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On tests for changes in persistence
Economics Letters 84 (2004) 107 – 115 www.elsevier.com/locate/econbase
On tests for changes in persistence Stephen Leybourne a, A.M. Robert Taylor b,* b
a Department of Economics, University of Nottingham, University Park, Nottingham NG7 2RD, UK Department of Economics, University of Birmingham, Edgbaston Park Road, Edgbaston, Birmingham B15 2TT, UK
Received 22 October 2003; accepted 19 December 2003 Available online 15 April 2004
Abstract There is mounting evidence that the parameters of autoregressive models fitted to many economic and financial time-series are unstable across time, often displaying changes in persistence between I(0) and I(1) behaviour. See, inter alia, Stock and Watson [J. Business Econ. Stat. 14 (1996) 11] and Garcia and Perron [Rev. Econ. Stat. 78 (1996) 111]. Tests for changes in persistence have recently been developed in Kim [J. Econ. 95 (2000) 97], based on ratio statistics which require no modifications for weak dependence in the driving shocks to facilitate asymptotically pivotal inference. However, the tests can be very badly over-sized against weakly dependent processes in finite samples, as we show in this paper. We propose simple modifications of these ratio statistics, which have the same (pivotal) limiting null distributions as the unmodified statistics but yield tests with much improved finite sample size properties under weakly dependent shocks. D 2004 Elsevier B.V. All rights reserved. Keywords: Changes in persistence; Ratio tests JEL classification: C22
1. Introduction In two recent papers, Kim (2000); Kim et al. (2002) develop residual-based ratio tests against changes in persistence in a time series, focusing on the case of a shift from stochastic stationarity, I(0), to difference stationarity, I(1), at some point in the sample. Busetti and Taylor (in press) [BT] develop this material further, proposing additional ratio tests which are consistent against I(0) to I(1) changes. BT also propose tests which are designed for the case where the direction of the change is unknown; * Corresponding author. Tel.: +44-121-414-6657; fax: +44-121-414-7377. E-mail address: [email protected] (A.M.R. Taylor). 0165-1765/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2003.12.015
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that is, ratio-based tests which are consistent against series which display either shifts from I(0) to I(1) or from I(1) to I(0). An interesting feature of the ratio tests is that they require no form of long run variance correction in order to obtain pivotal limiting null distributions even when the driving shocks display weak dependence. However, and as noted in BT, this does not guarantee that the tests will have good finite sample size properties in such cases. In this paper we investigate the size distortions seen in the ratio tests when the shocks display weak parametric autocorrelation and propose modifications to the original tests which are shown to greatly improve the finite sample size properties of the resulting tests, without sacrificing too much finite sample power against persistence change alternatives. The paper is organised as follow. In Section 2 we provide a brief review of the ratio-based tests against changes in persistence. In Section 3 we discuss non-parametric modifications of the ratio tests designed to improve the finite-sample size properties of the tests, affording the practitioner a size/power trade-off not available using the standard ratio tests. The relative finite sample size and power properties of the ratio tests and their non-parametrically modified counterparts are investigated by Monte Carlo simulation in Section 4. Section 5 concludes.
2. Ratio tests for change in persistence Kim (2000), considers the null hypothesis that the scalar time-series process yt is formed as the sum of a purely deterministic component, dt (e.g. an intercept, or an intercept and linear trend), and a short memory component; that is, yt ¼ dt þ ut ;
t ¼ 1; . . . ; T
ð2:1Þ
where ut satisfies the usual strong mixing conditions of, inter alia, Phillips and Perron (1988, p. 336) 2 . In what follows, reference to a series as being with strictly positive and bounded long run variance rl I(0) is taken to imply that these conditions hold. Kim (2000) considers two alternatives: firstly, that yt displays a shift from I(0) to I(1) behaviour1 at time t = ts0Tb , where tb denotes the integer part of its argument, and secondly that there is a shift from I(1) to I(0) behaviour at time t = ts0Tb. These may be expressed within the model, yt ¼ dt þ zt;1 ;
t ¼ 1; . . . ts0 T b;
yt ¼ dt þ zt;2 ;
t ¼ ts0 T b þ 1; . . . ; T :
s0 að0; 1Þ
In the first case zt,2 = zt 1,2 + ut, with ut and zt,1 both I(0), while in the second case zt,1 = zt 1,1 + ut with ut and zt,2 both I(0). 1
An I(1) series is one formed from the accumulation of an I(0) series.
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BT and Kim et al. (2002) have independently proposed tests for the null hypothesis that yt is a constant I(0) process against an I(0)–I(1) shift at some unknown point in the sample. These tests are constructed using the sequence of ratios t X
t¼tsT bþ1
i¼tsTbþ1 !2 t
ðT tsT bÞ2 KðsÞ ¼ ðtsT bÞ
!2
T X
2
tsT b X X t¼1
eˆ 2;i ;
ð2:2Þ
eˆ 1;i
i¼1
saT , where T is a given sub-interval of [0, 1]. In Eq. (2.2), eˆ 1,t are the OLS residualsˆ from the regression of yt on either an intercept or an intercept and linear trend, t = 1, . . . tsTb, and e2,t are the OLS residuals from the regression of yt on either an intercept or an intercept and linear trend, t = tsTb + 1, . . ., T. Tests are then based on taking an appropriate function of the sequence K(). These authors investigate three such functions: the maximum over the sequence of statistics, after Andrews (1993); Hansen’s (1991) mean score, and, finally, Andrews and Ploberger’s (1994) meanexponential: we denote these statistics as K j, j = 1, . . ., 3, respectively. In each case, the null is rejected for large values of these statistics, and the tests are consistent at rate Op(T 2) against the I(0)–I(1) change DGP. Suppose now that yt is generated by the I(1)–I(0) change DGP. Here BT show that the tests which reject for large values of K j, j = 1, . . ., 3, are all inconsistent. However, BT show that tests which reject for large values of the corresponding statistics, denoted K Rj , j = 1, 2, 3, formed from the sequence of reciprocals of K(s), are consistent at rate Op(T 2) in this case, but are inconsistent against I(0)–I(1) alternatives. BT also suggest a test based on the maximum of the K () and (K ()) 1 sequences; that is, Kj *umaxfKj ; KRj g;
j ¼ 1; 2; 3;
ð2:3Þ
in each case rejecting for large values of the statistics. BT show that the pairwise ratio-based tests are Op(T 2) under both I(1)–I(0) and I(0)–I(1) alternatives. Representations for the limiting null distributions of the foregoing statistics are given in Kim et al. (2002) and BT. Crucially, they show that these representations do not depend on the long run variance 2 , even though neither the numerator nor the denominator of K(s) of Eq. (2.2) is scaled by a long run rl variance estimator. BT also show that all of the ratio-based tests discussed in this section are inconsistent against constant I(1) processes; that is where yt = dt + zt*, t = 1, . . ., T, where zt* = z*t 1 + ut, with ut an I(0) process. This is a useful result, implying that these tests will only diverge against series, which display changes in persistence. BT also propose LBI-based tests against changes in persistence but, unfortunately, these are consistent against constant I(1) processes. Interestingly, these LBI-based tests involve the use of a longrun variance estimator, of exactly the form used in Kwiatkowski et al. (1992) [KPSS], to obtain pivotal limiting null distributions.
