Testing for a serial correlation in VaR failures through the exponential autoregressive conditional duration model

Małecka, Marta Article Testing for a serial correlation in VaR failures through the exponential autoregressive conditional duration model Statistics in Transition New Series Provided in Cooperation with: Polish Statistical Association Suggested Citation: Małecka, Marta (2021) : Testing for a serial correlation in VaR failures through the exponential autoregressive conditional duration model, Statistics in Transition New Series, ISSN 2450-0291, Exeley, New York, Vol. 22, Iss. 1, pp. 145-162, https://doi.org/10.21307/stattrans-2021-008 This Version is available at: https://hdl.handle.net/10419/236820 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ STATISTICS IN TRANSITION new series, March 2021 Vol. 22, No. 1 pp. 145–162, DOI 10.21307/stattrans-2021-008 Received – 08.04.2019; accepted – 07.09.2020 Testing for a serial correlation in VaR failures through the exponential autoregressive conditional duration model Marta Małecka1 ABSTRACT Although regulatory standards, currently developed by the Basel Committee on Banking Supervision, anticipate a shift from VaR to ES, the evaluation of risk models currently remains based on the VaR measure. Motivated by the Basel regulations, we address the issue of VaR backtesting and contribute to the debate by exploring statistical properties of the exponential autoregressive conditional duration (EACD) VaR test. We show that, under the null, the tested parameter lies at the boundary of the parameter space, which can profoundly affect the accuracy of this test. To compensate for this deficiency, a mixture of chi-square distributions is applied. The resulting accuracy improvement allows for the omission of the Monte Carlo simulations used to implement the EACD VaR test in earlier studies, which dramatically improves the computational efficiency of the procedure. We demonstrate that the EACD approach to testing VaR has the potential to enhance statistical inference in most problematic cases – for small samples and for those close to the null. Key words: VaR backtesting, exponential autoregressive conditional duration, boundary of the parameter space, test size, test power. 1. Introduction Value-at-Risk (VaR) and Expected Shortfall (ES) are two measures of market risk that dominate contemporary banking regulation. Since its original inception in business (JP Morgan, 1994) and incorporation to regulatory standards (Basel Committee on Banking Supervision, 1996), VaR has become an industry standard in market risk management. Its constantly widening range of applications include new types of risk and new markets. Despite its widespread use, however, it has several flows. It does not take account of losses beyond a designated threshold as well as lacks subadditivity, which means that diversification does not, necessarily, imply reduction of risk. Therefore, ES, which remedies this problems, seems to be emerging as a new standard. In the light of the major reform of global supervisory standards, pursued 1 Department of Statistical Methods, University of Łódź, Poland. E-mail: [email protected]. ORCID: http://orchid.org/0000-0003-4465-9811. 146 M. Małecka: Testing for a serial correlation in VaR failures… by the Basel Committee since 2012 (Basel Committee on Banking Supervision, 20122017), ES is recommended for reporting exposures to market risk. Nevertheless, ES fails to satisfy a different mathematical principle – elicitability. Although this criterion has been shown to be erroneously deemed essential to backtesting (Gneiting, 2011), the question about ES-based statistical tests remains open (Acerbi and Szekely, 2014, Chen, 2014, Fissler and Ziegel, 2015, Fissler et al., 2016). No consensus on relevant procedures has yet been reached, either in academic studies or in business practice. Therefore, evaluation of risk models still relies on VaR. In an attempt to include most extreme losses, the regulator has recommended testing VaR on two low coverage levels – 1% and 2.5%. These Basel regulations motivate academics to review, develop and enhance statistical methods of backtesting VaR. VaR backtesting procedures commonly refer to two criteria: the postulate of unconditional coverage, which treats the overall fraction of VaR violations, and the postulate of conditional coverage, which addresses their serial dependence. Perhaps of greater practical importance is detecting serial correlation of VaR failures, for their clustering may result in a series of catastrophic losses occurring one by one. This, in turn, seriously increases the risk of bankruptcy of a financial institution. The Markov