Tests for Parameter Constancy in Regression Models
MEI-YUAN CHEN Department of Economics National Chung Cheng University E-mail:
[email protected] May, 7, 2002
c Mei-Yuan Chen. The LATEX file is constan.tex.
1
Introduction
Testing structural breaks have been an important issue in theoretical and empirical econometrics. Recently, tests for structural breaks are developed emphatically in dynamic models, especially in models with nonstationary time series. It is a well known result that the existence of nonstationary variable(s) in a regression model can induce spurious structural breaks. Nunes, Newbold, and Kuan (1996) shows that spurious number of breaks occurs when regression models with nonstationary variables but without breaks. Gregory, Nason, and Watt (1996) investigates the performance of Hansen (1992)’s tests in detecting structural breaks in a cointegrating relationship.
2
Tests for Structural Breaks
Consider the following time-varying regression model yt = x′t βt + ǫt ,
t = 1, 2, . . . , T,
(1)
where xt is p × 1 vector, and {ǫt } is a stationary error terms. The null hypothesis of constant coefficients (no structural breaks) is H0 : βt = β0 ,
∀t.
In general, conventional tests for structural breaks at unknown points could be classified into two classes, one is based on parameter estimates, such as the recursive-estimates test (denoted as RE test hereafter) of Ploberger et al. (1989) and the moving-estimates test (denoted as ME test hereafter) of Chu et al. (1995a) and the other is based on regression residuals, such as the CUSUM test of Brown et al. (1975) and the MOSUM test of Bauer and Hackl (1978) in which recursive residuals are used; and the OLS-CUSUM test of Ploberger and Kramer (1992) and the OLS-MOSUM test of Chu et al. (1995b) in which OLS residuals are used.
2.1
RE and ME Tests
Let the j-th recursive and moving estimate of β0 be, respectively, βˆj
j X
j+[T h]
=
β˜j,h =
t=1
−1
xt x′t
X
t=j+1
j X t=1
−1
xt x′t
xt yt ,
j+[T h]
X
t=j+1
j = k, · · · , T ;
xt yt ,
j = 0, · · · , T − [T h] + 1,
where 0 < h < 1 characterizes the bandwidth of moving windows. Writing j = [T z] we have Q[T z] =
[T z] 1 X xt x′t , [T z] t=1
Q[T r],h =
1 [T h]
[T r]+[T h]
X
t=[T r]+1
xt x′t ,
where 0 < z ≤ 1, 0 ≤ r < 1, and [a] is the integer part of a. In what follows we let σ ˆT2 2 denote a consistent estimator of σ , ⇒ denote weak convergence, k·k denote the maximum norm, and W and W0 denote the k-dimensional Wiener process and Brownian bridge, respectively. Under the null hypothesis, Ploberger et al. (1989) show that the RE test is
RET = max k≤j≤T
j ˆ ˆ √ Q1/2 max kW0 (t)k, T (βj − βT ) ⇒ 0≤t≤1 σ ˆT T
and Chu et al. (1995a) show that the ME test is M ET,h
[T h] 1/2
˜ ˆ
√ = max QT (βj,h − βT )
⇒ 0≤j≤T −[T h]+1
σ ˆT T
max kW0 (t + h) − W0 (t)k.
0≤t≤1−h
While, the type 1 and type 2 modified RE and ME tests are written as RE1T
j √ Q1/2 (βˆj − βˆT )
, [T z] k≤j≤T σ ˆT T
[T h] 1/2
˜ ˆ
√ Q[T r],h(βj,h − βT ) = max
0≤j≤T −[T h]+1 σ ˆT T
j
√ Q−1/2 = max Q[T z] (βˆj − βˆT )
, T k≤j≤T σ ˆT T
[T h] −1/2
˜ ˆ
√ QT Q[T r],h(βj,h − βT ) = max
. 0≤j≤T −[T h]+1 σ ˆT T
=
M E1T,h RE2T M E2T,h
max
Since the functional central limit theorem and weak law of large number still hold after the modification, the RE1, ME1, RE2 and ME2 tests have same limiting distributions as the RE and ME tests do.
2.2
Residuals-Based Tests
Instead of focusing on the fluctuation of estimates, the other class of tests is constructed by looking at the fluctuation of residuals. Usually, two popular types of residuals are used: the recursive and OLS residuals. 2.2.1
The Recursive-CUSUM Test
Let Xr be a r × p submatrix of X. The recursive residuals are defined as wr = q
yr − x′r β˜r−1
′ 1 + x′r (Xr−1 Xr−1 )−1 xr
,
r = p + 1, . . . , T,
where x′r is the r-th row vector of X. Let Mr =
r 1X wj . σ ˆ p+1
2
Approximating the crossing boundary at significance level α by a linear boundary, p
p
y = bα T − p + 2bα T − p(r − p), the CUSUM test statistic suggested by Brown, Durbin, and Evan (1975) is CST =:
max
p+1 bα , 0 ≤ t ≤ 1 − h}.
T →∞
Choosing h = 0.5, Chu et al. (1995b) calculate the asymptotic critical values at different significant levels, such as b0.90 = 2.55047, b0.95 = 2.83338, and b0.99 = 3.39125. Replacing the recursive residuals with the OLS residuals, Chu et al. (1995b) suggest another test, the OLS-MOSUM test. The OLS-MOSUM statistic is defined as OMT,h =
max
0≤j≤T −[T h]
σ ˆT
1 √
j+[T h] X d u ˆ n = T n=j+1
max W 0 (t + h) − W 0 (t) .
