A robust closed-form estimator for the GARCH(1,1) model

In this paper we extend the closed-form estimator for the generalized autoregressive conditional heteroscedastic (GARCH(1,1)) proposed by Kristensen and Linton [A closed-form estimator for the GARCH(1,1) model. Econom Theory. 2006;22:323–337] to deal with additive outliers. It has the advantage that is per se more robust that the maximum likelihood estimator (ML) often used to estimate this model, it is easy to implement and does not require the use of any numerical optimization procedure. The robustification of the closed-form estimator is done by replacing the sample autocorrelations by a robust estimator of these correlations and by estimating the volatility using robust filters. The performance of our proposal in estimating the parameters and the volatility of the GARCH(1,1) model is compared with the proposals existing in the literature via intensive Monte Carlo experiments and the results of these experiments show that our proposal outperforms the ML and quasi-maximum likelihood estimators-based procedures. Finally, we fit the robust closed-form estimator and the benchmarks to one series of financial returns and analyse their performances in estimating and forecasting the volatility and the value-at-risk.


Introduction
Return series of financial assets typically exhibit high kurtosis, higher order dependence and volatility clustering. The generalized autoregressive conditional heteroscedastic model (GARCH) is the most popular specification in parameterizing the higher order dependence and the evolution of volatility. Since its proposal in the literature by Bollerslev, [1] it has been extended in several directions. The first extension, proposed by Bollerslev, [2] allows the error of the GARCH model to follow a Student-t distribution in order to accommodate the high kurtosis of the data. However, it has been observed that the estimated residuals from this extended model still register excess kurtosis. [3,4] One possible reason for this occurrence is that some observations on returns are not fitted by a Gaussian GARCH model and not even by a t-distributed GARCH model. These observations may be influential (see [5], for a detailed definition of influential observation) since they can affect undesirably the estimation of parameters (see, e.g. [6][7][8]), the tests of conditional homoscedasticity, [9,10] the out-of-sample volatility forecasts (see for instance [11][12][13][14]) and the risk measures. [15] When this is the case, some authors denote them *Corresponding author. Email: mhveiga@est-econ.uc3m.es by outliers and distinguish between additive and innovational (or innovations) outliers. The first type is classified into two categories: additive level outliers (ALO), which exert an effect on the level of the series but not on the evolution of the underlying volatility, and additive volatility outliers (AVO), that also affect the conditional variance. [16,17] Innovational outliers contrarily to additive outliers are outliers that may have a long-run effect on modelling the volatility. Less studies have focus on the effect of innovational outliers due mainly to the fact they are transmitted by the same dynamics of the series which makes their effect less relevant (see [9,18]).
In the literature, there are two ways of dealing with outliers, either identify these observations and correct them before estimating the parameters and the volatility of the GARCH model or use robust methods. In this paper, we follow the second alternative and deal with additive outliers by robustifying the closed-form estimator for the GARCH(1,1) proposed by Kristensen and Linton. [19] This estimator has several advantages in comparison with the ML estimator often used to estimate the GARCH(1,1) model that are: it is easy to implement, it does not require the use of any numerical optimization procedure and initial starting values. The use of starting values might be a drawback since it can generate different outputs across popular packages. [20,21] Our proposal follows that of Kristensen and Linton [19] and it is based on the autorregressive moving average (ARMA) representation of the squared GARCH model and on the use of the implied autocovariance and autocorrelation functions to obtain closed-form estimators of the parameters. The difference regarding the original estimator is that we replace the sample autocorrelation function by a robust estimator of this function. Very recently, Prono [22] proposed a closed-form estimator that is based on second-order covariances and cross-order covariances that are also affected by the presence of outliers. Therefore, the robustification of Prono [22]'s closed-form estimator would require robust measures of the cross-order covariances. We use the proposal of Teräsvirta and Zhao [23] that is based on applying Huber and Ramsay's weights to the sample variance and autocovariances. In the time series literature, it is well known the importance of using robust estimators for measuring the time series dependence. The volatility is estimated by using robust filters proposed by Muler and Yohai [24] and Carnero et al. [25] Furthermore, the small sample properties of our proposal in the estimation of parameters and volatility are analysed via intensive Monte Carlo experiments and compared to those of the existing alternatives in the literature. The results of these experiments show that our proposal is robust to the presence of additive outliers and outperforms the alternatives in terms of volatility estimation independently if the outliers are either isolated or patches, large or small. Finally, we illustrate our results empirically by fitting the robust closed-form estimator and the benchmarks to one series of financial returns in order to forecast the volatility and compute the value-at-risk (VaR) forecasts.
The remainder of the paper is organized as follows. In Section 2 we robustify the closed-form estimator by Kristensen and Linton [19] and in Section 3 we propose to estimate its volatility by applying a robust filter. In Section 4, we perform intensive Monte Carlo experiments and show that in the presence of additive outliers the robust closed-form estimator provides more accurate estimates of the volatilities. Section 5 illustrates the performance of the new estimator in an empirical application, and finally, Section 6 summarizes our conclusions.

