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Testing for Smooth Structural Changes in Time Series Models via Nonparametric Regression

Bin Chen Department of Economics University of Rochester and Yongmiao Hong Department of Economics and Statistical Science Cornell University, Wang Yanan Institute for Studies in Economics Xiamen University June, 2008

We would like to thank Mark Bils, Tim Bollerslev, Zongwu Cai, George H. Jakubson, Robert Jarrow, Nick Kiefer, Chung-ming Kuan, Oliver Linton, Joon Park, Jean-Francois Richard, Alan Stockman, George Tauchen and seminar participants at Cornell University, Duke University, National University of Singapore, University of Pittsburgh, University of Rochester and the 2007 Sino-Korean Econometrics Workshop in Xiamen, China for their comments. All remaining errors are solely ours. Hong acknowledges support from the National Science Foundation of China and the Cheung Kong Visiting Scholarship by the Chinese Ministry of Education and Xiamen University, China. Correspondences: Bin Chen, Department of Economics, University of Rochester, Rochester, NY 14627; email: [email protected]; Yongmiao Hong, Department of Economics & Department of Statistical Science, Cornell University, Ithaca, NY 14850, USA, and Wang Yanan Institute for Studies in Economics (WISE), Xiamen University, Xiamen 361005, China; email: [email protected].

Testing for Smooth Structural Changes in Time Series Models via Nonparametric Regression Abstract Checking parameter stability of economic models is a long-standing problem in time series econometrics. A classical econometric procedure is Chow’s (1960) test, which checks for the existence of a structural change on a known date. Various extensions have been made to test multiple changes with known or unknown change points. However, almost all existing structural change tests in econometrics are deigned to detect abrupt breaks. Little attention has been paid to smooth structural changes, which may be more realistic in economics. This paper proposes two consistent tests for smooth structural changes as well as abrupt structural breaks with known or unknown change points. The idea is to estimate the smooth time-varying parameters by local smoothing and compare them with the OLS parameter estimator. The …rst test compares the sums of squared residuals of the restricted constant parameter model and the unrestricted nonparametric time-varying parameter model, in a spirit similar to Chow’s (1960) F test. The second test compares the …tted values of the restricted and unrestricted models, which can be viewed as a generalization of Hausman’s (1978) test. Both tests have a convenient asymptotic N(0,1) distribution and do not require any prior information about the alternatives. Interestingly, unlike Chow’s (1960) test, the generalized Chow test is no longer optimal; it is asymptotically less powerful than the generalized Hausman test. A simulation study highlights the merits of the proposed tests in comparison with a variety of popular tests for structural changes. JEL Classi…cations: C1, C4, E0. Key words: Kernel, Chow’s test, Hausman’s test, Model stability, Nonparametric regression, Parameter constancy, Smooth structural change

1. INTRODUCTION Detection of structural changes in economic relationships is a long-standing problem in time series 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. At the individual level, economic agents may change their production, consumption, savings and other patterns of behavior. This will lead to structural changes in an economic system, because econometric models consist of optimal decision rules of economic agents. A second driving force for structural changes are “shocks” induced by institutional changes, such as changes of government policies from a faith in monetarist policies to reliance on more Keynesian policies. Indeed, as Lucas (1976) points out, “any change in policy will systematically alter the structure of econometric models”, given that the structure of an econometric model depends crucially on agents’expectations, which in turn vary systematically with changes in the structure of time series relevant to decision makers. Third, technological progress, which embraces the process of innovations and the long term determinants of capital accumulation, has dramatically changed economic fundamentals and economic structures in the past decades. The prevalence of structural instability in macroeconomic 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. Ben-David and Papell (1998) …nd statistically signi…cant evidence of a slowdown in the GDP growth in 46 countries. McConnell and Perez-Quiros (2000) and Kim, Nelson and Piger (2004) …nd overwhelming evidence of a substantial decrease in the volatility of the U.S. GDP growth rates in the early 1980s. 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. Econometric models usually specify a relationship between economic fundamentals. Economists always hope that econometric models can capture important features of these relationships using a constant structure. Model stability is crucial for statistical inference, out-of-sample forecasts, and any policy implications drawn from the model. Moreover, the existence of relatively constant linear relationships between economic variables is important for model parameters to be regarded as marginal propensities or elasticities. Such economic interpretations will be invalid in the presence of structural changes. Therefore, detection and identi…cation of structural changes are important and can be regarded as a …rst step to build a nonstationary time series model in view of modeling strategy. Not surprisingly, a large portion of the literature has been devoted to testing structural changes. See, e.g., Hansen (2001), Banerjeea and Urga (2005) and Perron (2006) for excellent 1

recent surveys. A classical econometric procedure is Chow’s (1960) F test, which checks for the existence of an abrupt structural break at a known break date. This test has optimal power in the classical normal linear regression context for testing constancy of regression parameters across two or more regimes, each of which has enough data points to allow reliable estimation of the regression parameters, conditional on the error variances being constant across regimes. However, the F test becomes invalid even asymptotically when conditional heteroskedasticity exists, which has been documented as an empirical stylized fact in many economic and …nancial time series (e.g., Engle 1982, Bollerslev 1986). Another important limitation of Chow’s (1960) test is that the break date must be known ex-ante, which is usually unattainable.1 In practice, a break date may be decided on some known feature of the data. However, whether or not the obtained p-value takes into account the pre-test examination of the data can make a dramatic di¤erence: the signi…cance levels are usually much higher than the nominal ones that assume that the break point was chosen exogenously. There has been some research on testing structural breaks with unknown break dates. For example, Quandt (1960) proposes taking the largest Chow statistic over all possible break dates. If the break date is unknown a priori, however, the Chisquare critical values are inappropriate. This is a non-standard problem as the break date is only identi…ed under the alternative hypothesis. This problem is treated under various degrees of speci…city by Davies (1977), Hawkins (1987), Horvath (1995) and generalized by Andrews (1993). Andrews (1993) and Andrews and Ploberger (1994) consider a class of the supremum and the average of exponential Lagrange multiplier (LM ), Wald, and likelihood ratio (LR) tests for structural breaks with unknown break dates. The test statistics have nonstandard asymptotic distributions and simulation methods are needed to obtain critical values. All these are speci…cally designed for the case of a single break although some have power against multiple breaks. The problem of multiple structural changes has received considerably less but increasing attention. Andrews, Lee and Ploberger (1999) develop the average and exponential F tests for the existence of changepoints at unknown times in a normal linear multiple regression. Bai and Perron (1998) promote the maximum of a sequence of F statistics for multiple breaks. The average, exponential and maximum F tests require the computation of the F test over all permissible partitions of the sample. Therefore the number to be evaluated is of the order O(T m ) for a sample with T observations and m possible breaks. Moreover, Vogeslang (1999) concludes that most tests have nonmontonic power if the number of true breaks is greater than the number of breaks explicitly accounted for in the construction of the tests. For cases where the number of breaks is unknown, Bai and Perron (1998, 2003) suggest a double maximum tests given some pre1 In rare cases, there are some natural canadidates for breakdates, such as the Black Monday, October 19, 1987.

2

speci…ed upper bound and Qu and Perron (2007) extend this analysis to a system of equations. However prior information on the upper bound may not be available and a large upper bound will lead to very involved computation. Another problem of the class of the supremum and average tests is that the statistics must be calculated in a trimmed range of the sample and hence power loss is inevitable. In addition to Chow’s (1960) test, many existing tests for structural changes in the literature are also based on the analysis of residuals. One which has played an important historical role is CUSUM test proposed by Brown, Durbin and Evans (1975) and further studied by Kramer, Ploberger and Alt (1988), Ploberger and Kramer (1992), among many others. They work on the maximum of partial sums of recursive or OLS residuals. A drawback of CUSUM test is that it exhibits only trivial power against alternatives in certain directions, as shown by Kramer et al. (1988) using asymptotic local power and by Garbade (1977) using simulations. Also, Hackl (1980) proposes MOSUM tests using the maximum of moving sums of recursive residuals and generalized by Kuan and Hornik (1995). Several additional tests in the literature such as tests of Leybourne and McCabe (1989), Nyblom (1989) and Hansen (1992) are designed for alternatives with stochastic trend and hence, have a di¤erent focus. Most existing tests for structural changes in time series econometrics are designed to detect abrupt structural breaks. Two exceptions are tests of Farley, Hinich and McGuire (1975) and Lin ::

and Terasvirta (1994), who consider testing for parametric smooth structural changes. Smooth structural changes have received little attention in the literature. In fact, as Hansen (2001) points out, “it may seem unlikely that a structural break could be immediate and might seem more reasonable to allow a structural change to take a period of time to take e¤ect”. Technological progress, preference change and policy switch are some leading driving forces of structural changes that usually exhibit evolutionary changes in the long term. In economic modeling, parameters re‡ect economic agents’ decision rules. Because it takes time for economic agents to collect and digest information needed for making decisions and for markets to reach some consensus due to heterogeneous beliefs, economic agents usually update their beliefs continuously given their information. On the other hand, even if individual agents could respond immediately to sudden changes, the aggregated economic variables over many individuals may become smooth. Aggregated data are measured by weighting the relative importance of heterogeneous subsets of economic units, where aggregating weights usually change smoothly with time. Moreover, di¤erent economic agents may switch their behaviors or expectations at di¤erent time. Hence, the aggregated economic variables will change smoothly rather than jump abruptly over time. In fact, evolutionary changes have been found as a rule rather than an exception in developing countries. For example, Sen (1989) shows that the poverty line, income gap ratio and Gini coe¢ cient change smoothly since governments in developing countries impose direct or indirect taxes immediately 3

