ARCH and GARCH Models

Transcription

1 By Torben Andersen, 1 Tim Bollerslev, 2 and Ali Hadi 3 1 Introduction Many time series display time-varying dispersion, or uncertainty, in the sense that large (small) absolute innovations tend to be followed by other large (small) absolute innovations. A natural way to model this phenomenon is to allow the variance to change through time in response to current developments of the system. Specifically, let {y t }denote the observable univariate discrete-time stochastic process of interest. Denote the corresponding innovation process by {, t }, where, t y t E t 1 (y t ), and E t 1 ( ) refers to the expectation conditional on time-(t 1) information. A general specification for the innovation process that takes account of the time-varying uncertainty would then be given by, t = z t F t, (1) where {z t }is an i.i.d. mean-zero, unit-variance stochastic process, and F t represents the time-t latent volatility; i.e., E(, 2 t F t) = Ft 2. Model specifications in which F t in 1 depends nontrivially on the past innovations and/or some other latent variables are referred to as stochastic volatility (SV) models. The historically first, and often most convenient, SV representations are the autoregressive conditionally heteroscedastic (ARCH) models pioneered by Engle [21]. Formally the ARCH class of models are defined by 1, with the additional restriction that F t must be measurable with respect to the time-(t 1) observable information set. Thus, in the ARCH class of models var t 1 (y t ) E t 1 (, 2 t ) = F t 2 is predetermined as of time t 1. 2 Volatility Clustering The ARCH model was originally introduced for modeling inflationary uncertainty, but has subsequently found especially wide use in the analysis of financial time series. To illustrate, consider the plots in Figures 1 and 2 for the daily Deutsche-mark U.S. dollar (DM/$) exchange rate and the Standard and Poor s 500 composite stock-market index (S&P 500) from October 1, 1979, through September 30, It is evident from panel (a) of the figures that both series display the long-run swings or trending behavior that are characteristic of unit-root, or I(1), nonstationary processes. On the other hand, the two return 1 Northwestern University 2 Duke University 3 American University in Cairo This article was originally published online in 2010 in Encyclopedia of Statistical Sciences, c John Wiley & Sons, Inc. and republished in Wiley StatsRef: Statistics Reference Online, Copyright c 2010 by John Wiley & Sons, Inc. All rights reserved. 1

2 series, r t = 100ln(P t /P t 1 ), in panel (b) appear to be covariance-stationary. However, the tendency for large (and for small) absolute returns to cluster in time is clear. Many other economic and financial time series exhibit analogous volatility clustering features. This observation, together with the fact that modern theories of price determination typically rely on some form of a risk reward tradeoff relationship, underlies the very widespread applications of the ARCH class of time series models in economics and finance over the past decade. Simply treating the temporal dependencies inf t as a nuisance would be inconsistent with the trust of the pertinent theories. Similarly, when evaluating economic and financial time series forecasts it is equally important that the temporal variation in the forecast error uncertainty be taken into account (See Forecasting). The next section details some of the most important developments along these lines. For notational convenience, we shall assume that the {, t }process is directly observable. However, all of the main ideas extend directly to the empirically more relevant situation in which, t denotes the time-t innovation of another stochastic process, y t, as defined above. We shall restrict discussion to the univariate case; most multivariate generalizations follow by straightforward analogy. 3 GARCH The definition of the ARCH class of models in 1 is extremely general, and does not lend itself to empirical investigation without additional assumptions on the functional form, or smoothness, of F t. Arguably, the two most successful parameterizations have been the generalized ARCH, or GARCH (p, q), model of Bollerslev [7] and the exponential GARCH, or EGARCH (p, q), model of Nelson [46]. In the GARCH (p, q) model, the conditional variance is parametrized as a distributed lag of past squared innovations and past conditional variances, F 2 t =T + q p " i, 2 t i + $ j Ft 2 j i=1 j=1 T +"(B), 2 t +$(B)F 2 t, (2) where B denotes the backshift (lag) operator (See Backward and forward shift operators); i.e., B i y t y t i. For " i > 0, this parametrization directly captures the tendency for large (small), 2 t i s to be followed by other large (small) squared innovations. Of course, for the conditional variance in 2 to be positive almost surely, and the process well-defined, the coefficients in the corresponding infinite ARCH representation for Ft 2, expressed in terms of {,2 t i } i=1, must all be nonnegative, i.e., [1 $(B)] 1 "(B), where all of the roots of 1 $(x) = 0 are assumed to be outside the unit circle. On rearranging the terms in 2, we obtain [1 "(B) $(B)], 2 t = T + [1 $(B)]< t, (3) where < t, 2 t F 2 t. Since E t 1(< t ) = 0, the GARCH(p, q) formulation in 3 is readily interpreted as an ARMA(max{p, q}, p) model for the squared innovation process {, 2 t }; see Milhøj [43] and Bollerslev [9]. Thus, if the roots of 1 "(x) $(x) = 0 lie outside the unit circle, then the GARCH(p, q) process for {, t }is covariance-stationary, and the unconditional variance equals F 2 = T[1 "(1) $(1)] 1. Furthermore, standard ARMA-based identification and inference procedures may be directly applied to the process in 3, although the heteroscedasticity in the innovations,{< t }, renders such an approach inefficient. In analogy to the improved forecast accuracy obtained in traditional time-series analysis by utilizing the conditional as opposed to the unconditional mean of the process, ARCH models allow for similar improvements when modeling second moments. To illustrate, consider the s-step-ahead (s 2) minimum 2 Copyright c 2010 by John Wiley & Sons, Inc. All rights reserved.

