The conditional heteroscedastic models have become a standard family of nonlinear time series models since the introduction of Engle’s (1982) autoregressive conditional heteroscedastic (ARCH) and Bollerslev’s (1986) generalized autoregressive conditional heteroscedastic (GARCH) models. Among existing conditional heteroscedastic models, the double autoregressive (AR) model recently has attracted more and more attentions; see Ling (2004, 2007), Ling and Li (2008), Zhu and Ling (2013), Li et al. (2016), Li et al. (2017), Zhu et al. (2018) and references therein. This model has the form of
where , with , and are identically and independently distributed (1986), and will reduce to Engle’s (1982) ARCH model when all ’s are zero. The double AR model has two novel properties. First, it has a larger parameter space than that of the commonly used AR model. For example, when , the double AR model may still be stationary even as (Ling, 2004)
, whereas this is impossible for AR-ARCH models. Secondly, no moment condition onis needed to derive the asymptotic normality of the Gaussian quasi-maximum likelihood estimator (QMLE) (Ling, 2007). This is in contrast to the ARMA-GARCH model, for which the finite fourth moment of the process is unavoidable in deriving the asymptotic distribution of the Gaussian QMLE (Francq and Zakoian, 2004), resulting in a much narrower parameter space (Li and Li, 2009).
In the meanwhile, conditional heteroscedastic models are considered mainly for modeling volatility and financial risk. Some quantile-based measures, such as the value-at-risk (VaR), expected shortfall and limited expected loss, are intimately related to quantile estimation, see, e.g., Artzner et al. (1999), Wu and Xiao (2002), Bassett et al. (2004) and Francq and Zakoian (2015). Therefore, it is natural to consider the quantile estimation for conditional heteroscedastic models. Many researchers have investigated the conditional quantile estimation (CQE) for the (G)ARCH models; see Koenker and Zhao (1996) for linear ARCH models, Xiao and Koenker (2009) for linear GARCH models, Lee and Noh (2013) and Zheng et al. (2018) for quadratic GARCH models. Chan and Peng (2005) considered a weighted least absolute deviation estimation for double AR models with the order of , and Zhu and Ling (2013) studied the quasi-maximum exponential likelihood estimation for a general double AR model. It is still open to perform the CQE for double AR models.
Moreover, for the double AR process generated by model (1.1), its th conditional quantile has the form of
and the coefficients of ’s are all -independent, where is the -field generated by , and is the th quantile of . However, when modeling the closing prices of S&P500 Index by the CQE, we found that the estimated coefficients of ’s depend on the quantile level significantly, while those of ’s also slightly depend on the quantile level; see Figures 3 and 4 in Section 6 for empirical evidences. Actually this phenomenon can also be found in many other stock indices. Koenker and Xiao (2006) proposed a quantile AR model by extending the common autoregression, and the corresponding coefficients are defined as functions of quantile levels. By adapting the method in Koenker and Xiao (2006), this paper attempts to introduce a new conditional heteroscedastic model, called the quantile double autoregression, to better interpret the financial time series with the above phenomenon. Particularly, this paper has three main contributions below.
A direct extension of Koenker and Xiao’s method will result in a strong constraint that the coefficients of ’s will be zero at a certain quantile level simultaneously; see Section 2. A novel transformation of is first introduced to the conditional scale structure in Section 2, and hence this drawback can be removed. Section 2 also establishes the strict stationarity and ergodicity of the newly proposed model, and the first novel property of double AR models is shown to be preserved.
The finite third moment on will be inevitable for the asymptotic normality of the CQE (Zhu and Ling, 2011), and this will make the resulting parameter space narrower; see Section 2. When modeling the infinite variance AR model, Ling (2005) proposed a self-weighted least absolute deviation method to avoid the possible moment condition. Motivated by Ling (2005), Section 3 considers a self-weighted CQE, and only a fractional moment on the process is needed. As a result, the second novel property of double AR models is preserved. Moreover, the objective function in this paper is non-differentiable and non-convex, and this causes the challenges in asymptotic derivations. Section 3 overcomes the difficulty by adopting the bracketing method in Pollard (1985), and this is another important contribution of this paper.
In addition, Section 5 conducts simulation studies to evaluate the finite-sample performance of self-weighted estimators and portmanteau tests, and Section 6 presents an empirical example to illustrate the usefulness of the new model and its inference tools. Conclusion and discussion are made in Section 7. All the technical details are relegated to the Appendix. Throughout the paper, denotes the convergence in distribution, and
denotes a sequence of random variables converging to zero in probability.