0≤t≤1−h
(5)
That is, the limiting process of the OLS-MOSUM test is the increment of a Brownian bridge process which is the same as that of the moving-estimates test of Chu et al. (1995a). 4
3
Tests for Structural Breaks in Cointegrating Vector
Gregory, Nason and Watt (1996) utilize Hansen (1992)’s tests to investigate the test performance in cointegrating model with serial correlated disturbances which results from linear quadratic models. A brief description of the Hansen’s tests is presented as follows. Recall that the regression model we are considering is, under H0 , yt = x′t β + ǫt , and that zt = µz + zt−1 + ξt , where xt = (1, zt′ )′ , zt is a (p − 1) × 1 vector. Define the vector ut = (ǫt , ξt′ )′ and the following matrices (the long-run variance matrices): T X T 1X IE[uj u′t ], T →∞ T t=1 j=1
T X t 1X IE[uj u′t ], T →∞ T t=1 j=1
Ω = lim
Λ = lim
partitioned in conformity with u: Ω=
"
#
Ωǫǫ Ωǫξ Ωξǫ Ωξξ
,
Λ=
"
Λǫǫ Λǫξ Λξǫ Λξξ
#
.
+ −1 Also define Ωǫ·ξ = Ωǫǫ − Ωǫξ Ω−1 ξξ Ωξǫ and Λξǫ = Λξǫ − Λξǫ − Λξξ Ωξξ Ωξǫ . ǫt , (△zt − Now estimate (1) by OLS and obtain residual ǫˆ = yt − x′t βˆ and define u ˆt = (ˆ △¯ z )′ )′ , where △¯ z is the sample mean of △zt . With the u ˆt , form estimates of Ω and Λ, ˆ and Λ. ˆ The estimators of these long-run covariance matrices cab be obtained denoted by Ω ˆ and Λ ˆ as Ω and Λ from Andrews (1991) and Andrews and Monahan (1992). Partition Ω −1 + −1 ˆ Ω ˆ ˆ ǫ·ξ = Ω ˆ ǫǫ − Ω ˆ ǫξ Ω ˆ ˆ ˆ ˆ ˆ as: Ω ξξ ξǫ and Λξǫ = Λξǫ − Λξξ Ωξξ Ωξǫ . Define the transformed dependent variable:
ˆ ǫξ Ω ˆ −1 (△zt − △¯ z ). yt+ = yt − Ω ξξ The fully modified (FM) estimator of β is βˆ+ =
" T X t=1
(yt+ zt′ − (0
#" T X
ˆ ) Λ ξǫ +′
t=1
zy zt′
#−1
with the associated residual vector: + ′ ˆ+ ǫˆ+ t = yt − xt β .
From the “score” st : sˆt =
"
xt ǫˆ+ t
−
0 ˆ Λ+ ξǫ
!#
. 5
,
To improve the FM estimator, a new dependent variable suggested by Bewley (1979) is constructed as ˆ ˆ − λ)]△y ytBew = yt + [λ/(1 t
ˆ can to correct the problem resulting from the serial correlation of yt . The estimate of λ, λ, ˆ is the estimated coefficient of be obtains by regressing y on z, lag y, and lags of △z and λ + the lagged dependent variable. Taking yt as the dependent variable and estimate (1) by OLS and following the procedures described above, we have another “score”, sˆBew . t Three tests are discussed in this paper. The first test assumes there is a structural break at unknown point. Suppose the break point is at t = k, denote the statistic ˆ ǫ·ξ VT k ]−1 ST k , Fk = ST′ k [Ω
(6)
where ST k =
k X
VT k = MT k − MT k MT−1 T MT k ,
sˆt ,
t=1
and MT k =
k X
xt x′t .
t=1
Actually, the F test (6) is a test for a structural break at a known point and which is asymptotically distributed under H0 as χ2 with p degree of freedom. However, when the break point is unknown, then the test is Fsup = sup Fk , k∈ℑ
where ℑ is some compact subset of (0, 1). The second and third tests treat βt as a martingale sequence: βt = βt−1 + ωt ,
IE[ωt |Ht−1 ] = 0,
IE[ωt ωt′ ] = κ2 Gt ,
where Ht is some increasing sequence of σ-fields to which βt is adapted and Gt is some known covariance array which measures the parameter stability in the tth period. That is, the null hypothesis of constant parameter becomes H0 : κ2 = 0. One possible alternative is H1 : κ2 > 0,
ˆ ǫ·ξ VT k ]−1 . GT k = [Ω
The test statistic is 1 X Fmean = ∗ F , T τ ∈ℑ [T τ ]
T∗ =
X
1,
τ ∈ℑ
6
where τ = k/T . Another alternative is H2 : κ > 0,
ˆ ǫ·ξ MT T ]−1 , GT k = [Ω
with the test statistic T X 1 ˆ −1 S ′ . St Ω Lc = tr MT−1 T ǫ·ξ t T t=1
#
"
The limiting distributions and critical values of these two test statistics can be found in Hansen (1992).