3 Figure 1. Daily deutsche-mark U.S. dollar exchange rate. Panel (a) displays daily observations on the DM/U.S. $ exchange rate, s t, over the sample period October 1, 1979 through September 30, Panel (b) graphs the associated daily percentage appreciation of the U.S. dollar, calculated as r t 100ln(s t /s t 1 ). Panel (c) depicts the conditional standard-deviation estimates of the daily percentage appreciation rate for the U.S. dollar implied by each of the three volatility model estimates reported in Table 1. Copyright c 2010 by John Wiley & Sons, Inc. All rights reserved. 3

4 Figure 2. Daily S&P 500 stock-market index. Panel (a) displays daily observations on the value of the S&P 500 stock-market index, P t, over the sample period October 1, 1979 through September 30, Panel (b) graphs the associated daily percentage appreciation of the S&P 500 stock index excluding dividends, calculated as r t 100ln(P t /P t 1 ). Panel (c) depicts the conditional standard-deviation estimates of the daily percentage appreciation rate for the S&P 500 stock-market index implied by each of the three volatility-model estimates reported in Table 2. 4 Copyright c 2010 by John Wiley & Sons, Inc. All rights reserved.

5 mean square error forecast for the conditional variance in the simple GARCH(1,1) model, E t (, 2 t+s ) = E t (F 2 t+s ) s 2 = T (" 1 +$ 1 ) i + (" 1 +$ 1 ) s 1 Ft+1 2. i=0 (4) If the process is covariance-stationary, i.e.," 1 +$ 1 < 1, it follows that E t (F 2 t+s ) = F 2 + (" 1 +$ 1 ) s 1 (F 2 t+1 F 2 ). Thus, if the current conditional variance is large (small) relative to the unconditional variance, the multistep forecast is also predicted to be above (below) F 2, but converges to F 2 at an exponential rate as the forecast horizon lengthens. Higher-order covariance-stationary models display more complicated decay patterns [3]. 4 IGARCH The assumption of covariance stationarity has been questioned by numerous studies which find that the largest root in the estimated lag polynomial 1 ˆ"(x) ˆ$(x) = 0 is statistically indistinguishable from unity. Motivated by this stylized fact, Engle and Bollerslev [22] proposed the so-called integrated GARCH, or IGARCH(p, q), process, in which the autoregressive polynomial in 3 has one unit root; i.e., 1 "(B) $(B) (1 B)N(B), where N(x) 0 for x 1. However, the notion of a unit root is intrinsically a linear concept, and considerable care should be exercised in interpreting persistence in nonlinear models. For example, from 4, the IGARCH(1,1) model with" 1 +$ 1 = 1 behaves like a random walk, or an I(1) process, for forecasting purposes. Nonetheless, by repeated substitution, the GARCH(1,1) model may be written as F 2 t =F 2 0 t (" 1 zt i 2 +$ 1) i=1 t 1 +T 1 + j=1 i=1 j (" 1 zt i 2 +$ 1). Thus, as Nelson [44] shows, strict stationarity and ergodicity of the GARCH(1,1) model requires only geometric convergence of {" 1 z 2 t +$ 1 }, ore[ln(" 1 z 2 t +$ 1 )] < 0, a weaker condition than arithmetic convergence, or E(" 1 z 2 t +$ 1 ) = " 1 +$ 1 < 1, which is required for covariance stationarity. This also helps to explain why standard maximum likelihood based inference procedures, discussed below, still apply in the IGARCH context [39, 42, 51]. 5 EGARCH While the GARCH(p, q) model conveniently captures the volatility clustering phenomenon, it does not allow for asymmetric effects in the evolution of the volatility process. In the EGARCH(p, q) model of Nelson [46], the logarithm of the conditional variance is given as an ARMA(p, q) model in both the absolute Copyright c 2010 by John Wiley & Sons, Inc. All rights reserved. 5