A robust closed-form estimator
In this section, first we present the closed-form estimator of Kristensen and Linton [19] that is based on the ARMA representation of the GARCH(1,1) model and on the use of the implied autocovariance and autocorrelation functions. Second, we propose to robustify this estimator by replacing the sample autocorrelation function by a robust estimator of this function. We use the robust estimator of the autocorrelations proposed by Teräsvirta and Zhao [23] which is based on applying Huber and Ramsay's weights to the sample variance and autocovariances.

An ARMA representation of the GARCH(1,1) model
Let the GARCH(1,1) process be defined as where ε t is an i.i.d process with mean zero and variance 1. Defining x t ≡ y 2 t we can write that where η t = x t − σ 2 t is a martingale difference sequence, φ = α 1 + β 1 > 0 and θ = −β 1 < 0. From Equation (3), we observe that x t is an ARMA(1,1) with parameters φ and θ , respectively. In order the ARMA process to be stationary it is imposed that φ < 1.

The robustification
Our proposal is to replace the sample estimate of ρ(·) by a robust estimator that is the weighted autocorrelation function of x t provided by Teräsvirta and Zhao [23] that is, whereγ We use as weighting function that proposed by Ramsay [26] that has the form where [23,27], for a similar value of a). Now the robust closed-form estimator for the GARCH(1,1) model are:α KLr 0 ,α KLr whereφ w =ρ w (2)/ρ w (1) and We have also tried other robust estimator by replacing the means in the sample autocorrelations and autocovariances by the corresponding medians, but we have decided not present them here due to the poorness of the results. 2 We analyse all the methods of estimation using the meansquared error (MSE) as in [19] and the biases in the estimation of the volatility as in [25].

Robust estimation of the volatility
There are in the literature some robust volatility estimators for the GARCH(1,1) model. The first is provided by Muler and Yohai [24] and is given bŷ where r c = x, |x| < c c, |x| ≥ c andα 0 ,α 1 andβ 1 are estimated using the BM estimator (see [24], for details about this estimator).
Iqbal and Mukherjee [28] propose a large class of M-estimators for estimating the parameters of an asymmetric GARCH model. We have not considered this class of estimators since it is based on an iterative algorithm that depends crucially on the starting values and according to Tinkl [29] performs similarly to that proposed by Muler and Yohai [24] in terms of asymptotic convergence.
The third is proposed by Carnero et al. [25] and replaces the r c in Equation (16) by The parameters of the GARCH model are estimated, in this case, by maximizing the Student-t log-likelihood. We denote this estimator QMLE-t.
Our proposal is to estimate the volatility usinĝ where the parameters are estimated with the robust closed-form estimator.

