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Testing for misspecification in the short-run component of GARCH-type models

Thomas Chuffart, Emmanuel Flachaire and Anne Péguin-Feissolle

Abstract

In this article, a misspecification test in conditional volatility and GARCH-type models is presented. We propose a Lagrange Multiplier type test based on a Taylor expansion to distinguish between (G)ARCH models and unknown GARCH-type models. This new test can be seen as a general misspecification test of a large set of GARCH-type univariate models. It focuses on the short-term component of the volatility. We investigate the size and the power of this test through Monte Carlo experiments and we compare it to two other standard Lagrange Multiplier tests, which are more restrictive. We show the usefulness of our test with an illustrative empirical example based on daily exchange rate returns.

Acknowledgement

We would like to thank the anonymous referee for very useful comments and suggestions that helped us to prepare the final version of our paper, and Marjorie Sweetko for language verification of the manuscript. Finally, the paper has also benefited from the comments of Cyril Dell’Eva on the empirical application and Bilel Sanhaji. Previous versions of this paper have circulated under the title “Testing for Misspecification in GARCH-type models”.

Appendices

A Computation of the LM statistic

I(θ), given in Equation (10) asI(θ)=E[2lt(θ)θθ], can be written:

I(θ)=E[2lt(θ)λλ2lt(θ)λϕ2lt(θ)λB(2lt(θ)λϕ)2lt(θ)ϕϕ2lt(θ)ϕB(2lt(θ)λB)(2lt(θ)ϕB)2lt(θ)BB]=[I11(θ)I12(θ)I12(θ)I22(θ)]

with

(28){I11(θ)=E[2lt(θ)λλ2lt(θ)λϕ(2lt(θ)λϕ)2lt(θ)ϕϕ]I12(θ)=E[2lt(θ)λB2lt(θ)ϕB]I22(θ)=E[2lt(θ)BB]

Using lt(θ) given by (5), we have

2lt(θ)θθ=1htmtθmtθ+12ht2htθhtθ.

Thus, the elements in (28) can be derived as

I11(θ)=E[1htmtλmtλ+12ht2htλhtλ12ht2htλhtϕ12ht2htϕhtλ12ht2htϕhtϕ]I12(θ)=I21(θ)=E[2htλB2htϕB]=E[1ht2htλhtB1ht2htϕhtB],

and

I22(θ)=E[2lt(θ)BB]=E[12ht2htBhtB].

Under conditional symmetry of the error term, I11 becomes diagonal. The first derivative of the log-likelihood with respect to the parameters θ, lt(θ)θ, is defined as follows:

lt(θ)θ=[lt(θ)λlt(θ)ϕlt(θ)B].

Under the null hypothesis, lt(θ)λ=0 and lt(θ)ϕ=0, thus we have:

lt(θ)θ|θ=θ^R=[00lt(θ)B].

We can derive the LM statistic:

LM=1T[00lt(θ)B]I(θ^R)1[00lt(θ)B]=1T[00lt(θ)B][E(1htmtλmtλ+12ht2htλhtλ01ht2htλhtB012ht2htϕhtϕ12ht2htϕhtB1ht2htλhtB(12ht2htϕhtB)12ht2htBhtB)]1[00lt(θ)B]

Using the properties of the partitioned matrix, the statistic is equal to:

LM=1Tlt(θ)B{E(12ht2htBhtB)[E(12ht2htϕhtB)E(12ht2htϕhtϕ)1E(12ht2htϕhtB)+E(1ht2htλhtB)E(1htmtλmtλ+12ht2htλhtλ)1E(1ht2htλhtB)]}1lt(θ)B

with

lt(θ)B=12t=1TB[ln(ht)+εt2ht]=12t=1T[Bln(ht)+εt2B1ht]=12t=1T1ht[εt2ht1]htB.

htϕ and htB are respectively given by equations (14) and (15). If we note ft=m(xt,λ)λ, ct=1hthtλ, vt=1hthtB and zt=1hthtϕ, the statistic will be computed as

LM=14(t=1T[εt2ht1]vt)V1(t=1T[εt2ht1]vt)

with

V=12t=1Tvtvt12t=1T(ztvt)t=1T(ztzt)1t=1T(ztvt)+t=1T(ctvt)t=1T(1htftft+12ctct)1t=1T(ctvt).