test, which embeds the iid Bernoulli hypothesis within a binary first-order Markov chain and utilizes the likelihood ratio framework, has become the industry standard for testing the conditional coverage property (Christoffersen, 1998). This standard test, however, has been shown to exhibit unsatisfactory power (Lopez, 1999, Christoffersen and Pelletier, 2004, Berkowitz et al., 2011, Pajhede, 2017), which boosted the debate on other possibilities of testing VaR conditional coverage. Among other directions, like spectral tests (Berkowitz et al., 2011, Gordy and McNeil, 2018) or multi-level tests (Berkowitz, 2001, Hurlin and Tokpavi, 2007, Colletaz et al., 2013, Leccadito, Boffelli and Urga, 2014, Wied, Wei and Ziggel, 2016, Kratz et al. 2018), the duration-based approach attracted much attention in the scientific community (Christoffersen and Pelletier, 2004, Candelon et al. 2011, Pelletier and Wei, 2016). In the duration-based framework the sequence of VaR violations is transformed into the duration series. The idea behind this approach follows from the observation that the time that has passed since a VaR violation (hit) should not contain any information about further duration of the no-hit sequence. This implies the memory-free property of the duration series. Within discrete distributions, this property characterizes the geometric distribution (Berkowitz et al., 2011), while the only memory-free continuous distribution is the exponential distribution. To test VaR by means of the exponential distribution it has been proposed to nest the memory free null in the exponential autoregressive conditional model (EACD model, Engle and Russel, 1998). The EACD VaR test has been shown to compare favourably, in terms of its power, to other duration-based tests like the Weibull of the gamma test, especially for small sample sizes (Christoffersen and STATISTICS IN TRANSITION new series, March 2021 147 Pelletier, 2004, Małecka, 2018). This test, however, suffers from significant size distortions, which means that the asymptotic distribution does not guarantee the correct test level. To make up for this deficiency it has been proposed to use the Monte Carlo method to simulate the null distribution of the test statistic (Christoffersen and Pelletier, 2004). The Monte Carlo approach, however, while ensuring the correct test level, impedes practical implementation of the procedure. Our work addresses applicability of the exponential autoregressive conditional model to testing for serial correlation in VaR failure series. The goals of the paper are twofold: firstly, we seek to handle the problem of EACD test size distortions without resorting to the use of Monte Carlo simulations and secondly we investigate its power in relation to the standard VaR backtesting procedure. To avoid p-value computation through simulations, we study the asymptotic properties of the test statistic. Exploiting the fact that, in the VaR testing framework, the null value of the parameter vector lies exactly at the boundary of the parameter space, we show that the test statistic does not converge to the standard likelihood ratio (LR) limiting distribution. Using results on asymptotic LR properties under non-regular conditions (Self and Liang, 1987), we suggest p-value computation from the mixture of two chi-square distributions. We experimentally demonstrate the size improvement obtained by the proposed approach. Given improved accuracy of the EACD VaR test, we investigate its power properties. To mimic a typical VaR failure correlation scheme, we adopt a GARCH model. The comparative evaluation of the EACD test power is conducted in relation to the Markov procedure, which has, so far, won widest recognition in the industry. We indicate cases where the EACD approach allows for power gains, which gives guidance as to practical application of the examined procedures. Our study is based on earlier works by Christoffersen and Pelletier (2004) and Małecka (2018). The results of Christoffersen and Pelletier are improved by using asymptotic LR properties under non-regular conditions and implementing the EACD VaR test with the limiting mixture distribution. Since this replaces Monte Carlo simulations, our approach improves computational effectiveness of the procedure and facilitates its practical implementation. The results of Christoffersen and Pelletier are also improved by replacing the historical simulation model in the power study with the GARCH-model-based experiment. In this way we obtain the realistic setting, which mimics the volatility clustering of real financial data. In this experiment the serial correlation of VaR failures, as in reality, results from the volatility clustering of the portfolio returns. The volatility clustering is measured by the correlation coefficient of the squared returns, which, in the model we use, can be calculated analytically. Therefore, we are able to study the power of the test as a function of a controlled parameter of the return distribution, which is not attainable with the historical simulation experiment. 