6 size and the sign of the lagged innovations, ln F 2 t =T + + p i=1 n i ln F 2 t i q R j g(z t 1 j ) j=0 T +n(b)ln F 2 t +R(B)g(z t ), (5) g(z t ) =2z t +([ z t E( z t )], (6) along with the normalizationr 0 1. By definition, the news impact function g( ) satisfies E t 1 [g(z t )] = 0. When actually estimating EGARCH models the numerical stability of the optimization procedure is often enhanced by approximating g(z t ) by a smooth function that is differentiable at zero. Bollerslev et al. [12] also propose a richer parameterization for this function that downweighs the influence of large absolute innovations. Note that the EGARCH model still predicts that large (absolute) innovations follow other large innovations, but if 2 < 0 the effect is accentuated for negative, t s. Following Black [6], this stylized feature of equity returns is often referred to as the leverage effect. 6 Alternative Parameterizations In addition to GARCH, IGARCH, and EGARCH, numerous alternative univariate parametrizations have been suggested. An incomplete listing includes: ARCH-in-mean, or ARCH-M [25], which allows the conditional variance to enter directly into the equation for the conditional mean of the process; nonlinear augmented ARCH, or NAARCH [37], structural ARCH, or STARCH [35] ; qualitative threshold ARCH, or QTARCH [31] ; asymmetric power ARCH, or AP-ARCH [19] ; switching ARCH, or SWARCH [16, 34] ; periodic GARCH, or PGARCH [14] ; and fractionally integrated GARCH, or FIGARCH [4]. Additionally, several authors have proposed the inclusion of various asymmetric terms in the conditional-variance equation to better capture the aforementioned leverage effect; see e.g., References [17, 26], and [30]. 7 Time-Varying Parameter and Bilinear Models There is a close relation between ARCH models and the widely used time-varying parameter class of models. To illustrate, consider the simple ARCH(q) model in 2, i.e., Ft 2 = T +" 1, 2 t 1 + +" q, 2 t q. This model is observationally equivalent to the process defined by, t = w t + q a i, t i, i=1 where w t, a 1,..., a q are i.i.d. random variables with mean zero and variances T, " 1,..., " q, respectively; see Tsay [54] and Bera et al. [5] for further discussion. Similarly, the class of bilinear time-series models discussed by Granger and Anderson [32] provides an alternative approach for modeling nonlinearities; see Weiss [56] and Granger and Teräsvirta [33] for a more formal comparison of ARCH and bilinear models. However, while time-varying parameter and bilinear models may conveniently allow for heteroskedasticity and/or nonlinear dependencies through a set of nuisance parameters, in applications in economics and 6 Copyright c 2010 by John Wiley & Sons, Inc. All rights reserved.

