GARCH Model Estimation Using Estimated Quadratic Variation

Transcription

1 GARCH Model Estimation Using Estimated Quadratic Variation John W. Galbraith, Victoria Zinde-Walsh and Jingmei Zhu Department of Economics, McGill University Abstract We consider estimates of the parameters of GARCH models obtained using auxiliary information on latent variance which may be available from higher-frequency data, for example from an estimate of the daily quadratic variation such as the realized variance. We obtain consistent estimates of the parameters of the infinite ARCH representation via a regression using the estimated quadratic variation, without requiring that it be a consistent estimate; that is, variance information containing measurement error can be used for consistent estimation. We obtain GARCH parameters using a minimum distance estimator based on the estimated ARCH parameters. With LAD estimation of the truncated ARCH approximation, we show that consistency and asymptotic normality can be obtained using a general result on LAD estimation in truncated models of infinite-order processes. We provide simulation evidence on small-sample performance for varying numbers of intra-day observations. Key words: GARCH, high-frequency data, quadratic variation, LAD JEL Classification number: C22 * The authors thank the Fonds pour la formation de chercheurs et l aide à la recherche (Quebec) and the Social Sciences and Humanities Research Council of Canada for financial support of this research, and CIRANO (Centre Inter-Universitaire de Recherche en Analyse des Organisations) for research facilities. We are grateful to Ilze Kalnina, John Maheu, Nour Meddahi, Tom McCurdy and Eric Renault, and to participants in Econometric Society and Canadian Econometric Study Group meetings, for valuable comments and discussions, and to Roger Koenker for kindly providing his Matlab code for estimation of constrained quantile regressions.

2 1. Introduction GARCH models are widely used for forecasting and characterizing the conditional variance of financial time series. Since the original contributions of Engle (1982) and Bollerslev (1986), the models have been estimated by Maximum Likelihood (or quasi- Maximum Likelihood, QML ) methods using observations at the frequency of interest. In the case of asset returns, the frequency of interest is often the daily fluctuation. However, financial data are usually recorded at frequencies much higher than the daily. Even where our interest lies in conditional variance at the daily frequency, these higherfrequency data may contain information which can be used to improve forecasts and characterization of daily conditional variance. In particular, higher-frequency data may be used to estimate the daily variance directly; these estimates based on higherfrequency data provide an auxiliary source of lower-frequency variance information. Numerous recent contributions in the financial econometrics literature have studied estimates of variance at one frequency using observations at a higher frequency. Andersen and Bollerslev (1998) used the sum of squared intra-day asset returns (realized variance) to estimate the integrated variance over a day, and several other estimators for the same quantity, such as that of Zhang et al. (2005), have subsequently been proposed. However, any such auxiliary variance estimates may contain non-trivial estimation error. Here therefore we allow explicitly for a noise term in the estimated daily conditional variance (or estimated quadratic variation). 1 The aim is to estimate GARCH models of daily conditional variance. We consider estimates of quadratic variation based on some higher frequency as an error-contaminated measure, similar to Maheu and McCurdy (2002) but without assuming moment existence restrictions. In fact, the measurement error is bounded on one side because of the non-negativity constraint, but may exhibit large outliers on the other side. This suggests the use of quantile (in particular Least Absolute Deviations, LAD ) estimation, which restricts only the signs but not the magnitudes of the measurement error. Where the estimates of daily quadratic variation contain non-negligible noise, we would normally expect biases if the estimated daily quantities were used as regressors (the standard problem of measurement error in regressors), but we reduce this particular source of bias here because we do not use direct regression on the estimated quantities. Instead we obtain estimates via ARCH approximation, using an estimator related to that of Galbraith and Zinde-Walsh (1997) (for ARMA models) to obtain estimates of GARCH parameters. The models may be estimated by a variety of techniques (including LS); by using the LAD estimator, it is possible to obtain consistent 1 We will refer in most instances to estimated quadratic variation, rather than to realized variance, to emphasize that estimators of the quadratic variation other than the realized variance are also admissible, and may indeed produce superior results. The realized variance is nonetheless one admissible estimator, and we use it in the simulation experiments below. 1