148 M. Małecka: Testing for a serial correlation in VaR failures… We extend the study by Małecka (2018) with respect to the contemporary international regulations in banking supervision. In addition to the typical 5% VaR, we include evaluation of test properties for two lower VaR coverage levels, indicated in the Basel rules. We discuss test accuracy in the context of the coverage level. We also extend the earlier study by depicting powers of the test as a function of volatility clustering. The shapes of the functions, compared to the power function of the standard Markov test, indicate cases where the EACD approach allows for more effective detection of incorrect risk models. The paper proceeds as follows. Section 2 introduces the notation and presents the duration-based approach to VaR backtesting in relation to the standard Markov procedure. It shows the applicability of the EACD model to testing VaR and discusses the asymptotic distribution of the test statistic. Section 3 provides the study of test properties. Firstly, it details the design of the Monte Carlo experiment, showing a way to control volatility clustering. Secondly, it addresses test accuracy and presents improvements obtained by the use of the asymptotic mixture distribution. Finally, it gives comparative evaluation of test power in relation to the Markov test. The final section summarizes and concludes. 2. Testing VaR Conditional Coverage: EACD vs. Markov-Chain Approach Let   t R be the asset or portfolio return process, for which VaR at time t , at the level of tolerance p , is defined as the p  quantile of the relevant return distribution:  󰇛 󰇜 , 1,..., . tt PR VaR p p t T  (1) Then, the VaR evaluation framework is based on the stochastic process of VaR failures:    1, 0, tt ttt R VaR p I R VaR p      , (2) whose realization is referred to as a hit sequence. The standard Christoffersen’s (1998) Markov test of VaR failure independence uses the framework of the binary Markov chain with the transition matrix: 00 01 10 11          , (3) where ij  denotes the probability of a single-step transition from state i to state , j ,󰇝0,1󰇞 ij . The null hypothesis of equal transition probabilities 001 11 : H    implies STATISTICS IN TRANSITION new series, March 2021 149 the iid Bernoulli process with probability of VaR violence 10111   . To verify the above parameter restriction it has been proposed to use the likelihood ratio statistics: 𝐿𝑅 2log  󰇛 󰇜  󰇛 󰇜 󰇛 󰇜 ~  𝜒󰇛󰇜 , (4) where 1 1 01 ˆ t tt    , 0 t is the number of non-exceptions, 1 t  the number of exceptions, 01 01 0 ˆ t t  , 11 11 1 ˆ t t   and ij t  the number of transitions form state i to state j . The construction of the Markov test implies that it only allows for detecting cases where the hit sequence follows a simple first-order Markov chain. A duration-based approach was proposed as means to capture more general forms of dependence. The duration-based tests use the transformation of the underlying 󰇝󰇞 t I process into the duration series 󰇝󰇞 i V defined as: 1, iii V tt   (5) where i t denotes the time of the -th i VaR violation. The independence of the 󰇝󰇞 t I process implies that the time that has passed since a VaR violation (hit) should not contain any information about the further duration of the no-hit sequence. This memory-free property of the duration series motivates the use of the exponential distribution. In the exponential autoregressive conditional test the memory free null is tested against the alternative of the exponential process with a conditional mean. Exploiting the fact that the serially dependent hit sequence is likely to produce an excessive number of relatively short no-hit durations and relatively long no-hit durations, the test checks the autoregression coefficient of the conditional mean of the duration. The EACD approach utilizes the regression of the form: 11 󰇛󰇜 ii i EV abV    (6) (Engle and Russel, 1998). It assumes the exponential distribution, which gives the following conditional pdf function of the duration i V : 1 1 . 