corresponding to such extraneous shocks as an "oil embargo" and the lowering of oil prices. Government policies work as a bumper to absorb some e¤ects of sudden shocks that may cause abrupt structural breaks. Greenaway, Leybourne and Sapsford (1997) reappraise the time series properties of long-run economic growth rates in a number of developing countries which have undertaken liberalization. They conclude that even if the liberalization is implemented in a “big bang”manner, any subsequent e¤ects on growth will typically be gradual and the speed of adjustment depends on the e¢ ciency of markets. During the past decade or so, time-varying time series models have appeared as a novel tool to model the evolutionary behavior of macroeconomic and …nancial time series. A leading example is the Smooth Transition Regression (STR) model, developed and popularized by Lin ::

and Terasvirta (1994). Thanks to the use of a transition function, the STR model allows both the intercept and slope to change smoothly over time. Parametric models for time-varying parameters lead to more e¢ cient estimation if the underlying coe¢ cient functions are indeed correctly speci…ed. However, economic theories usually do not suggest any concrete functional form for time-varying parameters; the choice of a functional form is somewhat arbitrary. As mentioned by Phillips (2001), models for trending time series are rather complicated and use of nonparametric methods may be one solution. A nonparametric time-varying parameter time series model is …rst introduced by Robinson (1989, 1991) and further studied by Orbe, Ferreira and Rodriguez-Poo (2000, 2005, 2006) and Cai (2007). One advantage of this nonparametric time-varying parameter model is that little restriction is imposed on the functional forms of both the time-varying intercept and slope, except for the condition that they evolve with time smoothly. Motivated by the ‡exibility of the nonparametric time-varying parameter model, we will use it as the alternative to test smooth structural changes for a linear regression model. To our knowledge, there are only two tests designed explicitly for smooth structural changes in the literature. Farley et al. (1975) construct an F test by comparing a linear time series model ::

with a parametric alternative whose slope is a linear function of time. Lin and Terasvirta (1994) develop LM type tests against a STR alternative, where the transition variable is a function of time. Simulations show that their tests have better power than CUSUM and ‡uctuation tests when the data generating process (DGP) has continuous or non-monotonic structural changes. An undesired feature of these tests is that they use a speci…c parametric time-varying parameter model. While these tests have best power against the assumed alternative, no prior information about the true alternative is usually available for practitioners. In such scenarios, it is highly desirable to develop consistent tests that have good power against all-round alternatives of structural changes. This paper proposes two new consistent tests for smooth structural changes as well as abrupt structural breaks. Such tests complement the existing tests for abrupt structural breaks and 4

avoid the di¢ culty associated with whether there are multiple breaks and/or whether the breakdates are unknown. We estimate smooth-changing parameters by local linear regressions and compare them with the OLS parameter estimator. Local linear smoothing is a generalization of standard kernel smoothing. It includes the standard kernel smoothing as a special case and has the advantage of e¤ective bias-reducing near the boundary regions (e.g., Fan and Yao 2003). The proposed …rst test is a generalized Chow’s (1960) F test, based on the comparison of the sums of squared residuals between the restricted constant parameter regression model and the unrestricted nonparametric regression model. The second test can be viewed as a generalized Hausman’s (1978) test, based on the direct comparison between the OLS regression estimator and the nonparametric regression estimator. Interestingly, unlike Chow’s (1960) test, which is optimal in the context of the classical linear regression model with i.i.d. normal errors, the generalized Chow test is no longer optimal. It is shown that the generalized Hausman test is asymptotically more powerful than the generalized Chow test. Compared with the existing tests for structural breaks in the literature, the proposed tests have a number of appealing features. First, the proposed tests are consistent against a large class of smooth time-varying parameter alternatives. They are also consistent against multiple sudden structural breaks with unknown break points. Second, no prior information on a structural change alternative is needed. In particular, we do not need to know whether the structural changes are smooth or abrupt, and in the cases of abrupt structural breaks, we do not need to know the dates or the number of breaks. Third, unlike most tests for structural breaks in the literature, which often have nonstandard asymptotic distributions, the proposed tests have a convenient null asymptotic N(0,1) distribution. The only inputs required are the OLS parameter estimator and the local linear time-varying parameter estimator. The latter is in fact a locally weighted least squares parameter estimator. Hence, any standard econometric software can carry out computational implementation easily. Fourth, the nonparametric time-varying parameter estimator is sensitive to the local behavior of time-varying parameters. Because only local information is employed in estimating parameters at each time point, the proposed tests have symmetric power against structural breaks that occur either in the …rst or second half of the sample period. This is di¤erent from some existing tests that have di¤erent powers against structural breaks that have same sizes but occur at di¤erent time points. Fifth, unlike some existing tests for structural breaks, no weighting function or trimming procedures are needed for the proposed tests. Moreover, as a by-product, the nonparametric local linear estimators of the potentially time-varying intercept and slope can provide insight into the economic relationship. 5

In Section 2, we introduce the framework and state the hypotheses of interest. Section 3 describes the nonparametric approach and Section 4 derives the forms of test statistics and their asymptotic null distribution. Section 5 investigates the asymptotic power properties of the test statistics. In Section 6, a simulation study is conducted to assess the reliability of the asymptotic theory in …nite samples. Section 7 provides concluding remarks and directions for future research. All mathematical proofs are collected in the appendix. A GAUSS code to implement the proposed tests is available from the author upon request. Throughout the paper, C denotes a generic bounded constant.

2. HYPOTHESES OF INTEREST Consider the following DGP Yt = X0t

t

+ "t ;

where Yt is a dependent variable, Xt is a d

(2.1)

t = 1; :::; T; 1 vector of explanatory variables,

t

is a d

1

possibly time-varying parameter vector, "t is an unobservable regression disturbance such that E("t jXt ) = 0 almost surely (a.s.), d is a …xed positive integer, and T is the sample size. The regressor vector Xt can contain exogenous explanatory variables and lagged dependent variables. Thus, both static and dynamic regression models are covered. Like the bulk of the literature on structure changes, we are interested in testing constancy of the regression parameter in (2.1). The null hypothesis of interest is H0 :

t

for some constant vector

=

and for all t:

The alternative hypothesis HA is that H0 is false. Under H0 ; the unknown constant parameter vector

can be consistently estimated by the OLS estimator ^ = arg min

2Rd

Statistical inferences on the parameter

T X

(Yt

X0t )2 :

(2.2)

t=1

can be made using the OLS estimator ^ : Various

inferential procedures have been available, depending on the assumption on the regression error "t (i.e., whether f"t g is i.i.d.(0;

2

); or whether there exists serial correlation and/or conditional

heteroskedasticity in f"t g).

Detection and identi…cation of structural changes have been a long-standing problem in time

series econometric modelling. Most existing works in the econometric literature are concerned

6

with abrupt structural breaks, either a single or multiple ones, and their locations. Little attention has been devoted to smooth structural changes. This is the main focus of the present paper. Under the alternative HA ; where t

t

t

is a time-varying parameter. We are interested in the alternatives

is a smooth deterministic function of t: A simple example of a smooth change parameter

is when Xt = 1: Then model (2.1) boils down to a deterministic trend model, which has been

considered by Hall and Hart (1990), Johnstone and Silverman (1997), and Robinson (1997) for time domain smoothing. Because deterministic trends need not be polynomial (see Phillips 2001), the nonparametric time trend model will allow more ‡exibility in capturing the trend dynamics of Yt :2 Another example of a smooth change parameter is a STR model, as developed by Granger ::

::

and Terasvirta (1993) and Lin and Terasvirta (1994), where t

0

and

1

are d

=

0

+

1 F (t);

1 parameter vectors, and F (t) is a smooth transition function. A choice of

F (t) is a logistic function, namely F (t) = f1 + exp [ where c and

(t

c)]g

1

;

are scalar parameters governing the threshold and speed of the transition.

By construction, tests for parametric smooth change alternatives have best power against the assumed alternative. Unfortunately, usually no prior information about the structural change alternative is available in practice. To cover a wide range of alternatives, we consider the following nonparametric time-varying parameter regression model: Yt = X0t where

t T

+ "t ;

t = 1; :::; T;

(2.3)

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

[0; 1]: Discontinuities of ( ) at a …nite number of points in [0; 1] allow abrupt structural changes.

This model is introduced by Robinson (1989, 1991) and its nonparametric estimation has been considered in Robinson (1989, 1991), Orbe et al. (2000, 2005, 2006) and Cai (2007).3 It 2

Juhl and Xiao (2005) propose a nonparametric test for structrual changes in a deterministic trend model, where the functional form of the trend is unknown. They …rst detrend the original data nonparametrically to avoid the bias in estimating parameters. Unlike existing tests in the literature, their test has monotonic power even when there are two changes in the trend. 3 These authors consider nonparametric estimation of time-varying parameters (t=T ) and 2 (t=T ); where var("t ) = 2 (t=T ): They do not consider testing parameter constancy.