3 and asymptotically normal estimates under quite general conditions. In section 2 we describe the models and estimators to be considered and give some relevant definitions and notation. Section 3 provides several asymptotic results for LAD estimation of these models, while section 4 presents simulation evidence on the finite-sample performance of regression estimators in comparison with that of standard QML GARCH estimates. 2. GARCH model estimation using higher-frequency data 2.1 Processes and definitions We begin by establishing notation for the processes of interest. Consider a driftless diffusion process {p t } such that p t = p 0 + t 0 σ s dw s, (2.1.1) where {W s } is a Brownian motion process and σs 2 is the instantaneous conditional variance. This is a special case of the structure used by, e.g., Nelson (1992), Nelson and Foster (1995). In a financial context, p t will represent the logarithm of an asset price, so that the difference p t p t s is a log-return on the asset over the period t s to t. The process is sampled discretely at an interval of time τ (e.g., each minute). We are interested in variance at a lower-frequency sampling, with sampling interval hτ (e.g., daily), so that there are h high-frequency observations per low-frequency observation. We index the lower-frequency observations by t, the size of the sample of low-frequency observations is T, and of the full set of high-frequency observations is ht, so that t = {1, 2,..., T } corresponds with {h, 2h,..., ht }. Following Andersen and Bollerslev (1998) and others, the lower-frequency (hereafter daily ) integrated variance at t is σ 2 t = An estimate of this quantity is given by ˆσ 2 t = t t 1 th σ 2 sds. j=(t 1)h+1 r 2 j, (2.1.2) with rj 2 = (p j p j 1 ) 2, p j indicating the discretely-sampled observations on the process. The estimate in (2.1.2) is referred to in the financial econometrics literature as the realized variance, or sometimes as the realized volatility, and its probability limit as h 0 is the quadratic variation. As we indicated above, we will use the 2

4 general term estimated quadratic variation (EQV) for such a quantity understood as an input to our GARCH estimation procedures below, to avoid suggesting that the realized variance is the only admissible estimator. See Andersen and Bollerslev (1998), Barndorff-Nielsen and Shephard (2002) and Andersen et al. (2003) for formal definitions of these quantities and for conditions under which convergence of ˆσ 2 t to σ 2 t takes places as intra-day frequency increases. Many types of models use dynamics at low (daily) frequency, including ARCH and GARCH, and variants such as non-linear GARCH. These models are useful for conditional variance prediction, and thus are objects of interest. Consider the ARCH and GARCH models at these lower-frequency observations: σ 2 t = ω + σ 2 t = ω + q α i ε 2 t i, (2.1.3) i=1 q α i ε 2 t i + i=1 p β i σt i, 2 (2.1.4) where ε t = r t µ t for a process with conditional mean µ t, or in the driftless case ε t = r t. However, direct estimation is not possible even for (2.1.3), since it involves latent σ 2 t. Additional assumptions are made for QMLE estimation, but the models could be directly estimable if one were to use measurements of σ 2 t. Here we propose to use the estimated quadratic variation as such a measurement. 2.2 Regression models with auxiliary high frequency information Consider now the EQV at some frequency, interpreted as an estimator of σ 2 t. However, we will not treat the estimate as exact. Instead, we introduce into the model the measurement error arising in estimation of σ 2 t via a daily EQV (such a specification was also used by Maheu and McCurdy, 2002). Let i=1 ˆσ 2 t = σ 2 t + e t. (2.2.1) The properties of (2.2.1) could follow from (2.1.2), as for example in the literature in which the Gaussian asymptotic distribution with variance depending on the quartic variation is derived; a recent review is provided by McAleer and Medeiros (2008). We will introduce some fairly general assumptions and treat e t as a measurement error. Introducing the measurement of variance into the models (2.1.3) and (2.1.4), we obtain (2.2.2) and (2.2.3) where both now can be viewed as regression models: ˆσ 2 t = ω + q α i ε 2 t i + e t, (2.2.2) i=1 3

5 ˆσ 2 t = ω + q α i ε 2 t i + i=1 p β iˆσ t i 2 i=1 p β i e t i + e t. (2.2.3) Both (2.2.2) and (2.2.3) are in principle estimable as regression models. Note that model (2.2.3) has an error term with an MA(p) form, with the coefficients of this moving average subject to the constraint that they are the same (up to sign) as the coefficients on lagged values of ˆσ t 2. Maheu and McCurdy (2002) find that the constrained model, estimated by QML, performs well on foreign exchange return data. Bollen and Inder (2002) estimate a model similar to (2.2.3) by standard QML methods (without accounting for the error autocorrelation structure), using intra-day data to obtain estimates of an unobservable sequence related to daily variance. This approach requires consistency of the estimates of the unobservable sequence as the number of intra-day observations per day increases without bound, to obtain consistency of the estimator; Bollen and Inder find good results on a sample of S&P Index futures with a large number of observations per day. Here we use the alternative strategy of estimation by using an ARCH approximation followed by computation of GARCH parameters from the ARCH approximation. Xiao and Koenker (2009) also use a long ARCH (LAD) regression in the first step of their estimation method for linear GARCH models; their second step however differs from our proposed estimation method in that they use predicted σt 2 to obtain the parameter estimates, while we use the estimated parameters from the first-stage ARCH approximation to solve for the GARCH parameter estimates. This strategy also has the advantage of producing immediately an estimated model which is directly usable for forecasting, and of allowing computation of parameters of any GARCH(p, q) model from a given estimated ARCH representation. Sufficient conditions for consistent and asymptotically normal estimation are given below; it is not necessary that the number of intra-day observations per day (h) increase without bound. Giraitis et al. (2000) give general conditions under which the ARCH( ) representation is possible for the strong GARCH(p, q) case; only the existence of the first moment and summability of the ARCH coefficients are required for the existence of a strictly stationary ARCH( ) solution as given below in (2.2.4). To obtain estimates of GARCH parameters from the ARCH representation we pursue an estimation strategy related to that of Galbraith and Zinde-Walsh (1994, 1997) or Galbraith, Ullah and Zinde-Walsh (2002), in which autoregressive models are used in estimation of MA, ARMA or VARMA models. Here, a high-order ARCH model is used, and estimates of GARCH (p, q) parameters are deduced from the patterns of ARCH coefficients. Consider first a case of known conditional variance σt 2. The GARCH process (2.1.4) has a form analogous to the ARMA(p, q); using standard results on representation of an ARMA (p, q) process in MA form (see, e.g., Fuller 1976), we can express (2.1.4) in 4 i=1

