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Chapter 49 ARCH MODELS” TIM BOLLERSLEV Northwestern University and N.B.E.R. ROBERT F. ENGLE University of California, San Diego and N.B.E.R. DANIEL B. NELSON University of Chicago and N.B.E.R. Contents 2961 2961 Abstract 1. Introduction 2. 2961 1.1. Definitions 1.2. Empirical regularities 1.3. Univariate parametric 1.4. ARCH 1.5. Nonparametric Inference 2963 of asset returns 2967 models 2912 in mean models and semiparametric methods 2.1. Testing for ARCH 2.2. Maximum 2.3. Quasi-maximum 2.4. Specification 2912 2974 procedures 2914 likelihood methods likelihood methods checks 2971 2983 2984 “The authors would like to thank Torben G. Andersen, Patrick Billingsley, William A. Brock, Eric Ghysels, Lars P. Hansen, Andrew Harvey, Blake LeBaron, and Theo Nijman for helpful comments. Financial support from the National Science Foundation under grants SES-9022807 (Bollerslev), SES9122056 (Engle), and SES-9110131 and SES-9310683 (Nelson), and from the Center for Research in Security Prices (Nelson), is gratefully acknowledged. Inquiries regarding the data for the stock market empirical application should be addressed to Professor G. William Schwert, Graduate School of Management, University of Rochester, Rochester, NY 14627, USA. The GAUSSTM code used in the stock market empirical example is available from Inter-University Consortium for Political and Social Research (ICPSR), P.O. Box 1248, Ann Arbor, MI 48106, USA, telephone (313)763-5010. Order “Class 5” under this article’s name. Handbook ofEconometrics, Volume IV, Edited by R.F. Engle and D.L. McFadden 0 1994 Elseuier Science B.V. All rights reserved T. Bollersku 3. 4. 5. 6. 7. 8. 9. Stationary et al. 2989 and ergodic properties 3.1. Strict stationarity 2989 3.2. Persistence 2990 Continuous 2992 time methods 4.1. ARCH models as approximations 4.2. Diffusions 4.3, ARCH as approximations to diffusions to ARCH models as filters and forecasters models 2994 2996 2991 Aggregation and forecasting 2999 5.1. Temporal aggregation 2999 5.2. Forecast error distributions Multivariate specifications 6.1. Vector ARCH 6.2. Factor 6.3. Constant conditional 6.4. Bivariate EGARCH 6.5. Stationarity 3002 and diagonal 3003 ARCH 3005 ARCH 3007 correlations 3008 3009 and co-persistence Model selection Alternative measures Empirical examples 9.1. U.S. Dollar/Deutschmark 9.2. U.S. stock prices 10. Conclusion References 3001 3010 3012 3014 for volatility exchange rates 3014 3017 3030 3031 Ch. 49: ARCH Models 2961 Abstract This chapter evaluates the most important theoretical developments in ARCH type modeling of time-varying conditional variances. The coverage include the specification of univariate parametric ARCH models, general inference procedures, conditions for stationarity and ergodicity, continuous time methods, aggregation and forecasting of ARCH models, multivariate conditional covariance formulations, and the use of model selection criteria in an ARCH context. Additionally, the chapter contains a discussion of the empirical regularities pertaining to the temporal variation in financial market volatility. Motivated in part by recent results on optimal filtering, a new conditional variance model for better characterizing stock return volatility is also presented. 1. Introduction Until a decade ago the focus of most macroeconometric and financial time series modeling centered on the conditional first moments, with any temporal dependencies in the higher order moments treated as a nuisance. The increased importance played by risk and uncertainty considerations in modern economic theory, however, has necessitated the development of new econometric time series techniques that allow for the modeling of time varying variances and covariances. Given the apparent lack of any structural dynamic economic theory explaining the variation in higher order moments, particularly instrumental in this development has been the autoregressive conditional heteroskedastic (ARCH) class of models introduced by Engle (1982). Parallel to the success of standard linear time series models, arising from the use of the conditional versus the unconditional mean, the key insight offered by the ARCH model lies in the distinction between the conditional and the unconditional second order moments. While the unconditional covariance matrix for the variables of interest may be time invariant, the conditional variances and covariances often depend non-trivially on the past states of the world. Understanding the exact nature of this temporal dependence is crucially important for many issues in macroeconomics and finance, such as irreversible investments, option pricing, the term structure of interest rates, and general dynamic asset pricing relationships. Also, from the perspective of econometric inference, the loss in asymptotic efficiency from neglected heteroskedasticity may be arbitrarily large and, when evaluating economic forecasts, a much more accurate estimate of the forecast error uncertainty is generally available by conditioning on the current information set. 1.I. Dejinitions Let {E,(O)} denote a discrete time stochastic process with conditional mean and variance functions parametrized by the finite dimensional vector OE 0 s R”, where T. Bolfersleu et al. 2962 8, denotes the true value. For notational simplicity we shall initially assume that s,(O)is a scalar, with the obvious extensions to a multivariate framework treated in Section 6. Also, let E,_ r(.) denote the mathematical expectation, conditional on the past, of the process, along with any other information available at time t - 1. The {E,(@,)}process is then defined to follow an ARCH model if the conditional mean equals zero, ~1-1MRJ))= 0 t= 1,2,..., (1.1) but the conditional variance. 