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Asymmetric linear double autoregression

This paper proposes the asymmetric linear double autoregression, which jointly models the conditional mean and conditional heteroscedasticity characterized by asymmetric effects. A sufficient condition is established for the existence of a strictly stationary solution. With a quasi-maximum likelihood estimation procedure introduced, a Bayesian information criterion and its modified version are proposed for model selection. To detect asymmetric effects in the volatility, the Wald, Lagrange multiplier and quasi-likelihood ratio test statistics are put forward, and their limiting distributions are established under both null and local alternative hypotheses. Moreover, a mixed portmanteau test is constructed to check the adequacy of the fitted model. Simulation studies indicate that the proposed inference tools perform well in finite samples, and an empirical application to NASDAQ Composite Index illustrates the usefulness of the new model.


Introduction
Volatility clustering is a major feature of financial time series, and the ability to forecast volatility is of vital importance for the pricing and risk management of financial assets. To capture the time-varying volatility, many conditional heteroscedastic models are proposed, and among them, the autoregressive conditional heteroscedastic (ARCH) and the generalized autoregressive conditional heteroscedastic (GARCH) models (Engle, 1982;Bollerslev, 1986) are very successful specifications. However, empirical facts indicate that the autocorrelation and volatility dynamics usually coexist in time series; see, for example, the daily returns of NASDAQ Composite Index in Kuester et al. (2006) and the weekly or monthly returns of S&P500 index in Linton and Mammen (2005). As a result, to better capture the volatility dynamics in the presence of data autocorrelations, it is necessary to jointly model the conditional mean and volatility (Li et al., 2002). The autoregressive moving average models with GARCH errors (ARMA-GARCH) and double autoregressive (DAR) models are popular specifications for this purpose.
In financial applications, ARMA-GARCH models are commonly used to fit return series (Francq and Zakoian, 2019). Many researchers have studied the inference tools in the framework of ARMA-GARCH models; see among others, Francq and Zakoian (2004) and Zhu and Ling (2011). Meanwhile, the DAR model proposed by Ling (2007) has recently attracted growing attention. It is defined as where ω > 0, β i ≥ 0 for 1 ≤ i ≤ p, and {ε t } are independent and identically distributed (i.i.d.) innovations with zero mean and unit variance. It is notable that, the Gaussian quasi-maximum likelihood estimator (QMLE) of model (1.1) is asymptotically normal provided y t has a fractional moment (Ling, 2007), while a finite fourth moment is required for the ARMA-GARCH model (Francq and Zakoian, 2004). This important property makes model (1.1) suitable for handling heavy-tailed data in application. Many variants of DAR models are widely proposed and studied, such as the threshold DAR (Li et al., 2016), the mixture DAR (Li et al., 2017), the linear DAR (Zhu et al., 2018) and the augmented DAR (Jiang et al., 2020) models. Specifically, the linear DAR model has the form of Model (1.2) assumes the conditional standard deviation rather than the conditional variance of y t in a linear structure, which can lead to more robust inference than model (1.1); see Taylor (2008) and Zhu et al. (2018). Moreover, the asymptotic normality of Gaussian QMLE can also be established for model (1.2) without any moment restriction on y t . As a result, the linear DAR model enjoys the important property of DAR models and hence can also be used to fit heavy-tailed data. Financial time series are usually characterized by asymmetry (leverage) effects, in the sense that the volatility of financial returns tends to be higher after a decrease than an equal increase. The leverage effect was documented by many authors as a stylized fact of stock returns; see, for example, Black (1976), Rabemananjara and Zakoian (1993) and Francq and Zakoian (2013). To account for the leverage phenomenon, many variants of classical GARCH models are introduced and studied, such as the exponential GARCH (Nelson, 1991), the threshold GARCH (Zakoian, 1994) and the power GARCH (Pan et al., 2008) models. Specifically, Engle and Ng (1993) defined the news impact curve to measure how new information is incorporated into volatility estimates, and based on this curve they provided diagnostic tests to detect asymmetric effects of news on volatility. However, limited literatures investigate the leverage effect in the presence of the conditional mean structure, even rare in the framework of DAR models. To fill this gap, we propose an asymmetric linear DAR model, which can be regarded as a modification of the linear DAR model along the lines of the threshold GARCH model. Particularly, this paper has four main contributions below.
(i) By allowing for the positive and negative parts of y t having different coefficients in the conditional standard deviation, Section 2 proposes an asymmetric linear DAR model to capture the leverage effect. The strict stationarity and ergodicity of the newly proposed model is established.
(ii) Section 3 proposes the QMLE for the new model and establishes its consistency and asymptotic normality. Based on the QMLE, a Bayesian information criterion (BIC) is proposed for model selection, and a modified BIC is introduced to improve the selection performance for small and moderate samples. It is shown that both BICs enjoy the selection consistency.
(iii) To detect the asymmetry in volatility, the Wald, Lagrange multiplier (LM) and quasi-likelihood ratio (QLR) tests are constructed in Section 4. The Wald and LM test statistics are shown to have the same limiting distributions under the null and local alternative hypotheses, while the QLR test statistic converges to weighted sums of i.i.d. central and noncentral chi-squared random variables, respectively.
(iv) The portmanteau test is commonly constructed using the sample autocorrelation functions (ACFs) of residuals for pure mean models (Ljung and Box, 1978), while the ACFs of squared or absolute residuals are used for volatility models (Li and Li, 2008). Following Wong and Ling (2005), this paper proposes a mixed portmanteau test based on the ACFs of residuals and absolute residuals to detect misspecifications in the fitted model, and establishes its asymptotics for diagnosis.
In addition, Section 6 conducts simulation studies to evaluate the finite-sample performance of all proposed inference tools, and Section 7 presents an empirical example to illustrate the usefulness of the new model. Conclusion and discussion are made in Section 8. All the technical details are relegated to the Appendix. Throughout the paper, N denotes the integer, → p and → L denote the convergences in probability and in distribution, respectively, and o p (1) denotes a sequence of random variables converging to zero in probability.