4
Monte Carlo Experiments
To compare performance of the estimates-based tests, residuals-based tests and Hansen’s tests, we consider the following regression model: yt = β0 + β1 x1t + β2 x2t + ut , and the DGP (data generating process): yt = 1.0 + 0.5x1t + 0.4x2t + ut , ut = ρut−1 + et ,
u0 = 0.0,
t = 1, . . . , [T × sb],
= 1.0 + (0.5 + △1 )x1t + (0.4 + △2 )x2t + et ,
x1t = ρ1 x1t + e1t ,
x10 = 0.0,
x2t = ρ2 x2t + e2t ,
x20 = 0.0,
(et , e1t , e2t ) ∼ N (0, Σ),
t = [T × sb] + 1, . . . , T,
1.0 γ1 γ2 Σ = γ1 1.0 0.0 . γ2 0.0 1.0
Notice that sb = 1.0 implies the null hypothesis of constant parameter is being considered. Experiment I: ρ1 = 1.0, ρ2 = 1.0, γ1 = 0.0, γ2 = 0.0, and ρ = 0.0 In this setting, two of regressors are I(1) processes with zero initial values. Typically, this setup stands for a cointegrating relation with cointegrating vector (1.0, 0.5, 0.4)′ and the regressors are uncorrelated to the regression errors. et , e1t , and e2t are all normally distributed with mean zero and unity variance. The break points under consideration are sb = 0.1, 0.3, 0.5, 0.7, 0.9 and sb = 1.0 for the null. The sample sizes we considered are 100, 200, and 400. The total number of replications is 1000. Our Monte Carlo simulations are implemented via GAUSS 3.2 and the GAUSS code of Hansen’s tests are from the module COINT 2.0. The simulation results are reported in Table 1. Under H0 , sb = 1.0, it is surprising that Hansen’s tests Lc, Fmean, and Fsup have empirical sizes smaller than the 7
theoretical size 0.05. It is surprise that all of the residual-based tests, RC, OC, RM, and OM, have appropriate sizes, especially the recursive residual-based tests, RC and RM, have empirical sizes closer to 0.05. However, the original (unmodified) estimates-based tests, ME0 and RE0, have empirical sizes distorted seriously. On the other hand, the modified tests, ME1, RE1, and RE2, have conservative empirical sizes and ME2 test has smaller size distortion. The empirical powers of these tests under the alternatives of one structural break at points sb = 0.1, 0.3, 0.5, 0.7 and 0.9 with the coefficient of the first regressor, x1 , changing from 0.5 to 1.0 are also shown in Table 1. Among tests with appropriate empirical size, they have the largest empirical powers when the break occurs at the middle of the sample, i.e., sb = 0.5. It is not surprising to have Hansen’s tests performing well in detecting structural break in a cointegrating relationship because they are originally designed for this purpose. It is very surprising to have the residuals-based tests performing well, especially the recursive residuals-based tests. Besides, the type I modified moving-estimates test and recursive-estimates test also perform well, conservative empirical sizes and acceptable empirical powers. Finally, the empirical power is getting larger as the sample size is getting larger. Alternatively, we allow the coefficient of the second regressor, x2 , changing from 0.4 to 0.9. The simulation results are presented in Table 2. Briefly, the results are the same as those described above. To summarize, Hansen’s tests, type I modified ME and RE tests, and residuals-based tests can be used to detect structural break in a cointegrating relationship. This conclusion is drawn when the nonstationary regressors are independent to the regression errors and et , e1t , and e2t are all independent, identical normally distributed. Is this conclusion also valid when the regressors are not independent to the regression errors, when the regression errors are serial correlated or when the regressors are not nonstationary? The answers will be found in the following Monte Carlo experiments. Experiment II: γ1 = 0.0, γ2 = 0.0, and ρ = 0.0 In this Monte Carlo experiment, we set γ1 = γ2 = 0.0 which implies the regressors are independent to the regression errors but we allow ρ1 and ρ2 deviate from 1.0. We set ρ1 and ρ2 increasing from 0.0 to 1.0 at an increment 0.2. It is worth to notice that the coefficients of the regressors are not changed. That is, the DGP is under the null hypothesis of constant parameters so that the conservative empirical sizes of tests should be expected. The simulation results are presented in Table 3, 4 and 5 for sample size 100, 200, and 400, respectively. It is easily found that Hansen’s tests perform worse as ρ1 and ρ2 deviate from 1.0 more. That is, the size distortions of Hansen’s tests are getting larger as ρ1 and ρ2 are smaller away from 1.0. However, the size distortions are smaller as the sample size getting larger. This indicates that Hansen’s tests are not robust to detect a structural change in dynamic models with stationary rather than nonstationary regressors. 8
The size distortions of the original ME and RE tests, ME0 and RE0, are larger as the serial correlation of the regressors getting larger and are smaller as the sample size getting larger. This is also indicated by Kuan and Chen (1994). The modified ME and RE tests have conservative empirical sizes, especially the type I modified ME and RE tests. This implies that the modified ME and RE tests can be used to detect a structural break in dynamic models with stationary and/or nonstationary regressors. All of the residuals-based tests have conservative empirical sizes at different values of ρ1 and ρ2 as expected. This implies that all the residuals-based tests are applicable to detect a structural break in dynamic regression models with stationary and/or nonstationary regressors. Experiment III: ρ1 = 1.0, ρ2 = 1.0 and ρ = 0.0 Now we consider the cointegrating regression model without structural break but the regressors are not independent to the regression errors. The covariances between et and e1t and e2t are γ1 and γ2 , respectively. In this experiment, we set ρ1 = 1.0 and ρ2 = 1.0 and γ1 and γ2 equal to 0.1, 0.3, 0.5, 0.7, and 0.9. The simulation results are presented in Table 6. No matter how e1t and e2t correlated to et , Hansen’s tests do have appropriate empirical sizes. The original and type II modified ME and RE tests have sizes distortion. However, the type I modified tests have conservative empirical sizes. As to the residualsbased tests, the recursive residuals-based tests have size distorted, especially when the correlations are large. For examples, when γ1 or γ2 equals to 0.9, the empirical sizes of RC and RM tests are 0.2 to 0.3 As to the OLS residuals-based tests, OC and OM, both of them have conservative empirical sizes. In summary, Hansen’s tests, type I modified Me and RE tests and OLS residuals-based tests do not have empirical size distorted. Experiment IV: ρ1 = 1.0, ρ2 = 1.0, γ1 = 0.0, γ2 = 0.0 Now we consider the influence of the serial correlation of the regression errors in a cointegrating regression on the performance of tests. The independence between regressors and regression errors is restricted, e.g., γ1 = γ2 = 0.0. The serial correlations of the regression errors under consideration are ρ = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. The simulation results are presented in Table 7. It is clear that the estimates-based and residuals-based tests have serious size distorted as the serial correlation getting larger. It is weird that Hansen’s tests have largest size distorted when the serial correlation is 0.5 but not 1.0. This phenomena motivates Gregory, Nason, and Watt (1996)’s study. In that paper, Hansen’s tests are modified by constructing a new dependent variable suggested by Bewley (1979). This is not an issue we pursuit here. The serial correlation of the regression errors indeed make the tests problematic.