7

avoids restrictive parameterization of ( ). The speci…cation that parameter ( ) is a function 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, Park and Hahn 1999 and Cai 2007). It might …rst appear a bit strange because it makes the time-varying parameter

t

depend on the sample size T: The

reason for this requirement is that a nonparametric estimator for

t

will not be consistent unless

the amount of data on which it depends increases, and merely increasing the sample size will not necessarily improve estimation of is imposed on

t:

t

The amount of local information must increase suitably if the variance and

bias of a nonparametric estimator of this is to regard

at some …xed point t; even if some smoothness condition

t

t

are to decrease suitably. A convenient way to achieve

as ordinates of smooth function

( ) on an equally spaced grid over [0; 1];

which becomes …ner as T ! 1; and consider estimation of

( ) at …xed points : Consistent

estimation of model (2.3) has been considered in Robinson (1991) and Cai (2007) using local constant smoothing and local linear smoothing respectively. The speci…cation of

t

= (t=T ) does not regard the sampling of (Yt ; X0t )0 as taking place on a

grid on [0; 1]; which would make the preservation of independence or weak dependence properties as T increases implausible. We note that the device of taking (Yt ; X0t )0 to be observations at intervals 1=T on a continuous process on [0; 1] that itself is independent of T would not work because it does not achieve the accumulation of new information as T increases, which is necessary for consistency. Making parameter 4

econometrics.

t

dependent on T is not unknown elsewhere in statistics and

A well-known example is the local power analysis, where local alternatives are

usually speci…ed as a function of the sample size T: Model (2.3) includes the class of locally stationary autoregressive models as considered in Dahlhaus (1996, 1997): Yt = where "t = (t=T )vt and vt

0

t T

+

p X j=1

j

t T

Yt

j

+ "t ;

i:i:d:N (0; 1): Intuitively, locally stationary processes are nonsta-

tionary time series whose behavior can be locally approximated by a stationary process. In time series econometrics, it is often assumed that nonstationary economic time series can be transformed, mainly by removing time trends and/or taking di¤erences, into a stationary process. In fact, the transformed time series may not be stationary, although trending behaviors have been removed. Locally stationary time series models nicely …ll this gap. They have attracted increasing attention in economics and …nance since they provide new insight into modelling such economic and …nancial variables as interest rates and stock prices (see, e.g., Polasek 1989, 4 It may be noted that any asymptotic analysis is open to the criticism that it is impossible in many applications to increase T; the motivation being merely to provide approximate justi…cation for an inferential procedure based on a …nite sample.

8

Stephan and Skander 2004). We will assume that

( ) is continuous except for a …nite number of points on [0; 1]: In

other words, we permit ( ) to have …nitely many discontinuities. Hence, single structural break or multiple breaks with known or unknown break points, which are often considered in this literature, are special cases of model (2.3). For example, suppose ( ) is a jump function, namely,

( )=

(

0;

if

1;

otherwise:

0;

Then we obtain the single break model considered in Chow (1960). 3. NONPARAMETRIC TESTING Almost all existing tests for structural changes in the literature consider an alternative of abrupt structural changes. A di¢ culty for testing H0 lies in the fact that the possible changepoints are usually unknown in practice so one needs to check for all t: Extending Chow’s (1960) test, which assumes a single break with a known break point, Andrews (1993), Andrews and Ploberger (1994), and Andrews et al. (1996) propose sup FT and exp FT tests, namely, sup FT

sup FT ( ) ; 2

and mv=2

exp FT = (1 + c)

X

exp

2

where

mv c FT ( ) J ( ) ; 2 1+c

is the discrete set of the unspeci…ed change points, FT ( ) is the standard F test of

structural change occurring at the time [T ]; c and J( ) are a constant and a weighting function respectively that have to be prespeci…ed by practitioners. Bai and Perron (1998, 2003) further generalize Andrews’(1993) test to cases where the exact number of breaks is unknown but the upper bound is known. They develop two double maximum tests. The …rst is an equal-weight version, namely, U D max FT = max

1 m M(

where FT ( 1 ;

2 ; :::;

m)

sup

FT ( 1 ;

2 ; :::;

m) ;

1 ;:::; m )2

is the F test of structural changes occurring at ([T

1 ]; [T

2 ]; :::; [T

m ])

and M is a prespeci…ed upper bound of the number of breaks. The second test applies weights to the individual tests so that the marginal p values are equal across values of m; namely, W D max FT = max

1 m M

c ( ; 1) c ( ; m) (

sup

FT ( 1 ;

where c( ; m) is the asymptotic critical value of the test sup( 9

2 ; :::;

m) ;

1 ; 2 ;:::; m )2

1 ; 2 ;:::; m )2

FT ( 1 ;

2 ; :::;

m)

for

a signi…cance level a: These tests are not most suitable for smooth structural changes, although they may have some power against them. By construction,

is a strict subset of f1=T; 2=T; :::; 1g;

so it is impossible to test structural changes for all t theoretically.5 In practice, the range from [0:15; 0:85] is commonly chosen (e.g., Stock and Watson 1996, Ben-David and Papell 1998 and McConnell and Perez-Quiros 2000). Obviously, a restricted range will lead to signi…cant power loss for the aforementioned tests. On the other hand, although a supremum function has good power in detecting sudden structural breaks, it may not be very sensitive to smooth changes. Below, we shall propose consistent tests for smooth structural changes, which will complement the existing tests for sudden structural breaks and avoid the di¢ culty associated with the possibility of multiple breaks and/or the existence of unknown break dates. Recall that under H0 ; we have a simple constant parameter regression model Yt = X0t + "t ; and HA ;

can be consistently estimated by the OLS estimator ^ in (2.2). Under the alternative t

= (t=T ) is changing over time. The OLS estimator ^ is no longer suitable under HA

because there exists no parameter

such that E(Yt jXt ) = X0t

estimator can consistently estimate the time-varying parameter

a.s.. However, a nonparametric t

= (t=T ): A consistent test

can be constructed by comparing the OLS estimator ^ with its nonparametric estimator ^ t . Various popular nonparametric methods could be used to estimate the time-varying parameter ( ). Robinson (1991) uses a kernel method, which is a local constant smoothing method. Here, we use local linear smoothing, which includes the kernel method as a special case. Local linear smoothing is introduced originally by Stone (1977) and subsequently studied by Cleveland (1979), Fan (1992, 1993), Ruppert and Wand (1994), Masry (1996a, 1996b) and Masry and Fan (1997), among many others. Local linear smoothing has signi…cant advantages over the conventional Nadaraya–Watson (NW) kernel estimator on the boundary of design points. Cai (2007) shows that although the NW and local linear estimators share the same asymptotic properties at the interior points, their boundary behaviors are very di¤erent, namely, the NW estimator su¤ers from boundary e¤ects but the local linear estimator does not. The use of local linear smoothing is quite suitable in the present context.6 In particular, structural changes near the boundary are notoriously di¢ cult to detect, as shown by many previous works in the literature (see, e.g., Chu, 5

The reason that is a strict subset of f1=T; 2=T; :::; 1g is to prevent the test statistic from diverging to in…nity under H0 : See (e.g.) Andrews (1993) for detailed discussion. 6 Local linear smoothing and the conventional kernel method are both local smoothing. Global smoothing is another class of nonparametric method. Local smoothing is more suitable in the present context. ( ) may not have a nice shape and many terms are needed when using serial expansion, which complicates the estimation. On the other hand, structural change is the local behavior of parameters and hence local smoothing is expected to have better power.

10

Hornik and Kuan 1995). Unlike many existing tests, no trimming is needed for the local linear smoother, which can estimate structural changes near boundary regions. Thus, it is expected to give better power in such cases. Put Zst =

1;

s

t T

0

; and kst =

1 k h

s t Th

;

where the kernel k( ) : [ 1; 1] ! R+ is a prespeci…ed symmetric bounded probability density, and h

h(T ) is a bandwidth such that h ! 0 and T h ! 1 as T ! 1. For notational simplicity,

we have suppressed the dependence of Zst and kst on the sample size T and h: Examples of k( ) include the uniform kernel

the Epanechniov kernel

1 k(u) = 1(juj 2 3 k(u) = (1 4

and the quartic kernel k(u) =

u2 )1(juj

15 (1 16

(3.1)

1);

u2 )2 1(juj

(3.2)

1);

(3.3)

1);

where 1( ) is the indicator function. We note that although the local linear smoothing can enhance the convergence rate of the asymptotic bias in the boundary regions [1; T h] [ [T

T h; T ] from h to h2 ; the scale or propor-

tionality is di¤erent from that of an interior point. As shown in Cai (2007), the asymptotic bias R1 at an interior point is proportional to h2 1 u2 k(u)du while that at a boundary point is proR1 i 2 portional to h2 b(c); where b(c) = ( 22c u k(u)du; i = 0; 1; 2; 3 1c 3c )=( 0c 2c 1c ); ic = c

and 0 < c < 1: Moreover, although the convergence rate is the same as in the interior region,

the asymptotic variance at a boundary point tends to be larger. This is because on a smaller interval, fewer observations contribute to the estimator. These di¤erences would complicate the form of the proposed test statistics. To make the behavior of the local linear estimator at boundary points similar to that at interior points, we follow Hall and Wehrly (1991) to re‡ect the data in the boundary regions, obtaining pseudodata Xt = X t ; Yt = Y

t

for

integer part of T h; and Xt = X2T t ; Yt = Y2T

t

bT hc

for T + 1

t t

2; where bT hc denotes the

T + bT hc: We use the overall

data (i.e., the union of the original data and the pseudodata) to estimate symmetric data are available in the original boundary regions [1; T h] [ [T

t:

By construction,

T h; T ]: Except in the

nonparametric regression estimation, this method has also been described as “re‡ection about the boundaries”or “boundary folding”by Schuster (1985), Silverman (1986) and Cline and Hart (1991) in a nonparametric density estimation framework. The local linear parameter estimator is obtained by minimizing the local sum of squared 11

residuals: t+bT hc

min

2R2d

where

=(

Qst = Zst

0 0;

X

0 0 Xs

kst Ys

s

0 1

0 0 1)

is a 2d

Xs is a 2d

1 vector,

j

is a d

1 vector, and

=

Xs

T

s=t bT hc

t+bT hc

2

t

X

kst (Ys

0

Qst )2 ;

(3.4)

s=t bT hc

1 coe¢ cient vector for ( sT t )j Xs ; j = 0; 1; and

is the Kronecker product. Note that the device of

using pseudodata does not a¤ect the estimation at interior points [T h; T

T h]: By solving the

optimization problem in (3.4), we can obtain the solution: 0

^ =@ t

1

t+bT hc

X

kst Qst Q0st A

s=t bT hc

1

t+bT hc

X

kst Qst Ys :

(3.5)

s=t bT hc

This is a locally weighted least square estimator. Indeed, as pointed out by Cai (2007), the local linear estimator can be regarded as the OLS estimator of the transformed model 1=2

s

1=2

1=2

kst Yt = kst X0s

0

+ kst

t T

X0s

1=2

1

+ kst "s ;

s = 1; :::; T:

Hence the estimation can be implemented by any standard econometric software. Put e1 = (1; 0)0 : Then the local linear estimator for ^ t = (e01

t

is given by

Id ) ^ t ;

(3.6)

which is the slope estimator. We note that one can also use the kernel method, which is equivalent to a restricted local linear estimation with the restriction

1

= 0: The test statistics and asymptotic distribution are

the same no matter whether the local constant or local linear estimation is used. The choice of the bandwidth h is generally believed to be more important than the choice of kernel function k( ) in the estimation of ( ): A small h tends to reduce the bias in ^ t ; and a large h tends to reduce the variance of ^ t : In practice, h is often chosen in an ad hoc manner. More sophistically, an automatic method such as cross-validation may be used. To describe this, de…ne a “leaveone-out”estimator ^ where ^

t

0

=@

t

= (e01

1

t+bT hc

X

s=t bT hc;s6=t

Id ) ^ t ;

kst Qst Q0st A 12

1

t+bT hc

X

s=t bT hc;s6=t

kst Qst Ys :

Then a possible choice of h is ^ CV = arg min CV (h); h h

where

T +bT hc

X

CV (h) =

X0t ^ t )2 :

(Yt

t= bT hc

With ^ t ; a generalization of Chow’s (1960) F test can be constructed. The idea is to compare the sum of squared residuals of the restricted constant parameter model with the sum of squared residuals of the unrestricted nonparametric time-varying parameter model. Let SSR0 denote the sum of squared residuals of the constant parameter model, that is, SSR0 =

T X

(Yt

X0t ^ )2 ;

t=1

where ^ is the OLS estimator in (2.2). Let SSR1 denote the sum of squared residuals of the nonparametric time-varying parameter model, that is, SSR1 =

T X

(Yt

X0t ^ t )2 ;

t=1

where ^ t is the nonparametric time-varying parameter estimator in (3.6). It is important to note that SSR0 and SSR1 are sums of squared residuals of the original data, namely, fXt ; Yt gTt=1 : The

pseudodata are only used for estimating

t

in the boundary regions. Hence it will not a¤ect the

asymptotic distribution of test statistics. Intuitively, the use of the pseudodata only has impact on the boundary regions [1; T h] [ [T

T h; T ] and its order of magnitude will vanish as h ! 0:

However, it improves the …nite sample performance of the proposed tests.

Then the generalized Chow’s (1960) F test statistic is based on the comparison of SSR0 and SSR1 : C^ =

p

h(SSR0

SSR1 ) ^C B

p

A^C

;

^C are some suitable centering and scaling factors. Their expressions, to be where A^C and B discussed in Section 4, depend on the assumption on the regression error f"t g as well as its relationship with the regressor fXt g. Under H0 ; SSR0 and SSR1 will be close to each other,

but under the alternative HA ; however, SSR0 will always be larger than SSR1 at least for large samples, giving the generalized Chow test C^ asymptotic unit power. Unlike Chow’s (1960) test, the generalized Chow test C^ does not follow an F distribution even if the regression error f"t g is i.i.d. N(0,1). Instead, it follows a convenient asymptotic N(0,1) distribution.

In addition to Chow’s (1960) test, many existing tests for structural changes in the literature 13

are also based on the analysis of residuals. For example, the most well-known CUSUM and CUSUM-SQ tests of Brown et al. (1975) and MOSUM tests of Hackl (1980) are based on the (standardized) recursive residual X0t a ^t 1 );

wt = rt (Yt where a ^t rt = [1 +

1

is the recursive OLS estimator of

X0t (U0t 1 Ut 1 ) 1 Xt ] 1=2 ;

Ut

1

t = d + 1; :::; T; based on the observations prior to t; and

= (X1 ; X2 ; :::; Xt 1 )0 : The weighting rt is approximately

the inverse of the standard deviation of the estimated recursive residual Yt X0t a ^t 1 : Like the ^ these tests do not require a prior speci…cation of when the structural generalized Chow test C, changes will take place. Nevertheless, the recursive residual is analogous to an integrating procedure that may smooth out structural changes to certain extent. Therefore, we may expect that local smoothing is more powerful than the tests based on the recursive residuals when structural changes are smooth and non-monotonic. In addition, it is well-known that the location of the structural change a¤ects the power of the CUSUM test. Break points near the left boundary of the sample period lead to a more powerful test, and the power is reduced dramatically if the break point is moved toward the end of the sampling period. In contrast, local smoothing is expected to have symmetric power no matter whether the change occurs in the left or right end of the sampling period. Another test for H0 can be developed by comparing the OLS regression estimator and the nonparametric regression estimator. This can be viewed as a generalized Hausman’s (1978) test. Hausman’s (1978) test is a convenient speci…cation test that compares two parameter estimators, where one is e¢ cient but inconsistent under the alternative, and the other is ine¢ cient but consistent under the alternative. White (1981) extends Hausman’s (1978) test by allowing both parameter estimators to be ine¢ cient estimators. Here we extend Hausman’s (1978) methodology from a parametric regression context to a nonparametric time-varying parameter regression context, where the OLS regression estimator X0t ^ can be viewed as an e¢ cient estimator for E(Yt jXt ), and the nonparametric time-varying parameter regression estimator X0t ^ t

can be viewed as an ine¢ cient but consistent estimator for E(Yt jXt ) under both H0 and HA . We can compare these two estimators via a sample quadratic form: T 1X 0 ^ Q= (X ^ t T t=1 t

X0t ^ )2 :

^ converges to zero under H0 , but it converges to a strictly positive constant The quadratic form Q ^ from 0 is under HA , giving the test asymptotic unit power. Any signi…cant departure of Q

14

evidence of structural changes.7 The generalized Hausman test is de…ned as p ^ A^H T hQ ^ = p H ; ^H B

^H are some suitable centering and scaling factors, and their forms, to be discussed where A^H and B in Section 4, depend on the assumption on the regression error f"t g. ^ di¤ers from the generalized Chow test C^ in that H ^ is based The generalized Hausman test H on the comparison of the …tted values of the restricted and unrestricted regression models, while C^ is based on the comparison of the sums of squared residuals of the restricted and unrestricted models. As will be seen below, although the classical Chow’s (1960) test has the optimal power under a classical linear regression model with i.i.d. normal errors, this is no longer the case for ^ It is shown in Section 5 that the generalized Hausman test H ^ is the generalized Chow test C. asymptotically more e¢ cient than the generalized Chow test C^ in terms of the Bahadur (1960) asymptotic e¢ ciency criterion, and this is con…rmed in the simulation study in Section 6. 4. ASYMPTOTIC DISTRIBUTION ^ we impose To derive the null asymptotic distribution of the proposed test statistics C^ and H; the following regularity conditions. Assumption A.1: fX0t ; "t g0 is a (d + 1) 1 stationary -mixing process with mixing coe¢ cients P 2 (j) 1+ < C for some 0 < < 1. f (j)g satisfying 1 j=1 j d matrix M = E(Xt X0t ) is …nite and positive de…nite; (ii)

Assumption A.2: (i) The d

E(X8ti ) < 1 for i = 1; :::; d and for all t:

p

Assumption A.3: ^ is a parameter estimator such that p limT !1 ^ and

=

under H0 ; where

T (^

) = OP (1); where

=

is given in H0 :

Assumption A.4: The kernel function k : [ 1; 1] ! R+ is a symmetric bounded probability density function.