Monte Carlo experiments
In this section we present the results of two intensive Monte Carlo experiments. The aim of the first experiment is to evaluate the finite sample properties of the robust closed-form estimator in comparison to those of the benchmarks in estimating the parameters and the marginal variance.
The second experiment has as aim to evaluate the performance of all estimators in estimating the volatility.
Regarding the first experiments, we have generated 1000 series of sizes T = 500, 1000 and 5000 with a GARCH(1,1) with parameters α 0 = 0.1, α 1 = 0.1 and β 1 = 0.8 (see [25], for similar parameter values). The simulated series are contaminated either with one isolated ALO or multiple isolated ALOs or patches of three outliers of sizes ω AO = 0, 5, 10 and 15 standard deviations of the original simulated series. In this experiment the outliers are placed randomly in the simulated series. Notice that the conditional mean equation of the GARCH(1,1) contaminated by ALOs is defined as where ε t is defined as before, ω AO represents the magnitude (or size) of the ALO and I T (t) = 1 for t ∈ T and 0 otherwise, representing the presence of the outlier at a set of times T. Equation (2) of the conditional variance remains the same, since this type of outliers only affect the level of the series. Tables 1-3 reports the Monte Carlo means, standard deviations of the parameter and marginal variance (mv = α 0 /(1 − α 1 − β 1 )) estimates, obtained with the closed-form estimators and the several alternatives based on QMLE, and the Monte Carlo MSE. Here, QMLE is equivalent to the maximum likelihood estimator since we assume that the error is Gaussian. In order to estimate the GARCH(1,1) by QMLE we use the Econometrics toolbox of Matlab R2013b. Regarding the QMLE-based methods and the parameters α 1 and β 1 , the methods tend to overestimate these parameters and the QMLE-t provides the smallest MSEs. The closed-form estimators provide the largest biases and MSEs for the parameter estimates. However, if we increase the sample size above T = 5000, these biases almost disappear (see Note 2). On the other hand, if we focus on the estimates of the marginal variance the closed-form estimators beat always the  QMLE-based estimators for T = 5000. The exceptions are when we contaminate the series with multiple ALOs or patches of outliers of moderate and large sizes and the sample size is small. In these cases, the robust QMLE-based estimators provide estimates of the marginal variance with smaller bias. Carnero et al. [25] provide evidence that the estimator that reports the smallest biases for the parameters is not necessarily the one that leads to good volatility estimates. The authors argue that the most relevant key for the accurate estimation of the volatility is the accurate estimation of the marginal variance. As in [25], the error in the estimation of σ 2 t when there is an isolated   [23] and QMLE-t for the robust QMLE-t proposed by Carnero et al. [25] outlier at time t = τ is given by   [19] MY for the robust estimator proposed by Muler and Yohai, [24] R TZ KL for the robust closed-form estimator using the proposal of Teräsvirta and Zhao [23] and QMLE-t for the robust QMLE-t proposed by Carnero et al. [25] The expected error depends on the parameter biases, covariances, expectations of nonlinear functions of the estimator and the initial estimate of σ 2 t ,σ 2 1 , that it is often set equal to the estimate of the marginal variance. Given that the presence of outliers affects the estimation of the autocorrelation function of the squared observations, the closed-form estimators of the parameters are affected and consequently the estimate of the marginal variance (see [9], for more details). In the simulations reported in Tables 1-3, we observe that, in general, closed-form estimators tend to overestimate α 0 and α 1 and underestimate β 1 , as the ML parameter estimators. Yet, for moderate and large sample sizes, the closed-form estimators estimate better the marginal variance. So, we expected that (a) the closed-form estimators and, in particular, the robust closed-form estimator would perform the best in estimating the volatility for these sample sizes and (b) the expected error in the estimation of σ 2 t is positive at the moment that occurs the outlier. In fact, at period τ + 1, y * 2 τ is large but σ 2 τ is still not affected by the outlier and given the parameter biases, we expect that the error τ is positive.
The second experiment provides evidence on this issue by generating 1000 series of size 1000 with parameter values similar to those used in the first Monte Carlo experiment. The series are contaminated at t = 500 and we consider isolated ALOs and patches of ALOs of size ω AO = 0, 5, 10 and 15 standard deviations of the original simulated series.
In Figure 1, we plot the Monte Carlo means of the volatility biases (σ t − σ t ). From the figure we observe that all robust methods are better in estimating the volatility than the QMLE. The closed-form estimator of Kristensen and Linton [19] presents also small volatility biases than the QMLE, and the QMLE-t performs better than the procedure proposed by Muler and Yohai [24] (see [25], for similar results). Finally, the robust closed-form estimator performs the best in the presence of ALOs and patches of ALOs, except in one situation when the patch of ALOs is of size 15. In this case, the volatility estimated by the QMLE-t reacts less around the location of the patch of outliers. However, when the sample size increases till T = 5000 the robust closed-form estimator beats all the estimators in estimating the volatility (see Note 2). Therefore, our findings reinforce the conclusions of Carnero et al. [25] that the estimation methods that lead to accurate volatility estimates are those that estimate better the marginal variance.