9
5
Conclusions and Suggestions
Detecting structural breaks is an important issue in time series analysis. Especially, how to make a test robust not only to usual dynamic models but also to cointegrating regression models. Hansen’s tests, estimates-based tests, and residuals-based tests are investigated in the following situations: (1) a usual dynamic model; (2) a cointegrating regression model; (3) a cointegrating regression model with regressors correlated with regression errors; (4) a cointegrating regression model with serial correlated regression errors. All investigations are implemented through four Monte Carlo simulation experiments. Our conclusions are as follows. (1) Hansen’s tests work only in cointegrating regression models but not in stationary dynamic regression models. Type I modified ME and RE tests and the residuals-based tests can be robust to stationary dynamic and cointegrating regression models. (2) In cointegrating regression model where regressors being correlated with regression errors, Hansen’s tests, type I modified ME and RE tests, and the OLS-residuals-based tests are applicable but other tests are not. (3) In cointegrating regression model with serial correlated errors, All tests are not applicable. According to the above conclusions, we suggest the type I modified RE and ME tests and the OLS-residuals-based tests, OLS-CUSUM and OLS-MOSUM to detect structural breaks in dynamic regression models. These tests not only have conservative empirical sizes under the null in usual dynamic and cointegrating regression models but also have good empirical powers in detecting structural breaks. Besides, these tests are also applicable in a cointegrating regression model where regressors are correlated with the regression errors. However, they still have empirical size distortions in a cointegrating regression model with serial correlation. This problem may be overcome by modifying these tests with a new dependent variable suggested as Bewley (1979). This is worth a further study in the future. References
10
Andrews, D. K. (1991), ”Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimaton,” Econometroca, 59, 817–858. Andrews, D. K. and J. C. Monahan (1992), ”An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator, Econometrica, 60, 953–966. Bauer, P. and P. Hackl (1978), ”The Use of MOSUMS for Quality Control,” Technometrics, 20, 431–436. Bewly, R. A. (1979), ”The Direct Estimation of the Equilibrium Response in Linear Models,” Economics Letters, 3, 375–381. Brown, R. L., J. Durbin, and J. M. Evans (1975), ”Techniques for Testing the Constancy of Regression Relationships over Time,” Journal of the Royal Statistical Society, B 37, 149–163. Chu, C.-S. J., K. Hornik, and C.-M. Kuan (1995a), ”The Moving-Estimates Test for Parameter Stability,” Econometric Theory, 11, 699–720 Chu, C.-S. J., K. Hornik, and C.-M. Kuan (1995b), ”MOSUM Tests for Parameter Constancy,” Biometrika, 82, 603–617. Dufour, J.-M. (1982), ”Recursive Stability Analysis of Linear Regression Relationship,” Journal of Econometrics, 19, 31–76. Gregory, A. W., J. M. Nason, and D. G. Watt (1996), ”Testing for Structural Breaks in Cointegrated Relationships,” Journal of Econometrics, 71, 321–341. Hansen, Bruce E. (1992), ”Tests for Parameter Instability in Regressions with I (1) Processes,” Journal of Business & Economic Statistics, 10, 321–335. Hendry, D. F. and A. J.Neale (1991), ”A Monte Carlo Study of the Effects of Structural Breaks on Tests for Unit Roots,” Econometric Structural Change: Analysis and Forecasting, Chapter 8, 95–119. Kuan, C.-M. and M.-Y. Chen (1994), ”Implementing the Fluctuation and Moving-Estimates Tests in Dynamic Econometric Models,” Economics Letters, 44, 235–239. Kr¨ amer, W., W. Ploberger, and R. Alt (1988), ”Testing for Structural Change in Dynamic Models,” Econometrica, 56, 1355–1369. Nunes, L., P. Newbold, and C.-M. Kuan (1996), ”Spurious Number of Breaks”, Economics Letters, 50, 175–178. Perron, Pierre (1989), ”The Great Crash, the Oil Price Shock, and the Unit Root Hypoth-
11
esis,” Econometrica, 57, 1361–1401. Ploberger, W., W. Kr¨ amer, and K. Kontrus (1989), ”A New Test for Structural Stability in the Linear Regression Model,” Journal of Econometrics, 40, 307–318. Ploberger, W. and W. Kr¨ amer (1992), ”The CUSUM Test with OLS Residuals,” Econometrica, 60, 271–285.
