

i P (A) ;

Theorem 1. Suppose Assumptions A.1-A.4 hold, and hj = cT 0 < c < 1 and

1

>

2:

Then for any

( )

where B( )= 00

= @2

for 0 <

j

< 1; j = 1 ; 2 ;

j

for 0 <

j

< 1; j = 1 ; 2 ;

2 (0; 1); as T ! 1;

^( )

and

j

h22

B ( ) = OP

00

R

p

T h2 ;

u2 l (u) du + oP h22 2

=@ 2 .

Theorem 2. Suppose Assumptions A.1-A.4 hold, and hj = cT 0 < c < 1 and

1

>

2:

Then for any p

2 (0; 1); as T ! 1;

T h2 [ ^ ( )

B ( )] !d N (0;

( )

);

R ; S = | Mzz ( ) ; = 2 | Mz" ( ) and 2 = l2 (u) du: Theorem 1 shows that ^ ( ) is a consistent estimator of ( ) and the asymptotic bias depends on the curvature of the curve ( ): If ( ) is not time-varying, ^ ( ) is still consistent but less where

=S

1

S

1

e¢ cient than the parametric estimator. Given h1 =h2 ! 0; the bias of the …rst stage estimation is

dominated by that of the second stage and B( ) solely depends on h2 : The asymptotic distribution of ^ ( ) is asymptotically equivalent to that of the local linear estimator ~ ( ) for the following arti…cial nonparametric regression ~ t| (t=T ) + "t ; Yt = X ~t = where X

| t Zt :

(5)

This feature is similar to the 2SLS estimator of a regression model with

endogeneous regressors. As shown in the appendix, we have p T h2 [ ^ ( ) ( ) X p ~ i "i h2 =T lit X

B( )]

i

X p ~i v| + h2 =T lit X i

1X | Zi Mzz1 (i=T )Zj vj| t kji T j

i

i

!

;

where the …rst term determines the asymptotic variance and the second term vanishes to 0 as T ! 1: This issue is related to the nonparametric estimation with nonparametrically estimated regressors. Mammen, Rothe and Schienle (2012) derive some general results under an i:i:d: assumption, but no such results are available in the time series context to our knowledge. Theorem 2 shows how to construct pointwise con…dence bands and the pointwise asymptotic

7

mean squared error (AMSE) of ^ ( ) is R h42 [ u2 l(u)du]2 AM SE = k 4

00 2

k +

trace ( ) : T h2

Hence, the optimal bandwidth in the second stage is hopt 2

Z ) =f[ u2 l(u)du]2 k

= ftrace (

00 2

k gg1=5 T

1=5

;

by minimizing the AMSE. One could obtain some preliminary estimate of

00

and use a plug-in

method. Alternatively, we consider the following cross-validation method, which has been very popular in the literature. (see, e.g., Cai, Das, Xiong and Wu 2006 and Zhang and Wu 2012). = (e|1 Im )^ t ; where ^ t = ( s=t bT1 h1 c;s6=t kst Lst L|st ) 1 ^ 1;CV = arg minc T 1=5 h c T 1=5 CV (h1 ); kst Lst X | : Then a data-driven choice of h1 is h

De…ne a “leave-one-out”estimator ^ t+bT h1 c s=t bT h1 c;s6=t

where CV (h1 ) =

s T t=1

t+bT h c

t

1

kXt

|

1

2

2

^ t Zt k and c1 ; c2 are two prespeci…ed constants: Similarly, for the t+bT h c t+bT h c second stage, let ^ t = I(d+q) )^ t ; ^ t = ( s=t bT2 h2 c;s6=t lst P^st P^st| ) 1 s=t bT2 h2 c;s6=t lst P^st Ys : ^ 2;CV = arg minh T c3
t

t=1

To construct pointwise con…dence intervals, we need to estimate the covariance matrix S

1

S

1

=

: Let S^ = T

1

t+bT h2 c ^ ^| s=t bT h2 c Xs Xs lts ;

^ s = ^ |s Zs and where X ^ =T where ^"t = Yt

Xt| ^ t : Hence,

1

t+bT h2 c ^ ^ | "2s lts ; s=t bT h2 c Xs Xs ^

can be estimated by its sample counterpart: ^ = S^

1^

S^ 1 :

A special case of model (1) is with constant correlation between Xt and "t : Then (2) is replaced by Xt = where

|

(6)

Zt + v t ;

is a constant matrix. Hall et al (2012) also consider a test for structural break with

endogeneous regressors. Their test essentially assumes the reduced form equation to be stable, which corresponds to (6).1 As

is not time-varying, OLS is su¢ cient for the …rst stage estimation:

^=

T X t=1

Zt Zt|

!

