JWPR026-Fabozzi c114-NP November 27, 2007 10:17 1 CHAPTER NP 2 3 4 5 6 ARCH/GARCH Models in Applied 7 8 Financial Econometrics 9 10 11 ROBERT F. ENGLE, PhD 12 Michael Armellino Professorship in the Management of Financial Services, Leonard N. Stern School 13 of Business, New York University 14 15 SERGIO M. FOCARDI 16 Partner, The Intertek Group 17 18 19 FRANK J. FABOZZI, PhD, CFA, CPA 20 Professor in the Practice of Finance, School of Management, Yale University 21 22 23 24 25 Review of Linear Regression and Autoregressive Integration of First, Second, and Higher Moments 7 26 Models 2 Generalizations to High-Frequency Data 7 27 ARCH/GARCH Models 3 Multivariate Extensions 9 28 Application to Value at Risk 5 Summary 10 29 Why ARCH/GARCH? 5 References 10 30 Generalizations of the ARCH/GARCH Models 7 31 32 33 34 Abstract: Volatility is a key parameter used in many financial applications, from deriva- 35 tives valuation to asset management and risk management. Volatility measures the size 36 of the errors made in modeling returns and other financial variables. It was discov- 37 ered that, for vast classes of models, the average size of volatility is not constant but 38 changes with time and is predictable. Autoregressive conditional heteroskedasticity 39 (ARCH)/generalized autoregressive conditional heteroskedasticity (GARCH) models 40 and stochastic volatility models are the main tools used to model and forecast volatil- 41 ity. Moving from single assets to portfolios made of multiple assets, we find that not 42 only idiosyncratic volatilities but also correlations and covariances between assets are 43 time varying and predictable. Multivariate ARCH/GARCH models and dynamic fac- 44 tor models, eventually in a Bayesian framework, are the basic tools used to forecast 45 correlations and covariances. 46 47 48 Keywords: autoregressive conditional duration, ACD-GARCH, autoregressive 49 conditional heteroskedasticity (ARCH), autoregressive models, conditional 50 autoregressive value at risk (CAViaR), dynamic factor models, generalized 51 autoregressive conditional heteroskedasticity (GARCH), exponential 52 GARCH (EGARCH), F-GARCH, GARCH-M, heteroskedasticity, 53 high-frequency data, homoskedasticity, integrated GARCH (IGARCH), 54 MGARCH, threshold ARCH (TARCH), temporal aggregation, 55 ultra-high-frequency data, value at risk (VaR), VEC, volatility 56 57 58 In this chapter we discuss the modeling of the time be- ers know that errors made in predicting markets are not 59 havior of the uncertainty related to many econometric of a constant magnitude. There are periods when unpre- 60 models when applied to financial data. Finance practition- dictable market fluctuations are larger and periods when 61 1 JWPR026-Fabozzi c114-NP November 27, 2007 10:17 2 ARCH/GARCH Models in Applied Financial Econometrics 1 they are smaller. This behavior, known as heteroskedastic- time series are considered to be realizations of stochastic 2 ity, refers to the fact that the size of market volatility tends processes. That is, each point of an economic time series 3 to cluster in periods of high volatility and periods of low is considered to be an observation of a random variable. 4 volatility. The discovery that it is possible to formalize and We can look at a stochastic process as a sequence of vari- 5 generalize this observation was a major breakthrough in ables characterized by joint-probability distributions for 6 econometrics. In fact, we can describe many economic and every finite set of different time points. In particular, we 7 financial data with models that predict, simultaneously, can consider the distribution ft of each variable Xt at each 8 the economic variables and the average magnitude of the moment. Intuitively, we can visualize a stochastic process 9 squared prediction error. as a very large (infinite) number of paths. A process is 10 In this chapter we show how the average error size can called weakly stationary if all of its second moments are 11 be modeled as an autoregressive process. Given their au- constant. In particular this means that the mean and vari- µ = µ σ 2 = σ 2 12 toregressive nature, these models are called autoregressive ance are constants t and t that do not depend 13 conditional heteroskedasticity (ARCH) or generalized autore- on the time t. A process is called strictly stationary if none 14 gressive conditional heteroskedasticity (GARCH). This dis- of its finite distributions depends on time. A strictly sta- 15 covery is particularly important in financial econometrics, tionary process is not necessarily weakly stationary as its 16 where the error size is, in itself, a variable of great interest. finite distributions, though time-independent, might have 17 infinite moments. µ σ 2 18 The terms t and t are the unconditional mean and 19 variance of a process. In finance and economics, however, 20 REVIEW OF LINEAR REGRESSION we are typically interested in making forecasts based on 21 past and present information. Therefore, we consider the AND AUTOREGRESSIVE MODELS 22 distribution ft2 x It1 of the variable Xt2 at time t2 con- Let’s first discuss two examples of basic econometric mod- 23 ditional on the information It1 known at time t1.Based 24 els, the linear regression model and the autoregressive on information available at time t − 1, It−1, we can also 25 model, and illustrate the meaning of homoskedasticity or define the conditional mean and the conditional variance heteroskedasticity in each case. µ | , σ 2 | 26 ( t It−1 ) t It−1 . 