1 󰇛󰇜 i i v abv EACD i i fv e abv     (7) Under the null hypothesis 0:0 Hb  the conditional distribution becomes the exponential distribution with a constant mean. 150 M. Małecka: Testing for a serial correlation in VaR failures… By using the regression of the durations on their past values this test incorporates the information about the ordering of VaR failures. This offers potential power gains over other duration based procedures like the Weibull test or the gamma test, that simply nest the exponential distribution in wider distribution families and verify relevant restrictions. The EACD-based VaR test verifies the parameter restriction through the likelihood ratio statistic, which requires computation of the loglikelihood function for the unrestricted and restricted case. Taking account of possible presence of censored durations at the beginning and at the end of the series, the loglikelihood takes the form: 1 11 1 1 2 log 󰇛 , 󰇜 log 󰇛 󰇜 󰇛1 󰇜log 󰇛 󰇜 log 󰇛 󰇜 log 󰇛 󰇜 󰇛1 󰇜log 󰇛 󰇜, N i i NN N N LV C SV C fV fV CSV C fV         (8) where i C is 1 if the duration i V is censored and 0 otherwise, S is the survival function of the variable i V , N is the number of VaR failures and  is the vector of parameters (Christoffersen and Pelletier, 2004). Assuming parameter values in the interior of the parameter space, the likelihood ratio statistic for one parameter restriction has the chi-square distribution with one degree of freedom 2 1  . However, if the tested parameter value lies at or near the boundary of the parameter space, the asymptotic convergence to the chi-square distribution ceases to hold true. This is the case with the EACD VaR test since the null hypothesis imposes the zero value of the autoregression coefficient, and, at the same time, the coefficient satisfies the nonnegativity condition. This means that the vector of ECAD model parameters 󰇟,󰇠 ab   belongs to the space 󰇛0,󰇜󰇟0,󰇜,   which, under the null, reduces to 0󰇛0, 󰇜 󰇝0󰇞  . In such a case statistical inference based on the asymptotic 2 1  may be inaccurate. To overcome the problem of potential size distortions, the EACD VaR test has been originally implemented with the use of the Monte Carlo simulated p-values. Instead, using asymptotic results on the likelihood ratio distribution under non-standard conditions (Self and Liang, 1987), we propose to compute the p-values from the 50:50 mixture of chi-square distributions, with zero and one degrees of freedom: 𝐿𝑅~𝑎𝑠0.5𝜒  0.5𝜒 . (9) Using the fact that the chi-square distribution with zero degrees of freedom reduces to the distribution with all its mass cumulated at zero, we get that the value of the test with 50% probability takes the value of 0 and with 50% probability is drawn from the chi-square distribution with one degree of freedom 2 1.  STATISTICS IN TRANSITION new series, March 2021 151 3. Monte Carlo Study of Test Properties The tests described in Section 2 verify the conditional coverage property of VaR failures referring to the Markov chain framework or, after the transformation of the hit sequence into durations, to the exponential autoregressive conditional duration model. Since the two tests exploit different approaches and make use of different variables, they are likely to differ in power properties. Moreover, as they rely on asymptotic distributions, their finite sample properties are unknown. In the present section, using a finite sample setting, we evaluate and compare the statistical properties of the two tests through the Monte Carlo study. The comparative analysis includes their size and power. We discuss practical implications of the power properties, presenting conclusions as to when to prefer which of the two tests and indicating cases when the two approaches may complement each other. The finite-sample statistical properties of the tests are evaluated for sample sizes chosen to be realistic for applications in finance: 250, 500,..., 1500. T  Such samples roughly correspond to daily data covering periods from one year to six years. The size and the power of the tests are approximated by rejection frequencies under the null and under the alternative, respectively. The size study includes significance levels 0.01, 0.05 and 0.1. For powers of the tests, only rejection rates at 0.05 significance level are reported. The size and the power estimates are computed over 10000 Monte Carlo trials. The size study examines test rejection probabilities when the risk model is correct. We refer to a test as accurate if, under the correct model, the rejection probability corresponds to the assumed level of significance (nominal test size). Therefore, the size study requires generating 󰇝󰇞 t I series under the correct model, i.e. under the assumptions of the true failure probability and independence of VaR violations. To this end we use the Bernoulli distribution with the probability of success , p equal to the assumed level of VaR tolerance. The size estimates obtained from the Bernoulli experiment (Tables 1-3) show the accuracy improvement of the EACD test gained by replacing the 2 1  distribution by the mixture of distributions 22 01 0.5 0.5 .    