6 the form with ν 0 = 0 and σt 2 = κ + ν l ε 2 t l, (2.2.4) l=1 ν 1 = α 1 ν 2 = α 2 + β 1 ν 1 and finally. min(l,p) ν l = α l + β i ν l i, l q, ν l = min(l,p) i=1 i=1 κ = (1 β(1)) 1 ω = β i ν l i, l > q; ( 1 (2.2.5) ) 1 p β i ω. (2.2.6) i=1 In place of the latent quantity σt 2 we will use the estimated low-frequency conditional variance, for example the EQV from (2.1.2), defining the estimation error as in (2.2.1) and substituting into (2.2.4) to obtain ˆσ 2 t = κ + Σ l=1ν l ε 2 t l + e t. We estimate the model using a truncated version ˆσ 2 t = κ + Σ k l=1ν l ε 2 t l + u t. (2.2.7) The error u t embodies the measurement error e t and truncation error. The rate of increase of the truncation parameter with sample size will be discussed below. The model (2.2.7) may be estimated by LS or, to obtain results robust to less restrictive conditions on the integrated variance estimation errors, LAD. Asymptotic properties of the LAD estimator are considered in Section 3. Suppose that ˆγ (k) = (ˆκ, ˆν 1,..., ˆν k ) is the estimator of the truncated ARCH parameters in (2.2.7). Denote ( ) Ik 0 by Ω k the k k partitioned matrix, where the zero entries are understood 0 0 to be of appropriate dimension for Ω k to be of dimension k k, and do not appear at all if k = k. 5

7 Define also ˆγ = Ω k ˆγ (k), so that ˆγ is a sub-vector of length k of the full k vector of estimated parameters, ˆγ (k). Let the full set of (p+q +1) parameters of the GARCH model be δ = (ω, α, β ). Then we can compute estimates of these GARCH parameters δ using the subset of k of ˆγ (k). It may be optimal to use a subset because we will require a long regression (large k) in (2.2.7) to reduce approximation errors, while the higher-lag parameter estimates may be relatively poorly estimated, so that basing the computation only on the first k of the ν l estimates may produce more efficient estimators. The notational distinction between k and k allows us to represent this case in the exposition of the estimator. Denote by F (δ) the transformation from δ to γ = (κ, ν 1,...ν k ) defined recursively by equations ( ), selecting k k components. Then define the minimum distance estimator ˆδ = arg min (ˆγ F (δ)) (ˆγ F (δ)). Estimation proceeds by first obtaining estimates of β = (β 1, β 2,..., β p ) from (2.2.5) for l > q, followed by estimation of the q parameters of α from the first q relations of (2.2.5), and of ω from (2.2.6). Begin by defining, for any k such that p+q+1 k k, v(0) = ν q+1 ν q+2. ν k, and v( i) = ν q+1 i ν q+2 i. ν k i. (2.2.8) Next define the (k q) p matrix C = [v( 1)v( 2)... v( p)] = ν q ν q 1... ν q p+1 ν q+1 ν q... ν q p+2... ν k 1 ν k 2... ν k p where ν r = 0 for r 0. It follows from (2.2.5) that v(0) = β C. The p 1 vector of estimates of the parameters β is defined by ˆβ = (Ĉ Ĉ) 1 Ĉ ˆv(0), (2.2.9) where the circumflex indicates replacement of ν l with the estimated (by OLS or LAD) values ˆν l in the definitions above. An estimate of α can then be obtained using the 6