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Table 1. Empirical Powers of Tests for Structural Breaks in Cointegrating Vector ρ1 = ρ2 = 1.0, γ1 = γ2 = ρ = 0.0, △1 = 0.5, △2 = 0.0 sb Lc Fmean Fsup ME0 ME1 ME2 RE0 RE1 RE2 RC T = 100 0.1 .140 .123 .099 .930 .042 .323 .973 .052 .231 .189 0.3 .473 .579 .647 .989 .506 .800 .999 .490 .873 .540 0.5 .642 .705 .686 1.00 .815 .983 .998 .717 .955 .619 0.7 .619 .670 .631 .994 .783 .922 .989 .753 .921 .596 0.9 .543 .430 .351 .975 .684 .831 .977 .693 .829 .244 1.0 .052 .041 .021 .900 .004 .174 .941 .000 .084 .033 T = 200 0.1 .369 .356 .404 .957 .277 .543 .999 .305 .513 .481 0.3 .852 .965 .986 .999 .907 .978 1.00 .878 1.00 .854 0.5 .912 .938 .960 1.00 .989 1.00 1.00 .979 1.00 .879 0.7 .867 .908 .926 1.00 .983 .997 .999 .974 .994 .834 0.9 .777 .718 .768 .988 .866 .916 .981 .868 .916 .640 1.0 .055 .047 .035 .924 .007 .164 .942 .005 .078 .029 T = 400 0.1 .686 .700 .827 .990 .664 .831 1.00 .642 .886 .795 0.3 .969 .996 1.00 1.00 .984 .999 1.00 .978 1.00 .971 0.5 .981 .981 .990 1.00 1.00 1.00 1.00 1.00 1.00 .981 0.7 .940 .955 .957 1.00 1.00 1.00 1.00 .999 1.00 .959 0.9 .910 .876 .909 .998 .950 .970 .998 .952 .977 .835 1.0 .040 .040 .032 .924 .007 .180 .943 .000 .056 .038
13
OC
RM
OM
.030 .387 .593 .677 .643 .003
.154 .664 .795 .763 .550 .034
.046 .383 .602 .659 .561 .007
.243 .769 .890 .913 .837 .006
.428 .911 .934 .913 .782 .043
.215 .750 .871 .865 .780 .008
.561 .945 .993 .995 .930 .002
.748 .984 .991 .984 .911 .039
.537 .940 .976 .981 .901 .010
Table 2. Empirical Powers of Tests for Structural Breaks in Cointegrating Vector ρ1 = ρ2 = 1.0, γ1 = γ2 = ρ = 0.0, △1 = 0.0, △2 = 0.5 sb Lc Fmean Fsup ME0 ME1 ME2 RE0 RE1 RE2 RC T = 100 0.1 .135 .109 .084 .923 .038 .289 .982 .049 .224 .185 0.3 .203 .210 .158 .950 .081 .413 .973 .064 .352 .167 0.5 .511 .581 .515 .991 .534 .868 .990 .415 .803 .417 0.7 .627 .662 .604 .989 .685 .891 .989 .632 .884 .512 0.9 .442 .384 .297 .948 .441 .629 .962 .419 .625 .093 T = 200 0.1 .357 .358 .405 .960 .287 .576 .995 .305 .539 .504 0.3 .598 .750 .799 .994 .530 .802 .999 .512 .863 .565 0.5 .923 .983 .992 1.00 .965 .998 .999 .932 .996 .795 0.7 .363 .387 .337 .948 .247 .536 .963 .174 .484 .156 0.9 .796 .721 .787 .984 .741 .832 .983 .760 .835 .458 T = 400 0.1 .695 .710 .827 .986 .671 .838 1.00 .637 .887 .798 0.3 .937 .990 .997 .999 .908 .984 1.00 .894 .998 .882 0.5 .886 .945 .930 .999 .823 .967 .993 .758 .938 .646 0.7 .972 .993 .996 .999 .945 .985 .998 .934 .972 .793 0.9 .876 .856 .890 .994 .952 .971 .995 .952 .973 .838
14
OC
RM
OM
.026 .079 .388 .596 .336
.156 .221 .595 .694 .285
.033 .084 .400 .574 .351
.244 .441 .796 .209 .708
.455 .678 .901 .304 .625
.202 .429 .807 .255 .638
.570 .768 .637 .866 .931
.752 .929 .785 .873 .902
.527 .783 .678 .830 .899
Table 3. Empirical Sizes of Tests for Structural Breaks in a Dynamic Model: T = 100 ρ1 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.2 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.6 0.6 0.8 0.8 0.8 0.8 0.8 0.8 1.0 1.0 1.0 1.0 1.0 1.0
ρ2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Lc .309 .263 .269 .270 .201 .159 .261 .273 .247 .238 .217 .149 .266 .247 .250 .195 .202 .141 .243 .228 .194 .211 .178 .097 .220 .196 .190 .167 .146 .110 .152 .147 .129 .108 .084 .059
Fmean .123 .105 .115 .114 .079 .077 .109 .096 .087 .078 .076 .073 .090 .087 .080 .065 .077 .069 .099 .084 .050 .078 .062 .053 .082 .065 .066 .050 .046 .056 .071 .080 .068 .064 .058 .049
Fsup .104 .086 .087 .061 .056 .060 .091 .078 .056 .045 .064 .059 .084 .056 .052 .042 .044 .038 .069 .040 .025 .041 .041 .025 .054 .038 .039 .031 .025 .027 .047 .059 .051 .044 .028 .025