1

1

T X t=1

Zt Xt|

!

:

They …rst apply Bai and Perron’s (1998) test to the reduced form equation. If it is not stable, they just divide them to subsamples and focus on stable subsamples which have stable reduced-form equations.

8

^ t replaced by X ^t ; The second stage local linear estimation can be implemented via (4) with X ^ = ^ | Zt : We can derive the asymptotic normality of the estimator as follows. where X t Assumption A.5: Mzz (t=T ) = E (Zt Zt| ) and Mze (t=T ) = E (Zt Zt| e2t ), where et = vt|

t

+ "t

and Mzz ( ) and Mze ( ) are smooth functions such that their second order derivatives are continuous in [0; 1]. Corollary 1. Suppose Assumptions A.1, A.2, A.3(ii), A.4(i) and A.5 hold, and h2 = cT for 0 <

< 1; 0 < c < 1: Then for any p

2 (0; 1); as T ! 1;

T h2 [ ^ ( )

where B( ) =

B( )] !d N (0;

( )

h22

00

R

);

u2 l (u) du + oP h22 ; 2

Mzz ( ) ; = 2 | Mze ( ) ; Mze ( ) = E(Zt Zt| e2t ) and et = vt| t + "t : Corollary 1 shows that although ^ ( ) has the same convergence rate as ^ t ; they have di¤erent p asymptotic variance. When is constant, the …rst stage LS estimator is T consistent. The es=S

1

S

1

;S =

|

timation uncertainty has no impact on the second stage nonparametric estimation asymptotically. ~ t = | Zt and hence the second stage regression is We can proceed as if we knew X ~ t| (t=T ) + et ; Yt = X

(7)

where et = vt| (t=T ) + "t : Note that (7) is di¤erent from (5), where "t is the true innovation of model (1).

4. NONPARAMETRIC TESTING We shall propose a consistent test for smooth structural changes in regression models with endogeneity. The null hypothesis is H0 :

t

=

for some constant vector

and for all t:

and the alternative hypothesis is HA : H0 is false. Under HA ; over [0; 1]:

: [0; 1] ! Rd is an unknown smooth function except for a …nite number of points

9

It includes the sudden break as a special case: for example, ( )=

(

0;

if

1;

otherwise:

0;

More generally, we consider (1) as our alternative. With strict exogeneity, Chen and Hong (2012) propose the generalized Hausman test for smooth structural changes as well as abrupt structural breaks with known or unknown change points in regression models. We shall extend their test to allow for endogeneous regressors. As we allow for time-varying coe¢ cients in the …rst stage, we apply local linear method to estimate the model and generate the …tted values of Xt : ^ t = ^ |t Zt : X Then we estimate the restricted second stage regression parametrically (e.g., OLS) ^ t| + er ; Yt = X t and the unrestricted second stage regression nonparametrically ^ t| Yt = X

t

+ eut :

The generalized Hausman (1978) test compares the …tted values of Yt by a quadratic form ^ = H

"

p

h2

T X

^ t| ^ t (X

^ t| ^ )2 X

t=1

# q ^H ; A^H = B

(8)

^H are centering and scaling factors where A^H and B Z 1 1=2 ^ ^ 1 (u)]du AH = h2 CA trace[ ^ (u)M Z 1 0 ^H = 4CB ^ 1 (u) ^ (u)M ^ 1 (u)]du; B trace[ ^ (u)M 0

R1 2 bT h c 2 j )l ( ) = l (u) du+o(1); CB = T 1 h2 1 Tj=11 (1 h2 1 j= 2bT h2 c (1 jjj T T h2 1 R R 1 1 j )du]2 = 0 [ 1 l(u)l(u + v)du]2 dv + o(1); M (t=T ) = |t E(Zt Zt| ) t ; (t=T ) = T h2 ^ under H0 : We now state the asymptotic distribution of H

CA = T

1

R1 j )[ l(u)l(u+ T 1 | | 2 t E("t Zt Zt ) t :

Theorem 3: Suppose Assumptions A.1 A.3(i)(iii), A.4 hold and hj = cT j for 12 + 42 > d 1 2 ^ ! ; 0 < 2 < 1; 1 > 2 and 0 < c < 1: Then under H0 ; H N (0; 1) as T ! 1: 4