Asymmetric linear double autoregression
Consider the asymmetric linear double autoregressive (DAR) model, where ω > 0, β i+ , β i− ≥ 0 for 1 ≤ i ≤ p, y + variables with mean zero and variance one. The asymmetric linear DAR model in (2.1) is an extension of the linear DAR model (Zhu et al., 2018) along the lines of the threshold GARCH model (Zakoian, 1994). Although the linear DAR model can be extended to allow for asymmetries in both the conditional mean and conditional heteroscedasticity, this paper focuses on model (2.1) to take account for the asymmetry in volatilities. The real example of stock index returns in Section 7 provides evidence for this motivation. For general distributions of η t , it is difficult to derive a necessary and sufficient condition for the strict stationarity due to the nonlinearity of model (2.1); see also Li et al. (2016) and Zhu et al. (2018). Alternatively, a sufficient condition is provided below.
Assumption 1. The density function of η t is continuous and positive everywhere on R, and E(|η t | κ ) < ∞ for some κ > 0.
Theorem 1. Under Assumption 1, if either of the following conditions holds: ; then there exists a strictly stationary solution {y t } to model (2.1), and this solution is unique and geometrically ergodic with E (|y t | κ ) < ∞.
The stationarity region in Theorem 1 depends on the distribution of η t and implies a moment condition on y t . In addition, when η t has a symmetric distribution and the asymmetric linear DAR model reduces to a linear DAR model, that is Because the stationarity region is at least three-dimensional, for illustration, we consider model (2.1) of order one and β 1+ = dβ 1− with the constant d being different positive values. Figure 1(a) indicates that model (2.1) of order one can be stationary if |α 1 | ≥ 1. As shown in Figure 1(b), a larger value of κ in Theorem 1 leads to a higher moment of y t , and hence results in a narrower stationarity region. Moreover, Figure 1(c) shows that the stationarity region gets smaller as the asymmetry in volatilities becomes greater.
Assumption 3. The density function of η t is continuous and positive everywhere on R.
A sufficient condition for Assumption 2 is given in Theorem 1. Assumption 3 is imposed for identifying the unique maximizer of E[ t (θ)] at θ 0 . Assumption 4 is required to ensure the log-likelihood function, score function and information matrix to be bounded without moment restrictions on y t ; see also Ling (2007). As a result, the model based on the QMLE can be applied to heavy-tailed data.