Assumption A.5: The bandwidth h = cT

for 0
< 1 2k (v) = k (u) k (u + v) du dv > 1 ^ 1 : C^1 ) = RE(H : i2 R 1 hR 1 > > ; : k (u) k (u + v) du dv 1 1

Suppose the bandwidth rate parameter

= 1=5; which gives the optimal bandwidth rate for esti-

mating (t=T ): Then for most commonly used kernels, such as the uniform kernel, the Epanech^ 1 : C^1 ) = 6:401=1:8 ; 3:601=1:8 ; niov kernel and the quartic kernel in (3.1)–(3.3), we have RE(H ^ 2 to C^2 3:451=1:8 = 2:80; 2:04; 1:99 respectively. We note that the Bahadur relative e¢ ciency of H ^ 1 : C^1 ): is the same as RE(H Caution may be taken when the generalized Chow and Hausman tests reject the null hypothesis H0 of parameter constancy. It is possible that the rejection of structural changes is due to nonlinear relationship or other model misspeci…cations rather than structural changes. For example, the choice of an inappropriate functional form and omitted variables can result in apparent structural changes. Of course, this feature is not particular to the proposed tests here, but relevant to all existing tests for structural breaks. 6. FINITE SAMPLE PERFORMANCE We compare the …nite sample performance of our tests with tests of Brown et al. (1975), ::

Hackl (1980), Lin and Terasvirta (1994), Andrews (1993), Andrews and Ploberger (1994) and Bai and Perron (1998, 2003), using Monte Carlo methods. We examine both size and power by testing structural stability against a number of alternatives. 6.1. Simulation design To examine the size of all tests under H0 , we consider the following DGP: 23

DGP S.1 [No Structural Change]: 8 > > < Yt = 1 + 0:5Xt + "t ;

Xt = 0:5Xt 1 + vt ; > > : " i:i:d:N (0; 1) and t

i:i:d:N (0; 1) :

t

We generate 1; 000 data sets of a random sample fXt ; Yt gTt=1 for T = 250; 500 and 1; 000 respectively, using the GAUSS Windows Version 4.0 random number generator on a personal computer.

To investigate the power of all tests in detecting structural changes, we consider four classes of alternatives: (i) a single break, (ii) multiple breaks, (iii) non-persistent temporal breaks, and (iv) smooth structural changes. DGP P.1 [Single Structural Break]:

Yt =

(

if t

1 + 0:5Xt + "t ;

t1 ;

1:2 + 0:7Xt + "t ; otherwise,

where t1 = 0:1T; 0:5T; 0:9T: DGP P.2 [Monotonic Multiple Structural Breaks]:

Yt =

8 > > < 1 + 0:5Xt + "t ;

if t

t1 ;

1:2 + 0:7Xt + "t ; if t1 < t < t2 ; > > : 1:4 + 0:9X + " ; otherwise, t t

where (t1 ; t2 ) are (0:2T; 0:4T ) ; (0:2T; 0:6T ) ; (0:4T; 0:6T ) : DGP P.3 [Non-monotonic Multiple Structural Breaks]:

Yt =

where (

01 ;

11 ;

8 > >
> : 1 + 0:5X + " ; otherwise, t t

02 ;

12 )

0:2T or 0:7T

t

0:8T;

0:5T ,

are (1:2; 0:7; 0:8; 0:3); (0:8; 0:3; 1:2; 0:7); (1:2; 0:7; 1:2; 0:7):

DGP P.4 [Non-persistent Temporal Structural Breaks]:

Yt =

(

1 + 0:5Xt + "t ;

if t

t1 or t

1:2 + 0:7Xt + "t ; otherwise,

where (t1 ; t2 ) are (0:2T; 0:4T ) (0:4T; 0:6T ) and (0:6T; 0:8T ) :

24

t2 ;

DGP P.5 [Smooth Structural Changes]: Yt = F ( ) (1 + 0:5Xt ) + "t ; where

=

t T

and three types of F ( ) are considered:

Case 1 [Monotonic Smooth Structural Changes]: F ( ) = 1 + exp

1500

3

3

2

+3

1

1

:

Case 2 [Non-monotonic Smooth Structural Changes]: F( )=1

exp

3(

0:5)2 :

Case 3 [High-frequency Smooth Structural Changes]: F ( ) = 0:5 + sin (25

):

The single break has been a structural change with classical importance. Under DGP P.1, an abrupt break occurs at t1 ; where t1 = 0:1T; 0:5T; 0:9T respectively: By considering di¤erent break points, we can investigate whether the power of various tests is sensitive to the location of the break. Another class of alternatives is multiple breaks. We consider two types of multiple breaks. The …rst is a class of monotonic multiple breaks over time. Under DGP P.2, Yt has two monotonic breaks at times t1 and t2 respectively, where the break sizes are larger at t2 than at t1 ; and t1 < t2 . We consider di¤erent combinations of t1 and t2 : The second is a class of non-monotonic multiple breaks, where the break sizes at a later period are not necessarily larger than at an earlier period. Under DGP P.3, Yt either jumps up, goes down and rebounds, or jumps down, then up and returns. A third class of alternatives is non-persistent temporal breaks, which only last for a short period of time. This may occur when a large shock (e.g., a war) only lasts for some period and consequently the behavior of economic agents will return to the pre-shock condition. Under DGP P.4, Yt jumps up, stays at that level for a period of time and then returns. The …nal class of alternatives is smooth structural changes. It includes monotonically increasing smooth structural changes, non-monotonically increasing smooth structural changes, and smooth but high-frequency structural changes. The …rst two cases of DGP P.5 are two STR ::

models considered by Lin and Terasvirta (1994), who use third-order and second-order logistic 25

functions respectively as the transition functions. These two processes are parametric alterna::

tives that Lin and Terasvirta’s LM 2; LM 3; F 2 and F 3 tests are designed for.8 In Case 3 of DGP P.5, Yt exhibits smooth but high-frequency structural changes. The sine function changes periodically over time, which can mimic some dynamic structures of business cycles. For each of DGPs P.1-P5, we generate 500 data sets of the random sample fXt gTt=1 for

T = 250; 500 and 1; 000 respectively. Both asymptotic critical values (ACV) and empirical

critical values (ECV) are used. ECVs are obtained under the null hypothesis (DGP S.1). They provide a fair ground to compare all tests under study. 6.2 Monte Carlo Evidence For the proposed generalized Chow and Hausman tests, we use the quartic kernel in (3.3) and our simulation experience suggests that the choice of k( ) has little impact on the performance p p 1 of tests. For simplicity, we choose h = (1= 12)T 5 ; where 1= 12 is the standard deviation of U (0; 1); which could be viewed as the limiting distribution of the grid points

t ; T

t = 1; :::; T; as

T ! 1: This simple bandwidth rule attains the optimal rate for local linear …tting. We compare the proposed tests with a variety of popular tests in the literature, namely, Brown et al.’s (1975) ::

CUSUM test, Hackl’s (1980) MOSUM test, Lin and Terasvirta’s(1994) LM 1 LM 3 and F 1 F 3 tests, Andrews’(1993) supremum LM and F tests, Andrews and Ploberger’s (1994) exponential and average LM and F tests, and Bai and Perron’s (1998, 2003) double maximum tests. For Hackl’s (1980) MOSUM test, we choose a window size of 0.5, following the recommendations of Chu et al. (1995). Following Andrews (1993), we choose

= [0:15; 0:85] for Andrews’(1993),

Andrews and Ploberger’s (1994) and Bai and Perron’s (1998, 2003) tests. Following Bai and Perron (2003), we set the upper bound of the number of breaks at 5: The rejection rates of all tests under DGP S.1 at the 10% and 5% signi…cance levels, using the asymptotic theory, are reported in Table 1. These rates show that the generalized Chow test tends to a bit overreject when T = 250, but improves as T increases. In contrast, the generalized Hausman test tends to a bit underreject when T = 250 , but improves as T increases. Not surprisingly, the …nite sample versions of the generalized Chow and Hausman tests perform better than their simpli…ed versions. For other tests, CUSUM, MOSUM, Andrews’(1993) supremum F and LM tests are a bit conservative, while Lin et al.’s LM 1

LM 3, F 1

F 3 and Bai and

Perron’s U D max and W D max tests show a bit overrejection at the 5% level. Overall, all tests have reasonable size performance, and parametric tests have better sizes in small samples.9 Next, we examine the power. The rejection rates of all tests under DGPs P.1-P.5 at the 5% level are reported in Tables 2-6. We use both ACVs and ECVs. The latter give size-corrected 8

To be consistent with our theory, we use the ratio rather than t to obtain a monotonically increasing deviation from the null model. 9 Using a bootstrap procedure, a better size performance can be obtained for the proposed tests, but with higher computational cost.