8 estimate of β and the relations (2.2.5): that is ˆα 1 = ˆν 1 ˆα 2 = ˆν 2 ˆβ 1ˆν 1. (2.2.10) min(q,p) ˆα q = ˆν q i=1 ˆβ iˆν q i. Finally, ( ) p ˆω = ˆκ(1 ˆβ(1)) = ˆκ 1 ˆβ i. i=1 Note the role of k in ( ): to solve ( ) the number of equations used could be as small as p + q + 1; if we set this number k above this value the solution is by minimum distance. For the commonly-used GARCH(1,1) model, the mapping is ˆα = ˆν 1, k 1 ˆβ = i=1 ˆν iˆν i+1 k, ˆω = ˆκ(1 ˆβ). (2.2.11) 1 i=1 ˆν2 i In the GARCH model all parameters are non-negative; this implies non-negative parameters in the infinite ARCH representation, and so it is useful to restrict estimators to satisfy the non-negativity constraint. This can be done in various ways. One is to proceed to estimation of the GARCH parameters and simply to truncate any estimator at zero. Another way is to truncate at zero the estimators in the long ARCH approximation and then adjust estimation of the GARCH parameters to ensure nonnegativity. For example in the GARCH(1,1) case, by the expression for ˆβ in (2.2.11), non-negativity of the ν s ensures that 0 ˆβ 1. A more computationally-intensive alternative is to use an estimation algorithm that produces constrained LAD or OLS estimates, guaranteeing positive ARCH parameter estimates. The following theorem shows that we can conduct inference on the GARCH parameters δ = (ω, α, β ) conditional on an asymptotically normal set of estimates of the ARCH-approximation parameters γ. Below, Theorem 2 establishes the latter condition on the ARCH parameters. Theorem 1. Suppose that ˆγ is such that T (ˆγ γ) N(0, Γ). 7

9 ( ) 1 F Then for J = F δ δ F δ evaluated at δ, T (ˆδ δ) N(0, J ΓJ). Proof: follows from the standard asymptotic result for minimum distance estimation. The next section is devoted to the estimation of ˆγ(k) as k with sample size and to establishing asymptotic normality for ˆγ used in the Theorem. 3. LAD estimation of the ARCH regression and asymptotic properties In this section we consider estimation of the parameters of (2.2.4) via the truncated model (2.2.7). To avoid restrictive moment assumptions on the measurement error we use quantile estimation; here we consider only estimation at the 0.5th quantile, i.e. LAD. This can of course be generalized to other quantiles. Note that similar models could be written for, e.g., linear GARCH. Quantile estimation in a linear GARCH model was considered by Xiao and Koenker (2009). They propose a two-step procedure; in the first step the weighted regression estimation of the approximating long ARCH model, and in the second step quantile estimation of the GARCH parameters using the estimates obtained from the first step for the unobserved σ. They do not, however, consider measurements of σ based on the information in high frequency observations. The LAD regression approach used here could be applied for the linear GARCH model as well. Results for QML estimation of the ARCH model were established by Weiss (1986), using the assumption of finite fourth moments of the unnormalized data. Lumsdaine (1996) established consistency and asymptotic Normality of the QMLE for GARCH models by imposing conditions on the re-scaled data, z t = ε t /σ t, including the IID assumption and the existence of high-order moments. Lee and Hansen (1994) generalized these results to {z t } which are not IID, but simply strictly stationary and ergodic. Consider a true process (2.2.4), which could be a conditional mean process or a process at some q th conditional quantile. Here we focus on the median for LAD estimation: q = 0.5, where the conditional median of the error e t is zero. The left-hand side variable is an observed measurement of conditional variance, and so we can treat this in a general regression context. The proof here uses a more general theorem of Zernov, Zinde-Walsh and Galbraith (2009) (hereafter ZZG), that is proven using fairly high-level conditions. This theorem is applicable to the general problem of LAD regression with a truncation of an infinite representation, not only to such problems in the context of conditional variance estimation. For the case of conditional variance models, other models which can be represented as an infinite-order ARCH with summable coefficients could also be handled using this theorem. For the specific case of semi-strong GARCH models, we then show that consistency and asymptotic normality follow for the estimator of Section 2, 8