Model selection
This subsection considers the selection of order p for model (2.1) in practice. We first introduce the Bayesian information criterion (BIC) below to select the order p, where θ p n is the QMLE when the order is set to p, and L n ( θ p n ) is the log-likelihood evaluated at θ p n . However, model (2.1) is fitted by the Gaussian QMLE, hence the model misspecification should be considered in deriving the asymptotic expansion of the Bayesian principle, which leads to the modified BIC below BIC 2 (p) = −2L n ( θ p n ) + (3p + 1) ln where Σ p is a consistent estimator of Σ defined as in (3.2) at order p, and det( Σ p ) is its determinant. In practice, Σ p can be calculated with θ 0 replaced by θ p n and the expectation approximated by the sample average. The modified BIC in (3.4) is adapt to the generalized BIC proposed by Lv and Liu (2014), where the addtional term of their generalized BIC is introduced to account for model misspecifications. For more details of the derivation of the above BICs, please refer to Appendix A.3.1.
Let p in = arg min 1≤p≤pmax BIC i (p) for i = 1 and 2, where p max is a predetermined positive integer. Note that, when the sample size n is sufficiently large, the additional terms −(3p + 1) ln(2π) and ln(det( Σ p )) can be ignored as they are O(1). As a result, BIC 1 (p) and BIC 2 (p) are asymptotically equivalent in order selection. Simulation results in Section 6 indicate that the modified BIC in (3.4) performs better than the original BIC in (3.3) for moderate and small samples, although the two BICs have very similar performance for large samples. Hence, for moderate and small samples, we suggest to use (3.4). The following theorem verifies their selection consistency.
Theorem 3. Under the assumptions of Theorem 2, if p max ≥ p 0 , then as n → ∞, where p 0 is the true order, and p max is a predetermined positive integer.

Testing for asymmetry
This section studies the Wald, Lagrange multiplier (LM) and quasi-likelihood ratio (QLR) tests to detect the asymmetry (leverage) effect of news on volatilities.

Asymmetry Tests
For model (2.1), the asymmetry testing is of the form where i ∈ {1, · · · , p}. Let R = (0 p×(p+1) , I p , −I p ) be the p × (3p + 1) matrix, where 0 m×n is the m × n zero matrix and I p is the p × p identity matrix. Then the null hypothesis can be represented as H 0 : Rθ 0 = 0 p , where θ 0 is the true parameter vector and 0 p is a p-dimensional zero vector. Hence, the Wald, LM and QLR test statistics are defined as respectively, where θ n is the restricted QMLE under H 0 while θ n is the unrestricted QMLE, Ξ is the sample estimate of Ξ with θ 0 estimated by θ n and the expectation replaced by sample average, while Ξ (or Σ) is the sample estimate of Ξ (or Σ) with θ 0 estimated by θ n and the expectation replaced by sample average. Let χ 2 ν be the chi-squared distribution with ν degrees of freedom. Define the p × p matrix Ψ = ∆ −1/2 RΞR ∆ −1/2 with ∆ = RΣ −1 R . For j = 1, . . . , p, let e j 's be the eigenvalues of Ψ, and x j 's be the i.i.d. random variables following the χ 2 1 distribution. The following theorem gives the limiting distributions of three test statistics under H 0 .
Theorem 4. Suppose the assumptions of Theorem 2 hold. Then, under H 0 , as n → ∞, where Q = p j=1 e j x j .
Theorem 4 shows that the limiting null distribution of Q n is not the usual χ 2 p distribution but a distribution of the weighted sum of i.i.d. χ 2 1 random variables. This is because Σ = Ω in the absence of normality assumption on η t ; see also MaCurdy (1981). If η t is normally distributed, then e j = 1 for j = 1, . . . , p and Q reduces to a χ 2 p distribution, and hence Q n has the standard limiting null distribution as W n and L n . For general cases of η t , we adopt the Pearson's three-moment central chi-square approach (Pearson, 1959) to approximate p-values of the QLR test; see also Imhof (1961) and Liu et al. (2009).