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3. Non-parametrically modified ratio-based tests In discussing the properties of the ratio tests, BT argue that . . . in the presence of serially correlated innovations they do not require the arguably arbitrary decisions over the lag truncation parameter. . .. However. . . the finite sample size properties of the ratio-based tests are not satisfactory in practical situations. Although unnecessary for the purpose of achieving statistics with pivotal limiting null distributions, it is nonetheless perfectly feasible to scale the numerator and denominator of Eq. (2.2) by appropriate sub-sample (long run) variance estimators. A reason for considering such a modification is that if we individually studentise the numerator and denominator terms, we might expect the finite sample distribution of their ratio to more closely approximate the limiting null distribution. To that end, consider replacing K(s) of Eq. (2.2), for each saT , by the modified statistic Kðs; mÞ ¼
rˆ 21 KðsÞ rˆ 22
where, following KPSS (Eq. (10), p. 164), rˆ 22 ¼ ðT tsT bÞ1
T X
eˆ 22;t þ 2ðT tsT bÞ1
m X i¼1
t¼tsT bþ1
T X
wði; mÞ
eˆ 2;t eˆ 2;ti ;
t¼iþtsT bþ1
and rˆ 21 ¼ ðtsTbÞ1
tsTb X t¼1
eˆ 21;t þ 2ðtsTbÞ1
m X i¼1
wði; mÞ
tsT b X
eˆ 1;t eˆ 1;ti ;
t¼iþ1
with w(i, m) = 1 i/(m + 1), and proceed as above replacing K(s) by K(s, m) throughout. We denote the resulting statistics as Kj (m), K Rj (m) and K j*(m), j = 1, 2, 3, with an obvious notation. Notice that the numerator (denominator) of K(s, m) is nothing other than a standard KPSS stationarity test statistic applied to the second (first) sub-sample of the data. As a consequence, this modified statistic affords the practitioner a certain degree of control over the finite-sample properties of the ratio-based testing procedure via the usual size-power trade-off in the bandwidth (lag truncation) parameter m observed for these tests in KPSS. In the standard KPSS framework m is chosen to satisfy the conditions that m ! l and m = o(T1/2) as T ! l; cf. KPSS. The same can be done here, although in fact m can be set to any non-negative integer value (including zero) since under the null rˆ12 and rˆ22 will converge in probability to the same constant as T ! l. As a consequence the non-parametrically modified statistics will possess identical limiting null
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distributions to the corresponding unmodified statistic so that the asymptotic critical values provided in BT and Kim et al. (2002) may therefore still be used. Where the unmodified tests were consistent at rate Op(T2) above, the corresponding modified test will be consistent at rate Op(T/m). This because any variance estimator constructed from data containing any of the I(1) segment of the sample will diverge at rate Op(Tm), while variance estimators constructed over purely I(0) data will converge in probability to some finite constant.
4. Simulation results In this section we use Monte Carlo simulation methods to investigate the finite-sample size and power properties of the non-parametrically modified ratio-based tests vis-a`-vis the corresponding unmodified tests, highlighting the role of the lag truncation parameter m in the implied size-power trade-off for a given sample size. First, in Tables 1 and 2 we report the empirical rejection frequencies of the tests against data generated by Eq. (2.1) with dt = 0, and the following ARMA(1,1) parametric form on ut, ut ¼ /ut1 þ et het1 ;
t ¼ 200; . . . ; T ;
ð4:1Þ
Table 1 Empirical rejection frequencies of modified and unmodified ratio-based persistence change tests: de-meaned data T
/
h
K1
K1(0)
K1(1)
K1(2)
K1(mT)
K1*
K1*(0)
K1*(1)
K1*(2)
K1*(mT)
50
0.85 0.90 0.95 0.00 0.00 0.00 0.00 0.85 0.90 0.95 0.00 0.00 0.00 0.00 0.85 0.90 0.95 0.00 0.00 0.00 0.00
0 0 0 0.5 0.5 0.8 0.8 0 0 0 0.5 0.5 0.8 0.8 0 0 0 0.5 0.5 0.8 0.8
0.35 0.41 0.48 0.01 0.08 0.00 0.09 0.27 0.33 0.43 0.01 0.07 0.00 0.08 0.17 0.22 0.33 0.02 0.06 0.00 0.07
0.10 0.11 0.14 0.04 0.04 0.03 0.04 0.08 0.09 0.11 0.03 0.05 0.02 0.05 0.05 0.06 0.08 0.04 0.05 0.02 0.05
0.08 0.10 0.12 0.04 0.05 0.03 0.04 0.07 0.08 0.10 0.03 0.05 0.01 0.05 0.05 0.06 0.07 0.03 0.05 0.02 0.05
0.07 0.08 0.10 0.04 0.06 0.04 0.06 0.07 0.08 0.10 0.03 0.05 0.01 0.05 0.05 0.06 0.08 0.03 0.05 0.02 0.05
0.05 0.06 0.06 0.04 0.06 0.05 0.06 0.06 0.07 0.08 0.03 0.05 0.01 0.05 0.06 0.06 0.08 0.03 0.05 0.02 0.05
0.49 0.57 0.66 0.00 0.09 0.00 0.10 0.37 0.47 0.60 0.01 0.07 0.00 0.08 0.23 0.31 0.47 0.02 0.06 0.00 0.07
0.12 0.14 0.17 0.04 0.04 0.03 0.05 0.09 0.11 0.14 0.03 0.05 0.01 0.05 0.05 0.06 0.09 0.04 0.05 0.02 0.05
0.10 0.12 0.14 0.04 0.05 0.03 0.05 0.08 0.10 0.14 0.03 0.05 0.01 0.05 0.06 0.06 0.08 0.03 0.05 0.02 0.05
0.07 0.09 0.11 0.03 0.06 0.03 0.06 0.08 0.09 0.12 0.03 0.06 0.01 0.05 0.06 0.07 0.09 0.03 0.06 0.01 0.05
0.05 0.06 0.07 0.04 0.05 0.04 0.06 0.07 0.08 0.09 0.03 0.06 0.01 0.05 0.06 0.07 0.09 0.03 0.06 0.02 0.05
100
200
DGP Eq. (2.1), Eq. (4.1).