ME0 .060 .067 .096 .095 .145 .666 .061 .068 .072 .090 .185 .716 .070 .098 .098 .113 .193 .647 .088 .080 .096 .139 .199 .713 .151 .149 .196 .197 .287 .745 .685 .689 .695 .715 .743 .921
15
ME1 .017 .028 .028 .024 .024 .008 .021 .023 .014 .014 .030 .012 .015 .026 .023 .024 .012 .008 .026 .017 .013 .015 .011 .013 .014 .007 .021 .016 .010 .007 .020 .009 .006 .011 .006 .005
ME2 .055 .067 .085 .067 .058 .121 .065 .063 .061 .068 .098 .116 .071 .072 .061 .071 .078 .107 .078 .064 .058 .064 .075 .132 .072 .056 .078 .070 .082 .139 .126 .128 .118 .132 .117 .187
RE0 .032 .036 .061 .057 .136 .725 .041 .043 .050 .060 .154 .742 .037 .059 .049 .071 .159 .725 .056 .071 .071 .101 .180 .743 .124 .135 .143 .165 .243 .777 .738 .730 .724 .756 .775 .939
RE1 .002 .009 .007 .008 .002 .002 .004 .003 .003 .003 .005 .002 .002 .006 .005 .002 .002 .002 .002 .004 .003 .005 .004 .000 .004 .002 .004 .003 .000 .002 .003 .002 .002 .002 .004 .002
RE2 .032 .032 .044 .045 .032 .062 .036 .037 .027 .038 .052 .070 .033 .041 .030 .036 .035 .066 .036 .029 .032 .036 .029 .082 .040 .043 .039 .042 .032 .068 .082 .085 .062 .065 .064 .087
RC .018 .021 .016 .017 .024 .026 .019 .018 .017 .019 .033 .038 .022 .014 .014 .028 .025 .043 .015 .011 .017 .021 .026 .035 .028 .029 .030 .030 .020 .046 .034 .025 .030 .034 .025 .031
OC .035 .043 .044 .027 .032 .015 .029 .028 .033 .038 .029 .010 .032 .034 .030 .035 .023 .012 .036 .028 .024 .023 .022 .010 .035 .022 .038 .017 .018 .008 .008 .012 .008 .009 .007 .006
RM .034 .044 .042 .040 .043 .037 .045 .038 .034 .034 .047 .045 .044 .037 .035 .046 .037 .054 .042 .034 .033 .035 .041 .039 .050 .030 .047 .041 .039 .054 .036 .034 .038 .038 .036 .046
OM .030 .041 .036 .031 .029 .025 .024 .032 .029 .022 .033 .007 .027 .032 .022 .031 .023 .013 .024 .029 .022 .028 .030 .012 .029 .014 .020 .028 .014 .013 .009 .016 .007 .014 .008 .006
Table 4. Empirical Sizes of Tests for Structural Breaks in a Dynamic Model: T = 200 ρ1 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.2 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.6 0.6 0.8 0.8 0.8 0.8 0.8 0.8 1.0 1.0 1.0 1.0 1.0 1.0
ρ2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Lc .269 .251 .238 .223 .202 .149 .246 .235 .223 .206 .199 .114 .232 .217 .202 .208 .206 .117 .213 .193 .184 .194 .189 .113 .205 .189 .179 .177 .155 .091 .133 .113 .124 .114 .105 .050
Fmean .077 .088 .078 .068 .072 .072 .079 .088 .067 .059 .065 .054 .068 .076 .052 .057 .061 .048 .061 .053 .045 .055 .067 .060 .075 .052 .054 .058 .064 .040 .061 .061 .054 .062 .060 .041
Fsup .062 .053 .060 .041 .041 .042 .059 .049 .052 .039 .042 .043 .053 .042 .034 .038 .043 .032 .039 .025 .028 .039 .043 .043 .045 .031 .033 .028 .029 .037 .044 .045 .032 .033 .028 .047
ME0 .053 .064 .063 .074 .102 .730 .077 .059 .065 .085 .122 .711 .063 .057 .057 .085 .117 .711 .072 .057 .070 .093 .133 .724 .110 .096 .098 .110 .162 .722 .696 .716 .712 .731 .738 .923
ME1 .030 .033 .032 .031 .032 .020 .037 .028 .030 .043 .028 .021 .036 .025 .026 .030 .029 .013 .031 .024 .022 .026 .021 .016 .035 .024 .031 .022 .026 .013 .027 .021 .014 .012 .015 .016
16
ME2 .051 .057 .059 .059 .057 .135 .070 .061 .062 .082 .061 .120 .056 .050 .045 .061 .070 .130 .059 .044 .053 .045 .063 .131 .063 .051 .063 .060 .054 .124 .124 .124 .126 .136 .119 .191
RE0 .020 .020 .022 .027 .055 .749 .032 .022 .026 .031 .077 .746 .022 .028 .024 .034 .073 .743 .031 .025 .024 .041 .079 .746 .058 .061 .057 .064 .117 .777 .736 .750 .759 .767 .766 .957
RE1 .010 .008 .004 .007 .006 .008 .009 .013 .009 .012 .007 .002 .012 .008 .009 .008 .010 .004 .009 .004 .000 .009 .007 .007 .011 .010 .007 .007 .008 .002 .004 .006 .004 .001 .001 .002
RE2 .017 .015 .017 .020 .024 .062 .026 .027 .018 .028 .022 .049 .029 .018 .014 .018 .027 .070 .021 .012 .016 .027 .030 .064 .024 .020 .017 .021 .023 .054 .052 .048 .064 .065 .044 .067