27 The linear regression model is the workhorse of eco- A process can be weakly stationary but have time- 28 nomic modeling. A univariate linear regression represents varying conditional variance. If the conditional mean is 29 a proportionality relationship between two variables: constant, then the unconditional variance is the uncondi- 30 y = α + βx + ε tional expectation of the conditional variance. If the con- 31 ditional mean is not constant, the unconditional variance 32 The preceding linear regression model states that the ex- is not equal to the unconditional expectation of the condi- 33 pectation of the variable y is β times the expectation of the tional variance; this is due to the dynamics of the condi- 34 variable x plus a constant α. The proportionality relation- tional mean. 35 ship between y and x is not exact but subject to an error In describing ARCH/GARCH behavior, we focus on 36 ε. In standard regression theory, the error ε is assumed to the error process. In particular, we assume that the errors 37 have a zero mean and a constant standard deviation σ . are an innovation process, that is, we assume that the 38 The standard deviation is the square root of the variance, conditional mean of the errors is zero. We write the error 2 2 39 which is the expectation of the squared error: σ = E ε . process as: εt = σt zt where σt is the conditional standard 40 It is a positive number that measures the size of the error. deviation and the z terms are a sequence of independent, 41 We call homoskedasticity the assumption that the expected zero-mean, unit-variance, normally distributed variables. 42 size of the error is constant and does not depend on the size Under this assumption, the unconditional variance of the 43 of the variable x. We call heteroskedasticity the assumption error process is the unconditional mean of the conditional 44 that the expected size of the error term is not constant. variance. Note, however, that the unconditional variance 45 The assumption of homoskedasticity is convenient from of the process variable does not, in general, coincide with 46 a mathematical point of view and is standard in regression the unconditional variance of the error terms. 47 theory. However, it is an assumption that must be verified In financial and economic models, conditioning is often 48 empirically. In many cases, especially if the range of vari- stated as regressions of the future values of the variables 49 ables is large, the assumption of homoskedasticity might on the present and past values of the same variable. For 50 be unreasonable. For example, assuming a linear relation- example, if we assume that time is discrete, we can express 51 ship between consumption and household income, we conditioning as an autoregressive model: 52 can expect that the size of the error depends on the size of 53 household income. But, in fact, high-income households Xt+1 = α0 + β0 Xt +···+βn Xt−n + εt+1 54 have more freedom in the allocation of their income. 55 In the preceding household-income example, the lin- The error term εi is conditional on the information Ii 56 ear regression represents a cross-sectional model without that, in this example, is represented by the present and the 57 any time dimension. However, in finance and economics past n values of the variable X. The simplest autoregressive 58 in general, we deal primarily with time series, that is, se- model is the random walk model of the logarithms of 59 quences of observations at different moments of time. Let’s prices pi : 60 call Xt the value of an economic time series at time t. Since 61 the groundbreaking work of Haavelmo (1944), economic pt+1 = µt + pt + εt JWPR026-Fabozzi c114-NP November 27, 2007 10:17 PLEASE SUPPLY PART TITLE 3 1 In terms of returns, the random walk model is simply: Thus, the statement that ARCH models describe the time 2 evolution of the variance of returns is true only if returns rt = pt = µ + εt 3 have a constant expectation. 4 A major breakthrough in econometric modeling was the ARCH/GARCH effects are important because they are 5 discovery that, for many families of econometric mod- very general.