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Table 2 Empirical rejection frequencies of modified and unmodified ratio-based persistence change tests: de-meaned and de-trended data T
/
h
K1
K1(0)
K1(1)
K1(2)
K1(mT)
K1*
K1*(0)
K1*(1)
K1*(2)
K1*(mT)
50
0.85 0.90 0.95 0.00 0.00 0.00 0.00 0.85 0.90 0.95 0.00 0.00 0.00 0.00 0.85 0.90 0.95 0.00 0.00 0.00 0.00
0 0 0 0.5 0.5 0.8 0.8 0 0 0 0.5 0.5 0.8 0.8 0 0 0 0.5 0.5 0.8 0.8
0.58 0.63 0.68 0.01 0.13 0.03 0.15 0.48 0.56 0.67 0.01 0.09 0.01 0.09 0.31 0.42 0.58 0.01 0.06 0.00 0.08
0.20 0.23 0.27 0.06 0.04 0.11 0.04 0.15 0.19 0.24 0.04 0.04 0.05 0.04 0.09 0.12 0.18 0.03 0.05 0.02 0.05
0.11 0.13 0.16 0.10 0.05 0.23 0.04 0.12 0.16 0.22 0.04 0.05 0.06 0.04 0.08 0.11 0.16 0.03 0.05 0.03 0.05
0.02 0.02 0.03 0.15 0.04 0.28 0.04 0.10 0.14 0.18 0.05 0.05 0.10 0.05 0.08 0.11 0.16 0.04 0.05 0.02 0.05
0.01 0.01 0.01 0.10 0.04 0.17 0.04 0.04 0.05 0.08 0.07 0.05 0.22 0.05 0.07 0.08 0.13 0.04 0.05 0.04 0.05
0.74 0.79 0.83 0.01 0.15 0.02 0.17 0.63 0.74 0.84 0.01 0.10 0.01 0.11 0.44 0.58 0.76 0.01 0.07 0.00 0.09
0.24 0.28 0.33 0.06 0.04 0.11 0.05 0.18 0.24 0.30 0.04 0.04 0.04 0.05 0.10 0.14 0.22 0.03 0.05 0.01 0.05
0.12 0.15 0.18 0.11 0.05 0.26 0.05 0.15 0.20 0.28 0.04 0.05 0.05 0.05 0.09 0.12 0.21 0.03 0.05 0.01 0.05
0.02 0.01 0.02 0.17 0.04 0.33 0.04 0.11 0.16 0.23 0.04 0.05 0.09 0.06 0.09 0.12 0.20 0.03 0.05 0.02 0.05
0.01 0.01 0.01 0.10 0.04 0.17 0.04 0.05 0.06 0.09 0.08 0.05 0.24 0.05 0.07 0.09 0.16 0.03 0.05 0.03 0.05
100
200
DGP Eq. (2.1), Eq. (4.1).
with et f NIID(0, 1), and the design parameters /a{0.85, 0.90, 0.95} and ua{ F 0.5 F 0.8}2. Table 1 reports results for the case where the residuals used in Eq. (2.2) are obtained from regressions on an intercept (de-meaned data) and Table 2 the corresponding results from regressions on an intercept and trend (de-meaned and de-trended data). Second, in Table 3 (de-meaned data) and Table 4 (de-meaned and de-trended data) we report the empirical rejection frequencies of the tests when the data are generated according to the I(0)–I(1) switch DGP yt ¼ qt yt1 þ et ;
t ¼ 200; . . . ; T ;
ð4:2Þ
et f NIID(0, 1), with qt = 0, t = 200, . . ., ts0Tb, and qt = 1, t = ts0Tb + 1, . . ., T. We consider the following values of the breakpoint parameter, s0a{0.3, 0.4, 0.5, 0.6, 0.7}. We have set the autoregressive parameter to zero in the first sub-sample to avoid biasing the results via the inherent over-sizing problems that result (in particular in the case of the unmodified tests) from a positive first-order autoregressive coefficient; cf. Tables 1 and 2.
2
Other values of the design parameters were considered but the results for these cases neither add to nor contradict the qualitative conclusions drawn from the cases reported.
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Table 3 Empirical rejection frequencies of modified and unmodified ratio-based persistence change tests: de-meaned data T
s0
K1
K1(0)
K1(1)
K1(2)
K1(mT)
K1*
K1*(0)
K1*(1)
K1*(2)
K1*(mT)
50
0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7
0.89 0.90 0.90 0.87 0.80 0.98 0.98 0.98 0.97 0.94 1.00 1.00 1.00 1.00 0.99
0.70 0.75 0.77 0.75 0.70 0.87 0.90 0.91 0.91 0.87 0.96 0.97 0.97 0.97 0.96
0.52 0.58 0.61 0.60 0.56 0.74 0.79 0.81 0.80 0.76 0.89 0.91 0.92 0.92 0.89
0.37 0.43 0.46 0.46 0.43 0.64 0.69 0.71 0.71 0.67 0.82 0.85 0.87 0.87 0.84
0.21 0.27 0.30 0.32 0.30 0.47 0.52 0.56 0.56 0.51 0.66 0.70 0.73 0.73 0.69
0.90 0.89 0.87 0.84 0.76 0.98 0.98 0.98 0.96 0.93 1.00 1.00 1.00 1.00 0.98
0.66 0.69 0.70 0.67 0.61 0.84 0.87 0.87 0.86 0.82 0.94 0.95 0.96 0.95 0.93
0.49 0.52 0.53 0.50 0.46 0.70 0.74 0.75 0.73 0.69 0.85 0.88 0.89 0.86 0.85
0.32 0.36 0.36 0.35 0.32 0.60 0.64 0.65 0.64 0.59 0.78 0.81 0.83 0.82 0.78
0.17 0.20 0.21 0.21 0.20 0.43 0.46 0.48 0.46 0.41 0.62 0.65 0.67 0.66 0.61
100
200
DGP Eq. (4.2).