8

10

1

>

Di¤erent from Chen and Hong (2012), the current testing problem involves a …rst-stage nonparametric estimation and the estimation uncertainty does have an impact on the limiting distribution of our test statistic. Therefore, the theoretical derivation is much more involved than Chen and Hong (2012). Similar to Chen and Hong (2012), the use of the restricted estimator ^ in place ^ Conunder H0 has no impact on the limiting distribution of H: ^ is solely determined by the nonparametric estimator sequently, the asymptotic distribution of H ^ . In small samples, the distribution of H ^ may not be well approximated by N(0,1). Accurate t of the regression parameter

…nite sample critical values can be obtained via bootstrap; see Section 5 for more discussion. ^ under HA ; we impose the following assumption: To study the asymptotic power of H Assumption A.6: The coe¢ cient function

: [0; 1] ! Rd is continuous except for a …nite

number of discontinuity points on [0; 1] and sup v2(0;1) klimu!v+

(u)

limu!v

(u)k

C:

This allows both smooth structural changes and abrupt structural breaks with known or unknown break points. For abrupt structural breaks, the break size is bounded. Theorem 4: Suppose Assumptions A.1 A.3(i)(iii), A.4, A.6 hold and hj = cT

j

for 0 <

> 2 and 0 < c < 1. Then for any sequence of nonstochastic constants fCT = ^ > CT ) ! 1 under HA as T ! 1: o(T h2 )g; P (H 1;

2

p

< 1;

1

The restriction on h1 and h2 in Theorem 4 is weaker than that in Theorem 3, as we allow for ^ slower convergence rate of the …rst-stage nonparametric estimation. Theorem 4 implies that H is consistent against all alternatives to H0 at any given signi…cance level, subject to Assumption A.5. This is quite appealing because no prior information about the alternative is available in practice. It avoids the blindness of searching for possible alternatives of structural changes. Caution may be taken when the generalized Hausman test rejects H0 . It is possible that the rejection is due to model misspeci…cation rather than structural changes. For example, the choice of an inappropriate functional form may result in spurious structural changes. Of course, this is not particular to the proposed tests, but relevant to all existing tests for structural breaks. ^ we consider two classes of local alternatives. To gain more insight into the power property of H, Alternative 1: H1A (jT ) :

(u) =

+ jT g (u) ; u 2 [0; 1];

where g : [0; 1] ! Rd . The term jT g(u) characterizes the departure of the smooth-changing coe¢ cient

(u) from the constant

at each point u 2 [0; 1] and jT is the speed at which the

departure vanishes to 0 as T ! 1. This alternative can capture the local smooth structural change.

Assumption A.7: The vector function g : [0; 1] ! Rd is twice continuously di¤erentiable: Alternative 2: 11

H2A (bT ; rT ) :

(u) =

+ bT f [(u

u0 ) =rT ] ; u 2 [0; 1];

where u0 is a given point in [0; 1]: This alternative can be viewed as the local sharp structural change at some point u0 : Under H2A (bT ; rT ); the coe¢ cient function (u) becomes a temporary jump around u0 as T ! 1; due to the existence of the shrinking width parameter rT : Here,

rT controls the sharpness of the structural change around u0 , and bT is the speed at which the departure of (u) from

at each point u 2 [0; 1] vanishes to 0 as T ! 1:

Assumption A.8: The vector function f : R ! Rd is twice continuously di¤erentiable with

supz2R kf (z)k4

C and supz2R kd2 f (z) =dz 2 k

C:

Theorem 5: (i) Suppose Assumptions A.1 A.3(i)(iii), A.4 and A.7 hold and hj = cT

j

for 1=4

1 2

+ 42 > 1 > 14 82 ; 0 < 2 < 1; 1 > 2 and 0 < c < 1: Under H1A (jT ) with jT = T 1=2 h2 ; R1 R1 d ^ ! H N ( 1 ; 1) as T ! 1; where 1 = f 0 g (u)| M (u) g (u) du g (u)| M (u) du 0 R1 R1 p [ 0 M (u)du] 1 0 M (u)g (u) dug= BH ; and BH and M ( ) are de…ned in (8): (ii) Suppose Assumptions A.1 A.3(i)(iii), A.4, and A.8 hold and hj = cT

j

for

1 2

+

2

4

>

1

>

1 4

2

; 0<

8 1=2

and 0 < c < 1. Under H2A (bT ; rT ) with bT ! 0; rT ! 0; b2T rT = T 1 h2 and R1 p d | ^ ! N ( 2 (u0 ); 1) as T ! 1; where 2 (u0 ) = [ h2 = o (rT ) ; H f (z) M (u0 )f (z) dz]= BH . 1 2

< 1;