In the case of the 2 1  the procedure is very conservative with the true test level leaning towards zero. This size distortion indicates that practical application of this test should not be based on the asymptotic 2 1  distribution. Employment of the mixture 22 01 0.5 0.5    has the effect that the true test level approaches the nominal size. The test still tends to underreject the null, however the discrepancies between the simulated and the nominal size markedly decrease and the simulated rejection frequencies seem to converge to the desired level with lengthening the sample. The improvement in the accuracy of the test is demonstrated through the fit of the asymptotic and the empirical distribution function, based on a 1500 observation sample (Figure 1). 152 M. Małecka: Testing for a serial correlation in VaR failures… Table 1. Size estimates for Markov and EACD 1% VaR tests* Test Significance level 0.01   Series length 250 500 750 1000 1250 1500 Ind LR 0.0111 0.0122 0.0116 0.0124 0.0128 0.0128 Chi square EACD LR 0.0000 0.0000 0.0009 0.0010 0.0018 0.0019 Mixture EACD LR 0.0000 0.0008 0.0024 0.0034 0.0051 0.0047 Test Significance level 0.05   Series length 250 500 750 1000 1250 1500 Ind LR 0.0234 0.0248 0.0293 0.0269 0.0209 0.0210 Chi square EACD LR 0.0002 0.0016 0.0085 0.0110 0.0131 0.0146 Mixture EACD LR 0.0013 0.0077 0.0203 0.0248 0.0298 0.0358 Test Significance level 0.1   Series length 250 500 750 1000 1250 1500 Ind LR 0.0294 0.0418 0.0474 0.0480 0.0425 0.0453 Chi square EACD LR 0.0013 0.0077 0.0203 0.0248 0.0298 0.0358 Mixture EACD LR 0.0080 0.0280 0.0487 0.0616 0.0683 0.0796 * Chi square EACD LR denotes the cases when the EACD LR test size was estimated under the 2 1  distribution, while Mixture EACD LR – the cases when the size was estimated under the mixture distribution 22 01 0.5 0.5   . Source: Own work. Table 2. Size estimates for Markov and EACD 2.5% VaR tests* Test Significance level 0.01   Series length 250 500 750 1000 1250 1500 Ind LR 0.0265 0.0293 0.0310 0.0284 0.0274 0.0274 Chi square EACD LR 0.0002 0.0007 0.0011 0.0009 0.0019 0.0022 Mixture EACD LR 0.0008 0.0016 0.0024 0.0030 0.0042 0.0047 Test Significance level 0.05   Series length 250 500 750 1000 1250 1500 Ind LR 0.0393 0.0447 0.0443 0.0448 0.0424 0.0430 Chi square EACD LR 0.0032 0.0052 0.0079 0.0097 0.0104 0.0119 Mixture EACD LR 0.0077 0.0126 0.0189 0.0234 0.0241 0.0253 STATISTICS IN TRANSITION new series, March 2021 159 4. Conclusion The paper tackled the issue of evaluating risk models with respect to the contemporary changes in international banking regulation. In accordance with the Basel recommendations, we inquired into ways of assessing risk models based on the VaR measure. In this context we studied applicability of the EACD model. We considered the EACD test as means of testing conditional coverage property of VaR violations. We addressed the construction, asymptotic distribution as well as the finite sample size and power properties of the test. With reference to the accuracy of backtesting, we sought to handle the problem of EACD test size distortions without resorting to the use of Monte Carlo simulations. Based on the observation that the conditional coverage property implies the parameter restriction that lies at the boundary of the parameter space, we suggested p-value computation from the mixture of chi-square distributions. In this way we obtained the procedure which is both accurate and computationally effective as it replaces the originally proposed Monte Carlo method. Since its construction is based on the duration series instead of the hit sequence, it also has the potential to exhibit power against more general forms of dependence than the standard VaR test, which operates within the framework of the first order Markov chain. Via simulations we showed improvement in the test accuracy owned to replacing the asymptotic likelihood ratio distribution with a mixture of chi-square distributions. We confirmed the convergence of the true test level to the nominal size of the test. With the use of the GARCH model we designed the experiment, which enabled us to study the power of the tests against various levels of volatility clustering in return data. The estimated power functions showed that the EACD test outperforms the benchmark Markov procedure at the null and its power grows faster close to the null. Thus, this procedure may be useful to detect low-scale correlations and in this sense it may complement the standard Markov test. 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