zbMATH — the first resource for mathematics

Asymptotic filtering theory for multivariate ARCH models. (English) Zbl 0845.62080
Summary: ARCH models are widely used to estimate conditional variances and covariances in financial time series models. How successfully can ARCH models carry out this estimation when they are misspecified? How can ARCH models be made robust to misspecification? D. B. Nelson and D. P. Foster [Econometrica 62, No. 1, 1-41 (1994; Zbl 0804.62085)] employed continuous record asymptotics to answer these questions in the univariate case. This paper considers the general multivariate case. Our results allow us, for example, to construct an asymptotically optimal ARCH model for estimating the conditional variance or conditional beta of a stock return given lagged returns on the stock, volume, market returns, implicit volatility from options contracts, and other relevant data. We also allow for time-varying shapes of conditional densities (e.g., ‘heteroskewticity’ and ‘heterokurticity’). Examples are provided.

62P20 Applications of statistics to economics
62M20 Inference from stochastic processes and prediction
GAUSS; Mathematica
Full Text: DOI
[1] Andersen, T. G.: Volatility. Working paper (1992)
[2] Anderson, B. D. O.: Second-order convergent algorithms for the steady state Riccati equation. International journal of control 28, 295-306 (1978) · Zbl 0385.49017
[3] Anderson, B. D. O.; Moore, J. B.: Linear optimal control. (1971) · Zbl 0321.49001
[4] Anderson, B. D. O.; Moore, J. B.: Optimal filtering. (1979) · Zbl 0688.93058
[5] Anderson, T. W.: An introduction to multivariate statistical analysis. (1984) · Zbl 0651.62041
[6] Inc., Aptech Systems: Gauss. (1992)
[7] Arnold, L.: Stochastic differential equations: theory and applications. (1973) · Zbl 0296.34001
[8] Baillie, R. T.; Bollerslev, T.; Ole, H.; Mikkelsen, A. E.: Fractionally integrated generalized autoregressive conditional heteroskedasticity. Working paper (1993) · Zbl 0865.62085
[9] Bates, D. S.: The crash of ’87: was it expected? the evidence from options markets. Journal of finance 46, 1009-1044 (1991)
[10] Bates, D. S.: Jumps and stochastic volatility: exchange rate processes implicit in PHLX deutschemark options. Working paper (1993)
[11] Bellman, R.: Introduction to matrix analysis. (1970) · Zbl 0216.06101
[12] Bera, A. K.; Higgins, M. L.: A survey of ARCH models: properties, estimation, and testing. Journal of economic surveys 7, 305-366 (1993)
[13] Billingsley, P.: Probability and measure. (1986) · Zbl 0649.60001
[14] Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. Journal of econometrics 31, 307-327 (1986) · Zbl 0616.62119
[15] Bollerslev, T.: Modelling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH approach. Review of economics and statistics 72, 498-505 (1990)
[16] Bollerslev, T.; Chou, R. Y.; Kroner, K.: ARCH modeling in finance: A review of the theory and empirical evidence. Journal of econometrics 52, 5-60 (1992) · Zbl 0825.90057
[17] Bollerslev, T.; Engle, R. F.; Nelson, D. B.: ARCH models. Handbook of econometrics 4 (1994)
[18] Bollerslev, T.; Engle, R. F.; Wooldridge, J. M.: A capital asset pricing model with time-varying covariances. Journal of political economy 96, 116-131 (1988)
[19] Brenner, R. J.; Harjes, R. H.; Kroner, K. F.: Another look at alternative models of the short-term interest rate. Working paper (1994)
[20] Cambanis, S.; Huang, S.; Simons, G.: On the theory of elliptically contoured distributions. Journal of multivariate analysis 11, 368-385 (1981) · Zbl 0469.60019
[21] Chan, K. C.; Karolyi, G. A.; Longstaff, F. A.; Sanders, A. B.: An empirical comparison of alternative models of the short-term interest rate. Journal of finance 47, 1209-1227 (1992)
[22] Chiras, D. P.; Manaster, S.: The information content of option prices and a test of market efficiency. Journal of financial economics 6, 213-234 (1978)
[23] Christie, A. A.: The stochastic behavior of common stock variances: value, leverage and interest rate effects. Journal of financial economics 10, 407-432 (1982)