7 finance the temporal dependencies in F t are often of primary interest. ARCH models have a distinct advantage in such situations by directly parameterizing this conditional variance. 8 Estimation and Inference ARCH models are most commonly estimated via maximum likelihood. Let the density for the i.i.d. process z t be denoted by f(z t ;v), where v represents a vector of nuisance parameters. Since F t is measurable with respect to the time (t 1) observable information set, it follows by a standard prediction-error decomposition argument that, apart from initial conditions, the log likelihood function for ǫ T {, 1,, 2,...,, T }equals log L(, T ;>,v) = T t=1 [ ln f (, t Ft 1 ; v) 1 ] 2 ln F t 2, where > denotes the vector of unknown parameters in the parameterization for F t. Under conditional normality, f (z t ; v) = (2B) 1/2 exp( 1 2 z2 t ). (8) By Jensen s inequality, E(, 4 t ) = E(z4 t ) E(F t 4) E(z4 t )E(F t 2)2 = E(zt 4)E(,2 t )2. Thus, even with conditionally normal innovations, the unconditional distribution for, t is leptokurtic. Nonetheless, the conditional normal distribution often does not account for all the leptokurtosis in the data, so that alternative distributional assumptions have been employed; parametric examples include the t-distribution in Bollerslev [8] and the generalized error distribution (GED) in Nelson [46], while Engle and Gonz alez-rivera [23] suggest a nonparametric approach. However, if the conditional variance is correctly specified, the normal quasiscore vector based on 7 and 8 is a martingale difference sequence when evaluated at the true parameters, n 0 ; i.e., E t 1 [ 1 2 ( >Ft 2)F 2 t (, 2 t F 2 t 1)] = 0. Thus, the corresponding quasi-maximum-likelihood estimate (QMLE), ˆ>, generally remains consistent, and asymptotically valid inference may be conducted using an estimate of a robustified version of the asymptotic covariance matrix, A(n 0 ) 1 B(n 0 )A(n 0 ) 1, where A(n 0 ) and B(n 0 ) denote the Hessian and the outer product of the gradients, respectively [55]. A convenient form of A(ˆ>) with first derivatives only is provided in Bollerslev and Wooldridge [15]. Many of the standard mainframe and PC computer-based packages now contain ARCH estimation procedures. These include E-VIEW, RATS, SAS, TSP, and a special set of time series libraries for the GAUSS computer language. (7) 9 Testing Conditional moment (CM)-based misspecification tests are easily implemented in the ARCH context via simple auxiliary regressions [50, 53, 57, 58]. Specifically, following Wooldridge [58], the moment condition E t 1 [(8 t F 2 t )(, 2 t F 2 t )F 2 t ] = 0 (9) (evaluated at the true parameter n 0 ) provides a robust test in the direction indicated by the vector k t of misspecification indicators. By selecting these indicators as appropriate functions of the time (t 1) information set, the test may be designed to have asymptotically optimal power against a specific alternative; e.g., the conditional variance specification may be tested for goodness of fit over subsamples by letting Copyright c 2010 by John Wiley & Sons, Inc. All rights reserved. 7