10 by demonstrating that the estimator fulfills the assumptions of the theorem of ZZG under a more specific set of conditions which is directly applicable to the GARCH model. Define y t = ˆσ t 2 and X ( ) = (X 0t, X 1t,...) with X 0,t = 1, X tl = ε 2 t l. For any k <, X t ( ) can be partitioned as X t ( ) = ( ) X t (k), Xt R ; analogously, partition the column vector of coefficients γ( ) as ( γ(k), (γ R ) ) ; the superscript R indicates a remainder, the elements or coefficients corresponding with indices k + 1,...,. Next consider the data generation process y t = γ 0 + X t l γl + e t = X t ( )γ + e t (3.1.1) l=1 with a summable sequence of parameters γ. Using the partitions just defined we obtain the representation We truncate the model (3.1.1)/(3.1.2) at order k : y t = X t (k)γ(k) + X R t γ R t + e t. (3.1.2) y t = X t (k)γ(k) + u t. (3.1.3) This represents (2.2.7) in the more general regression notation. We now examine LAD estimation of (3.1.3) as k increases; we use the result of ZZG on quantile estimation for the case where the truncated model (3.1.3) is used, but where the true process has the infinite representation (3.1.1). The result in ZZG uses a scaling matrix V T (k), the value of the error density at the quantile, p e (0), and a sequence k = ω(t ) to provide an asymptotic normality result for the vector of the first k coefficients (where k could increase with sample size and k). Denote by Ω k the k k partitioned matrix ( Ik ). Theorem 1 from ZZG (reproduced here as Theorem ZZG-1) considers the values of γ q (k) for the q th quantile and is proved under the assumption that is reproduced here in the Appendix as Assumption 0 (A0). Theorem ZZG-1. Under Assumption 0, as T, k = ω (T ), Ω k V T (k) (ˆγ q (k) γ q (k)) N ( ) q(1 q) 0, p e (0) Ω k. Proof: See ZZG. 9

11 The conditions of this theorem build on results (e.g. Pollard, 1991) that do not require moment existence in the errors or regressors. As long as the high-level Assumption 0 of ZZG is satisfied at the median of ˆσ t 2 in the GARCH( ) regression, the result holds without any moment assumptions. The assumptions required include stationarity and ergodicity, but not necessarily independence as in Xiao and Koenker (2009). Theorem ZZG-1 is a general result on consistency and asymptotic Normality of LAD parameter estimates in a truncation of an infinite-order process, where the truncation parameter k increases with sample size. Typically, a slower growth rate in k is needed to satisfy Assumption 0(d and e) on the regressors, but k needs to increase fast enough for Assumption 0(f) on the truncated part to hold. We show below that for a GARCH model with bounded fourth moments of ε 2 t, the condition k4 T 0 is sufficient for Assumption 0(d,e), and Assumption 0(f) holds for any k, so that ω(t ) could be proportional to either T c, c < 1 4, or ln(t ). Note that for the GARCH case the condition on the order of the approximating model is more stringent than the well-known k = o(t 1 3 ) condition for ARMA approximation. We next propose assumptions on the LAD regression for the infinite representation of the GARCH model considered here. Denote by I t the σ field generated by { ε 2 t l } l>0. Assumption 1: (a) The squared innovations { ε 2 t } form a strictly stationary ergodic sequence. (b) 2 The conditional variance is in the class of stationary, invertible semi-strong GARCH models that have the ARCH ( ) representation (2.2.4) (c) E ( ε 4 t ) is finite; (d) the measurement error {e t } is (i) a stationary ergodic sequence independent of F t ; (ii) the median of e t is zero; (iii) the pdf of e t, p e, exists, p e (0) 0 and p e is continuous at 0. 2 Definitions of Strong, Semi-strong and Weak GARCH given in Drost and Nijman (1993). In strong GARCH, {ε t } is such that z t ε t /σ t IID(0, 1); semi-strong GARCH holds where {ε t } is such that E[ε t ε t 1,...] = 0 and E[ε 2 t ε t 1,...] = σ 2 t ; weak GARCH holds where {ε t } is such that P [ε t ε t 1,...] = 0 and P [ε 2 t ε t 1,...] = σ 2 t, where P [ε 2 t ε t 1,...] denotes the best linear predictor of ε 2 t given a constant and past values of both ε t and ε 2 t. To obtain consistent estimation by QML, it will be necessary that the process is semi-strong GARCH: the standard Quasi-ML estimator of the GARCH model will in general be inconsistent in weak GARCH models (as noted by Meddahi and Renault 2004 and Francq and Zakoïan 2000). 10

12 Define W T (k) = T t=k+1 X t(k)x t (k); then for i j the elements of the symmetric matrix W T (k) are T k for i = j = 1; {W T (k)} ij = ε 2 t i for j = 1; ε 2 t i ε 2 t j otherwise. Under Assumption 1(a,c), by the Ergodic Theorem, elementwise as long as k = o (T ), Define V T (k) = (T k) 1 2 W T (k). (T k) 1 W T (k) p W (k) E ( X (k) X (k) ). Theorem 2. Under Assumption 1 and Assumption 0(d,e) (given in the Appendix) for V T (k) as just defined, and k = o (ln T ), Proof. See the Appendix. (T k) 1 2 Ωk (ˆγ (k) γ (k)) N(0, 1 4p e (0) Ω k W 1 (k) Ω k ). Together with Theorem 1, Theorem 2 allows asymptotic inference on the parameters of the GARCH model estimated by this ARCH-approximation method. This theorem still relies on the high level Assumption 0(d,e) that are similar to the usual high level assumptions for this type of regression (see e.g., Xiao and Koenker, 2009). Assumption 1 is sufficient to bound the tail sum in the infinite regression (Assumption 0f). Below we list additional assumptions on the ε 2 t that can provide the remaining Assumptions 0(d,e) and thus the result of the Theorem. Lemma 1. If Assumption 1 is satisfied and additionally the fourth moment of ε 2 t is ( ) bounded and k = o T 1 4, then the conditions in Assumption 0(d) and (e) hold and Theorem 2 holds with this k. Proof. See the Appendix. The proof for the conditions of Assumption 0(d,e) uses the rates of convergence (T k) 1 2 for the covariances to determine the appropriate rate of k and builds on the results of Berk (1974) and Pollard (1991). 4. Simulation evidence In this section we present some evidence on the finite-sample performance of the ARCH-regression estimator of Section 2.2, using daily variance information calculated 11

