×

Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes. (English) Zbl 0679.60029

Let \(\{Y_ n\}\), \(n\geq 1\), be a sequence of random variables such that \(Y_ n=A_ nY_{n-1}+B_ n\), \(n\geq 1\), \(Y_ 0\geq 0\), where \((A_ n,B_ n)'s\) are i.i.d. \({\mathbb{R}}^ 2_+\)-valued random pairs. An example of such a sequence is \(\{X^ 2_ n\}\) where \(\{X_ n\}\) is the first order autoregressive conditional heteroscedastic (ARCH) process used in econometric modelling.
Now let \(M_ n=\max (Y_ 1,...,Y_ n)\) and for any \(C\subset {\mathbb{R}}_+\) define \(N_ n(C)=\#\{(k/n)\in C:\) \(Y_ k>u_ n\}\) where \(u_ n\) is a suitably chosen high level. In addition to some mild conditions on \((A_ 1,B_ 1)\), suppose there exists an \(\alpha >0\) such that \(E(A_ 1^{\alpha})=1\), and let \(u_ n=xn^{1/\alpha}\). Main results are:
(1) \(P(M_ n\leq u_ n)\to \exp (-c\theta x^{-\alpha})\), \(x>0\), as \(n\to \infty\) and (2) \(N_ n\to^{d}N\), as \(n\to \infty\), where N is a compound Poisson process with intensity parameter \(c\theta x^{- \alpha}.\)
Expressions for \(\theta\) and the compounding probabilities of N are given. Simulation procedures to compute them are described. They are applied to compute these parameters for the ARCH process. (The authors remark that the value of c is known only for integer values of \(\alpha\).)
Reviewer: H.N.Nagaraya

MSC:

60F05 Central limit and other weak theorems
60H99 Stochastic analysis
60G10 Stationary stochastic processes
60G17 Sample path properties
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bollerslev, T., Generalized autoregressive conditional heteroskedasticity, J. Econometrics, 31, 307-327 (1986) · Zbl 0616.62119
[2] Bollerslev, T., A conditionally heteroscedastic time series model for speculative prices and rates of return, Rev. Econ. Statist., 69, 542-547 (1987)
[3] Bollerslev, T.; Engle, R. F., Modelling the persistence of conditional variances, Econ. Rev., 27, 1-50 (1986) · Zbl 0619.62105
[4] Domowitz, I.; Hakkio, C. S., Conditional variance and the risk premium in the foreign exchange market, J. Internat. Economics, 19, 47-66 (1985)
[5] Engle, R. F., Autoregressive conditional heteroscedastic models with estimates of the variance of United Kingdom inflation, Econometrica, 50, 987-1007 (1982) · Zbl 0491.62099
[6] Engle, R. F.; Lilien, D. M.; Robins, R. P., Estimating time varying risk premia in the term structure: the ARCH model, Econometrica, 55, 391-407 (1987)
[7] Feller, W., An Introduction to Probability Theory and its Applications, II (1971), Wiley: Wiley New York · Zbl 0219.60003
[8] Flood, R. P.; Garber, P. M., Collapsing exchange rate regimes: some linear examples, J. Internat Economics, 17, 1-13 (1984)
[9] Goldie, C. M., Implicit renewal theory and tails of solutions of random equations (1988), Sussex University: Sussex University Brighton, UK, Preprint
[10] Hsieh, D. A., The statistical properties of daily foreign exchange rates: 1974-1983, J. Internat. Economics, 24, 129-145 (1988)
[11] Kesten, H., Random difference equations and renewal theory for products of random matrices, Acta Math., 131, 207-248 (1973) · Zbl 0291.60029
[12] Leadbetter, R.; Lindgren, G.; Rootzén, H., Extremes and Related Properties of Random Sequences and Processes (1983), Springer: Springer New York · Zbl 0518.60021
[13] McCulloch, J. H., Interest rate risk and capital adequacy for traditional bank and financial intermediaries, (Maisel, S. J., Risk and Capital Adequacy in Commerical Banks (1981), Univ. of Chicago Press: Univ. of Chicago Press Chicago)
[14] Milhøj, A., The moment structure of ARCH processes, Scand. J. Statist., 12, 281-292 (1985) · Zbl 0595.62089
[15] Rootzén, H., Extreme value theory for moving average processes, Ann. Probob., 14, 612-652 (1986) · Zbl 0604.60019
[16] Rootzén, H., Maxima and exceedances of stationary Markov chains, J. Appl. Probob., 20, 371-390 (1988) · Zbl 0654.60023
[17] Vervaat, W., On a stochastic difference equation and a representation of non negative infinitely divisible random variables, Adv. in Appl. Probab., 11, 750-783 (1979) · Zbl 0417.60073
[18] Weiss, A., Asymptotic theory for ARCH models: estimation and testing, Econometric Th., 2, 107-131 (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.