26

power. The discussion below is mainly based on ECVs. We …rst consider the deterministic single break (DGP P.1), namely, a single break with a given break point and size. For the interior break point, all tests have power against DGP P.1, although Andrews and Ploberger’s (1994) exponential and average LM and F tests have best power. At the left and right boundaries, the ::

generalized Hausman and Lin and Terasvirta’s LM and F tests perform symmetrically well and are more powerful than Andrews’(1993) and Andrews and Ploberger’s (1994) LM and F tests, and Bai and Perron’s double maximum tests. Moreover, the power of CUSUM and MOSUM is reduced dramatically when the break point moves toward the end of the sample period. When t1 = 0:9T , both CUSUM and MOSUM have no power, even with sample size T = 1; 000. We note that the generalized Hausman tests are more powerful than the generalized Chow tests under DGP P.1. Next, we consider a class of multiple breaks (DGPs P.2-P.3). DGP P.2 is such an alternative with monotonic increasing break sizes over time. When sample sizes are small, our new tests ::

are slightly less powerful than Lin and Terasvirta’s and Andrews’ (1993) and Andrews and Ploberger’s (1994) LM and F tests, and Bai and Perron’s double maximum tests. However, the rejection rates of the new tests increase quickly with the sample size and attain unity for all cases considered, which is not attainable by Andrews’(1993) and Andrews and Ploberger’s (1994) tests. CUSUM and MOSUM continue to be the least powerful, although their rejection rates also increase with the sample size T . Interestingly, using ACVs, Andrews and Ploberger’s (1994) exponential and average LM and F tests exhibit nonmonotonic power for all combinations of t1 and t2 considered. For example, when t1 = 0:4T and t2 = 0:6T; the rejection rates of average LM; exponential LM and F tests decrease dramatically to 38.2%, 79.0% and 67.6% respectively with sample size T = 1000; although they are all close to unity with T = 500: This undesirable nonmontonicity in power has been documented in Vogelsang (1999), which suggests that substantial power loss can result from using single break tests for the DGPs with multiple breaks. Under DGP P.3, which is an alternative of multiple breaks with non-monotonic break sizes. The new tests proposed dominate all other tests. For example, in the …rst case of DGP P.3, the new tests have rejection rates around 18% at the 5% level when T = 250, while all other tests have almost no power. When T = 1; 000, the rejection rates of the new tests are around 78% and 82% respectively, while the second best tests, Bai and Perron’s U D max and W D max tests have ::

rejection rates around 50.8% and 69.2% respectively. Lin and Terasvirta’s LM 1 and F 1 tests have no power even when T = 1; 000, while their LM 2, LM 3, F 2 and F 3 tests have rejection rates around 20%, a bit worse than those of Andrews’ (1993) and Andrews and Ploberger’s (1994) tests. Bai and Perron’s double maximum tests improve a lot upon Andrews’(1993) and Andrews and Ploberger’s (1994) single break tests, which con…rms Perron’s (2006) observation 27

that "while the test for one break is consistent against alternatives involving multiple changes, its power in …nite samples can be rather poor". The third class considered is a class of non-persistent temporal breaks. Under DGP P.4, the break only lasts for some period of time. The new tests proposed and Bai and Perron’s tests ::

outperform other tests for all sample sizes and all di¤erent break points. Lin and Terasvirta’s ::

LM 3 and F 3 tests perform second best, but Lin and Terasvirta’s LM 1 and F 1 have low or little power. Andrews’(1993) and Andrews and Ploberger’s (1994) tests do not perform as well as Lin ::

and Terasvirta’s LM 3 and F 3 tests, but outperform CUSUM and MOSUM. Finally, we consider the class of smooth structural changes (DGP P.5). Case 1 of DGP P.5 is a monotonically increasing smooth structural change process. It is a STR alternative considered ::

by Lin and Terasvirta (1994) with a third-order logistic function as the transition function. Lin ::

and Terasvirta’s LM 3 and F 3 tests have the best power while the new tests have very close rejection rates. Case 2 is an alternative with non-monotonic smooth structural changes. This is ::

another STR model considered by Lin and Terasvirta (1994), where the transition function is a ::

second-order logistic function. Lin and Terasvirta’s LM 2, LM 3, F 2 and F 3 tests have the best power. The new tests and Bai and Perron’s tests perform slightly worse than these tests but better than Andrews’(1993) and Andrews and Ploberger’s (1994) tests. As expected, Lin et al.’s LM 1 and F 1, which are based on the …rst order Taylor approximation, have no power against this alternative, even when T = 1; 000: Case 3 is an alternative with smooth but high-frequency structural changes. The new tests dominate all tests and Bai and Perron’s tests, Andrews’(1993) supremum LM and F tests, and Andrews and Ploberger’s (1994) exponential LM and F tests ::

outperform Lin and Terasvirta’s tests, which are designed for smooth structural changes. In summary, we observe: The generalized Chow and Hausman tests have reasonable sizes in …nite samples. The new tests have reasonable all-around power against both smooth and abrupt structural changes. In most cases considered, they outperform all other tests in detecting smooth structural changes; they have good power against various monotonic and non-monotonic structural breaks, including the alternatives where the break occurs at the boundary of the sample period. The new tests are not always the most powerful in detecting each of the alternatives considered. However, they have relatively omnibus power against all …ve DGPs, provided the sample size is su¢ ciently large. Tests for parametric smooth structural changes are powerful against the speci…ed alternatives but they may have low power if the polynomial order is not high enough; tests with the trimmed range also have a danger of omitting breaks occurring near the boundary of the sample period; the power of tests with recursive 28

residuals depends crucially on the location of break points; tests for single break may have rather poor or nonmonotonic power against alternatives involving multiple breaks. The generalized Hausman test is more powerful than the generalized Chow test in most cases, which is consistent with the theory on asymptotic e¢ ciency analysis. 7. CONCLUSION Detection and identi…cation of structural breaks have attracted a great amount of attention in econometrics over the past few decades. We have contributed to this literature by proposing two new nonparametric tests for smooth structural changes as well as abrupt structural breaks. Our tests have intuitive appeal because they can be regarded as the generalization of Chow’s (1960) and Hausman’s (1978) tests from a parametric context to a nonparametric context. They have a convenient null asymptotic N(0,1) distribution, do not require trimming data, do not require prior information on the possible alternative, and are consistent against all smooth structural changes as well as multiple abrupt structural breaks. Simulation studies show that the proposed tests perform reasonably in …nite samples, which demonstrates the nice features of the nonparametric tests. Extensions of the proposed nonparametric method to linear regression models with nonstationary regressors or serially correlated errors with time-varying variances are possible. Moreover, the approach developed in this paper can be adopted to test whether an ARMA model or a GARCH model has smooth structural changes, using the log-likelihood criterion. Also, it can be used to test whether a time trend follows a polynomial of time, with the stochastic component being a weakly stationary but not necessarily martingale di¤erence sequence. It can be generalized to test whether smooth structural changes exist in panel data models as well. All these are left for future research. REFERENCES

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33

Table 1.1 Sizes of Tests Under DGP S.1 T

α Cˆ1 ~ C1 Hˆ 1 ~ H1 CUSUM MOSUM LM1 LM2 LM3 F1 F2 F3 Sup-LM Exp-LM Ave-LM Sup-F Exp-F Ave-F UDMax WDMax

250

500

1000

.10

.05

.10

.05

.10

.05

.147

.086

.104

.065

.091

.047

.083

.053

.080

.044

.070

.036

.066

.040

.069

.044

.074

.042

.135

.087

.112

.071

.101

.057

.078 .077 .111 .105 .097 .105 .103 .096 .076 .105 .109 .085 .115 .111 .117 .127

.043 .043 .062 .052 .053 .061 .052 .053 .042 .049 .058 .048 .052 .064 .066 .070

.095 .088 .083 .084 .085 .082 .083 .085 .087 .086 .080 .091 .090 .080 .106 .105

.042 .038 .040 .039 .048 .040 .039 .048 .045 .047 .046 .051 .051 .047 .050 .053

.082 .085 .101 .085 .103 .101 .085 .103 .094 .105 .099 .098 .111 .100 .098 .105

.043 .047 .045 .047 .045 .045 .047 .045 .051 .050 .050 .058 .053 .051 .055 .047

~ Notes: (1) Cˆ1 and C1 are the generalized Chow test and its simplified version, given in ~ equations (4.1-4.3) and (4.4-4.5) respectively. Hˆ 1 and H 1 are the generalized Hausman test and its simplified version, given in equations (4.6-4.8) and (4.9-4.10); CUSUM and MOSUM are Brown et al.'s (1975) CUSUM and Hackl's (1980) MOSUM tests respectively; LM1-3 and F1-3 are Lin and Teräsvirta’s (1994) LM and F tests respectively; Sup-LM and Sup-F are Andrew's (1993) supremum LM and Ftests respectively; Exp-LM, Exp-F and Ave-LM, and Ave-F are Andrew and Ploberger's (1994) exponential and average LM and F tests; UDMax and WDMax are Bai and Perron’s (1998, 2003) double maximum tests. (2) p values are based on 1000 iterations.