1

>

2

Our test has nontrivial power against the class of smooth alternatives H1A (jT ) with rate

jT = T

1=2

h2

1=4

; which is slightly slower than the parametric rate T

1=2

as h2 ! 0: But our

test can have better power than some parametric tests against the class of temporary sharp alternatives H2A (bT ; rT ) for suitable sequences of bT and rT :2 Therefore, even for the model with a stable reduced form, which is not our main focus, our test could be more powerful. Unlike Chen

and Hong (2012), we allow for time-varying second moments and hence the noncentrality term 2 (u0 )

depends on the location parameter u0 : 5. NUMERICAL RESULTS

5.1 Simulation study 5.1.1 Nonparametric estimation In this section we conduct simulations to examine the …nite sample performance of our proposed estimator in comparison with Cai’s (2007) estimator: We consider the following DGP: 8 > > < Yt = 2

0

(t=T ) +

1

(t=T ) Xt + "t ;

Xt = (t=T ) Zt + t ; > > : Z = 1 + 0:5Z + u ; t t 1 t

See Chen and Hong (2012) for an example.

12

where 0

(u) = 0:2 exp ( 0:7 + 3:5u) ;

1

(u) = 2u + exp (u) = 3:5 exp

"t

!

i:i:d:N vt functional forms of

1 1 0 ( );

!

;

1(

0:5)2

16 (u (4u

1;

1)2 + 3:5 exp

(4u

3)2

1:5;

i:i:d:N (0; 1) ; ut is independent of "t and vt : The

= 0; 0:4; 0:8; ut

) and ( ) are from Cai (2007). We use

to control the degree of

endogeneity. We generate 1; 000 data sets of the random sample fYt ; Xt ; Zt gTt=1 for T = 100; 250; 500 and 1; 000 respectively.

We use the uniform kernel for both k( ) and l( ) and our experience suggests that the choice of k( ) has little impact on the performance of our estimator. We employ the two step cross validation method discussed in Section 3 for the bandwidth selection. The performance of the proposed estimator is evaluated by the mean squared error (MSE)

M SE = 1=T

T X

2

^t

t

^t

t

t=1

and the mean absolute deviation (MAD)

M AD = 1=T

T X

:

t=1

Tables 1 and 2 report the mean and standard deviation of MSE and MAD of both estimators. For all cases considered, both MSE and MAD of our 2SLL estimator decrease substantially with the increase of the sample size. When T = 100; our 2SLL estimator has larger MSE or MAD in most cases. It is mainly due to the fact that our estimator involves two stages of nonparametric estimation. With endogeneity, our 2SLL estimator generally performs better than Cai’s (2007) estimator in terms of MSE and MAD. When

= 0:4 or 0.8, the MSE and MAD of Cai’s (2007)

estimator even increase with the sample size. The stronger endogeneity, the more gains to apply our 2SLL estimation method. And as expected, in the absence of endogeneity ( = 0), our 2SLL estimator may not outperform Cai’s (2007) estimator in terms of either MSE or MAD. 5.1.2 Nonparametric testing Next, we study the …nite sample performance of our test. Theorem 3 provides the null as^ Thus, one can implement our test for H0 by comparing H ^ ymptotic N (0; 1) distribution of H: with a N (0; 1) critical value. However, like many other nonparametric tests in the literature, the ^ in …nite samples may di¤er signi…cantly from the prespeci…ed asymptotic signi…cance size of H 13

level. Our analysis suggests that the asymptotic theory may not work well even for relatively large ^ are close in order of sample sizes, because the asymptotically negligible higher order terms in H ^ To overcome magnitude to the dominant U -statistic that determines the limit distribution of H: this problem, we consider a wild bootstrap procedure following Davidson and MacKinnon (2010): Step (i): Use the sample fYt ; Xt| ; Zt| gTt=1 to estimate the restricted and unrestricted models ^ statistic and the nonparametric residuals ^"t = Yt Xt| ^ t and respectively and compute the H ^ |t Zt ;

v^t = Xt

Step (ii): Obtain wild bootstrap residuals ^"t and v^t from the centered nonparametric residual T "t t=1 ^

1

T

and vt = v^t

1

T ^t t=1 v

respectively and construct a bootstrap sample |^ where Xt = ^ t Zt + v^t and residuals are # " Yt #= X"t px + ^"t : The wild bootstrap T = (T m)vt ut v^t ; where ut = a with generated according to the formula that = "t ut ^"t p p p probability 1 a= 5 and ut = 1 a with probability a= 5 and a = (1 + 5)=2: ^ , in the same way as H; ^ with fY ; Xt | ; Zt| gT Step (iii): Compute the bootstrap statistic H "t = ^"t

T

|

fYt ; Xt ; Zt| gTt=1 ;

t

replacing the original sample

t=1

fYt ; Xt| ; Zt| gTt=1 ;

^ gB ; Step (iv): Repeat steps (ii) and (iii) B times to obtain B bootstrap test statistics fH l l=1

where B is su¢ ciently large;

Step (v): Compute the bootstrap p-value p

B

1

B ^ l=1 1(Hl

^ where 1( ) is the indicator > H);

function. To examine the size of our test under H0 , we consider the following DGP: DGP S.1 [No Structural Change]: 8 > > < Yt =

0

+

1 Xt

+ "t ;

Xt = (t=T ) Zt + t ; > > : Z = 1 + 0:5Z + u ; t t 1 t where

0

and 1

1

are the average of

0

(t=T ) and

!