13 from the EQV in higher-frequency data; we present results relative to the standard Quasi-Maximum Likelihood (QML) estimator (Bollerslev 1986) based on the lowerfrequency data alone. A primary point of interest is the change in precision of the estimates based on EQV as the number of high-frequency (intra-day) observations per low-frequency (daily) observation is increased. We record relative precision graphically, in Figures 1-3, using the root mean squared error (RMSE). The standard estimator s results are generated by the widely-used QML GARCH procedure provided in MATLAB (R2009a); in the cases where we use normal innovations in generating the simulated data (upper panels in the figures), these are in fact true ML estimates, so that conditions are as favorable as possible for the standard QML estimator. Other cases (lower panels) use t 40 innovations in the high-frequency process. 3 LAD estimates of the truncated approximating ARCH model (2.2.7) are obtained using EQV computed according to (2.1.2), followed by computation of GARCH(1,1) parameter estimates via (2.2.9)- (2.2.10). Although the LAD estimator has the advantage over OLS of relative robustness, we also report results from the OLS estimator for comparison. All LAD and OLS estimates are computed with k = 12 lags in the approximating ARCH regression, and k = k. A relatively low value of k is chosen for computational speed; performance of the regression-based estimators would likely be enhanced (particularly for the first set of parameter values) by a higher value of k. As well, we present LAD and OLS estimates based on unconstrained estimates of the truncated ARCH regression, and the resulting GARCH parameter estimates are not constrained to be positive. Improvements in LAD-ARCH or OLS-ARCH estimates may be available through such constraints. Our purpose here however is to illustrate the effect of improved estimates of integrated variance with increasing h (the number of high-frequency observations per day) rather than to investigate optimal form for the ARCH-regression estimators. The simulations use 25,000 replications for Normal innovations in the high- frequency process, 50,000 replications for t 40 at the high frequency, and in all cases use a low-frequency sample size of T = 1200 observations (corresponding with roughly five years of daily returns in the context of financial data). The high-frequency sample size is ht with h = {10, 20,..., 100}. For comparison, the commonly-used five-minute interval for computation of realized variance (which we use to estimate quadratic variation) in daily financial returns implies h = 78 in an equity market in which trading takes place over the hour interval from 0930 to A high-frequency (strong) GARCH log-return process is simulated as ε (h)i = σ (h)i ν (h)i and σ 2 (h)i = ω h + α h ε 2 (h)i + β hσ 2 (h)i, for i = {1,... ht }, with the {ν (h)i} 3 This distribution allows us to generate many replications without encountering any instances of unstable squared returns; low degrees of freedom would at least occasionally produce explosive sequences of squared returns. 12