Power analysis
We next discuss the efficiency of the proposed asymmetry tests through Pitman analysis.
n such that θ n ∈ Θ for sufficiently large n. Consider the local alternatives, that is, for each n, the observed time series {y p+1,n , . . . , y n,n } are generated by where the subscript n is used to emphasize the dependence of y t,n on n, η t is defined as in the model (2.1), Y t,n = (y t,n , . . . , y t−p+1,n ) , X t,n = (1, Y t+,n , −Y t−,n ) with Y t+,n = (y + t,n , . . . , y + t−p+1,n ) and Y t−,n = (y − t,n , . . . , y − t−p+1,n ) . {y t,n } satisfies the condition below.
Assumption 5. There exists a positive integer n 0 such that for n ≥ n 0 , {y t,n : t ∈ N } is strictly stationary and geometrically ergodic with E(|y t,n | κ ) < ∞ for some κ > 0.
Based on {y 1,n , . . . , y n,n }, the QMLE under H 1n can be defined as Denote P n,h as the law of y t,n . The asymptotic distribution of θ n,h under sequences of local alternatives is given below.
Theorem 5 verifies that model (4.3) is locally asymptotically normal (van der Vaart, 2000) at θ 0 . If η t follows a normal distribution, we can show Theorem 5 by Le Cam's third lemma. However, when η t is not normal, the sequences P n,0 and P n,h are not mutually contiguous; see Example 6.5 in van der Vaart (2000) for more discussions. Therefore, we show Theorem 5 in a direct way.
Theorem 6. Suppose the assumptions of Theorem 5 hold and Rθ 0 = 0 p . Then, under P n,h , as n → ∞, where δ = h R (RΞR ) −1 Rh, and x j,v 2 j 's are independent random variables following the Theorem 6 obtains the asymptotic distributions of three test statistics under the local alternatives, which shows that the Wald and LM tests have the same local asymptotic powers. If η t is normally distributed, then p j=1 e * j x j,v 2 j reduces to a χ 2 p (δ) distribution, and the QLR test is as efficient as the Wald and LM tests. Moreover, note that P ( ) holds for p = 1, then it follows that the proposed three tests are equivalent in the local asymptotic power when p = 1. For general cases of η t with p > 1, it is difficult to compare the local asymptotic power of the QLR test with the other two tests. Alternatively, the simulation study in Section 6.3 compares the local power of all three tests in finite samples, and it is found that three tests perform very similarly when the sample size is as large as 2000.

Model checking
To check adequacy of the fitted asymmetric linear DAR model, we construct a mixed pormanteau test to detect misspecifications in the conditional mean and standard deviation jointly; see Wong and Ling (2005). In the literature, diagnostic checking the conditional mean and standard deviation, can be conducted by checking the significance of sample autocorrelation functions (ACFs) of residuals and absolute residuals, respectively.
The ACFs of {η t } and {|η t |} at lag k can be defined by ρ k = cov(η t , η t−k )/ var(η t ) and γ k = cov(|η t |, |η t−k |)/ var(|η t |), respectively. If the data generating process is correctly specified by model (2.1), then {η t } and {|η t |} are i.i.d. such that ρ k = 0 and γ k = 0 hold for any k ≥ 1. For model (2.1) fitted by the QMLE, the corresponding residuals can be defined as η t = (y t − α n Y t−1 )/( β n X t−1 ), and then the residual ACF and absolute residual ACF at lag k can be calculated as Clearly, ρ k (or γ k ) is the sample version of ρ k (or γ k ). Accordingly, if the value of ρ k (or γ k ) deviates from zero significantly, it indicates that the conditional mean (or standard deviation) structure in model (2.1) is misspecified.
Denote the 2M × (2M + 3p + 1) matrix below Theorem 7. Suppose the assumptions of Theorem 2 hold. If model Theorem 7 can be used to check the significance of ρ k or γ k individually. We can construct consistent estimators of V and G using sample averages, which are denoted by G and V , respectively. Then we can approximate the asymptotic distribution in Theorem 7, and obtain confidence intervals for ρ k and γ k .
To check the first M lags jointly, we construct a portmanteau test statistic below Theorem 7 and the continuous mapping theorem imply that, Q(M ) → L χ 2 2M as n → ∞. Therefore, we reject the null hypothesis that ρ k and γ k (1 ≤ k ≤ M ) are jointly insignificant at level τ , if Q(M ) exceeds the (1 − τ )th quantile of χ 2 2M distribution.