[24] Cox, J. C.; Jr., J. E. Ingersoll; Ross, S. A.: A theory of the term structure of interest rates. Econometrica 53, 385-407 (1985) · Zbl 1274.91447
[25] Cox, J. C.; Ross, S. A.; Rubinstein, M.: Options pricing: A simplified approach. Journal of financial economics 7, 229-263 (1979) · Zbl 1131.91333
[26] Danielsson, J.: Stochastic volatility in asset prices: estimation with simulated maximum likelihood. Journal of econometrics 64, 375-400 (1994) · Zbl 0825.62953
[27] Davis, P. J.: Gamma function and related functions. Handbook of mathematical functions, 253-293 (1964)
[28] Day, T. E.; Lewis, C. M.: Stock market volatility and the information content of stock index options. Journal of econometrics 52, 267-288 (1992)
[29] Ding, Z.; Granger, C. W. J.; Engle, R. F.: A long memory property of stock returns and a new model. Journal of empirical finance 1, 83-106 (1993)
[30] Engle, R. F.: Autoregressive conditional heteroskedasticity with estimates of the variance of united kingdom inflation. Econometrica 50, 987-1008 (1982) · Zbl 0491.62099
[31] Engle, R. F.: Comment on ’Bayesian analysis of stochastic volatility models’. Journal of business and economic statistics 12, 395-396 (1982)
[32] Engle, R. F.; Kroner, K. F.: Multivariate simultaneous generalized ARCH. Econometric theory (1994)
[33] Engle, R. F.; Ng, V.: Measuring and testing the impact of news on volatility. Journal of finance 48, 1749-1778 (1993)
[34] Engle, R. F.; Ng, V.; Rothschild, M.: Asset pricing with a factor-ARCH covariance structure: empirical estimates for treasury bills. Journal of econometrics 45, 213-238 (1990)
[35] Ethier, S. N.; Kurtz, T. G.: Markov processes: characterization and convergence. (1986) · Zbl 0592.60049
[36] Fisher, R. A.: On the probable error of a coefficient of correlation deduced from a small sample. Metron 1, No. no. 4, 1 (1921)
[37] Foster, D. P.; Nelson, D. B.: Continuous record asymptotics for rolling sample variance estimators. Econometrica (1995) · Zbl 0860.62101
[38] Friedman, A.: Stochastic differential equations with applications. 1 (1975) · Zbl 0323.60056
[39] Gallant, A. R.; Rossi, P. E.; Tauchen, G.: Stock prices and volume. Review of financial studies 5, 199-242 (1992)
[40] Gallant, A. R.; Rossi, P. E.; Tauchen, G.: Nonlinear dynamic structures. Econometrica 61, 871-907 (1993) · Zbl 0780.62100
[41] Garman, M. B.; Klass, M. J.: On the estimation of security price volatilities from historical data. Journal of business 53, 67-78 (1980)
[42] Glosten, L. R.; Jagannathan, R.; Runkle, D.: On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of finance 48, 1779-1801 (1993)
[43] Hansen, B. E.: Autoregressive conditional density estimation. International economic review 35, 705-730 (1994) · Zbl 0807.62090
[44] Hardy, G.; Littlewood, J. E.; Pólya, G.: Inequalities. (1952) · Zbl 0047.05302
[45] Harvey, A.; Ruiz, E.; Shephard, N.: Multivariate stochastic variance models. Review of economic studies 61, 247-264 (1994) · Zbl 0805.90026
[46] Heiland, I. S.: Central limit theorems for martingales with discrete or continuous time. Scandinavian journal of statistics 9, 79-94 (1982) · Zbl 0486.60023
[47] Horn, R. A.; Johnson, C. R.: Matrix analysis. (1985) · Zbl 0576.15001
[48] Hull, J.; White, A.: The pricing of options on assets with stochastic volatilities. Journal of finance 42, 281-300 (1987)
[49] Jacquier, E.; Poison, N. G.; Rossi, P. E.: Bayesian analysis of stochastic volatility models. Journal of business and economic statistics 12, 371-389 (1994)
[50] Johnson, N. L.; Kotz, S.: Continuous multivariate distributions. (1972) · Zbl 0248.62021
[51] Karatzas, I.; Shreve, S. E.: Brownian motion and stochastic calculus. (1988) · Zbl 0638.60065
[52] Karpoff, J.: The relation between price changes and trading volume: A survey. Journal of financial and quantitative analysis 22, 109-126 (1987)
[53] Kim, S.; Shephard, N.: Stochastic volatility: new models and optimal likelihood inference. Working paper (1993)
[54] Kroner, K. F.; Ng, V. K.: Modelling the time varying comovement of asset returns. Working paper (1993)