2 Quantile double autoregression
This section introduces the quantile double autoregression, a new conditional heteroscedastic model, to accommodate the phenomenon that the financial time series may have varying structures at different quantile levels.
Consider the th conditional quantile of a double AR process at (1.2). By reparameterizing it and letting the corresponding coefficients depend on the quantile level , we may naturally have
see Zhu et al. (2018). However, if there exists such that , all ’s will then disappear from the quantile structure at this level, i.e. the contributions of all ’s are zero at a certain quantile level simultaneously. Moreover, both coefficient functions and are related to the same term of .
This paper attempts to tackle the problem by looking for a way to move at (1.2) inside the square root, and it leads to a transformation of , which is an extension of the square root function with the support from to , where is the sign function. As a result, we define the quantile double AR process to be a time series with its th conditional quantile having the form of
where , ’s and ’s with are continuous functions . Let be a sequence of standard uniform random variables. As in Koenker and Xiao (2006), we have an equivalent definition below,
According to the definition of conditional quantile functions, the right hand side of (2.2) is an increasing random function with respect to , and this implies that is an increasing function. We may further assume that is an increasing function with respect to (Koenker and Xiao, 2006), and ’s with are all increasing functions. As a result, it is guaranteed that (2.2) defines a qualified conditional quantile function. Moreover, it is not necessary to assume that and ’s with are all equal to zero at a certain quantile level, and the drawback of the definition at (2.1) is then removed.
When , for all , let with , with , and , where . Then model (2.2) will reduce to (1.2), i.e. the quantile double AR model at (2.3) includes the double AR model at (1.1) as a special case. Moreover, when and ’s with are zero for all , model (2.3) will reduce to the quantile AR model in Koenker and Xiao (2006).
Let , where is a quantile double AR process generated by model (2.3). It can be verified that
is a homogeneous Markov chain on the state space, where is the class of Borel sets of , and is the Lebesgue measure on . Denote by and the distribution and density functions of , respectively. It holds that and for any .
The density function is positive and continuous on , and for some .
Since is a standard uniform random variable, the condition in the above theorem is equivalent to . Moreover, for the double AR model at (1.1), Ling (2007) derived its stationarity condition for the case with being a standard normal random variable, while it is still unknown for the other distributions of . From Theorem 1 and Remark 1, we can obtain a stationarity condition for a general double AR model below.
Suppose that has a positive and continuous density on its support, and for some . If , then there exists a strictly stationary solution to the double AR model at (1.1), and this solution is unique and geometrically ergodic with .
For the case with the order of , when is symmetrically distributed, the above condition can be simplified to . Moreover, if the normality of is further assumed, the necessary and sufficient condition for the strict stationarity is then (Ling, 2007). The comparison of the above stationarity regions is illustrated in the left panel of Figure 1. It can be seen that a larger value of in Corollary 1 leads to a higher moment of , and hence results in a narrower stationarity region. Figure 1 also gives the stationarity regions with different distributions of . As expected, the parameter space of model (1.1) becomes smaller as gets more heavy-tailed.
3 Self-weighted conditional quantile estimation
Let , , and . Denote by
the parameter vector of the quantile double AR model. From (2.2), we then can define the conditional quantile function below,
and this paper considers a self-weighted conditional quantile estimation (CQE),
When for all with probability one, the self-weighted CQE will become the common CQE. Since , it is necessary to assume to achieve the consistency, and higher order moment will be needed for the asymptotic normality; see, e.g., Gross and Steiger (1979), An and Chen (1982) and Davis et al. (1992) for the least absolute deviation estimation of infinite variance AR models. As a result, it will lead to a much narrower stationarity region; see Figure 1 for the illustration.
Denote the true parameter vector by , and it is assumed to be an interior point of the parameter space , which is a compact set. Moreover, let and be the distribution and density functions of conditional on , respectively.
is strictly stationary and ergodic with for some .
is strictly stationary and ergodic, and is nonnegative and measurable with respect to with .
With probability one, and its derivative function are uniformly bounded, and is positive on the support .
Theorem 1 in the previous section gives a sufficient condition for Assumption 2, and the proposed self-weighted CQE can handle very heavy-tailed data since only a fractional moment is needed. Assumption 4 is mainly to simplify the technical proofs. The strong consistency requires the positiveness and continuity of , while the boundedness of and is needed for the -consistency and asymptotic normality.