All reported simulation experiments were performed using 40,000 Monte Carlo replications and the RNDN function of Gauss 3.1, for T = 50, 100, 200 with the first 200 observations discarded to control for initial effects. All tests were run at the nominal exact 5% level, with T =[0.2, 0.8] as in Kim (2000); Kim et al. (2002) and BT. Results are reported for the K1, K1*, and K1(m) and K1*(m)
Table 4 Empirical rejection frequencies of modified and unmodified ratio-based persistence change tests: de-meaned and de-trended data T
s0
K1
K1(0)
K1(1)
K1(2)
K1(mT)
K1*
K1*(0)
K1*(1)
K1*(2)
K1*(mT)
50
0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7
0.83 0.81 0.79 0.75 0.68 0.98 0.98 0.98 0.97 0.94 1.00 1.00 1.00 1.00 0.99
0.70 0.72 0.72 0.71 0.65 0.92 0.92 0.93 0.93 0.90 0.98 0.99 0.99 0.99 0.98
0.37 0.43 0.44 0.44 0.42 0.75 0.79 0.80 0.81 0.78 0.92 0.94 0.95 0.96 0.94
0.05 0.07 0.08 0.09 0.09 0.57 0.63 0.65 0.67 0.65 0.84 0.88 0.89 0.90 0.89
0.01 0.01 0.01 0.01 0.01 0.19 0.26 0.28 0.29 0.30 0.57 0.63 0.66 0.68 0.67
0.85 0.84 0.80 0.75 0.63 0.98 0.98 0.97 0.96 0.92 1.00 1.00 1.00 1.00 0.99
0.65 0.67 0.66 0.63 0.55 0.89 0.90 0.90 0.89 0.85 0.97 0.98 0.98 0.98 0.97
0.32 0.36 0.36 0.33 0.30 0.71 0.75 0.75 0.75 0.70 0.89 0.92 0.92 0.93 0.91
0.02 0.03 0.03 0.04 0.04 0.53 0.59 0.60 0.58 0.55 0.81 0.85 0.86 0.86 0.83
0.01 0.01 0.01 0.01 0.01 0.17 0.22 0.23 0.20 0.21 0.55 0.61 0.64 0.62 0.60
100
200
DGP Eq. (4.2).
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tests3 and where in relation to the non-parametrically modified ratio tests the following values of the lag truncation parameter were used: m = 0, 1, 2, mT, where mT = t4(T/100)1/4b; that is, both fixed and sample size dependent values were considered. Consider first the results in Tables 1 and 2. It is clear that although asymptotically pivotal, the unmodified ratio-based statistics yield tests, which can be very badly over-sized in finite samples. Empirical size does tend towards the nominal level as T is increased, ceteris paribus, albeit at a rather slow rate. To illustrate, for the de-meaned and de-trended case (Table 2) we see that for / = 0.95 the unmodified K1 and K1* tests have empirical sizes of (0.68, 0.83) for T = 50, improving only marginally to (0.58, 0.76) for T = 200. In this case the corresponding modified tests with m = 0 and m = mT have sizes (0.27, 0.33) and (0.01, 0.01), respectively, for T = 50, and (0.18, 0.22) and (0.13, 0.16), respectively, for T = 200. As with the foregoing example, it is clear that most of the size improvements are obtained simply by setting m = 0 in the variance estimators. This is apparent for both the de-meaned and de-trended cases. Increasing m beyond zero tends to introduce rather smaller additional improvements; indeed, for T = 50, in the de-trending case increasing m tends to cause under (over) sizing with autoregressive (negative moving average) errors, the antithesis of what is seen for the unmodified tests. Size distortions tend to be worse, ceteris paribus, for all tests in the case of de-meaned and de-trended vis-a`-vis de-meaned data, echoing the findings of KPSS. Overall, however, it is clear that the non-parametrically modified tests perform very well in re-dressing size, taken relative to the performance of the unmodified tests. All of the tests tend to display their greatest distortions in the cases of very large positive autoregressive errors (over-sizing) and very large negative moving average errors (under-sizing), with the exceptions noted above. Tables 3 and 4 allow us to investigate the degree of power loss incurred by using the nonparametrically modified tests. The power losses incurred from using the modified tests is greatest, ceteris paribus, where T is small, and also appear rather smaller in the de-meaned and de-trended case. As would be expected (again compare with KPSS), other things being equal, power decreases as m is increased. The power loss from using m = mT is particularly large, especially so in the de-meaned and detrended case for T = 50. In general, however, the difference in power between the unmodified tests and the corresponding modified test with m = 0 are not too substantial, particularly in the de-meaned and detrended case. This is very encouraging given the observation from Tables 1 and 2 that this choice of m delivers most of the size improvements seen when using the modified statistics. Finally, it is worth noting that, other things equal, power is not necessarily lower in the de-meaned and de-trended case than for the de-meaned case; this phenomenon is also apparent in the simulation results presented in Table 4 (p. 172) of KPSS. For the modified tests then, the usual trade-off between size and power in m seen in residual-based tests is apparent here. From our results we recommend setting m = 0 in the modified tests, this choice
3
Results for tests based on the mean-score and mean-exponential functions gave qualitatively similar results to those reported and, hence, are omitted. Moreover, results are not reported for the KR1 and KR1 (m) tests, since their size properties were almost identical to those reported for the K1 and K1(m) tests, while they are not consistent against the I(0) – I(1) switch DGP considered, Eq. (4.2). We also considered the corresponding I(1) – I(0) switch DGP: the tests of Eq. (2.3) and the corresponding non-parametric modifications behaved almost identically to the results in Tables 3 and 4, while the KR1 and KR1 (m) tests display similar power to the results reported for the K1 and K1(m) tests in Tables 3 and 4 if one exchanges s0 with 1 s0.
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delivering most of the available size improvements under a given scenario yet not sacrificing too much test power against persistence change processes.
5. Conclusions In this paper we have proposed modifications to the ratio-based tests for persistence change developed in Kim (2000), BT and Kim et al. (2002). These modifications involve scaling the ratio statistics by (long run) variance estimators constructed across sub-samples of the data. We show that these modifications can deliver dramatic improvements in the finite-sample size properties of the tests without altering the limiting null distributions of the statistics. For a given sample size, the larger the lag truncation parameter used in constructing the variance estimators the larger the power losses seen against processes displaying persistence change. However, a very low-order (often zero) fixed lag truncation appears to give a useful pragmatic balance between re-dressing the size properties of the tests and keeping the degree of power loss against persistence change processes relatively small.