14 distributed Normally with unit variance, or t 40. Low-frequency returns and EQV are then computed by aggregation of the high-frequency values as ε t = th j=(t 1)h+1 ε (h)j and σ 2 t = th j=(t 1)h+1 ε 2 (h)j, yielding a weak GARCH daily process. The expressions in Drost and Nijman (1993) allow us to compute directly lowfrequency parameters of an aggregated process from the parameters of the high- frequency process. In order to compare processes with different values of h yielding the same low-frequency ( daily ) parameters, we need conversely to solve for the particular high-frequency (intra-day) parameters (ω h, α h, β h ) that will yield a given set of low-frequency values (ω, α, β) at different values h. We solve for these numerically, using the high-to-low-frequency functional relationship given by Drost and Nijman, and so compare results from different numbers of high-frequency observations per day, h = {10, 20,..., 100}, all aggregating to the same low-frequency (daily) process. We present results for low-frequency GARCH parameters (ω, α, β) equal to (1, 0.05, 0.9) and (1, 0.1, 0.4). The simulations of Figures 1-3 compare the methods by recording ratios of RMSE s among the three methods: LAD-ARCH to QML, OLS-ARCH to QML, and LAD- ARCH to OLS-ARCH respectively. In each graphic, a horizontal line at the value 1.0 shows the point at which performance is equal for two estimators. Figures 1 and 2 show the anticipated improvement in relative performance of the ARCH-regression estimators as h increases, reflecting the fact that ˆσ t 2 approximates σt 2 more closely, so that the measurement error variance in (2.2.1) is reduced. The improvement is much less clear in the second parameter set (right-hand panels) where these estimators perform relatively well even for small h. Despite the fairly large number of replications used, substantial experimental variability appears to remain, reflected in particular in variation in the relative RMSE s for ω. Figure 1 compares LAD with QML. For the first parameter set (left-hand panels) the ARCH-regression estimators come to dominate QML for some value of h in the interval depicted, while for the second set (β is farther from unity) performance of the LAD regression estimators is better throughout the interval (note the differences in vertical scales between left- and right-hand panels). A fairly similar pattern appears in Figure 2, comparing OLS with QML; the OLS regression estimates are more erratic than LAD but nonetheless on the first parameter set come to dominate LAD for a large number of intra-day observations; on the second parameter set, while variable, the OLS results show lower RMSE than QML in almost all experiments. Of course, the QML and ARCH-regression estimators are evaluated here using different information sets the LAD-ARCH and OLS-ARCH use the additional higher-frequency data, while ML 13

15 does not and the outcome of any comparison depends upon the information content of the additional ( intra-day ) data as well as on the parameters. The top panels of Figure 3 indicates that OLS estimates tend to perform better than LAD on the first parameter set, and similarly in the second. Recall, however, that we have used Normal errors in the these (top-panel) simulations, implying that OLS is not disadvantaged by any relative lack of robustness. The lower panels, with t errors in the high-frequency innovations, show better relative performance of the LAD estimates, particularly as the number of intraday observations increases. These results, while very limited, illustrate nonetheless that the ARCH-regression estimators may be competitive with QML, and potentially more precise if sufficient intra-day information is available. 5. Concluding remarks Consistent estimation of GARCH model parameters using auxiliary information on daily variance, for example from the realized variance as an estimate of quadratic variation, is feasible even where substantial noise is present in these daily variance estimates. These estimates have at least two features which make the method a useful supplement to the standard QML GARCH procedure. First, in some contexts the estimates may simply be more precise. While we have provided simulation evidence on relative performance only for a limited number of cases, it is clear both a priori and from simulation results that the performance of the LAD/OLS-ARCH estimates relative to QML improves as the precision of intra-day variance estimates improves, because error variance in the estimating model declines correspondingly. The use of methods other than realized variance to estimated daily quadratic variation may provide further gains. In any event, however, the method is robust to the presence of measurement error in the daily variance estimates. Second, it is well known in forecasting contexts that results can often be improved by combination of forecasts from different methods. The existence of an alternative to QML estimates offers the possibility of forecast combination methods that could improve upon the either method alone, even in cases where one of the alternatives produces clearly lower forecast error variance. In addition to the potential for improved robustness of estimates through the use of LAD estimation for the truncated ARCH representation, the computational methods used to produce LAD estimates can also easily be used to estimate quantiles of conditional variance other than the median, which may be of substantial value in risk-management contexts. 14

16 Figure 1 Estimated relative RMSE of estimators of GARCH(1,1) parameters RMSE(LAD-ARCH) / RMSE(QML); 25k replications (Normal), 50k replications (t 40 ) (ω, α, β) = (1, 0.05, 0.9) (left panels); (ω, α, β) = (1, 0.1, 0.4) (right panels) RMSE ratio, Normal high-frequency innovations RMSE ratio, t 40 high-frequency innovations 15

17 Figure 2 Estimated relative RMSE of estimators of GARCH(1,1) parameters RMSE(OLS-ARCH) / RMSE(QML); 25k replications (Normal), 50k replications (t 40 ) (ω, α, β) = (1, 0.05, 0.9) (left panels); (ω, α, β) = (1, 0.1, 0.4) (right panels) RMSE ratio, Normal high-frequency innovations RMSE ratio, t 40 high-frequency innovations 16

18 Figure 3 Estimated relative RMSE of estimators of GARCH(1,1) parameters RMSE(LAD-ARCH) / RMSE(OLS-ARCH); 25k replications (Normal), 50k replications (t 40 ) (ω, α, β) = (1, 0.05, 0.9) (left panels); (ω, α, β) = (1, 0.1, 0.4) (right panels) RMSE ratio, Normal high-frequency innovations RMSE ratio, t 40 high-frequency innovations 17