The nonzero restriction on and with is due to two reasons: (1) the first order derivative of does not exist at , and we need to bound the term of away from zero; and (2) this term will also be used to reduce the moment requirement on or in the technical proofs and, without it, a moment condition on will be required.
Let , and denote the first derivative of by . Define symmetric matrices
Suppose that the conditions of Theorem 2 hold. If is positive definite, then
The technical proof of the above theorem is nontrivial since the objective function of the self-weighted CQE is non-convex and non-differentiable. The main difficulty is to prove the -consistency at Theorem 3 (i), and we overcome it by adopting the bracketing method in Pollard (1985). For random weights , there are many choices satisfying Assumption 3, and the selection of optimal weights was discussed by Ling (2005), Zhu and Ling (2011) and Zhu et al. (2018). However, it becomes much more complicated for a quantile model as in (2.3). As a result, this paper suggests to simply use .
where is the fitted th conditional quantile. For bandwidth , we employ two commonly used choices, proposed by Bofinger (1975) and Hall and Sheather (1988), in the literature; see Koenker and Xiao (2006) and Li et al. (2015). The two matrices and can then be approximated by the sample averages below,
where . Consequently, a consistent estimator of the asymptotic variance matrix can be constructed.
From the self-weighted CQE , the th quantile of conditional on can be estimated by . The following corollary provides the theoretical justification for one-step ahead forecasting, and it is a direct result from Taylor expansion and Theorem 3.
Under the conditions of Theorem 3, it holds that
In practice, we may consider multiple quantile levels simultaneously, say . Although from the proposed procedure may not be monotonically increasing in , it is convenient to employ the rearrangement method in Chernozhukov et al. (2010) to solve the quantile crossing problem after the estimation.
4 Diagnostic checking for conditional quantiles
To check the adequacy of fitted conditional quantiles, we construct two portmanteau tests to detect possible misspecifications in the conditional location and scale, respectively.
Let be the conditional quantile error. In line with the estimating procedure in the previous section, it is natural to introduce the self-weighted quantile autocorrelation function (QACF), which actually is a combination of the concept of QACF in Li et al. (2015) and the idea of self-weighting in Ling (2005). Specifically, the self-weighted QACF of at lag is defined as
where , are random weights used in Section 3, and . By replacing with , a variant of can be defined as
where and . Note that if is correctly specified by model (2.2), then and for all .
Accordingly, denote by the conditional quantile residuals, where . The self-weighted residual QACFs at lag can then be defined as
where , , and . For a predetermined positive integer , let and . We first derive the asymptotic distributions of and .
Let , , and . For and 2, denote the matrices and , and the matrices and
Suppose the conditions of Theorem 3 hold and , then we have and as .
The finite second moment of is required in the above theorem, while the condition of is unavoidable if we set for all . This paper tried many other approaches, such as transforming the residuals by a bounded and strictly increasing function (Zhu et al., 2018), however, the condition of is unavoidable.
As in Section 3, we first employ the difference quotient method to estimate the the quantity of , and then approximate the matrices in and by sample averages with being replaced by . Consequently two consistent estimators, denoted by and , for the asymptotic variances in Theorem 4 can be constructed, respectively. We then can check the significance of ’s and
’s individually by establishing their confidence intervals.
From Theorem 4, the Box-Pierce type test statistics can be designed below,
where and are multivariate normal random vectors with zero mean vectors and variance matrices and , respectively. The test statistic (or ) can be used to check the significance of (or ) with jointly. To calculate the critical value or -value of (or ), we generate a sequence of, say , multivariate random vectors with the same distribution of (or ), and then use the empirical distributions to approximate the corresponding null distribution. From the simulation experiments in the next section, is more powerful in detecting the misspecification in the conditional location, while has a better performance in detecting the misspecification in the conditional scale. As in Li and Li (2008), these two test statistics should be used in conjunction to check the adequacy of the fitted conditional quantiles.
5 Simulation studies
This section conducts three simulation experiments to evaluate the finite-sample performance of the proposed self-weighted CQE in Section 3 and diagnostic tools in Section 4. For all experiments, we consider two sample sizes, and 1000, and there are 1000 replications for each sample size.