1

(t=T ) respectively with t = 1; :::; T;

"t vt

!

; = 0; 0:4; 0:8; ut is independent of "t and vt : 1 To check the robustness of our ! test, we also consider ! the heteroscadasticity case, where "t = et 1 0:4 p wt et ; wt = 0:2 + 0:5Xt2 ; i:i:d:N : vt 0:4 1 To investigate the power of our test in detecting structural changes, we consider four alter-

i:i:d:N

natives: (i) a single break, (ii) multiple breaks, (iii) non-persistent temporal breaks, (iv) smooth structural changes, respectively:

14

DGP P.1 [Single Structural Break]: (

Yt =

0

(

+ 0

1 Xt

if t

+ "t ;

+ 0:2) + (

1

0:3T;

+ 0:2)Xt + "t ; otherwise.

DGP P.2 [Multiple Structural Breaks]:

Yt =

8 > > < ( > > :

( 0

0

0:3) + (

1

0:3)Xt + "t ; if 0:1T

t

0:2T or 0:7T

0

+ 0:3) + (

1

+ 0:3)Xt + "t ; if 0:4T

t

0:5T ,

+

1 Xt

t

0:8T;

otherwise.

+ "t ;

DGP P.3 [Non-persistent Temporal Structural Breaks]:

Yt =

(

(

0 0

+ 0:5) + (

+

1 Xt

1

+ 0:5)Xt + "t ; if 0:4T

t

0:6T;

otherwise.

+ "t ;

DGP P.4 [Smooth Structural Changes]: Yt = where

0

+

0

(t=T )

0

+

1

+

1

(t=T )

1

Xt + "t ;

= 0:15:

For each of DGPs P.1-P.4, we generate 500 data sets of the random sample fYt ; Xt ; Zt gTt=1 for

T = 100; 250 and 500 and use B = 99 bootstrap iterations for each simulated data set. The two step cross validation method discussed in Section 2 is used for the bandwidth selection. ^ under DGP S.1 at the 10% and 5% signi…cance levels, Table 3 reports the rejection rates of H ^ overjects H0 ; but not excessively. The using bootstrap critical values (BCVs). When "t is i:i:d:; H size performance enjoys similar pattern when "t is conditional heteroscadastic, which suggests that our test is robust to potential heteroscadasticity. For comparison, we apply Hall et al.’s (2012) test as well. Under DGP S.1, their test has very high rejection rates and rejection rates increase with the increase of the sample size, suggesting that their approach is not applicable to models

with smooth structural changes in the reduced form. ^ test under HA . Table 4 reports the rejection Next, we turn to the power performance of the H ^ with BCVs under DGPs P.1-P.4 at the 5% level. The H ^ test has reasonable all-around rates of H power against smooth and abrupt structural changes. The rejection rate is about 65% at the 5% level even when the sample size T is as small as 100; and attains unity when T = 500: To con…rm that our test has the right power when the distance between the null and alternative ^ as a function of for DGP hypotheses is increased, we plot in Figure I the empirical power of H P.4. When

= 0; we are back to our null model DGP S.1. Figure 1 shows that the power function

15

increases monotonically with : When the distance

is increased to a larger extent, the power of

our test is reaching unity. ^ test has good size in …nite samples when the wild bootstrap To sum up, we observe that the H is used. It also has reasonable power against both sudden structural breaks and smooth structural changes. 5.2 Application to the New Keynesian Phillips Curve To illustrate the application of our method, we estimate the US NKPC nonparametrically and assess its stability with our test. An important di¤erence between the NKPC and the traditional Phillips curve is how to model core in‡ation (i.e., the in‡ation rate when the output gap is zero). Gali and Gertler (1999) suggest that core in‡ation can be formulated as a weighted sum of lagged in‡ation and expected in‡ation. A basic speci…cation is t

where

t

=c+

is the in‡ation rate, Et

t+1

f Et t+1

+

b t 1

+

y yt

(9)