19 Appendix We begin by reproducing the assumptions underlying Theorem 1 of ZZG. Denote by χ q (y t I t ) the q th quantile of the conditional distribution. The assumption uses a scaling matrix V T (k). If the second moment matrix Σ(k) of X t (k) exists then V T (k) V T (k) = T Σ(k). We include the constant 1 as the first (degenerate) component of the random vector X t. Assumption 0 (from ZZG). For a sequence of (possibly random) non-singular matrices {V T (k)}, - (a)x t (k)v T (k) 1 is I t -measurable for all T, k - (b) χ q (y t I t ) = X t ( ) γ q ( ), where γ q ( ) is the coefficient vector, which depends on the quantile q; γ q ( ) = sup 1 i< γ q ( ) i <. - (c) e qt = y t X t ( )γ q ( ) is such that. (i) {e qt, X t } is a stationary ergodic sequence. (ii) the p.d.f. of e q, p e (x), exists and is continuous at x = 0. (iii) { f q(e qt ), I t } is a martingale difference (m.d.) sequence - (d) 4 sup 1 t T max Xt (k)v 1 T (k) = op (1) - (e) T max V T (k) 1 X t (k) X t (k)v T (k) 1 I k+1 = o p (1) t=1 - (f) There exists a monotonically increasing function ω(x) such that k = ω(t ) as T and sup 1 t T X R t γq R = op (T 1 2 ). Denote by ˆγ q (k) the quantile estimator of γ q (k): ˆγ q (k) = arg min γ T f q (y t X t (k)γ). t=1 4 For any matrix X, max X denotes in this paper the absolute value of the largest component of the matrix. 18

20 Proof of Theorem 2. We can now show that Assumption 0 is satisfied for the median under the conditions of Theorem 2. Recall that V T (k) = (T k) 1 2 W T (k); Assumption 0 (a) holds from measurability of every ε t with respect to I t and the fact that V T is a constant matrix; Assumption 0(b) is a consequence of Assumption 1 (b) and med e t = 0; Assumption 0(c) is a straightforward consequence of Assumption 1(c). We show that Assumption 0(f) holds for a stationary ergodic {ε 2 t }, with E(ε 2 t ) <. First note that for our model all γ i 0, and also that from (2.2.5) that the usual exponential decline follows: l=k+1 as k. Then as T, B, Pr ( Pr sup 1 t T ( T 1 2 γ l = O (exp( k)) l=k+1 T t=1 l=k+1 γ l ε 2 t l > T 1 2 B ) γ l ε 2 t l > B ) ( T 1 2 E T t=1 l=k+1 γ lεt l) 2 B ( E(ε 2 T 1 1 ) l=k+1 γ ) l 2 B ) = O (T 3 2 exp( k). Note that the first inequality replaces the sup by a sum, the second applies the Markov inequality to the sum of non-negative summands, and the third uses stationarity and the existence of moments. Choose k = w (T ) such that T 3 2 +δ γ k 0. For the ARCH expansion of a GARCH model the coefficients decline exponentially, so that k = c ln T will be sufficient for a suitable c. 5 5 For example, in the GARCH(1,1), γ l = β l 1 α, so that for c > 3 2 +δ ln β ) have ln (β k T 3 2 +δ = ( c ln β δ) ln T. and k = c ln T we 19

21 Proof of Lemma. For Assumption 0(e) define = ΣX t (k) X t (k) V T (k) V T (k) ; then the expression that needs to be evaluated is V T (k) 1 V T (k) 1. By the Ergodic Theorem each element of the k k matrix 1 T converges to zero a.s. Similarly to Berk (1974), if fourth moments of ε 2 t exist, then this convergence is at rate (T k) 1 2 Partition V T (k) V T (k) as ( and thus = O p ((T k) 1 2 k). T T E X(k) T E X(k) T E X(k) X(k) where the constant first term in X t (k) is partitioned from the remaining X t (k). Then using partitioned inverse ( V T (k) V T (k) ) 1 = ), ( ) T 1 1 E 1 X(k) E X(k) E X(k) X(k) and denoting E X(k) X(k) E X(k)E X(k) by Σ, Note that as k the element = T 1 ( 1 + E X(k) Σ 1 E X(k) E X(k) Σ 1 Σ 1 E X(k) Σ E X(k) Σ 1 E X(k) = O(k), and then the rate for this inverse matrix is O(T 1 k). It follows that V T (k) 1 V T (k) 1 = O p (T 1 2 k 2 ) ; therefore it is o p (1) if k = o(t 1 4 ). To show Assumption 0(d) write ε 2 t via its AR( ) representation corresponding to the representation of the ARMA for ε 2 t (obtained from the GARCH model) then as in ) Pollard (1991, example 2) the supremum of ε 2 t can be bounded by o p (T 1 4 (here using existence of fourth moments rather than just the second to improve the bound). Then for k = o(t 1 2 ) ). sup V T (k) 1 X T (k) = O p (T 1 2 k 1 2 T 1 4 ) = Op (T 1 4 k 1 2 ) = op (1). 20

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