8 8 t be the relevant indicator function, or for asymmetric effects by letting 8 t, t 1 I{, t 1 < 0}, where I{ }denotes the indicator function for, t 1 < 0. Lagrange multiplier type tests that explicitly recognize the one-sided nature of the alternative when testing for the presence of ARCH have been developed by Lee and King [40]. 10 Empirical Example As previously discussed, the two time-series plots for the DM/$ exchange rate and the S&P 500 stock market index in Figures 1 and 2 both show a clear tendency for large (and for small) absolute returns to cluster in time. This is also borne out by the highly significant Ljung Box [41] portmanteau tests for up to 20th-order serial correlation in the squared residuals from the estimated AR(1) models, denoted by Q 2 20 in panel (a) of Tables 1 and 2. To accommodate this effect for the DM/$ returns, Panel (b) of Table 1 reports the estimates from an AR(1) GARCH(1,1) model. The estimated ARCH coefficients are overwhelmingly significant, and, judged by the Ljung Box test, this simple model captures the serial dependence in the squared returns remarkably well. Note also that ˆ" 1 + ˆ$ 1 is close to unity, indicative of IGARCH-type behavior. Although the estimates for the corresponding AR(1) EGARCH(1, 0) model in panel (c) show that the asymmetry coefficient2 is significant at the 5% level, the fit of the EGARCH model is comparable to that of the GARCH specification. This is also evident from the plot of the estimated volatility processes in panel (c) of Figure 1. The results of the symmetric AR(1) GARCH(2,2) specification for the S&P 500 series reported in Table 2 again suggest a very high degree of volatility persistence. The largest inverse root of the autoregressive polynomial in 3 equals 2 1 {ˆ" 1 + ˆ$ 1 + [(ˆ" 1 + ˆ$ 1 ) 2 + 4(ˆ" 2 + ˆ$ 2 )] 1/2 } = 0.984, which corresponds to a half-life of 43.0, or approximately two months. The large differences between the conventional standard errors reported in parentheses and their robust counterparts in square brackets highlight the importance of the robust inference procedures with conditionally nonnormal innovations. The two individual robust standard errors for" 2 and$ 2 suggest that a GARCH(1,1) specification may be sufficient, although previous studies covering longer time spans have argued for higher-order models [27, 52]. This is consistent with the results for the EGARCH(2,1) model reported in panel (c), where both lags of g(z t ) and ln F1 2 are highly significant. On factorizing the autoregressive polynomial for ln Ft 2, the two inverse roots equal and Also, the EGARCH model points to potentially important asymmetric effects in the volatility process. In summary, the GARCH and EGARCH volatility estimates depicted in panel (c) of Figure 2 both do a good job of tracking and identifying periods of high and low volatility in the U.S. equity market. 11 Future Developments We have provided a very partial introduction to the vast ARCH literature. In many applications a multivariate extension is called for; see References [10, 11, 13, 18, 24, 48] for various parsimonious multivariate parameterizations. Important issues related to the temporal aggregation of ARCH models are addressed by Drost and Nijman [20]. Rather than directly parametrizing the functional form for F t in 1, Gallant and Tauchen [29], and Gallant et al. [28] have developed flexible nonparametric techniques for analysis of data with ARCH features. Much recent research has focused on the estimation of stochastic volatility models in which the process for F t is treated as a latent variable [1, 36, 38]. For a more detailed discussion of all of these ideas, see the many surveys listed in the refereence section below. A conceptually important issue concerns the rationale behind the widespread empirical findings of IGARCH-type behavior, as exemplified by the two time series analyzed above. One possible explanation is provided by the continuous record asymptotics developed in a series of papers by Nelson [45, 47] and 8 Copyright c 2010 by John Wiley & Sons, Inc. All rights reserved.