Table 2. Powers of Tests under DGP P.1: Single Structural Break t=0.5T t₁=0.1T 250

Cˆ1 ~ C1 Hˆ 1 ~ H1 CUSUM MOSUM LM1 LM2 LM3 F1 F2 F3 Sup-LM Exp-LM Ave-LM Sup-F Exp-F Ave-F UDMax WDMax

500

1000

250

500

1000

t=0.9T

250

500

1000

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

.134

.080

.168

.148

.258

.286

.292

.188

.520

.468

.770

.782

.162

.092

.194

.168

.254

.264

.094

.080

.144

.148

.238

.270

.196

.178

.438

.472

.744

.780

.102

.088

.146

.168

.228

.258

.072

.084

.220

.172

.328

.354

.206

.228

.538

.568

.766

.876

.106

.114

.162

.182

.332

.360

.144

.084

.160

.184

.378

.366

.364

.246

.642

.578

.800

.872

.164

.110

.218

.182

.378

.368

.088 .088 .082 .118 .114 .082 .118 .116 .070 .088 .082 .082 .100 .084 .110 .122

.109 .096 .070 .118 .112 .070 .118 .112 .074 .084 .064 .088 .010 .072 .082 .072

.158 .116 .124 .184 .178 .124 .184 .178 .128 .130 .102 .140 .140 .102 .156 .164

.184 .136 .144 .212 .188 .144 .212 .188 .128 .130 .108 .140 .140 .118 .158 .142

.326 .278 .254 .388 .438 .254 .386 .438 .310 .280 .226 .316 .282 .226 .316 .304

.370 .292 .266 .426 .478 .266 .426 .478 .302 .280 .226 .310 .280 .226 .298 .288

.084 .218 .418 .326 .290 .414 .326 .290 .364 .432 .440 .412 .456 .450 .424 .394

.096 .234 .372 .322 .286 .372 .322 .286 .382 .430 .400 .418 .450 .412 .358 .278

.138 .426 .728 .602 .614 .728 .602 .616 .740 .784 .766 .744 .788 .770 .762 .750

.158 .452 .762 .646 .626 .762 .646 .626 .738 .784 .782 .744 .788 .786 .766 .718

.346 .740 .960 .930 .926 .960 .930 .926 .954 .970 .964 .958 .972 .964 .954 .946

.382 .756 .964 .938 .932 .964 .938 .932 .954 .970 .966 .954 .968 .966 .952 .940

.048 .056 .088 .112 .126 .088 .112 .126 .076 .080 .092 .090 .090 .092 .120 .128

.056 070 .076 .112 .124 .076 .112 .124 .086 .080 .072 .092 .086 .076 .100 .064

.036 .036 .130 .190 .208 .130 .190 .208 .154 .148 .112 .158 .152 .116 .170 .186

.048 .050 .160 .228 .222 .160 .228 .222 152 .148 .124 .158 .152 .126 .170 .160

.046 .066 .278 .358 .438 .278 .356 .438 .338 .292 .216 .342 .294 .218 .314 .300

.062 .066 .286 .388 .476 .286 .388 .476 .336 .292 .216 .336 .290 .216 .304 .290

~ Notes: (1) Cˆ1 and C1 are the generalized Chow test and its simplified version, given in equations (4.1-4.3) and (4.4-4.5) respectively. Hˆ 1 and ~ H 1 are the generalized Hausman test and its simplified version, given in equations (4.6-4.8) and (4.9-4.10); CUSUM and MOSUM are Brown et al.'s (1975) CUSUM and Hackl's (1980) MOSUM tests respectively; LM1-3 and F1-3 are Lin and Teräsvirta’s (1994) LM and F tests respectively; Sup-LM and Sup-F are Andrews’ (1993) supremum LM and F tests respectively; Exp-LM, Exp-F and Ave-LM, and Ave-F are Andrews and Ploberger's (1994) exponential and average LM and F tests; UDMax and WDMax are Bai and Perron’s (1998, 2003) double maximum tests. (2) p values are based on 500 iterations. (3) T=sample size; ACV=asymptotic critical value; ECV=empirical critical value.

Table 3. Powers of Tests under DGP P.2: Monotonic Multiple Structural Breaks t₁=0.2T, t₂=0.4T t₁=0.2T, t₂=0.6T t₁=0.4T, t₂=0.6T

250 Cˆ1 ~ C1 Hˆ 1 ~ H1 CUSUM MOSUM LM1 LM2 LM3 F1 F2 F3 Sup-LM Exp-LM Ave-LM Sup-F Exp-F Ave-F UDMax WDMax

500

1000

250

500

1000

250

500

1000

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

.652

.552

.922

.904

1.00

1.00

.586

.476

.880

.846

1.00

1.00

.752

.672

.988

.982

1.00

1.00

.562

.546

.898

.910

1.00

1.00

.492

.472

.838

.848

1.00

1.00

.682

.662

.982

.984

1.00

1.00

.628

.642

.944

.952

1.00

1.00

.564

.580

.912

.922

1.00

1.00

.786

.794

.994

.996

1.00

1.00

.760

.662

.962

.956

1.00

1.00

.692

.596

.942

.930

1.00

1.00

.856

.804

.998

.996

1.00

1.00

.364 .536 .828 .834 .782 .824 .834 .782 .808 .864 .862 .832 .870 .870 .856 .816

.380 .568 .808 .830 .780 .808 .830 .780 .844 .870 .832 .844 .870 .842 .796 .700

.686 .842 .986 .992 .984 .986 .992 .984 .994 .994 .994 .994 .994 .994 .992 .988

.730 .862 .990 .992 .984 .990 .992 .984 .994 .994 .994 .994 .994 .994 .992 .986

.946 .988 1.00 1.00 1.00 1.00 1.00 1.00 1.00 .958 .840 1.00 .932 .798 1.00 1.00

.952 .992 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

.260 .402 .834 .754 .688 .834 .752 .688 .730 .818 .818 .764 .826 .824 .774 .748

.274 .432 .818 .748 .682 .818 .748 .682 .774 .824 .810 .774 .824 .812 .720 .632

.548 .686 .992 .986 .962 .992 .986 .962 .988 .994 .996 .990 .994 .996 .988 .986

.594 .712 .992 .988 .968 .992 .988 .968 .990 .994 .996 .990 .994 .996 .988 .986

.888 .954 1.00 1.00 1.00 1.00 1.00 1.00 1.00 .994 .884 1.00 .988 .850 1.00 1.00

.914 .954 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

.358 .626 .946 .900 .862 .946 .900 .862 .892 .936 .946 .900 .942 .950 .936 .920

.382 .650 .934 .896 .858 .934 .896 .858 .904 .942 .938 .904 .942 .938 .922 .872

.706 .886 1.00 1.00 .998 1.00 1.00 .998 1.00 1.00 .998 1.00 .998 .956 1.00 1.00

.736 .910 1.00 1.00 .998 1.00 1.00 .998 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 .960 1.00 1.00 1.00 1.00 1.00 1.00 1.00 .790 .382 1.00 .676 1.00 1.00 1.00

.968 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00


Table 4. Powers of Tests under DGP P.3: Non-monotonic Multiple Structural Breaks ( α 01, α 11, α 02, α 12)=(1.2,0.7,0.8,0.3) ( α 01, α 11, α 02, α 12)=(0.8,0.3,1.2,0.7) ( α 01, α 11, α 02, α 12)=(1.2,0.7,1.2,0.7)


500

1000

250

500

1000

250

500

1000

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

.256

.176

.506

.464

.770

.778

.260

.192

.442

.408

.786

.794

.198

.122

.286

.264

.544

.552

.180

.178

.446

.458

.730

.772

.204

.182

.398

.414

.772

.796

.126

.112

.244

.266

.502

.546

.160

.174

.450

.486

.794

.824

.180

.202

.384

.410

.800

.812

.086

.098

.206

.228

.516

.542

.258

.186

.550

.482

.828

.820

.266

.202

.480

.420

.826

.818

.178

.112

.290

.224

.556

.546

.064 .086 .058 .074 .074 .056 .072 .072 .066 .078 .074 .072 .090 .078 .140 .192

.098 .078 .048 .072 .072 .048 .072 .072 .074 .090 .064 .074 .088 .064 .094 .122

.070 .106 .044 .102 .092 .044 .102 .092 .142 .144 .094 .158 .152 .100 .292 .410

.090 .122 .052 .126 .098 .052 .126 .098 .158 .154 .116 .158 .152 .116 .292 .386

.116 .214 .060 .152 .188 .060 .152 .188 .300 .286 .196 .308 .292 .198 .542 .704

.136 .220 .064 .176 .214 .064 .176 .214 .300 .286 .198 .300 .286 .198 .508 .692

.052 .074 .056 .076 .084 .056 .076 .084 .078 .092 .080 .100 .100 .086 .166 .214

.066 .084 .050 .068 .082 .050 .068 .082 .100 .100 .064 .100 .098 .064 .128 .138

.078 .096 .042 .060 .084 .042 .060 .084 .120 .128 .090 .132 .138 .092 .258 .346

.088 .112 .048 .092 .090 .048 .092 .090 .130 .140 .102 .130 .138 .102 .258 .320

.118 .222 .054 .150 .226 .054 .150 .226 .304 .330 .204 .322 .334 .208 .584 .722

.152 .234 .062 .174 .248 .062 .174 .248 .304 .332 .206 .304 .328 .206 .546 .714

.046 .054 .064 .064 .062 .062 .062 .062 .054 .062 .064 .066 .066 .064 .096 .130

.054 .070 .052 .062 .062 .052 .062 .062 .068 .066 .054 .068 .064 .054 .078 .084

.050 .048 .048 .058 .052 .048 .058 .052 .088 .084 .062 .092 .088 .064 .114 .174

.064 .060 .064 .076 .056 .064 .076 .056 .092 .086 .068 .092 .086 .068 .114 .150

.078 .088 .078 .112 .108 .078 .112 .108 .154 .142 .100 .162 .146 .104 .192 .338

.100 .094 .086 .128 .124 .086 .128 .124 .154 .142 .102 .154 .142 .102 .176 .326


Table 5. Powers of Tests under DGP P.4: Non-persistent Temporal Structural Breaks t₁=0.2T, t₂=0.4T t₁=0.4T, t₂=0.6T t₁=0.6T, t₂=0.8T