References Andrews, D.W.K., 1993. Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821 – 856. Andrews, D.W.K., Ploberger, W., 1994. Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383 – 1414. Busetti, F., Taylor, A.M.R., 2004. Tests of stationarity against a change in persistence. Journal of Econometrics (in press). Hansen, B.E., 1991. Testing for Structural Change of Unknown Form in Models with Nonstationary Regressors, Mimeo. Department of Economics, University of Rochester. Kim, J.Y., 2000. Detection of change in persistence of a linear times series. Journal of Econometrics 95, 97 – 116. Kim, J.Y., Belaire Franch, J., Badilli Amador, R., 2002. Corrigendum to ‘‘Detection of change in persistence of a linear times series’’. Journal of Econometrics 109, 389 – 392. Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., Shin, Y., 1992. Testing the null hypothesis of stationarity against the alternative of a unit root: how sure are we that economic time series have a unit root? Journal of Econometrics 54, 159 – 178. Phillips, P.C.B., Perron, P., 1988. Testing for a unit root in time series regression. Biometrika 75, 335 – 346.
On tests for changes in persistence Stephen Leybourne a, A.M. Robert Taylor b,* b
a Department of Economics, University of Nottingham, University Park, Nottingham NG7 2RD, UK Department of Economics, University of Birmingham, Edgbaston Park Road, Edgbaston, Birmingham B15 2TT, UK
Received 22 October 2003; accepted 19 December 2003 Available online 15 April 2004
Abstract There is mounting evidence that the parameters of autoregressive models fitted to many economic and financial time-series are unstable across time, often displaying changes in persistence between I(0) and I(1) behaviour. See, inter alia, Stock and Watson [J. Business Econ. Stat. 14 (1996) 11] and Garcia and Perron [Rev. Econ. Stat. 78 (1996) 111]. Tests for changes in persistence have recently been developed in Kim [J. Econ. 95 (2000) 97], based on ratio statistics which require no modifications for weak dependence in the driving shocks to facilitate asymptotically pivotal inference. However, the tests can be very badly over-sized against weakly dependent processes in finite samples, as we show in this paper. We propose simple modifications of these ratio statistics, which have the same (pivotal) limiting null distributions as the unmodified statistics but yield tests with much improved finite sample size properties under weakly dependent shocks. D 2004 Elsevier B.V. All rights reserved. Keywords: Changes in persistence; Ratio tests JEL classification: C22
1. Introduction In two recent papers, Kim (2000); Kim et al. (2002) develop residual-based ratio tests against changes in persistence in a time series, focusing on the case of a shift from stochastic stationarity, I(0), to difference stationarity, I(1), at some point in the sample. Busetti and Taylor (in press) [BT] develop this material further, proposing additional ratio tests which are consistent against I(0) to I(1) changes. BT also propose tests which are designed for the case where the direction of the change is unknown; * Corresponding author. Tel.: +44-121-414-6657; fax: +44-121-414-7377. E-mail address: [email protected] (A.M.R. Taylor). 0165-1765/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2003.12.015
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that is, ratio-based tests which are consistent against series which display either shifts from I(0) to I(1) or from I(1) to I(0). An interesting feature of the ratio tests is that they require no form of long run variance correction in order to obtain pivotal limiting null distributions even when the driving shocks display weak dependence. However, and as noted in BT, this does not guarantee that the tests will have good finite sample size properties in such cases. In this paper we investigate the size distortions seen in the ratio tests when the shocks display weak parametric autocorrelation and propose modifications to the original tests which are shown to greatly improve the finite sample size properties of the resulting tests, without sacrificing too much finite sample power against persistence change alternatives. The paper is organised as follow. In Section 2 we provide a brief review of the ratio-based tests against changes in persistence. In Section 3 we discuss non-parametric modifications of the ratio tests designed to improve the finite-sample size properties of the tests, affording the practitioner a size/power trade-off not available using the standard ratio tests. The relative finite sample size and power properties of the ratio tests and their non-parametrically modified counterparts are investigated by Monte Carlo simulation in Section 4. Section 5 concludes.
2. Ratio tests for change in persistence Kim (2000), considers the null hypothesis that the scalar time-series process yt is formed as the sum of a purely deterministic component, dt (e.g. an intercept, or an intercept and linear trend), and a short memory component; that is, yt ¼ dt þ ut ;
t ¼ 1; . . . ; T
ð2:1Þ
where ut satisfies the usual strong mixing conditions of, inter alia, Phillips and Perron (1988, p. 336) 2 . In what follows, reference to a series as being with strictly positive and bounded long run variance rl I(0) is taken to imply that these conditions hold. Kim (2000) considers two alternatives: firstly, that yt displays a shift from I(0) to I(1) behaviour1 at time t = ts0Tb , where tb denotes the integer part of its argument, and secondly that there is a shift from I(1) to I(0) behaviour at time t = ts0Tb. These may be expressed within the model, yt ¼ dt þ zt;1 ;
t ¼ 1; . . . ts0 T b;
yt ¼ dt þ zt;2 ;
t ¼ ts0 T b þ 1; . . . ; T :
s0 að0; 1Þ
In the first case zt,2 = zt 1,2 + ut, with ut and zt,1 both I(0), while in the second case zt,1 = zt 1,1 + ut with ut and zt,2 both I(0). 1
An I(1) series is one formed from the accumulation of an I(0) series.
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BT and Kim et al. (2002) have independently proposed tests for the null hypothesis that yt is a constant I(0) process against an I(0)–I(1) shift at some unknown point in the sample. These tests are constructed using the sequence of ratios t X
t¼tsT bþ1
i¼tsTbþ1 !2 t
ðT tsT bÞ2 KðsÞ ¼ ðtsT bÞ
!2
T X
2
tsT b X X t¼1
eˆ 2;i ;
ð2:2Þ
eˆ 1;i
i¼1
saT , where T is a given sub-interval of [0, 1]. In Eq. (2.2), eˆ 1,t are the OLS residualsˆ from the regression of yt on either an intercept or an intercept and linear trend, t = 1, . . . tsTb, and e2,t are the OLS residuals from the regression of yt on either an intercept or an intercept and linear trend, t = tsTb + 1, . . ., T. Tests are then based on taking an appropriate function of the sequence K(). These authors investigate three such functions: the maximum over the sequence of statistics, after Andrews (1993); Hansen’s (1991) mean score, and, finally, Andrews and Ploberger’s (1994) meanexponential: we denote these statistics as K j, j = 1, . . ., 3, respectively. In each case, the null is rejected for large values of these statistics, and the tests are consistent at rate Op(T 2) against the I(0)–I(1) change DGP. Suppose now that yt is generated by the I(1)–I(0) change DGP. Here BT show that the tests which reject for large values of K j, j = 1, . . ., 3, are all inconsistent. However, BT show that tests which reject for large values of the corresponding statistics, denoted K Rj , j = 1, 2, 3, formed from the sequence of reciprocals of K(s), are consistent at rate Op(T 2) in this case, but are inconsistent against I(0)–I(1) alternatives. BT also suggest a test based on the maximum of the K () and (K ()) 1 sequences; that is, Kj *umaxfKj ; KRj g;
j ¼ 1; 2; 3;
ð2:3Þ
in each case rejecting for large values of the statistics. BT show that the pairwise ratio-based tests are Op(T 2) under both I(1)–I(0) and I(0)–I(1) alternatives. Representations for the limiting null distributions of the foregoing statistics are given in Kim et al. (2002) and BT. Crucially, they show that these representations do not depend on the long run variance 2 , even though neither the numerator nor the denominator of K(s) of Eq. (2.2) is scaled by a long run rl variance estimator. BT also show that all of the ratio-based tests discussed in this section are inconsistent against constant I(1) processes; that is where yt = dt + zt*, t = 1, . . ., T, where zt* = z*t 1 + ut, with ut an I(0) process. This is a useful result, implying that these tests will only diverge against series, which display changes in persistence. BT also propose LBI-based tests against changes in persistence but, unfortunately, these are consistent against constant I(1) processes. Interestingly, these LBI-based tests involve the use of a longrun variance estimator, of exactly the form used in Kwiatkowski et al. (1992) [KPSS], to obtain pivotal limiting null distributions.