[55] Kučera, V.: A contribution to matrix quadratic equations. IEEE transactions on automatic control 17, 344-356 (1972)
[56] Lancaster, P.; Rodman, L.: Existence and uniqueness theorems for the algebraic ricatti equation. International journal of control 32, 285-309 (1980) · Zbl 0439.49011
[57] Lancaster, P.; Tismenetsky, M.: The theory of matrices. (1985) · Zbl 0558.15001
[58] Lo, A. W.; Wang, J.: Implementing option pricing formulas when asset returns are predictable. Journal of finance 50, 87-129 (1995)
[59] Magnus, J. R.; Neudecker, H.: Matrix differential calculus. (1988) · Zbl 0651.15001
[60] Mcculloch, J. H.: On heteroscedasticity. Econometrica 53, 483 (1985)
[61] Mitchell, A. F. S.: The information matrix, skewness tensor and \({\sigma}\)-connections for the general multivariate elliptic distribution. Annals of the institute of statistical mathematics 41, 289-304 (1989) · Zbl 0691.62049
[62] Nelson, D. B.: The time series behavior of stock market volatility and returns. (1988)
[63] Nelson, D. B.: ARCH models as diffusion approximations. Journal of econometrics 45, 7-38 (1990) · Zbl 0719.60089
[64] Nelson, D. B.: Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347-370 (1991) · Zbl 0722.62069
[65] Nelson, D. B.: Filtering and forecasting with misspecified ARCH models I: Getting the right variance with the wrong model. Journal of econometrics 52, 61-90 (1992) · Zbl 0761.62169
[66] Nelson, D. B.; Foster, D. P.: Asymptotic filtering theory for univariate ARCH models. Econometrica 62, 1-41 (1994) · Zbl 0804.62085
[67] Nelson, D. B.; Foster, D. P.: Filtering and forecasting with misspecified ARCH models II: Making the right forecast with the wrong model. Journal of econometrics (1994) · Zbl 0820.62098
[68] Nelson, D. B.; Schwartz, B. A.: Filtering with ARCH: A Monte Carlo experiment. Proceedings of the American statistical association, business and economic statistics section, 1-6 (1992)
[69] Phillips, P. C. B.: Regression theory for near-integrated time series. Econometrica 56, 1021-1044 (1988) · Zbl 0744.62128
[70] Parkinson, M.: The extreme value method for estimating the variance of the rate of return. Journal of business 53, 61-65 (1980)
[71] Prudnikov, A. P.; Brychkov, Y. A.; Marichev, O. I.: Elementary functions. Integrals and series 1 (1986) · Zbl 0733.00004
[72] Schwartz, B. A.: Essays on ARCH filtering and estimation. (1994)
[73] Schwert, G. W.: Why does stock market volatility change over time?. Journal of finance 44, 1115-1154 (1989)
[74] Scott, L. O.: Option pricing when the variance changes randomly: theory, estimation, and an application. Journal of financial and quantitative analysis 22, 419-438 (1987)
[75] Serfling, R. J.: Approximation theorems of mathematical statistics. (1980) · Zbl 0538.62002
[76] Shephard, N.: Local scale models: state space alternatives to integrated GARCH processes. Journal of econometrics 60, 181-202 (1994) · Zbl 0800.62807
[77] Stroock, D. W.; Varadhan, S. R. S.: Multidimensional diffusion processes. (1979) · Zbl 0426.60069
[78] Taylor, S. J.: Modeling financial time series. (1986) · Zbl 1130.91345
[79] Taylor, S. J.: Stochastic volatility: A review and comparative study. Mathematical finance 4, 183-204 (1994) · Zbl 0884.90054
[80] Turner, A. L.; Weigel, E. J.: Daily stock market volatility: 1928–1989. Management science 38, 1586-1609 (1992)
[81] Watanabe, T.: Alternative approach to conditional heteroskedasticity in stock returns: approximate non-Gaussian filtering. Working paper (1992)
[82] Wiggins, J. B.: Option values under stochastic volatility: theory and empirical estimates. Journal of financial economics 19, 351-372 (1987)
[83] Wiggins, J. B.: Empirical tests of the bias and efficiency of the extreme-value variance estimator for common stocks. Journal of business 64, 417-432 (1991)
[84] Research, Wolfram; Inc.: Mathematica. (1992)
[85] Zakoian, J. M.: Threshold heteroskedastic models. Working paper (1990)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.