Comparison of Misspecification Tests Designed for Non-linear Time Series Models

Transcription

1 ömmföäfsäafaäsfassfassfas fffffffffffffffffffffffffffffffffff Discussion Papers Comparison of Misspecification Tests Designed for Non-inear Time Series Modes Leena Kaiovirta University of Hesinki and HECER Discussion Paper No. 309 December 200 ISSN HECER Hesinki Center of Economic Research, P.O. Box 7 (Arkadiankatu 7), FI-0004 University of Hesinki, FINLAND, Te , Fax , E-mai info-hecer@hesinki.fi, Internet

2 HECER Discussion Paper No. 309 Comparison of Misspecification Tests Designed for Non-inear Time Series Modes* Abstract We use GARCH and regime switching modes to compare the reiabiity of recenty proposed misspecification tests. Our simuations indicate that simpe moment based LM type tests are more reiabe than other moment based tests or tests that empoy the empirica distribution function or non-parametric methods. JEL Cassification: C22, C52 Keywords: quantie residua, LM type test, Khmaadze s martingae transformation, empirica process, non-parametric test. Leena Kaiovirta Department of Poitica and Economic Studies University of Hesinki P.O. Box 7 (Arkadiankatu 7) FI-0004 University of Hesinki FINLAND e-mai: eena.kaiovirta@hesinki.fi * The author thanks Pentti Saikkonen, Markku Lanne, Jukka Nybom, and Bent Niesen for hepfu comments that improved this work. The author is responsibe for any remaining errors. The author is gratefu to the Academy of Finand, the Okobank Group Research Foundation, and the Finnish Foundation for Advancement of Securities Markets for financia support.

3 Introduction The misspecification tests of Bai (2003), Duan (2003), Hong and Li (2005) and Kaiovirta (2006) are appicabe to a wide cass of modes incuding inear and noninear time series modes, and these tests propery take the uncertainty caused by parameter estimation into account. Because the tests empoy different methodoogies, they may perform differenty in finite sampes. We study their performance by simuating GARCH and regime-switching modes. 2 Compared tests A the tests considered here use quantie residuas. These residuas, unike traditiona residuas, are reiabe in non-inear modes aso (Kaiovirta 2006). They exist for any fuy specified parametric mode with continuous (conditiona) cumuative distribution function (CDF). Let θ 0 be the true parameter vaue that generates the observed T vector Y. The definition of uniformy distributed quantie residua for each t {,..., T } is U t = F t (θ 0, Y t ), where F t is the conditiona CDF impied by the mode. Simiary, the normay distributed quantie residua is R t = Φ (U t ), where Φ ( ) is the inversed CDF of the standard norma distribution. If the estimated mode is correcty specified and θ T is a consistent estimator for θ 0, then vector of Ût = F t ( θ T, Y t )s (or simiary ˆR t s) are asymptoticay i.i.d. This impies that the hypothesis of a correct specification and properties of quantie residuas are convenienty connected, which makes quantie residuas usefu in mode evauation.

4 Bai (2003) generaizes the Komogorov-Smirnov test by appying the Khmaadze s martingae transformation to remove the effect of parameter estimation in the empirica process based on Ûts. Duan (2003) considers four different tests based on moments of modified ˆR t s and removes the effect of parameter estimation by incuding first order approximations in the test statistics. A Cramér-von Mises type test statistics of Hong and Li (2005) use nonparametric estimates of the density function of Ûts. Kaiovirta (2006) uses first order approximations to correct the effect of parameter estimation in three LM type tests based on moments of ˆR t s. Henceforth, we denote the test of Bai (2003) with KS, the tests of Duan (2003) with D i, i =, 2, 3, 4, the tests of Hong and Li (2005) with HL i, i =,..., 20, and the tests of Kaiovirta (2006) with N, A, and H. In addition, the resuts of Bai (2003) aow us to introduce aternative test statistics based on the transformed empirica process ŴT (r), 2 r [0, ], of Ûts, see Bai (2003) for detais. Thus, we consider aso the foowing statistics: Anderson-Daring type test AD c,d = sup c r d Ŵ T (r) for some c < d (0, ); Cramér-von Mises type tests CM = 0 ŴT (r) dr and CM 2 = 0 ŴT (r) 2 dr; and Pearson s goodness-of-fit type test S = ] 2 [ŴT (r i ) ŴT (r i ) / (ri r i ) with 0 < r < < r < i=. 3 Furthermore, we compute the standard, (non-transformed) empirica process based, Pearson s goodness-of-fit test P. The abbreviations indicate the type of possibe misspecifications: N for non-normaity, A for autocorreation, and H for heteroscedasticity. 2 For detais on ŴT (r), see Bai (2003), page Bai (2003) shows that ŴT (r) converges weaky to a standard Brownian motion. Thus, S d χ 2. The critica vaues of the other statistics need to be simuated. For exampe, the 5% eve critica vaues, computed using 0 5 repications and 0 5 observations, are 2.24 for KS, 2.00 for AD 0.2,0.8,.4 for CM, and.67 for CM 2 tests. 2