+ "t ;

denotes expected in‡ation for time t + 1 given information

available up to time t, yt denotes the output gap or real marginal cost,

f

and

b

are viewed as

forward-looking and backward-looking parameters respectively. The time series plot of

and yt

t

are given in Figures II(1) and (2). Expected in‡ation and output gap are endogeneous variables. Let Xt = (Et

| t+1 ; yt )

and Xt =

| t Zt

(10)

+ vt ;

where Zt is the instrument containing one lag of the following variables: the expected in‡ation rate, the output gap, the unemployment rate, the growth rate of the money aggregate M2 and the short-term interest rate. The data we employ are quarterly US data covering the period 1968:IV–2012:II. As in Zhang et al. (2008), in‡ation is measured by the annualized quarterly growth rate of the GDP de‡ator, the output gap is the series constructed by the Congressional Budget O¢ ce, and the expected in‡ation is measured by the 1-quarter-ahead median forecasts from the Survey of Professional Forecasters. We estimate the NKPC via 2SLL estimation with h1 = 0:157 and h2 = 0:348; which are selected by the CV method discussed in Section 3. The estimated curves c( );

f(

);

b(

) and

y(

) (solid

lines) with 95% pointwise con…dence intervals (dashed lines) are plotted in Figures II(3) to (6). It is evident from Figures II(3) to (6) that these curves do vary over time smoothly. Interestingly, the forward-looking parameter

f

generally dominates the backward-looking parameter

b

before

2001, which supports Zhang et al. (2008)’s …nding from the recursive estimation. Unlike Zhang et al. (2008), we use a data-driven method to select the estimation window and let data speak for itself.

16

Furthermore, we test the stability of the NKPC with the generalized Hausman test. The wild bootstrap procedure is applied and the number of bootstrap iterations is 499. The p value based on BCV is 0.01. Therefore, we strongly reject the stability of the NKPC.

6. CONCLUSION

Modeling and detecting structural changes with nonparametric methods have attracted increasing attention in time series econometrics. We have contributed to this literature by establishing the asymptotic properties of a two stage local linear estimator for the time-varying coe¢ cient models with endogeneity and proposing a new test for smooth structural changes as well as abrupt structural breaks in regression models with potential endogeneous regressors. Existing works mainly focus on models with exogenous regressors. Moreover, no structural break test was available for regression models which allow for smooth structural changes in the …rst-stage reduced-form model. On the other hand, our test is intuitively appealing and straightforward to compute. It has a convenient null asymptotic N(0,1) distribution, does not require trimming data, does not require prior information on the possible alternative, and is consistent against all smooth structural changes as well as multiple abrupt structural breaks in linear regression models with endogeneity. To overcome the adverse impact of the nonparametric estimation of the timevarying parameters, we employ a wild bootstrap, which provides reasonable size and power for the proposed test in …nite samples. We estimate and test the New Keynesian Phillips curve with the proposed method and …nd strong evidence against model stability.

17

REFERENCES Ang, A. and D. Kristensen (2012), "Testing Conditional Factor Models," Journal of Financial Economics, 106, 132-156. Bai, J. and P. Perron (1998): "Estimating and Testing Linear Models with Multiple Structural Changes," Econometrica, 66, 47-78. Cai, Z. (2007): "Trending Time-Varying Coe¢ cient Time Series Models with Serially Correlated Errors," Journal of Econometrics, 136, 163-188. Cai, Z., M. Das, H. Xiong and X. Wu (2006): "Functional coe¢ cient instrumental variables models," Journal of Econometrics, 133, 207–241. Chen, B. and Y. Hong (2012): "Testing for Smooth Structural Changes in Time Series Models via Nonparametric Regression," Econometrica, 80, 1157-1183. Chen, N., R. Roll, and S. Ross (1986): "Economic forces and the stock market," Journal of Business, 59, 383–403. Clarida, C., J. Gali, M. Gertler (2000): "Monetary policy rules and macroeconomic stability: evidence and some theory," Quarterly Journal of Economics, 2, 147–180. Connor, G., and R. Korajczyk (1986): "Performance measurement with the arbitrage pricing theory," Journal of Financial Economics, 15, 373–94. Connor, G., and R. Korajczyk (1991): "The attributes, behavior and performance of U.S. mutual funds," Review of Quantitative Finance and Accounting, 1, 5–26. Davidson and MacKinnon (2010), "Wild Bootstrap Tests for IV Regression", Working paper, McGill University. Ferson, W., and C. Harvey, 1999, "Conditioning variables and the cross-section of stock returns", Journal of Finance, 54, 1325–60. Fuhrer, J. C., and Moore, G. R. (1995), “In‡ation Persistence,”Quarterly Journal of Economics, 110, 127–159. Gali, J., and Gertler, M. (1999): “In‡ation dynamics: a structural econometric analysis,”Journal of Monetary Economics, 44, 195-222. Hall, A., S. Han and O. Boldeac (2012), "Inference regarding multiple structural changes in linear models with endogenous regressors," Journal of Econometrics, 170, 281-302. Hansen, B. (2001): "The New Econometrics of Structural Change: Dating Breaks in U.S. Labor Productivity," Journal of Economic Perspectives, 15, 117-128. Hausman, J. (1978): "Speci…cation Tests in Econometrics," Econometrica, 46, 1251-1271. Juhl, T. and Z. Xiao (2012): "Nonparametric Tests of Moment Condition Stability," Econometric Theory, forthcoming. Kim, C.J. and C. Nelson (2006), "Estimation of a forward-looking monetary policy rule: A timevarying parameter model using ex post data", Journal of Monetary Economics, 53, 1949–1966. 18