44J = Var,- lWo)) = L l(~:(&)) t= 1,2,..., (1.2) depends non-trivially on the sigma-field generated by the past observations; i.e. {~t-l(~O)r~,-2(e0),...}.Wh en obvious from the context, the explicit dependence on the parameters, 8, will be suppressed for notational convenience. Also, in the multivariate case the corresponding time varying conditional covariance matrix will be denoted by f2,. In much of the subsequent discussion we shall focus directly on the {st} process, but the same ideas extend directly to the situation in which {st} corresponds to the innovations from some more elaborate econometric model. In particular, let {yl(O,)} denote the stochastic process of interest with cohditional mean PtwM= 4 - l(YJ t=l2 ) ).... (1.3) Note, by the timing convention both ~~(0,) and a:(O,) are measurable with respect to the time t - 1 information set. Define the {s,(e,)} process by de,) = Y, - au t= 1,2,.... (1.4) The conditional variance for (ct} then equals the conditional variance for the {y,} process. Since very few economic and financial time series have a constant conditional mean of zero, most of the empirical applications of the ARCH methodology actually fall within this framework. Returning to the definitions in equations (1.1) and (1.2), it follows that the standardized process, 2,(e,) s E,(e,)a:(e,)- 1’2 t= 1,2,..., (1.5) will have conditional mean zero, and a time invariant conditional variance of unity. This observation forms the basis for most of the inference procedures that underlie the applications of ARCH type models. If the conditional distribution for z, is furthermore assumed to be time invariant 2963 Ch. 49: ARCH Models with a finite fourth moment, E(&;) = E(zf’)E(a;) 2 it follows by Jensen’s inequality that E(z;)E(af)2 = E(zp)E(q2, where the equality holds true for a constant conditional variance only. Given a normal distribution for the standardized innovations in equation (1.5), the unconditional distribution for E, is therefore leptokurtic. The setup in equations (1.1) through (1.4) is extremely general and does not lend itself directly to empirical implementation without first imposing further restrictions on the temporal dependencies in the conditional mean and variance functions. Below we shall discuss some of the most practical and popular such ARCH formulations for the conditional variance. While the first empirical applications of the ARCH class of models were concerned with modeling inflationary uncertainty, the methodology has subsequently found especially wide use in capturing the temporal dependencies in asset returns. For a recent survey of this extensive empirical literature we refer to Bollerslev et al. (1992). 1.2. Empirical regularities of asset returns Even in the univariate case, the array of functional forms permitted by equation (1.2) is vast, and infinitely larger than can be accommodated by any parametric family of ARCH models. Clearly, to have any hope of selecting an appropriate ARCH model, we must have a good idea of what empirical regularities the model should capture. Thus, a brief discussion of some of the important regularities for asset returns volatility follows. 1.2.1. Thick tails Asset returns tend to be leptokurtic. The documentation of this empirical regularity by Mandelbrot (1963), Fama (1965) and others led to a large literature on modeling stock returns as i.i.d. draws from thick-tailed distributions; see, e.g. Mandelbrot (1963), Fama (1963,1965), Clark (1973) and Blattberg and Gonedes (1974). 1.2.2. Volatility As Mandelbrot clustering (1963) wrote, . . . large changes tend to be followed by large changes, changes tend to be followed by small changes.. of either sign, and small This volatility clustering phenomenon is immediately apparent when asset returns are plotted through time. To illustrate, Figure 1 plots the daily capital gains on the Standard 90 composite stock index from 1928-1952 combined with Standard and Daily Standard 2 ’ 1920 1930 1940 1950 and Poor’s Capital 1960 Gains 1970 1980 1990 2000 Figure 1 Poor’s 500 index from 1953-1990. The returns are expressed in percent, and are continuously compounded. It is clear from visual inspection of the figure, and any reasonable statistical test, that the returns are not i.i.d. through time. For example, volatility was clearly higher during the 1930’s than during the 1960’s, as confirmed by the estimation results reported in French et al. (1987). A similar message is contained in Figure 2, which plots the daily percentage Deutschmark/U.S. Dollar exchange rate appreciation. Distinct periods of exchange market turbulence and tranquility are immediately evident. We