Empirical application
In this section we analyse one daily financial time series of returns to illustrate the different volatility estimates of the volatility under the methods analysed before via simulation. The series  considered is the Nasdaq composite index. The data were collected from Yahoo Finance website (http://finance.yahoo.com) and spans the period of 2 January 1987-25 November 2014. Figure 2 depicts the return series, y t = (log p t − log p t−1 ) · 100, where p t is the value at time t of the corresponding index and Table 4 reports some summary statistics and the results of the Kiefer and Salmon [30] test, which is a formal test of normality in the context of conditional heteroscedastic series. 3 The test confirms the non-Gaussianity of the two return series.
Looking at Figure 2 and Table 4, we may appreciate several extreme observations that can be the cause of excess of kurtosis presented by the return series. Table 5 reports the estimation of the parameters and the in-sample MSE and MAE of the volatility obtained with each of the estimation methods. The method proposed by Kristensen and Linton [19] and its robustification provide quite similar parameters, although the second reports low estimated values for α 0 and α 1 and a high estimate for β 1 . Yet, the estimated marginal variance of y t is slightly smaller for the robust closed-form estimator. In order to compare the models' goodness-of-fit, we calculate the in-sample MSE and MAE of the volatility assuming the squared returns as a proxy of the true nonobservable σ 2 t . Regarding the in-sample MSE, the robustified closed-form estimator provides the smallest value of this measure followed by the closed-form estimator of Kristensen and Linton. [19] Yet, if we compare the in-sample MAEs, we observe that the closed-form estimator of Kristensen and Linton [19] provides the smallest MAE  followed closely by its robustified version. Therefore, we may conclude that the closed-form estimators provide the best fit to the data and the QMLE the worst.
On the other hand, when we consider the QMLE-based estimators, the estimates of α 0 and α 1 are larger for the basic QMLE in comparison to the robust QMLE-based estimators, while the estimate of β 1 is smaller. Figure 3 depicts the estimated volatilities and the biases regarding the volatility obtained with the robust closed-form estimator. Looking at the figure, we observe that the estimators KL and R TZ KL are those that provide smaller estimates of the volatility, being the volatility of the second slightly smaller (see panel seven of Figure 3). The volatility estimated by the basic QMLE is the largest. The other robust QMLE-based estimators provide smaller estimates of the volatility but larger than those of the closed-form estimators. The implications of these results are important either for options pricing or risk management given that estimating volatility with robust estimators leads to smaller volatility estimates and consequently lower risk. In the case of options pricing lower volatility estimates indicate smaller expected fluctuations in underlying price levels and consequently lower option premiums for puts and calls.

Forecasting performance
In this subsection we perform an out-of-sample comparison of several GARCH(1,1) estimation proposals to calculate the one-day-ahead VaR of the daily Nasdaq returns. We split the sample in an in-sample period that ranges from 2 January 1987 to 8 May 2002 and an out-of sample period that spans the period 9 May 2002 to 19 September 2006. We obtain 1100 one-day-ahead VaR forecasts as VaR α t+1|t =σ t+1|t q α , with α = 1%, q α is the 1% quantile of the standard Normal distribution andσ t+1|t is the one-dayahead volatility forecast. We use a recursive expanding window to calculate the VaRs. In order to evaluate the performance of the methods in forecasting the VaR, Table 6 reports the failure rates for the 1100 VaR forecasts, the p-value of the conditional coverage test by Christoffersen [31] 4 and the p-value of the dynamic quantile test (DQtest) by Engle and Manganelli [32]. 5 Since the calculation of the empirical failure rate defines a sequence of ones (VaR violation) and zeros (no VaR violation), we can test if the theoretical failure rate, f, is equal to 1%, i.e. H 0 : f = 1% vs. H 1 : f = 1%. According to Christoffersen, [31] testing for conditional coverage is important in the presence of higher order dynamics and this author proposed a procedure that is composed of three tests.
Looking at Table 6, we conclude that the VaR forecasts that are closer to the 1% nominal value are those obtained with the closed-form estimators. The VaR forecasts obtained with QMLEbased methods tend to overreject and therefore the null hypotheses of the Christoffersen [31] and Engle and Manganelli [32] tests are rejected. The main conclusion is that the closed-form estimators seem to perform quite well in forecasting the volatility and the VaR.