RC .033 .026 .034 .027 .034 .032 .026 .020 .025 .025 .022 .046 .029 .028 .026 .021 .043 .038 .029 .027 .015 .021 .024 .038 .029 .024 .031 .026 .037 .032 .034 .027 .041 .032 .046 .044
OC .039 .046 .044 .039 .036 .018 .038 .030 .042 .045 .045 .008 .050 .046 .034 .033 .040 .010 .023 .029 .023 .034 .033 .017 .029 .028 .044 .036 .030 .011 .017 .014 .011 .021 .014 .009
RM .049 .047 .044 .040 .048 .037 .044 .039 .047 .047 .052 .045 .051 .040 .038 .039 .060 .042 .033 .033 .035 .044 .038 .041 .039 .041 .053 .035 .047 .047 .042 .031 .051 .049 .040 .049
OM .041 .045 .030 .032 .032 .020 .035 .028 .041 .039 .038 .017 .049 .039 .030 .034 .029 .020 .027 .020 .025 .024 .033 .012 .037 .034 .031 .032 .030 .017 .022 .017 .015 .015 .016 .007
Table 5. Empirical Sizes of Tests for Structural Breaks in a Dynamic Model: T = 400 ρ1 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.2 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.6 0.6 0.8 0.8 0.8 0.8 0.8 0.8 1.0 1.0 1.0 1.0 1.0 1.0
ρ2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Lc .576 .224 .219 .215 .205 .104 .223 .221 .195 .212 .187 .119 .239 .208 .196 .194 .204 .124 .203 .194 .186 .179 .189 .116 .201 .201 .202 .184 .181 .107 .111 .119 .115 .114 .103 .068
Fmean .256 .064 .068 .056 .051 .054 .062 .062 .049 .064 .049 .054 .067 .060 .062 .059 .056 .048 .059 .049 .053 .046 .049 .061 .056 .058 .066 .055 .046 .046 .056 .047 .049 .054 .048 .067
Fsup .266 .039 .044 .031 .041 .035 .050 .044 .036 .034 .028 .046 .053 .040 .033 .047 .040 .033 .039 .029 .035 .029 .035 .040 .033 .032 .040 .029 .034 .044 .034 .047 .036 .034 .041 .054
ME0 .140 .056 .050 .052 .071 .743 .043 .055 .046 .060 .073 .707 .058 .063 .055 .081 .072 .720 .057 .073 .059 .070 .084 .731 .084 .087 .092 .088 .110 .721 .709 .739 .725 .733 .762 .941
17
ME1 .110 .046 .034 .024 .029 .023 .029 .031 .032 .036 .031 .026 .041 .040 .038 .045 .029 .032 .033 .035 .038 .038 .032 .018 .038 .036 .049 .041 .038 .022 .018 .020 .026 .024 .029 .027
ME2 .145 .054 .042 .038 .049 .129 .042 .047 .049 .053 .050 .137 .060 .059 .054 .066 .048 .126 .056 .050 .050 .057 .048 .109 .062 .060 .078 .057 .068 .117 .127 .133 .137 .150 .149 .216
RE0 .092 .019 .013 .017 .027 .775 .019 .014 .016 .022 .024 .766 .028 .019 .012 .026 .031 .764 .021 .013 .020 .029 .038 .774 .022 .027 .046 .028 .056 .761 .732 .774 .765 .745 .793 .959
RE1 .073 .013 .013 .008 .008 .007 .010 .012 .009 .011 .006 .010 .014 .012 .004 .015 .009 .008 .011 .008 .005 .012 .009 .004 .011 .011 .017 .007 .005 .009 .004 .005 .005 .005 .007 .012
RE2 .091 .016 .012 .013 .019 .047 .018 .015 .014 .019 .013 .056 .026 .016 .010 .022 .020 .053 .016 .015 .011 .020 .021 .055 .016 .016 .023 .015 .019 .053 .050 .043 .051 .047 .063 .112
RC .030 .028 .033 .026 .033 .047 .023 .033 .033 .030 .034 .054 .030 .035 .038 .034 .033 .045 .039 .032 .043 .039 .039 .043 .025 .041 .030 .029 .037 .053 .042 .030 .031 .039 .038 .096
OC .046 .044 .046 .041 .041 .020 .044 .036 .034 .040 .046 .014 .030 .040 .042 .043 .048 .018 .047 .036 .049 .040 .040 .015 .031 .038 .041 .030 .043 .016 .012 .024 .013 .018 .011 .007
RM .049 .045 .042 .042 .035 .045 .038 .047 .040 .032 .053 .046 .032 .049 .050 .038 .046 .055 .044 .045 .046 .039 .033 .042 .042 .053 .048 .035 .052 .048 .051 .038 .036 .046 .037 .080
OM .036 .049 .037 .029 .029 .028 .026 .046 .049 .036 .035 .022 .033 .042 .040 .047 .038 .017 .034 .039 .035 .039 .036 .016 .048 .047 .041 .039 .038 .018 .016 .021 .011 .023 .015 .016
Table 6 Empirical Sizes of Tests for Structural Breaks in Cointegrating Vector: γ1
γ2
Lc
Fmean
Fsup
0.0 0.1 0.3 0.5 0.7 0.9 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.3 0.5 0.7 0.9
.050 .043 .052 .055 .050 .051 .039 .056 .043 .047 .045
.045 .036 .043 .049 .041 .032 .048 .056 .041 .039 .040