Simulation experiments
This section presents four simulation experiments to evaluate the finite-sample performance of the proposed QMLE, model selection method, three asymmetry tests and the mixed pormanteau test.

Model estimation
The first experiment aims to examine the finite-sample performance of the quasi-maximum likelihood estimator θ n , for which the data generation process is where {η t } are standard normal, or follow standardized Student t 5 distribution with unit variance, or standardized skewed t distribution, denoted by st 5,−1.2 , with unit variance and skew parameter −1.2 (Jiang et al., 2020). The sample size is set to n = 500, 1000 or 2000, with 1000 replications for each sample size. Table 1 lists the biases, empirical standard deviations (ESDs) and asymptotic standard deviations (ASDs) of θ n for different innovation distributions and sample sizes. As the sample size increases, most of the biases, ESDs and ASDs become smaller, and the ESDs get closer to the corresponding ASDs. Moreover, when the distribution of η t gets more heavy-tailed or skewed, all ESDs and ASDs increase. This is as expected since either heavier tails or severer skewness of {η t } will lead to lower efficiency of the QMLE.

Model selection
In the second experiment, we evaluate the performance of the proposed model selection method in Section 3.2, and compare the BIC 1 and its modified version BIC 2 in finite samples. The data generating process is where the innovations {η t } are defined as in the previous experiment. Three sample sizes, n = 200, 500 and 1000, are considered, and 1000 replications are generated for each sample size. The BIC 1 in (3.3) and BIC 2 in (3.3) are employed to select the order p with p max = 5. For i = 1 or 2, the cases of underfitting, correct selection and overfitting by BIC i correspond to p in being 1, 2 and greater than 2, respectively. Table 2 reports the percentages of underfitted, correctly selected and overfitted models by the two information criteria. The performance of both information criteria gets better when the sample size increases, while that becomes slightly worse as the distribution of η t gets more heavy-tailed or more skewed. For the comparison between BIC 1 and BIC 2 , it can be seen that the modified BIC (BIC 2 ) selects the correct model in most of the replications when the sample size is as small as n = 200, while BIC 1 has comparable performance when the sample size is as large as n = 500. Overall, BIC 2 has better performance in model selection than that of BIC 1 , especially for small and moderate samples. This indicates the necessity of the modified BIC in finite samples.

Asymmetry tests
The third experiment examines the empirical size and power of the proposed asymmetry test statistics W n , L n and Q n . The data are generated from . . , 10} and n being the sample size, and the innovations {η t } are defined as in the first experiment. The null hypothesis of the asymmetry test is H 0 : k = 0, so that the case of k = 0 corresponds to the size of the tests, the cases of k = 0 correspond to the local power. Table 3 reports the empirical sizes of three tests at the significance level 5% with n = 500, 1000 and 2000. From this table, we can see that, all tests have accurate sizes when the sample size is large. In addition, W n and Q n are slightly oversized, especially when the sample size is small. We next compare the local power of all three tests in finite samples at 5% significance level. Figure 2 shows the empirical power of three tests for n = 500 and 2000. We have the following findings. First, the local powers of W n and Q n are very similar, and they are slightly higher than that of L n when the sample size is small (i.e. n = 500), especially when η t is not normal or |h| is not large. Second, the local power of three tests is close to each other when the sample size is large (i.e. n = 2000), which is consistent to the theoretical comparison in Theorem 6 for p = 1. Finally, the local power for all three tests gets smaller as the innovations become more heavy-tailed or more skewed. In addition, we also conduct simulation studies for the data generating process with p > 1, the general findings are unchanged for the empirical size and power.