Kristensen D. (2012), "Nonparametric Detection and Estimation of Structural Change," Econometrics Journal, 15, 420-461. Li, Y. and L. Yang (2012), "Testing Conditional Factor Models: a Nonparametric Approach" Journal of Empirical Finance, 2011, 18(5): 972-992. Mammen, Rothe and Schienle (2012), "Nonparametric Regression with Nonparametrically Generated Covariates," the Annals of Statistics, 40, 1132-1170. 0

Orbe, S., E. Ferreira and J. Rodriguez-Poo (2000): "A Nonparametric Method to Estimate Time Varying Coe¢ cients Under Seasonal Constraints," Journal of Nonparametric Statistics, 12, 779806. — — (2005): "Nonparametric Estimation of Time Varying Parameters under Shape Restrictions," Journal of Econometrics, 126, 53-77. — — (2006): "On the Estimation and Testing of Time Varying Constraints in Economic Models," Statistical Sinica, 16, 1313-1333. Orphanides, A. (2004): "Monetary policy rules, macroeconomic stability and in‡ation: a view from the trenches", Journal of Money, Credit, and Banking, 36, 151–175. Perron, P., and Y. Yamamoto (2012): "Using OLS to Estimate and Test for Structural Changes in Models with Endogenous Regressors," forthcoming in the Journal of Applied Econometrics. — — (2012): "A Note on Estimating and Testing for Multiple Structural Changes in Models with Endogenous Regressors via 2SLS," forthcoming in Econometric Theory. Phillips, P.C.B. and Hansen B.E. (1990): "Statistical Inference in Instrumental Variables Regression with I(1) Processes," Review of Economic Studies, 57, 99–125. Robinson, P.M. (1989): "Nonparametric Estimation of Time-Varying Parameters," in Hackl, P. eds., Statistical Analysis and Forecasting of Economic Structural Change, Springer, Berlin, 253264. — — (1991): "Time-Varying Nonlinear Regression," in Hackl, P. eds., Economic Structural Change Analysis and Forecasting, Springer, Berlin, 179–190. Stock, J.H. and M.W. Watson (1996): "Evidence on Structural Instability in Macroeconomic Time Series Relations," Journal of Business and Economic Statistics, 14, 11-30. Taylor, J. (1993): "Discretion Versus Policy Rules in Practice," Carnegie-Rochester Conference Series on Public Policy, 39, 195-214. Zhang, C., D. Osborn and C. Kim (2008): "The New Keynesian Phillips Curve: From Sticky In‡ation to Sticky Prices," Journal of Money, Credit and Banking, 40, 667-699. Zhang, T. and W.B. Wu (2012): "Inference of time-varying regression models," the Annals of Statistics, 40, 1376-1402.

19

TABLE 1 MSEs of 2SLL and Local Linear Estimators =0 T

0

= 0:4 1

0

= 0:8 1

0

1

2SLL Estimator

100

0:195(0:319) 0:060(0:156) 0:234(0:389) 0:065(0:179) 0:275(0:474) 0:072(0:235)

250

0:079(0:115) 0:033(0:082) 0:093(0:140) 0:036(0:098) 0:109(0:171) 0:038(0:120)

500

0:047(0:066) 0:020(0:052) 0:057(0:084) 0:021(0:061) 0:068(0:106) 0:022(0:071)

1000

0:027(0:039) 0:010(0:025) 0:032(0:050) 0:011(0:026) 0:037(0:060) 0:011(0:029) Cai’s (2007) Estimator

100

0:128(0:203) 0:036(0:070) 0:127(0:195) 0:030(0:058) 0:248(0:294) 0:055(0:086)