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3. Non-parametrically modified ratio-based tests In discussing the properties of the ratio tests, BT argue that . . . in the presence of serially correlated innovations they do not require the arguably arbitrary decisions over the lag truncation parameter. . .. However. . . the finite sample size properties of the ratio-based tests are not satisfactory in practical situations. Although unnecessary for the purpose of achieving statistics with pivotal limiting null distributions, it is nonetheless perfectly feasible to scale the numerator and denominator of Eq. (2.2) by appropriate sub-sample (long run) variance estimators. A reason for considering such a modification is that if we individually studentise the numerator and denominator terms, we might expect the finite sample distribution of their ratio to more closely approximate the limiting null distribution. To that end, consider replacing K(s) of Eq. (2.2), for each saT , by the modified statistic Kðs; mÞ ¼
rˆ 21 KðsÞ rˆ 22
where, following KPSS (Eq. (10), p. 164), rˆ 22 ¼ ðT tsT bÞ1
T X
eˆ 22;t þ 2ðT tsT bÞ1
m X i¼1
t¼tsT bþ1
T X
wði; mÞ
eˆ 2;t eˆ 2;ti ;
t¼iþtsT bþ1
and rˆ 21 ¼ ðtsTbÞ1
tsTb X t¼1
eˆ 21;t þ 2ðtsTbÞ1
m X i¼1
wði; mÞ
tsT b X
eˆ 1;t eˆ 1;ti ;
t¼iþ1
with w(i, m) = 1 i/(m + 1), and proceed as above replacing K(s) by K(s, m) throughout. We denote the resulting statistics as Kj (m), K Rj (m) and K j*(m), j = 1, 2, 3, with an obvious notation. Notice that the numerator (denominator) of K(s, m) is nothing other than a standard KPSS stationarity test statistic applied to the second (first) sub-sample of the data. As a consequence, this modified statistic affords the practitioner a certain degree of control over the finite-sample properties of the ratio-based testing procedure via the usual size-power trade-off in the bandwidth (lag truncation) parameter m observed for these tests in KPSS. In the standard KPSS framework m is chosen to satisfy the conditions that m ! l and m = o(T1/2) as T ! l; cf. KPSS. The same can be done here, although in fact m can be set to any non-negative integer value (including zero) since under the null rˆ12 and rˆ22 will converge in probability to the same constant as T ! l. As a consequence the non-parametrically modified statistics will possess identical limiting null
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distributions to the corresponding unmodified statistic so that the asymptotic critical values provided in BT and Kim et al. (2002) may therefore still be used. Where the unmodified tests were consistent at rate Op(T2) above, the corresponding modified test will be consistent at rate Op(T/m). This because any variance estimator constructed from data containing any of the I(1) segment of the sample will diverge at rate Op(Tm), while variance estimators constructed over purely I(0) data will converge in probability to some finite constant.
4. Simulation results In this section we use Monte Carlo simulation methods to investigate the finite-sample size and power properties of the non-parametrically modified ratio-based tests vis-a`-vis the corresponding unmodified tests, highlighting the role of the lag truncation parameter m in the implied size-power trade-off for a given sample size. First, in Tables 1 and 2 we report the empirical rejection frequencies of the tests against data generated by Eq. (2.1) with dt = 0, and the following ARMA(1,1) parametric form on ut, ut ¼ /ut1 þ et het1 ;
t ¼ 200; . . . ; T ;
ð4:1Þ
Table 1 Empirical rejection frequencies of modified and unmodified ratio-based persistence change tests: de-meaned data T
/
h
K1
K1(0)
K1(1)
K1(2)
K1(mT)
K1*
K1*(0)
K1*(1)
K1*(2)
K1*(mT)
50
0.85 0.90 0.95 0.00 0.00 0.00 0.00 0.85 0.90 0.95 0.00 0.00 0.00 0.00 0.85 0.90 0.95 0.00 0.00 0.00 0.00
0 0 0 0.5 0.5 0.8 0.8 0 0 0 0.5 0.5 0.8 0.8 0 0 0 0.5 0.5 0.8 0.8
0.35 0.41 0.48 0.01 0.08 0.00 0.09 0.27 0.33 0.43 0.01 0.07 0.00 0.08 0.17 0.22 0.33 0.02 0.06 0.00 0.07
0.10 0.11 0.14 0.04 0.04 0.03 0.04 0.08 0.09 0.11 0.03 0.05 0.02 0.05 0.05 0.06 0.08 0.04 0.05 0.02 0.05
0.08 0.10 0.12 0.04 0.05 0.03 0.04 0.07 0.08 0.10 0.03 0.05 0.01 0.05 0.05 0.06 0.07 0.03 0.05 0.02 0.05
0.07 0.08 0.10 0.04 0.06 0.04 0.06 0.07 0.08 0.10 0.03 0.05 0.01 0.05 0.05 0.06 0.08 0.03 0.05 0.02 0.05
0.05 0.06 0.06 0.04 0.06 0.05 0.06 0.06 0.07 0.08 0.03 0.05 0.01 0.05 0.06 0.06 0.08 0.03 0.05 0.02 0.05
0.49 0.57 0.66 0.00 0.09 0.00 0.10 0.37 0.47 0.60 0.01 0.07 0.00 0.08 0.23 0.31 0.47 0.02 0.06 0.00 0.07
0.12 0.14 0.17 0.04 0.04 0.03 0.05 0.09 0.11 0.14 0.03 0.05 0.01 0.05 0.05 0.06 0.09 0.04 0.05 0.02 0.05
0.10 0.12 0.14 0.04 0.05 0.03 0.05 0.08 0.10 0.14 0.03 0.05 0.01 0.05 0.06 0.06 0.08 0.03 0.05 0.02 0.05
0.07 0.09 0.11 0.03 0.06 0.03 0.06 0.08 0.09 0.12 0.03 0.06 0.01 0.05 0.06 0.07 0.09 0.03 0.06 0.01 0.05
0.05 0.06 0.07 0.04 0.05 0.04 0.06 0.07 0.08 0.09 0.03 0.06 0.01 0.05 0.06 0.07 0.09 0.03 0.06 0.02 0.05
100
200
DGP Eq. (2.1), Eq. (4.1).