shall return to a formal analysis of both of these two time series in Section 9 below. Volatility clustering and thick tailed returns are intimately related. As noted in Section 1.1 above, if the unconditional kurtosis of a, is finite, E(E~)/[E($)]~ 3 E(z:), where the last inequality is strict unless ot is constant. Excess kurtosis in E, can therefore arise from randomness in ol, from excess kurtosis in the conditional distribution of sI, i.e., in zl, or from both. 1.2.3. Leverage eflects The so-called “leverage effect,” first noted by Black (1976), refers to the for changes in stock prices to be negatively correlated with changes volatility. Fixed costs such as financial and operating leverage provide explanation for this phenomenon. A firm with debt and equity outstanding tendency in stock a partial typically Ch. 49: ARCH 2965 Models Dally U.S. Dollar-Deutschmark “p 1960 1962 1964 1966 1988 Appreciation 1990 1992 1994 Figure 2 becomes more highly leveraged when the value of the firm falls. This returns volatility if the returns on the firm as a whole are constant. however, argued that the response of stock volatility to the direction too large to be explained by leverage alone. This conclusion is also the empirical work of Christie (1982) and Schwert (1989b). 1.2.4. raises equity Black (1976), of returns is supported by Non-trading periods Information that accumulates when financial markets are closed is reflected in prices after the markets reopen. If, for example, information accumulates at a constant rate over calendar time, then the variance of returns over the period from the Friday close to the Monday close should be three times the variance from the Monday close to the Tuesday close. Fama (1965) and French and Roll (1986) have found, however, that information accumulates more slowly when the markets are closed than when they are open. Variances are higher following weekends and holidays than on other days, but not nearly by as much as would be expected if the news arrival rate were constant. For instance, using data on daily returns across all NYSE and AMEX stocks from 1963-1982, French and Roll (1986) find that volatility is 70 times higher per hour on average when the market is open than when it is closed. Baillie and Bollerslev (1989) report qualitatively similar results for foreign exchange rates. 1.2.5. Forecastable events Not surprisingly, forecastable releases of important information are associated with high ex ante volatility. For example, Cornell (1978) and Pate11 and Wolfson (1979, T Bollersleu et al. 2966 1981) show that individual firms’ stock returns volatility is high around earnings announcements. Similarly, Harvey and Huang (1991,1992) find that fixed income and foreign exchange volatility is higher during periods of heavy trading by central banks or when macroeconomic news is being released. There are also important predictable changes in volatility across the trading day. For example, volatility is typically much higher at the open and close of stock and foreign exchange trading than during the middle of the day. This pattern has been documented by Harris (1986), Gerity and Mulherin (1992) and Baillie and Bollerslev (1991) among others. The increase in volatility at the open at least partly reflects information accumulated while the market was closed. The volatility surge at the close is less easily interpreted. 1.2.6. Volatility and serial correlation LeBaron (1992) finds a strong inverse relation between volatility and serial correlation for U.S. stock indices. This finding appears remarkably robust to the choice of sample period, market index, measurement interval and volatility measure. Kim (1989) documents a similar relationship in foreign exchange rate data. 1.2.7. Co-movements Black (1976) observed in volatilities that . . . there is a lot of commonality in volatility changes across stocks: a 1% market volatility change typically implies a 1% volatility change for each stock. Well, perhaps the high volatility stocks are somewhat more sensitive to market volatility changes than the low volatility stocks. In general it seems fair to say that when stock volatilities change, they all tend to change in the same direction. Diebold and Nerlove (1989) and Harvey et al. (1992) also argue for the existence of a few common factors explaining exchange rate volatility movements. Engle et al. (1990b) show that U.S. bond volatility changes are closely linked across maturities. This commonality of volatility changes holds not only across assets within a market, but also across different markets. For example, Schwert (1989a) finds that U.S. stock and bond volatilities move together, while Engle and Susmel (1993) and Hamao et al. (1990) discover close links between volatility changes across international stock markets. The importance of international linkages has been further explored by King et al. (1994), Engle et al. (1990a), and Lin et al. (1994). That volatilities move together should be encouraging to model builders, since it indicates that a few common factors may explain much of the temporal variation in the conditional variances and covariances of asset returns. This forms the basis for the factor ARCH models discussed in Section 6.2 below. 