5 3 Modes Let ε t n.i.d.(0, ) and η t n.i.d.(0, ) be independent, and I A is the indicator function of a set A. We generate data using modes: ) N(0,), Y t = ε t ; 2) GARCH(,), Y t = σ t ε t with σ t = yt σ 2 t ; 3) MAR(3,,0), Y t = ( Y t )I ηt <0 + ( Y t )I 0<ηt + ( Y t )I ηt > + ε t ; and 4) MAR(3,,0)-GARCH(,), Y t = 0.24I ηt < I 0.5<ηt I ηt > Y t + σ t ε t with σ t = yt σ 2 t. The MAR mode can generate positivey or negativey skewed, peaked or fat, and muti-moda distributions. For more detais, see Kaiovirta (2006). In power comparisons, we aso estimate submodes of MAR(3,,0): a) MAR(3,0,0), Y t = µ I ηt <c + µ 2 I c <η t c 2 + µ 3 I ηt >c 2 + ε t ; b) MAR(2,,0), Y t = (µ + φ Y t )I ηt <c + (µ 2 + φ 2 Y t )I c η t + ε t ; and c) MAR(2,0,0), Y t = µ I ηt <c + µ 2 I c η t + ε t. 4 Simuations Tabes, 2, and 3 report size and power of the tests at 5% nomina eve. Thus, we make no size corrections in misspecified modes. The sampe sizes vary from 00 to 3000, and resuts base on 2000 repications. We obtained the MLEs of the 3

6 Tabe : Size and power at 5% eve. Size Size Power N(,0) GARCH(,) GARCH(,) data N(µ,σ 2 ) estimated Sampe KS (a) AD (a) 0.2, CM (a) CM (a) S (a),(b) P (b) D D D D HL HL N A (c) H (c) NOTES: (a) We use the function ġ(r, θ T ) = (, Φ (r, θ T ), Φ 2 (r, θ T )), given in Bai(2003), in the Khmaadze s martingae transformation. (b) The bandwidth 3.49ŝtd(Ût)/T /3 determines the number of casses (Scott 979). (c) The subscript signifies the number of ags empoyed in the test statistics. 4

7 Tabe 2: MAR(3,,0) data, size and power at 5% eve. Size Power MAR(3,,0) MAR(2,,0) MAR(3,0,0) MAR(2,0,0) N(µ,σ 2 ) Sampe KS (d) AD (d) 0.2, S (b),(d) P (b) D D D D HL HL N A (c) H (c) NOTES: (b) and (c) See Tabe (d) We use an estimated function ḡ T (r, ˆθ T ), given in Bai (2003), in the Khmaadze s martingae transformation. parameters using the cm package in GAUSS. The behavior of the tests cassifies them roughy into three groups. One group is formed by the tests KS, AD, CM, CM 2, S, and P. These tests are unreiabe in size and exhibit no power (against GARCH in Tabe ) or occasiona power 5

8 Tabe 3: MAR(3,,0)-GARCH(,) data, size and power at 5% eve. Size Power Power MAR(3,,0)-GARCH(,) MAR(3,,0) GARCH(,) Sampe KS (d) AD (d) 0.2, S (b),(d) P (b) D D D D HL HL N A (c) H (c) NOTES: See Tabes and (Tabes 2 and 3) 4. This power is, however, exaggerated in KS and S by their oversizeness. Overa, a comparison to P shows that the Khmaadze s martingae transformation provides no improvement. The tests of Duan (2003) comprise another group that has adequate size in a inear mode (Tabe ), but are undersized, especiay D, in noninear modes. 4 The behavior of AD, CM, and CM 2 is very simiar, so we ony report resuts on AD in Tabes 2 and 3. 6

9 Further, these tests are rather poweress, especiay D and D 2. In the argest sampes D 3 and D 4 detect the misspecifications in dependence structure of the mean and variance. In contrast, the misspecification in the number of regimes (Tabe 2) remains undetected. Finay, the tests HL, HL 2, N, A, and H comprise the ast group, in which both size and power properties are acceptabe. Athough sizes are unadjusted, we can concude that H 3 has superior power to HL 5 in detecting GARCH (Tabe ). In practice one can ony use nomina eves. Therefore, one shoud prefer tests that are reiabe in size and exhibit best power at nomina eves. In this sense, we may concude that the tests of Kaiovirta (2006) perform best. In addition, these tests are abe to give hints where misspecification ies. For exampe, H 3 indicates heteroscedasticity in Tabes and 3, N wrong distribution in Tabe 2, and A autocorreation in Tabes 2 and 3. In comparison, the test HL detects misspecification in autocorreation structure, but ess so if misspecification is in conditiona heteroscedasticity or in distribution. Furthermore, HL impies no hints why a mode is rejected. 5 Concusions In our simuations, the tests based on Khmaadze s martingae transformation, incuding the test of Bai (2003), are unreiabe. The moment based tests of Duan (2003) exhibit undersizeness and insuffi cient power. The non-parametric tests of Hong and Li (2005) are sometimes undersized and ack power against GARCH, for exampe. The LM type tests of Kaiovirta (2006) have accurate size and the 5 We computed a HL i, i =,..., 20.They were simiar in size and HL aways had the best power among them. 7

10 best power at nomina eves against the misspecifications considered. References Bai, J.: 2003, Testing parametric conditiona distributions of dynamic modes, The Review of Economics and Statistics 85(3), Duan, J.-C.: 2003, A specification test for time series modes by a normaity transformation. working paper, University of Toronto. Hong, Y. and Li, H.: 2005, Nonparametric specification testing for continuoustime modes with appications to term stucture of interest rates, The Review of Financia Studies 8(), Kaiovirta, L.: 2006, Misspecification tests based on quantie residuas. HECER Discussion Paper, No 24. Scott, D. W.: 979, On optima and data-based histograms, Biometrika 66(3),