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Table 2 Empirical rejection frequencies of modified and unmodified ratio-based persistence change tests: de-meaned and de-trended data T
/
h
K1
K1(0)
K1(1)
K1(2)
K1(mT)
K1*
K1*(0)
K1*(1)
K1*(2)
K1*(mT)
50
0.85 0.90 0.95 0.00 0.00 0.00 0.00 0.85 0.90 0.95 0.00 0.00 0.00 0.00 0.85 0.90 0.95 0.00 0.00 0.00 0.00
0 0 0 0.5 0.5 0.8 0.8 0 0 0 0.5 0.5 0.8 0.8 0 0 0 0.5 0.5 0.8 0.8
0.58 0.63 0.68 0.01 0.13 0.03 0.15 0.48 0.56 0.67 0.01 0.09 0.01 0.09 0.31 0.42 0.58 0.01 0.06 0.00 0.08
0.20 0.23 0.27 0.06 0.04 0.11 0.04 0.15 0.19 0.24 0.04 0.04 0.05 0.04 0.09 0.12 0.18 0.03 0.05 0.02 0.05
0.11 0.13 0.16 0.10 0.05 0.23 0.04 0.12 0.16 0.22 0.04 0.05 0.06 0.04 0.08 0.11 0.16 0.03 0.05 0.03 0.05
0.02 0.02 0.03 0.15 0.04 0.28 0.04 0.10 0.14 0.18 0.05 0.05 0.10 0.05 0.08 0.11 0.16 0.04 0.05 0.02 0.05
0.01 0.01 0.01 0.10 0.04 0.17 0.04 0.04 0.05 0.08 0.07 0.05 0.22 0.05 0.07 0.08 0.13 0.04 0.05 0.04 0.05
0.74 0.79 0.83 0.01 0.15 0.02 0.17 0.63 0.74 0.84 0.01 0.10 0.01 0.11 0.44 0.58 0.76 0.01 0.07 0.00 0.09
0.24 0.28 0.33 0.06 0.04 0.11 0.05 0.18 0.24 0.30 0.04 0.04 0.04 0.05 0.10 0.14 0.22 0.03 0.05 0.01 0.05
0.12 0.15 0.18 0.11 0.05 0.26 0.05 0.15 0.20 0.28 0.04 0.05 0.05 0.05 0.09 0.12 0.21 0.03 0.05 0.01 0.05
0.02 0.01 0.02 0.17 0.04 0.33 0.04 0.11 0.16 0.23 0.04 0.05 0.09 0.06 0.09 0.12 0.20 0.03 0.05 0.02 0.05
0.01 0.01 0.01 0.10 0.04 0.17 0.04 0.05 0.06 0.09 0.08 0.05 0.24 0.05 0.07 0.09 0.16 0.03 0.05 0.03 0.05
100
200
DGP Eq. (2.1), Eq. (4.1).
with et f NIID(0, 1), and the design parameters /a{0.85, 0.90, 0.95} and ua{ F 0.5 F 0.8}2. Table 1 reports results for the case where the residuals used in Eq. (2.2) are obtained from regressions on an intercept (de-meaned data) and Table 2 the corresponding results from regressions on an intercept and trend (de-meaned and de-trended data). Second, in Table 3 (de-meaned data) and Table 4 (de-meaned and de-trended data) we report the empirical rejection frequencies of the tests when the data are generated according to the I(0)–I(1) switch DGP yt ¼ qt yt1 þ et ;
t ¼ 200; . . . ; T ;
ð4:2Þ
et f NIID(0, 1), with qt = 0, t = 200, . . ., ts0Tb, and qt = 1, t = ts0Tb + 1, . . ., T. We consider the following values of the breakpoint parameter, s0a{0.3, 0.4, 0.5, 0.6, 0.7}. We have set the autoregressive parameter to zero in the first sub-sample to avoid biasing the results via the inherent over-sizing problems that result (in particular in the case of the unmodified tests) from a positive first-order autoregressive coefficient; cf. Tables 1 and 2.
2
Other values of the design parameters were considered but the results for these cases neither add to nor contradict the qualitative conclusions drawn from the cases reported.
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113
Table 3 Empirical rejection frequencies of modified and unmodified ratio-based persistence change tests: de-meaned data T
s0
K1
K1(0)
K1(1)
K1(2)
K1(mT)
K1*
K1*(0)
K1*(1)
K1*(2)
K1*(mT)
50
0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7
0.89 0.90 0.90 0.87 0.80 0.98 0.98 0.98 0.97 0.94 1.00 1.00 1.00 1.00 0.99
0.70 0.75 0.77 0.75 0.70 0.87 0.90 0.91 0.91 0.87 0.96 0.97 0.97 0.97 0.96
0.52 0.58 0.61 0.60 0.56 0.74 0.79 0.81 0.80 0.76 0.89 0.91 0.92 0.92 0.89
0.37 0.43 0.46 0.46 0.43 0.64 0.69 0.71 0.71 0.67 0.82 0.85 0.87 0.87 0.84
0.21 0.27 0.30 0.32 0.30 0.47 0.52 0.56 0.56 0.51 0.66 0.70 0.73 0.73 0.69
0.90 0.89 0.87 0.84 0.76 0.98 0.98 0.98 0.96 0.93 1.00 1.00 1.00 1.00 0.98
0.66 0.69 0.70 0.67 0.61 0.84 0.87 0.87 0.86 0.82 0.94 0.95 0.96 0.95 0.93
0.49 0.52 0.53 0.50 0.46 0.70 0.74 0.75 0.73 0.69 0.85 0.88 0.89 0.86 0.85
0.32 0.36 0.36 0.35 0.32 0.60 0.64 0.65 0.64 0.59 0.78 0.81 0.83 0.82 0.78
0.17 0.20 0.21 0.21 0.20 0.43 0.46 0.48 0.46 0.41 0.62 0.65 0.67 0.66 0.61
100
200
DGP Eq. (4.2).