2961 Ch. 49: ARCH Models I .2.8. Macroeconomic variables and volatility Since stock values are closely tied to the health of the economy, it is natural to expect that measures of macroeconomic uncertainty such as the conditional variances of industrial production, interest rates, money growth, etc. should help explain changes in stock market volatility. Schwert (1989a, b) finds that although stock volatility rises sharply during recessions and financial crises and drops during expansions, the relation between macroeconomic uncertainty and stock volatility is surprisingly weak. Glosten et al. (1993), on the other hand, uncover a strong positive relationship between stock return volatility and interest rates. 1.3. 1.3.1. Univariate parametric models GARCH Numerous parametric specifications for the time varying conditional variance have been proposed in the literature. In the linear ARCH(q) model originally introduced by Engle (1982), the conditional variance is postulated to be a linear function of the past q squared innovations, o;=w+ 1 CLiE~_i-W+C((L)E:_l, (1.6) i=l,q where L denotes the lag to be well defined and parameters must satisfy Defining v, = E: - a:, Ef = w + or backshift operator, L’y, = Y,_~. Of course, for this model the conditional variance to be positive, almost surely the w > 0 and c(~ 3 0,. . . , a, > 0. the ARCH(q) model in (1.6) may be re-written as C((L)&f_1 + v,. (1.7) Since E, _ I (v,) = 0, the model corresponds directly to an AR(q) model for the squared innovations, E:. The process is covariance stationary if and only if the sum of the positive autoregressive parameters is less than one, in which case the unconditional variance equals Var(s,) = a2 = o/( 1 - U1 - ... - Uq). Even though the E,)Sare serially uncorrelated, they are clearly not independent through time. In accordance with the stylized facts for asset returns discussed above, there is a tendency for large (small) absolute values of the process to be followed by other large (small) values of unpredictable sign. Also, as noted above, if the distribution for the standardized innovations in equation (1.5) is assumed to be time invariant, the unconditional distribution for E, will have fatter tails than the distribution for z,. For instance, for the ARCH(l) model with conditionally normally distributed errors, E(sp)/E($)’ = 3( 1 - a:)/( 1 - 34) if 3c(T < 1, and E(.$)/E(E:)~ = co otherwise: both of which exceed the normal value of three. T Bollersleu et al. 2968 Alternatively the ARCH(q) model parameter MA(q) model for a,, E, = 0 + cc(J!& lE,_ may also be represented as a time varying (1.8) 1, where {i,} denotes a scalar i.i.d. stochastic process with mean zero and variance one. Time varying parameter models have a long history in econometrics and statistics. The appeal of the observational equivalent formulation in equation (1.6) stems from the explicit focus on the time varying conditional variance of the process. For discussion of this interpretation of ARCH models, see, e.g., Tsay (1987), Bera et al. (1993) and Bera and Lee (1993). In empirical applications of ARCH(q) models a long lag length and a large number of parameters are often called for. To circumvent this problem Bollerslev (1986) proposed the generalized ARCH, or GARCH(p, q), model, ~: = 0 + C C(iEf_i + C i=l,q j= ~jjb:_j ~ W + a(L)&:_ 1 + B(L)a:_ 1. (1.9) 1.~ For the conditional variance in the GARCH(p, q) model to be well defined all the coefficients in the corresponding infinite order linear ARCH model must be positive. Provided that a(L) and /J(L) have no common roots and that the roots of the polynomial /I(x) = 1 lie outside the unit circle, this positivity constraint is satisfied if and only if all the coefficients in the infinite power series expansion for a(~)/( 1 - /i(x)) are non-negative. Necessary and sufficient conditions for this are given in Nelson and Cao (1992). For the simple GARCH(l, 1) model almost sure positivity of 0: requires that w 3 0, c(i B 0 and /I1 > 0. Rearranging the GARCH(p, q) model as in equation (1.7), it follows that a; = 0 + [a(L) + B(L)l&:_ 1- P(L)v,-1+ v,, (1.10) which defines an ARMA[max(p,q),p] model for E:. By standard arguments, the model is covariance stationary if and only if all the roots of a(x) + b(x) = 1 lie outside the unit circle; see Bollerslev (1986) for a formal proof. In many applications with high frequency financial data the estimate for CI(1) + /I( 1) turns out to be very close to unity. This provides an empirical motivation for the so-called integrated GARCH (p, q), or IGARCH(p,q), model introduced by Engle and Bollerslev (1986). In the IGARCH class of models the autoregressive polynomial in equation (1.10) has a unit root, and consequently a shock to the conditional variance is persistent in the sense that it remains important for future forecasts of all horizons. Further discussion of stationarity conditions and issues of persistence are contained in Section 3 below. Just as an ARMA model often leads to a more parsimonious representation of the temporal dependencies in the conditional mean than an AR model, the GARCH (p, q) formulation in equation (1.9) provides a similar added flexibility over the linear