The LM statistic has an asymptotic χ2 distribution with N degrees of freedom (because the dimension of the vector B is N given by (9).)

B On Taylor expansions

In this appendix, we present the Taylor expansion when k is even. We also propose some examples where pa, qa and k take different values.

  • The Taylor expansion expression for k odd differs a bit from Equation (B):

    ht=α0+j=1qαjεtj2+j=1pγjhtj+j=q+1qaαjεtj2+j=p+1paγjhtj+j=1qaψjεtj+j1=1qaj2=j1+1qaϕj1j2εtj1εtj2+j1=1qai1=1paδj1j2εtj1htj2+j1=1paj2=j1paγj1j2htj1htj2++i1=1qai2=i1qaik=ik1qaϕi1ikεti1εtik+j1=1paj2=j1pajk=jk1paγj1jkhtj1htjk+j1=1paj2=j1pajk~2=jk~21pai1=1qai2=i1qaik¯2=ik¯21qaδj1jk~2i1ik¯2htj1htjk~2εti1εtik¯2+j1=1paj2=j1pajk¯2=jk¯21pai1=1qai2=i1qaik~2=ik~21qaδ~j1jk¯2i1ik~2htj1htjk¯2εti1εtik~2

    with k~=k1 and k¯=k+1.

  • Order k = 2 expansion of ht = α0 + G(εt−1, ht−1, ρ):

    ht=α0+αiεt12+ψ1εt1+j=1kγjht1j+δ11εt1ht1+R1
  • Order k = 3 expansion of ht = α0 + G(εt−1, ht−1, ρ):

    ht=α0+α1εt12+γ1ht1+ψ1εt1++δ11εt1ht1+ϕ111εt13+γ11ht12+γ111ht13+δ111ht1εt12+δ~111ht12εt1+R1
  • Order k = 3 expansion of ht = α0 + G(εt−1, εt−2, ht−1, ht−2, ρ):

    ht=α0+α1εt12+γ1ht1+ψ1εt1+ψ2εt2+γ2ht2+ϕ12εt1εt2+ϕ22εt22+δ11εt1ht1+δ12εt1ht2+δ21εt2ht1+δ22εt2ht2+γ11ht12+γ12ht1ht2+γ22ht22+ϕ111εt13+ϕ222εt23+γ111ht13+γ222ht13+δ111ht1εt12+δ112ht1εt1εt2+δ122ht1εt22+δ211ht2εt12+δ212ht2εt1εt2+δ222ht2εt22+δ~111ht1εt12+δ~112ht1εt1εt2+δ~122ht1εt22+δ~211ht2εt12+δ~212ht2εt1εt2+δ~222ht2εt22+R1

References

Amado, C., and T. Teräsvirta. 2017. “Specification and Testing of Multiplicative Time-Varying GARCH Models with Applications.” Econometric Reviews 36: 421–446.10.1080/07474938.2014.977064Search in Google Scholar

Beine, M., S. Laurent, and C. Lecourt. 2003. “Official Central Bank Interventions and Exchange Rate Volatility: Evidence from a Regime-Switching Analysis.” European Economic Review 47: 891–911.10.1016/S0014-2921(02)00306-9Search in Google Scholar

Berkes, I., L. Horváth, and P. Kokoszka. 2003. “GARCH Processes: Structure and Estimation.” Bernoulli 9: 201–227.10.3150/bj/1068128975Search in Google Scholar

Black, F. 1976. “Studies of Stock Price Volatility Changes.” In Proceedings of the 1976 Meetings of the American Statistical Association, Business and Economics Statistics Section, 177–181.Search in Google Scholar

Bollerslev, T. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31: 307–327.10.1016/0304-4076(86)90063-1Search in Google Scholar

Bonser-Neal, C., and G. Tanner. 1996. “Central Bank Intervention and the Volatility of Foreign Exchange Rates: Evidence from the Options Market.” Journal of International Money and Finance 15: 853–878.10.1016/S0261-5606(96)00033-2Search in Google Scholar