All reported simulation experiments were performed using 40,000 Monte Carlo replications and the RNDN function of Gauss 3.1, for T = 50, 100, 200 with the first 200 observations discarded to control for initial effects. All tests were run at the nominal exact 5% level, with T =[0.2, 0.8] as in Kim (2000); Kim et al. (2002) and BT. Results are reported for the K1, K1*, and K1(m) and K1*(m)
Table 4 Empirical rejection frequencies of modified and unmodified ratio-based persistence change tests: de-meaned and de-trended data T
s0
K1
K1(0)
K1(1)
K1(2)
K1(mT)
K1*
K1*(0)
K1*(1)
K1*(2)
K1*(mT)
50
0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7
0.83 0.81 0.79 0.75 0.68 0.98 0.98 0.98 0.97 0.94 1.00 1.00 1.00 1.00 0.99
0.70 0.72 0.72 0.71 0.65 0.92 0.92 0.93 0.93 0.90 0.98 0.99 0.99 0.99 0.98
0.37 0.43 0.44 0.44 0.42 0.75 0.79 0.80 0.81 0.78 0.92 0.94 0.95 0.96 0.94
0.05 0.07 0.08 0.09 0.09 0.57 0.63 0.65 0.67 0.65 0.84 0.88 0.89 0.90 0.89
0.01 0.01 0.01 0.01 0.01 0.19 0.26 0.28 0.29 0.30 0.57 0.63 0.66 0.68 0.67
0.85 0.84 0.80 0.75 0.63 0.98 0.98 0.97 0.96 0.92 1.00 1.00 1.00 1.00 0.99
0.65 0.67 0.66 0.63 0.55 0.89 0.90 0.90 0.89 0.85 0.97 0.98 0.98 0.98 0.97
0.32 0.36 0.36 0.33 0.30 0.71 0.75 0.75 0.75 0.70 0.89 0.92 0.92 0.93 0.91
0.02 0.03 0.03 0.04 0.04 0.53 0.59 0.60 0.58 0.55 0.81 0.85 0.86 0.86 0.83
0.01 0.01 0.01 0.01 0.01 0.17 0.22 0.23 0.20 0.21 0.55 0.61 0.64 0.62 0.60
100
200
DGP Eq. (4.2).
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tests3 and where in relation to the non-parametrically modified ratio tests the following values of the lag truncation parameter were used: m = 0, 1, 2, mT, where mT = t4(T/100)1/4b; that is, both fixed and sample size dependent values were considered. Consider first the results in Tables 1 and 2. It is clear that although asymptotically pivotal, the unmodified ratio-based statistics yield tests, which can be very badly over-sized in finite samples. Empirical size does tend towards the nominal level as T is increased, ceteris paribus, albeit at a rather slow rate. To illustrate, for the de-meaned and de-trended case (Table 2) we see that for / = 0.95 the unmodified K1 and K1* tests have empirical sizes of (0.68, 0.83) for T = 50, improving only marginally to (0.58, 0.76) for T = 200. In this case the corresponding modified tests with m = 0 and m = mT have sizes (0.27, 0.33) and (0.01, 0.01), respectively, for T = 50, and (0.18, 0.22) and (0.13, 0.16), respectively, for T = 200. As with the foregoing example, it is clear that most of the size improvements are obtained simply by setting m = 0 in the variance estimators. This is apparent for both the de-meaned and de-trended cases. Increasing m beyond zero tends to introduce rather smaller additional improvements; indeed, for T = 50, in the de-trending case increasing m tends to cause under (over) sizing with autoregressive (negative moving average) errors, the antithesis of what is seen for the unmodified tests. Size distortions tend to be worse, ceteris paribus, for all tests in the case of de-meaned and de-trended vis-a`-vis de-meaned data, echoing the findings of KPSS. Overall, however, it is clear that the non-parametrically modified tests perform very well in re-dressing size, taken relative to the performance of the unmodified tests. All of the tests tend to display their greatest distortions in the cases of very large positive autoregressive errors (over-sizing) and very large negative moving average errors (under-sizing), with the exceptions noted above. Tables 3 and 4 allow us to investigate the degree of power loss incurred by using the nonparametrically modified tests. The power losses incurred from using the modified tests is greatest, ceteris paribus, where T is small, and also appear rather smaller in the de-meaned and de-trended case. As would be expected (again compare with KPSS), other things being equal, power decreases as m is increased. The power loss from using m = mT is particularly large, especially so in the de-meaned and detrended case for T = 50. In general, however, the difference in power between the unmodified tests and the corresponding modified test with m = 0 are not too substantial, particularly in the de-meaned and detrended case. This is very encouraging given the observation from Tables 1 and 2 that this choice of m delivers most of the size improvements seen when using the modified statistics. Finally, it is worth noting that, other things equal, power is not necessarily lower in the de-meaned and de-trended case than for the de-meaned case; this phenomenon is also apparent in the simulation results presented in Table 4 (p. 172) of KPSS. For the modified tests then, the usual trade-off between size and power in m seen in residual-based tests is apparent here. From our results we recommend setting m = 0 in the modified tests, this choice
3
Results for tests based on the mean-score and mean-exponential functions gave qualitatively similar results to those reported and, hence, are omitted. Moreover, results are not reported for the KR1 and KR1 (m) tests, since their size properties were almost identical to those reported for the K1 and K1(m) tests, while they are not consistent against the I(0) – I(1) switch DGP considered, Eq. (4.2). We also considered the corresponding I(1) – I(0) switch DGP: the tests of Eq. (2.3) and the corresponding non-parametric modifications behaved almost identically to the results in Tables 3 and 4, while the KR1 and KR1 (m) tests display similar power to the results reported for the K1 and K1(m) tests in Tables 3 and 4 if one exchanges s0 with 1 s0.
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delivering most of the available size improvements under a given scenario yet not sacrificing too much test power against persistence change processes.
5. Conclusions In this paper we have proposed modifications to the ratio-based tests for persistence change developed in Kim (2000), BT and Kim et al. (2002). These modifications involve scaling the ratio statistics by (long run) variance estimators constructed across sub-samples of the data. We show that these modifications can deliver dramatic improvements in the finite-sample size properties of the tests without altering the limiting null distributions of the statistics. For a given sample size, the larger the lag truncation parameter used in constructing the variance estimators the larger the power losses seen against processes displaying persistence change. However, a very low-order (often zero) fixed lag truncation appears to give a useful pragmatic balance between re-dressing the size properties of the tests and keeping the degree of power loss against persistence change processes relatively small.
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