Conclusion
In the financial econometrics literature, it is well known that outliers affect the estimation of parameters and volatilities when using the traditional GARCH model and several robust alternatives have been actively investigated. All of them are based on QMLE methods and therefore are based on the use of numerical optimization procedures and starting values which might lead to different parameter and volatilities estimates.
In this paper we extend the closed-form estimator of the GARCH(1,1) proposed by Kristensen and Linton [19] for dealing with ALOs by replacing the estimators of the sample autocorrelations by robust estimators of these autocorrelations. Moreover, we also use robust filters that exist in the literature to estimate the underlying volatility.
The Monte Carlo experiments together with the empirical application show that the closedform estimators and in particular the robust closed-form estimator are more robust in terms of volatility estimation and VaR forecasting than the basic ML estimator and some based robust QMLE estimators.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
Financial support from the Spanish Ministry of Education and Science, research projects MTM2010-17323, ECO2012-32401 and Campus Iberus. The usual disclaimer applies. This work was supported by project FONDECYT 11121531. and y · t are the standardized returns. If the distribution of y · t is conditional N(0, 1), then KS S and KS K are asymptotically N(0, 1) and KS N is asymptotically χ 2 (2). 4. The first tests for the unconditional coverage (denoted LR uc ) and it is a standard likelihood ratio test (known also as [33]'s test) given by LR uc = −2 log L(p; I 1 , I 2 , . . . , I n ) L(π; I 1 , I 2 , . . . , I n ) asy ∼ χ 2 (1), where {I t } n t=1 is the indicator sequence, p is the theoretical coverage,π = n 1 /(n 0 + n 1 ) is the maximum likelihood estimate of the alternative failure rate π , n 0 is the number of zeros and n 1 is the number of ones in the sequence {I t } n t=1 . The second tests for the independence part of the conditional coverage hypothesis (denoted LR ind ) and it is also a likelihood ratio test n 00 (n 00 + n 01 ) n 01 (n 00 + n 01 ) n 10 (n 10 + n 11 ) n 11 (n 10 + n 11 ) ⎞ ⎟ ⎠ ,ˆ 2 = 1 −π 2π2 1 −π 2π2 , n ij is the number of observations with value i followed by j andπ 2 = (n 01 + n 11 )/(n 00 + n 10 + n 01 + n 11 ). Finally, the third is a joint test of coverage and independence (denoted LR cc ) given by LR cc = −2 log L(p; I 1 , I 2 , . . . , I n ) L(ˆ 1 ; I 1 , I 2 , . . . , I n ) asy ∼ χ 2 (1).
5. For computing [32]'s Dynamic Quantile test, H t (α) is defined as H t (α) = I t (α) − α where I(α) is a vector composed by ones (VaR violations) and zeros (VaR no violations). By the definition of VaR, we expect that the conditional expectation of H t (α) given the past information must be zero. This assumption can be tested with the following linear regression model: where ε t is an i.i.d process with zero mean and g(·) is a function of the past of variable z t . Consider H 0 : β 0 = β 1 = · · · = β P = γ 1 = · · · = γ K = 0, and denote = (β 0 , β 1 , . . . , β P , γ 1 , . . . , γ K ) T the vector of the P + K + 1 parameters of the model. The test statistics is given by where X denotes the covariates matrix in Equation (20). In our study, we select P = 4, K = 4 and g(z t ) = VaR t to account the influence of past exceedances up to four days (see [34], for more details).