The first experiment is to evaluate the self-weighted CQE . The data generating process is
where are standard uniform random variables. We set the coefficient functions below,
where is the inverse function of , and is the distribution function of the standard normal, the Student’s or the Student’s random variable. Note that the above data generating process is equivalent to a double AR model, with having the distribution of . Two bandwidths are used in the difference quotient method,
where and are the standard normal density and distribution functions, respectively, and with being set to ; see Bofinger (1975) and Hall and Sheather (1988). The bandwidth is selected by minimizing the mean square error of Gaussian density estimation, while is obtained based on the Edgeworth expansion for studentized quantiles.
presents the bias, empirical standard deviation (ESD) and asymptotic standard deviation (ASD) ofat quantile level or , and the corresponding values of are also given. It can be seen that, as the sample size increases, the biases, ESDs and ASDs decrease, and the ESDs get closer to their corresponding ASDs. Moreover, the biases, ESDs and ASDs get smaller as the quantile level increases from to . It is obvious since there are more observations as the quantile level gets near to the center, although the true values of and also become smaller as approaches . Finally, most biases, ESDs and ASDs increase as the distribution of gets heavy-tailed, and the bandwidth outperforms in general. We next consider a more general setting of coefficient functions,
The second experiment considers the self-weighted residual QACFs, and , and the approximation of their asymptotic distributions. The number of lags is set to , and all the other settings are the same as in the first experiment. For model (5.1) with coefficient function (5.2), Tables 4 and 5 give the biases, ESDs and ASDs of and , respectively, with lags and 6. We have four findings for both residual QACFs and : (1) the biases, ESDs and ASDs decrease as the sample size increases or as the quantile level increases from to ; (2) the ESDs and ASDs are very close to each other; (3) the biases, ESDs and ASDs get smaller as becomes heavy-tailed; (4) two bandwidths and perform similarly. For the case with coefficient functions at (5.3), similar results can be found, and hence are omitted.
In the third experiment, we study the proposed portmanteau tests and . The data generating process is
with being defined as in previous experiments, while a quantile double AR model with order one is fitted to the generated sequences. As a result, the case of corresponds to the size, the case of to the misspecification in the conditional location, and the case of to the misspecification in the conditional scale. Two departure levels, 0.1 and 0.3, are considered for both and , and we calculate the critical values by generating random vectors. Table 6 gives the rejection rates of and . It can be seen that the size gets closer to the nominal rate as the sample size increases to 1000 or the quantile level increases to 0.25, and almost all powers increase as the sample size or departure level increases. Moreover, two bandwidths and perform similarly in terms of both sizes and powers. In general, is more powerful than in detecting the misspecification in the conditional location. However, is more powerful in detecting the misspecification in the conditional scale, and actually even has no power at . This indicates that and should be used in conjunction. Finally, when the data are more heavy-tailed, becomes less powerful in detecting the misspecification in the conditional location, while is more powerful in detecting the misspecification in the conditional scale. This may be due to the mixture of two effects: the worse performance of the estimation and the larger value of in the conditional scale.
In sum, both the self-weighted CQE and diagnostic tools can be used to handle the heavy-tailed time series, and the two portmanteau tests are suggested to be used together to check the adequacy of fitted conditional quantiles. For the selection of bandwidth in estimating the quantity of , we recommend since it has a more stable performance in estimation, and a comparable performance in diagnostic checking. It is used in estimating the covariance matrices in the next section.
6 An empirical example
This section analyzes the weekly closing prices of S&P500 Index from January 10, 1997 to December 30, 2016. Figure 2 gives the time plot of log returns in percentage, denoted by , and there are 1043 observations in total. The summary statistics for are listed in Table 1
, and it can be seen that the data is negatively skewed and heavy-tailed.
The autocorrelation can be found from the sample autocorrelation functions (ACFs) of both and . We then consider a double AR model,
where the Gaussian QMLE is employed, standard errors are given in the corresponding subscripts, and the order is selected by the Bayesian information criterion (BIC) with. It can be seen that, at the 5% significance level, all fitted coefficients in the conditional mean are insignificant or marginally significant, while those in the conditional variance are significant.
We apply the quantile double AR model with order three to the sequence, and the quantile levels are set to with . The estimates of for , together with their 95% confidence bands, are plotted against the quantile level in Figure 3, and those of in the fitted double AR model (6) are also given for the sake of comparison. The confidence bands of and at lags 1 and 2 are not overlapped at the quantile levels around , while those at lag 3 are significantly separated from each other around . We may conclude the -dependence of ’s, and the commonly used double AR model is limited in interpreting such type of financial time series.