9 Table 1. Daily Deutsche-Mark U.S. dollar exchange-rate appreciation AR(1):(a) r t = r t 1 +, t F 2 t =0.585 (0.013) (0.017) [0.013] [0.019] (0.014) [0.022] Log1 = , b 3 = 0.25, b 4 = 5.88, Q 20 = 19.69, Q 2 20 = AR(1) GARCH(1,1): (b) r t = r t 1 +, t (0.012) (0.018) [0.012] [0.019] Ft 2 = , 2 2 t Ft 1 (0.004) (0.011)(0.012) [0.004] [0.015][0.015] Log1 = , b 3 = 0.10, b 4 = 4.67, Q 20 = 32.48, Q 2 20 = AR(1) EGARCH(1, 0): (c) r t = r t 1 +, t (0.012) (0.018) [0.012] [0.017] ln Ft 2 = [0.030 z t ( z t 1 2/B)] ln(ft 1 2 ) (0.081) (0.009)(0.019) (0.004) [0.108] [0.013][0.022] [0.008] Log1 = , b 3 = 0.16, b 4 = 4.54, Q 20 = 33.61, Q 2 20 = Notes: All the model estimates are obtained under the assumption of conditional normality; i.e., z t, t Ft 1 i.i.d. N(0, 1). Conventional asymptotic standard errors based on the inverse of Fisher s information matrix are given in parentheses, while the numbers in square brackets represent the corresponding robust standard errors as described in the text. The maximized value of the pseudo-log-likelihood function is denoted Log1. The skewness and kurtosis of the standardized residuals, ẑ t = ˆ, t ˆF t 1, are given by b 3 and b 4, respectively. Q 20 and Q 2 20 refer to the Ljung Box portmanteau test for up to 20th-order serial correlation in ẑ t and ẑt 2, respectively. Nelson and Foster [49]. Specifically, suppose that the discretely sampled observed process is generated by a continuous-time diffusion, so that the sample path for the latent instantaneous volatility process {Ft 2 } is continuous almost surely. Then one can show that any consistent ARCH filter must approach an IGARCH model in the limit as the sampling frequency increases. The empirical implications of these theoretical results should not be carried too far, however. For instance, while daily GARCH(1,1) estimates typically suggest ˆ" 1 + ˆ$ 1 1, on estimating GARCH models for financial returns at intraday frequencies, Andersen and Bollerslev [2] document large and systematic deviations from the theoretical predictions of approximate IGARCH behavior. This breakdown of the most popular ARCH parameterizations at the very high intraday frequencies has a parallel at the lowest frequencies. Recent evidence suggests that the exponential decay of volatility shocks in covariance-stationary GARCH and EGARCH parameterizations results in too high a dissipation rate at long horizons, whereas the infinite persistence implied by IGARCH-type formulations is too restrictive. Copyright c 2010 by John Wiley & Sons, Inc. All rights reserved. 9

10 Table 2. Daily S&P 500 stock-market index returns AR(1):(a) r t = r t 1 +, t F 2 t =1.044 (0.017) (0.017) [0.018] [0.056] (0.025) [0.151] Log1 = , b 3 = 3.20, b 4 = 75.37, Q 20 = 37.12, Q 2 20 = AR(1) GARCH(2, 2): (b) r t = r t 1 +, t (0.014) (0.018) [0.015] [0.019] Ft 2 = , 2 t ,2 2 2 t Ft Ft 2 (0.004) (0.018) (0.021) (0.101) (0.092) [0.008] [0.078] [0.077] [0.098] [0.084] Log1 = , b 3 = 0.58, b 4 = 8.82, Q 20 = 11.79, Q 2 20 = 8.45 AR(1) EGARCH(2, 1): (c) r t = r t 1 +, t (0.014) (0.018) [0.015] [0.017] ln F 2 t = ( B)[ z t ( z t 1 2/B)] (0.175) (0.031) (0.014) (0.018) [0.333] [0.046] [0.046] [0.058] ln Ft ln Ft 2 (0.062) (0.061) [0.113] [0.112] Log1 = , b 3 = 0.60, b 4 = 9.06, Q 20 = 8.93, Q 2 20 = 9.37 Note: See Table 1. The fractionally integrated GRACH, or FIGARCH, class of models [4] explicitly recognizes this by allowing for a low hyperbolic rate of decay in the conditional variance function. However, a reconciliation of the empirical findings at the very high and low sampling frequencies within a single consistent modeling framework remains an important challenge for future work in the ARCH area. 12 Related Articles Autoregressive-Moving Average (ARMA) Models, Autoregressive-Integrated Moving Average (ARIMA) Models, Exponential Autoregressive Models, Time series regression, Nonlinear Time Series 10 Copyright c 2010 by John Wiley & Sons, Inc. All rights reserved.

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