Cavicchioli, M. 2014. “Determining the Number of Regimes in Markov Switching VAR and VMA Models.” Journal of Time Series Analysis 35: 173–186.10.1002/jtsa.12057Search in Google Scholar

Chan, F., and M. McAleer. 2002. “Maximum Likelihood Estimation of STAR and STAR-GARCH Models: Theory and Monte Carlo Evidence.” Journal of Applied Econometrics 17: 509–534.10.1002/jae.686Search in Google Scholar

Chuffart, T. 2015. “Selection Criteria in Regime Switching Conditional Volatility Models.” Econometrics 3: 289.10.3390/econometrics3020289Search in Google Scholar

Davidson, R., and J. MacKinnon. 2000. “Bootstrap Tests: How Many Bootstraps?” Econometric Reviews 19: 55–68.10.1080/07474930008800459Search in Google Scholar

Ding, Z., C. W. Granger, and R. F. Engle. 1993. “A Long Memory Property of Stock Market Returns and a New Model.” Journal of Empirical Finance 1: 83–106.10.1016/0927-5398(93)90006-DSearch in Google Scholar

Dominguez, K. M. 1998. “Central Bank Intervention and Exchange Rate Volatility.” Journal of International Money and Finance 17: 161–190.10.1016/S0261-5606(97)98055-4Search in Google Scholar

Engle, R. F. 1982. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica 50: 987–1007.10.2307/1912773Search in Google Scholar

Engle, R. F., and V. K. Ng. 1993. “Measuring and Testing the Impact of News on Volatility.” Journal of Finance 48: 1749–1778.10.1111/j.1540-6261.1993.tb05127.xSearch in Google Scholar

Erdemlioglu, D., S. Laurent, and C. J. Neely. 2013. “Econometric Modeling of Exchange Rate Volatility and Jumps, chapter 16.” In Handbook of Research Methods and Applications in Empirical Finance, edited by Edward Elgar, 373–427. Cheltenham, UK: Edward Elgar Publishing.10.4337/9780857936097.00026Search in Google Scholar

Ferraro, D., K. Rogoff, and B. Rossi. 2015. “Can Oil Prices Forecast Exchange Rates? An Empirical Analysis of the Relationship Between Commodity Prices and Exchange Rates.” Journal of International Money and Finance 54: 116–141.10.1016/j.jimonfin.2015.03.001Search in Google Scholar

Francq, C., and J.-M. Zakoian. 2007. “Quasi-Maximum Likelihood Estimation in GARCH Processes When Some Coefficients are Equal to Zero.” Stochastic Processes and Their Applications 117: 1265–1284.10.1016/j.spa.2007.01.001Search in Google Scholar

Franses, P. H., and D. Van Dijk. 1996. “Forecasting Stock Market Volatility Using (non-linear) Garch Models.” Journal of Forecasting 15: 229–235.10.1002/(SICI)1099-131X(199604)15:3<229::AID-FOR620>3.0.CO;2-3Search in Google Scholar

Galì, J., and T. Monacelli. 2005. “Monetary Policy and Exchange Rate Volatility in a Small Open Economy.” Review of Economic Studies 72: 707–734.10.1111/j.1467-937X.2005.00349.xSearch in Google Scholar

Glosten, L. R., R. Jagannathan, and D. E. Runkle. 1993. “On the Relation Between the Expected Value and the Volatility of the Nominal Excess Return on Stocks.” Journal of Finance 48: 1779–1801.10.1111/j.1540-6261.1993.tb05128.xSearch in Google Scholar

Haas, M., S. Mittnik, and M. S. Paolella. 2004. “A New Approach to Markov-Switching Garch Models.” Journal of Financial Econometrics 2: 493–530.10.1093/jjfinec/nbh020Search in Google Scholar

Haas, M., and M. S. Paolella. 2012. “Mixture and Regime-Switching Garch Models.” In Handbook of Volatility Models and Their Applications, 71–102. Hoboken, New Jersey, États-Unis: Wiley-Blackwell.10.1002/9781118272039.ch3Search in Google Scholar