500

1000

250

500

1000

250

500

1000

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

.288

.142

.464

.298

.500

.516

.204

.142

.278

.250

.500

.512

.254

.170

.386

.346

.522

.540

.218

.132

.402

.302

.454

.510

.154

.128

.240

.250

.450

.510

.176

.156

.318

.346

.482

.532

.226

.158

.518

.302

.558

.588

.130

.142

.252

.284

.558

.596

.140

.160

.348

.374

.580

.616

.346

.168

.612

.296

.614

.598

.206

.148

.360

.304

.614

.600

.270

.172

.444

.382

.644

.624

.062 .082 .082 .104 .166 .082 .104 .166 .098 .120 .110 .114 .134 .114 .166 .206

.068 .092 .070 .104 .162 .070 .104 .162 .116 .132 .102 .116 .128 .102 .124 .128

.100 .124 .120 .114 .212 .118 .112 .214 .188 .204 .172 .196 .216 .176 .288 .348

.120 .158 .140 .148 .222 .140 .148 .222 .196 .216 .192 .196 .216 .190 .288 .310

.196 .232 .218 .232 .484 .218 .232 .484 .360 .380 .338 .362 .384 .338 .592 .662

.234 .236 .224 .250 .506 .224 .250 .506 .356 .380 .338 .356 .380 .338 .562 .656

.052 .070 .054 .112 .084 .054 .112 .084 .064 .084 .072 .078 .090 .078 .144 .206

.064 .084 .046 .106 .082 .046 .106 .082 .082 .088 .062 .082 .088 .062 .106 .126

.076 .102 .038 .172 .144 .038 .170 .144 .116 .130 .088 .128 .134 .092 .262 .348

.090 .122 .056 .210 .152 .056 .210 .152 .128 .136 .100 .128 .134 .100 .266 .314

.134 .228 .044 .346 .290 .044 .346 .290 .280 .270 .190 .294 .280 .190 .576 .650

.188 .232 .046 .380 .326 .046 .380 .326 .280 .270 .190 .280 .270 .190 .548 .636

.054 .082 .078 .092 .114 .078 .090 .114 .068 .098 .094 .080 .106 .098 .124 .170

.070 .094 .056 .086 .112 .056 .086 .112 .086 .108 .088 .086 .104 .088 .092 .108

.058 .126 .106 .116 .248 .102 .116 .248 .162 .190 .142 .178 .194 .144 .282 .374

.076 .150 .122 .142 .258 .122 .142 .258 .176 .194 .162 .176 .194 .158 .284 352

.100 .224 .212 .224 .454 .210 .224 .454 .356 .368 .348 .358 .374 .348 .594 .642

.122 .236 .218 .246 .472 .218 .246 .472 .356 .368 .348 .356 .368 .348 .566 .630


Table 6. Powers of Tests under DGP P.5: Smooth Structural Changes Monotonic Smooth Structural Changes


500

1000

Non-monotonic Smooth Structural Changes

250

500

1000

High-frequency Smooth Structural Changes

250

500

1000

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

.448

.362

.554

.502

.770

.778

.460

.374

.704

.664

.966

.966

.410

.302

.374

.332

.988

.992

.370

.350

.482

.502

.744

.768

.378

.356

.652

.666

.966

.966

.314

.302

.314

.336

.984

.990

.316

.334

.552

.578

.824

.854

.420

.452

.754

.788

.994

.994

.282

.310

.348

.378

.616

.662

.472

.352

.632

.588

.866

.854

.584

.450

.828

.792

.992

.992

.416

.326

.454

.390

.672

.658

.038 .064 .174 .266 .314 .174 .266 .316 .198 .212 .174 .228 .236 .180 .266 .284

.042 .068 .156 .262 .312 .156 .262 .312 .238 .226 .154 .238 .222 .154 .218 .192

.052 .106 .362 .518 .588 .362 .516 .590 .430 .396 .312 .440 .420 .318 .456 .416

.062 .134 .402 .568 .598 .402 .568 .598 .442 .422 .342 .442 .420 .340 .458 .392

.090 .230 .608 .830 .898 .606 .830 .898 .784 .734 .576 .784 .738 .580 .776 .758

.118 .238 .626 .842 .912 .626 .842 .912 .784 .734 .578 .784 .734 .578 .762 .752

.260 .386 .048 .630 .558 .048 .630 .558 .278 .296 .196 .318 .322 .206 .430 .476

.292 .424 .040 .624 .552 .040 .624 .552 .322 .324 .168 .322 .320 .170 .352 .340

.592 .712 .056 .920 .878 .056 .920 .878 .560 .580 .466 .580 .586 .476 .728 .776

.630 .734 .062 .936 .888 .062 .936 .888 .580 .588 .490 .580 .586 .490 .728 .756

.920 .958 .066 1.00 .998 .066 1.00 .998 .954 .962 .914 .956 .962 .918 .988 .996

.934 .958 .074 1.00 .998 .074 1.00 .998 .954 .962 .916 .954 .962 .916 .988 .996

.058 .044 .092 .140 .162 .092 .140 .162 .158 .162 .116 .174 .172 .126 .286 .360

.068 .056 .088 .136 .162 .088 .136 .162 .184 .166 .100 .184 .168 .102 .228 .242

.174 .082 .112 .208 .200 .112 .208 .200 .244 .250 .138 .258 .260 .142 .376 .472

.196 .112 .126 .234 .208 .126 .234 .208 .254 .260 .172 .254 .260 .170 .378 .434

.490 .202 .112 .320 .278 .112 .320 .278 .368 .324 .174 .378 .326 .176 .564 .702

.544 .210 .122 .338 .308 .122 .338 .308 .368 .324 .174 .368 .326 .176 .542 .690


Table 7. Powers of Tests under DGP P.6: Random Jumps Random Jump Time and Random Jump Size


500

1000

Random Jump Size and Deterministic Jump Time

250

500

1000

Random Jump Time and Deterministic Jump Size

250

500

1000

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

ACV

ECV

.448

.362

.554

.502

.770

.778

.460

.374

.704

.664

.966

.966

.410

.302

.374

.332

.988

.992

.370

.350

.482

.502

.744

.768

.378

.356

.652

.666

.966

.966

.314

.302

.314

.336

.984

.990

.316

.334

.552

.578

.824

.854

.420

.452

.754

.788

.994

.994

.282

.310

.348

.378

.616

.662

.472

.352

.632

.588

.866

.854

.584

.450

.828

.792

.992

.992

.416

.326

.454

.390

.672

.658

.038 .064 .174 .266 .314 .174 .266 .316 .198 .212 .174 .228 .236 .180 .266 .284

.042 .068 .156 .262 .312 .156 .262 .312 .238 .226 .154 .238 .222 .154 .218 .192

.052 .106 .362 .518 .588 .362 .516 .590 .430 .396 .312 .440 .420 .318 .456 .416

.062 .134 .402 .568 .598 .402 .568 .598 .442 .422 .342 .442 .420 .340 .458 .392

.090 .230 .608 .830 .898 .606 .830 .898 .784 .734 .576 .784 .738 .580 .776 .758

.118 .238 .626 .842 .912 .626 .842 .912 .784 .734 .578 .784 .734 .578 .762 .752

.260 .386 .048 .630 .558 .048 .630 .558 .278 .296 .196 .318 .322 .206 .430 .476

.292 .424 .040 .624 .552 .040 .624 .552 .322 .324 .168 .322 .320 .170 .352 .340

.592 .712 .056 .920 .878 .056 .920 .878 .560 .580 .466 .580 .586 .476 .728 .776

.630 .734 .062 .936 .888 .062 .936 .888 .580 .588 .490 .580 .586 .490 .728 .756

.920 .958 .066 1.00 .998 .066 1.00 .998 .954 .962 .914 .956 .962 .918 .988 .996

.934 .958 .074 1.00 .998 .074 1.00 .998 .954 .962 .916 .954 .962 .916 .988 .996

.058 .044 .092 .140 .162 .092 .140 .162 .158 .162 .116 .174 .172 .126 .286 .360

.068 .056 .088 .136 .162 .088 .136 .162 .184 .166 .100 .184 .168 .102 .228 .242

.174 .082 .112 .208 .200 .112 .208 .200 .244 .250 .138 .258 .260 .142 .376 .472

.196 .112 .126 .234 .208 .126 .234 .208 .254 .260 .172 .254 .260 .170 .378 .434

.490 .202 .112 .320 .278 .112 .320 .278 .368 .324 .174 .378 .326 .176 .564 .702

.544 .210 .122 .338 .308 .122 .338 .308 .368 .324 .174 .368 .326 .176 .542 .690


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