.023 .019 .022 .029 .025 .019 .031 .024 .020 .024 .019
0.0 0.1 0.3 0.5 0.7 0.9 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.3 0.5 0.7 0.9
.063 .048 .044 .049 .047 .047 .057 .049 .038 .046 .036
.057 .036 .042 .032 .041 .037 .047 .036 .035 .041 .034
.038 .027 .026 .021 .026 .028 .038 .025 .032 .028 .018
0.0 0.1 0.3 0.5 0.7 0.9 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.3 0.5 0.7 0.9
.057 .054 .059 .048 .050 .053 .053 .055 .043 .052 .054
.044 .047 .058 .043 .052 .042 .044 .051 .035 .052 .044
.033 .034 .042 .045 .038 .028 .038 .039 .022 .040 .038
ME0 T = 100 .908 .896 .924 .916 .937 .946 .916 .912 .923 .940 .956 T = 200 .922 .924 .934 .943 .951 .963 .927 .930 .940 .962 .956 T = 400 .904 .925 .932 .936 .962 .969 .940 .933 .947 .965 .960
18
ME1
ME2
RE0
RE1
RE2
RC
OC
RM
OM
.007 .005 .006 .011 .001 .007 .004 .005 .005 .016 .004
.169 .165 .192 .260 .320 .348 .167 .201 .260 .299 .367
.930 .945 .945 .960 .956 .956 .938 .946 .947 .956 .970
.002 .001 .001 .004 .003 .003 .001 .000 .001 .005 .003
.081 .099 .120 .163 .190 .257 .079 .091 .154 .223 .250
.047 .034 .052 .091 .142 .193 .048 .053 .089 .152 .205
.005 .001 .005 .006 .009 .018 .004 .007 .007 .010 .004
.052 .038 .065 .112 .152 .229 .055 .060 .121 .167 .253
.007 .003 .009 .014 .011 .018 .005 .009 .007 .017 .011
.016 .006 .004 .010 .011 .009 .010 .007 .012 .014 .011
.167 .158 .189 .272 .361 .352 .182 .206 .272 .332 .343
.942 .951 .955 .976 .970 .981 .946 .947 .971 .965 .964
.003 .001 .005 .002 .006 .004 .003 .001 .003 .006 .003
.076 .070 .099 .145 .213 .238 .082 .115 .160 .172 .227
.050 .044 .074 .109 .175 .242 .045 .068 .136 .189 .241
.006 .004 .005 .005 .012 .007 .005 .004 .008 .009 .011
.049 .048 .064 .119 .206 .272 .039 .069 .137 .188 .253
.011 .007 .009 .011 .015 .018 .011 .010 .013 .019 .022
.012 .010 .020 .012 .017 .010 .011 .017 .010 .007 .020
.182 .192 .238 .262 .345 .378 .162 .213 .262 .359 .382
.949 .967 .960 .961 .975 .976 .949 .964 .973 .978 .981
.001 .001 .008 .001 .006 .007 .004 .006 .006 .001 .009
.067 .070 .102 .151 .182 .207 .086 .110 .168 .194 .233
.039 .048 .084 .132 .189 .260 .043 .077 .126 .193 .282
.005 .002 .009 .009 .014 .011 .006 .002 .006 .004 .016
.046 .046 .069 .135 .204 .257 .050 .082 .136 .188 .292
.009 .004 .013 .009 .019 .013 .011 .011 .015 .012 .021
Table 7 Empirical Sizes of Tests for Structural Breaks in Cointegrating Vector: ρ
Lc
Fmean
Fsup
ME0
ME1
0.0 0.2 0.4 0.6 0.8 1.0
.063 .055 .051 .029 .015 .037
.045 .046 .035 .017 .003 .053
.026 .014 .005 .001 .000 .061
.892 .962 .984 .990 1.00 1.00
.004 .039 .154 .400 .745 .964
0.0 0.2 0.4 0.6 0.8 1.0
.044 .064 .070 .080 .024 .019
.048 .059 .055 .051 .014 .036
.024 .024 .021 .011 .000 .034
.922 .966 .980 .999 1.00 1.00
.006 .061 .242 .617 .898 .999
0.0 0.2 0.4 0.6 0.8 1.0
.044 .061 .196 .160 .057 .023
.042 .057 .186 .123 .036 .035
.038 .038 .163 .087 .001 .039
.927 .980 .997 1.00 1.00 1.00
.014 .100 .526 .806 .959 1.00
ME2 T = 100 .171 .406 .622 .858 .981 .997 T = 200 .189 .422 .707 .938 .993 1.00 T = 400 .182 .500 .846 .970 .999 1.00
19
RE0
RE1
RE2
RC
OC
RM
OM
.943 .962 .979 .998 .996 1.00
.001 .021 .066 .213 .559 .912
.092 .236 .465 .722 .937 .994
.036 .090 .198 .375 .557 .882
.004 .018 .065 .220 .527 .920
.034 .133 .262 .443 .677 .930
.010 .033 .121 .298 .572 .872
.949 .976 .996 1.00 1.00 1.00
.001 .035 .128 .481 .774 .994
.089 .229 .526 .854 .975 1.00
.034 .123 .317 .629 .752 .978
.003 .032 .098 .391 .662 .992
.040 .140 .351 .633 .776 .983
.009 .042 .163 .446 .693 .965
.957 .983 .999 1.00 1.00 1.00
.005 .040 .394 .644 .874 1.00
.076 .270 .717 .925 .995 1.00
.044 .183 .596 .758 .835 .999
.005 .043 .318 .561 .807 1.00
.058 .177 .553 .737 .875 .997
.009 .062 .338 .596 .821 .999