Halunga, A. G., and C. D. Orme. 2009. “First-Order Asymptotic Theory for Parametric Misspecification Tests of Garch Models.” Econometric Theory 25: 364–410.10.1017/S0266466608090129Search in Google Scholar

Krolzig, H.-M. 1997. Markov-Switching Vector Autoregressions, Lecture Notes in Economics and Mathematical Systems, volume 454. Berlin Heidelberg: Springer.10.1007/978-3-642-51684-9Search in Google Scholar

Lamoureux, C. G., and W. D. Lastrapes. 1990. “Persistence in Variance, Structural Change, and the GARCH Model.” Journal of Business & Economic Statistics 8: 225–234.Search in Google Scholar

Laurent, Sébastien, Christelle Lecourt, and Franz C. Palm. 2016. “Testing for Jumps in Conditionally Gaussian ARMA–GARCH Models, a Robust Approach.” Computational Statistics & Data Analysis 100: 383–400. DOI: 10.1016/j.csda.2014.05.015.Search in Google Scholar

Lebreton, M., and A. Péguin-Feissolle. 2007. “Robust Tests for Heteroscedasticity in a General Framework.” Annals of Economics and Statistics (85): 159–187.10.2307/20079184Search in Google Scholar

Lee, T.-H., H. White, and C. W. J. Granger. 1993. “Testing for Neglected Nonlinearity in Time Series Models: A Comparison of Neural Network Methods and Alternative Tests.” Journal of Econometrics 56: 269–290.10.1016/0304-4076(93)90122-LSearch in Google Scholar

Ling, S., and M. McAleer. 2003. “Asymptotic Theory for a Vector Arma-Garch Model.” Econometric Theory 19: 280–310.10.1017/S0266466603192092Search in Google Scholar

Loudon, G. F., W. H. Watt, and P. K. Yadav. 2000. “An Empirical Analysis of Alternative Parametric Arch Models.” Journal of Applied Econometrics 15: 117–136.10.1002/(SICI)1099-1255(200003/04)15:2<117::AID-JAE550>3.0.CO;2-4Search in Google Scholar

Lundbergh, S., and T. Teräsvirta. 2002. “Evaluating GARCH Models.” Journal of Econometrics 110: 417–435.10.1016/S0304-4076(02)00096-9Search in Google Scholar

Maheu, J. M., and T. H. McCurdy. 2000. “Identifying Bull and Bear Markets in Stock Returns.” Journal of Business & Economic Statistics 18: 100–112.Search in Google Scholar

Patton, A. J. 2011. “Volatility Forecast Comparison Using Imperfect Volatility Proxies.” Journal of Econometrics 160: 246–256.10.1016/j.jeconom.2010.03.034Search in Google Scholar

Péguin-Feissolle, A., and B. Sanhaji. 2016. “Tests of the Constancy of Conditional Correlations of Unknown Functional Form in Multivariate Garch Models.” Annals of Economics and Statistics (123/124): 77–101.10.15609/annaeconstat2009.123-124.0077Search in Google Scholar

Péguin-Feissolle, A., B. Strikholm, and T. Teräsvirta. 2013. “Testing the Granger Noncausality Hypothesis in Stationary Nonlinear Models of Unknown Functional Form.” Communications in Statistics - Simulation and Computation 42: 1063–1087.10.1080/03610918.2012.661500Search in Google Scholar

Storti, G., and C. Vitale. 2003. “BL-GARCH Models and Asymmetries in Volatility.” Statistical Methods & Applications 12: 19–39.10.1007/BF02511581Search in Google Scholar

Teräsvirta, T. 2012. “Nonlinear Models for Autoregressive Conditional Heteroskedasticity.” In Handbook of Volatility Models and Their Applications, 47–69. Hoboken, New Jersey, États-Unis: Wiley-Blackwell.10.1002/9781118272039.ch2Search in Google Scholar

Zhang, H. J., J. M. Dufour, and J. W. Galbraith. 2016. “Exchange Rates and Commodity Prices: Measuring Causality at Multiple Horizons.” Journal of Empirical Finance 36: 100–120. DOI: 10.1016/j.jempfin.2015.10.005.Search in Google Scholar


Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/snde-2017-0069).


Published Online: 2018-07-03

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