250

0:049(0:071) 0:020(0:037) 0:068(0:086) 0:020(0:041) 0:230(0:191) 0:059(0:091)

500

0:028(0:037) 0:009(0:016) 0:070(0:078) 0:020(0:038) 0:237(0:173) 0:069(0:100)

1000

0:016(0:022) 0:005(0:008) 0:051(0:050) 0:020(0:034) 0:216(0:126) 0:079(0:115)

Notes: (i) Mean and standard deviation (in parentheses) of MSEs are reported. (ii) 1,000 iterations. TABLE 2 MADs of 2SLL and Local Linear Estimators =0 T

0

= 0:4 1

0

= 0:8 1

0

1

2SLL Estimator

100

0:343(0:278) 0:176(0:170) 0:374(0:308) 0:183(0:178) 0:402(0:337) 0:190(0:190)

250

0:224(0:171) 0:131(0:126) 0:241(0:187) 0:134(0:133)

500

0:172(0:130) 0:100(0:100) 0:188(0:146) 0:102(0:103) 0:203(0:164) 0:103(0:107)

1000

0:129(0:100) 0:073(0:072) 0:139(0:112) 0:074(0:073) 0:148(0:123) 0:075(0:075)

0.258(0.206) 0:137(0:140)

Cai’s (2007) Estimator

100

0:283(0:220) 0:140(0:127) 0:279(0:221) 0:134(0:111) 0:417(0:271) 0:191(0:136)

250

0:176(0:133) 0:104(0:096) 0:214(0:149) 0:110(0:089) 0:435(0:203) 0:201(0:138)

500

0:135(0:097) 0:071(0:065) 0:221(0:144) 0:112(0:087) 0:450(0:185) 0:220(0:145)

1000

0:103(0:075) 0:049(0:047) 0:197(0:108) 0:111(0:086) 0:441(0:147) 0:235(0:153)

Notes: (i) Mean and standard deviation (in parentheses) of MADs are reported. (ii) 1,000 iterations.

TABLE 3 Empirical Sizes of Tests under DGP S.1 "t ~i:i:d:N (0; 1) =0 T

10%

5%

= 0:4 10%

= 0:8

5%

10%

5% ^ H

=0 10%

"t jXt ~N ( (Xt ) ; (Xt )) = 0:4

5%

10%

5%

= 0:8

10%

5%

100

0.096 0.056 0.096 0.044 0.112 0.048 0.114 0.056 0.120 0.062 0.120 0.086

250

0.112 0.048 0.118 0.066 0.128 0.060 0.122 0.074 0.126 0.072 0.120 0.074

500

0.114 0.054 0.118 0.070 0.136 0.084 0.122 0.050 0.096 0.048 0.130 0.058 Hall et al.’s (2012) test

100

0.374 0.276 0.324 0.204 0.258 0.166 0.318 0.254 0.330 0.250 0.310 0.238

250

0.980 0.950 0.942 0.886 0.910 0.790 0.900 0.828 0.884 0.778 0.852 0.760

500

0.998 0.996 0.996 0.996 0.994 0.982 0.968 0.942 0.964 0.924 0.954 0.916

^ denotes the generalized Hausman test; UDmax denotes Hall et al.’s (2012) test; Notes: (1) H (2) The p values are based on 500 iterations

TABLE 4 ^ Test Empirical Powers of the H 100 10%

250 5%

10%

500 5%

10%

5%

DGP P1-Single Structural Break

0.752 0.596 0.988 0.970 1.000

1.000

DGP P2- Multiple Structural Breaks

0.764 0.646 0.992 0.988 1.000

1.000

DGP P3-Non-persistent Temporal Structural Breaks

0.800 0.690 0.998 0.994 1.000

1.000

DGP P4-Smooth Structural Changes

0.840 0.726 0.994 0.984 1.000 Note: 500 iterations.

1.000

Figure 1: the inflation rate

Figure 2: the output gap

12

20

10

10

8

0

6 -10 4 -20

2

-30

0 -2 1970

1980

1990

2000

-40 1970

2010

Figure 3: the estimated intercept

1980

1990

2000

2010

Figure 4: the estimated coefficient for the expected inflation rate 2

8 6

1

4 0 2 -1 0 -2

-2 -4 1970

1980

1990

2000

2010

Figure 5: the estimated coefficient for the output gap 0.15

-3 1970

1980

1990

2000

2010

Figure 6: the estimated coefficient for lagged inflation rate 1.5

0.1

1 0.5

0.05

0

0

-0.5 -0.05 -0.1 1970

-1 1980

1990

2000

2010

-1.5 1970

1980

1990

2000

2010