Portmanteau test
In the fourth experiment, we study the proposed mixed portmanteau test Q(M ). The data are generated from where the innovations {η t } are defined as in the first experiment. We fit an asymmetric linear DAR model with p = 1 using the same method as in Section 3.1, so that the case of c = c + = c − = 0 corresponds to the size of the test, the case of c = 0 corresponds to misspecifications in the conditional mean, and the case of c + > 0 or c − > 0 corresponds to misspecifications in the conditional standard deviation. Two departure levels, 0.1 and 0.3, are considered for all c, c + and c − . Table 4 reports the rejection rates of Q(6) at 5% significance level based on 1000 replications, for sample size n = 500, 1000 and 2000. We have the following findings. First, all sizes are close to the nominal level as the sample size n increases, and most powers improve as n or the departure level increases. Second, Q(6) is more powerful in detecting the misspecification in the conditional mean (c = 0, c + = c − = 0) than that in the conditional standard deviation (c = 0, c + or c − > 0). Finally, the performance of Q(6) gets worse as the innovation distribution becomes more heavy-tailed or more skewed. This finding seems to be consistent with the result in the first experiment that, as the innovation distribution becomes more heavy-tailed or skewed, the estimation performance for all parameters tends to worsen.

An empirical example
We illustrate the proposed inference tools using the monthly closing prices of NAS-DAQ Composite Index, denoted as p t , span from February 1971 to December 2019, with 587 observations in total. The data is downloaded from the website of Yahoo Finance (https://hk.finance.yahoo.com). Let r t = 100 (ln p t − ln p t−1 ) be the log returns in percentage, and denote Y t = r t − n −1 n t=1 r t as the centered log returns in percentage. The time plot of Y t in Figure 3 suggests evident volatility clustering. Table 5 lists summary statistics of Y t , where the sample skewness −0.87 indicates possible asymmetries in the volatility, and the sample kurtosis 6.01 implies heavy-tailedness of Y t . Moreover, the PACFs of {Y t } and {|Y t |} are significant at the first few lags, which suggests that the autocorrelation coexists with conditional heteroscedasticity in {Y t }. The above findings motivate us to investigate {Y t } by our proposed model and inference tools.
Based on p max = 10, the proposed BIC 1 and BIC 2 both select p = 3. By the Quasimaximum likelihood estimation method in Section 3.1, the fitted model is where the subscripts are the standard errors of the estimated coefficients. It can be seen that the coefficients of y + t−i and y − t−i are clearly different for i = 1, 2, which suggests that there may be asymmetric effects in the conditional volatility of Y t . The Wald, LM and QLR tests in Section 4.1 are conducted for model (7.1) and all their p-values are less than 0.01, which corroborates the asymmetric effects in the volatility of Y t . To check the adequacy of the fitted model (7.1), we perform the mixed portmanteau test Q(M ) in Section 5 for M = 6, 12 and 18. The p-values of portmanteau tests are 0.28, 0.18 and 0.21, respectively, which suggests that the fitted model is adequate. In addition, as shown in Figure 4, most of the residual ACFs ρ k and γ k fall within their corresponding 95% confidence bounds at the first 18 lags.
Since Value-at-Risk (VaR) is an important risk measure for financial assets, we use the fitted model to forecast the conditional quantile of Y t , i.e. the negative VaR. To examine the forecasting performance, we conduct one-step-ahead predictions using a rolling forecasting procedure with a fixed moving window of size 400. Specifically, we fit an asymmetric linear DAR model of order three (ALDAR) for each moving window, and compute the forecast of the τ th conditional quantile of y t+1 , given by Q Yt (τ | F t−1 ) = µ t+1 + σ t+1 b τ , where µ t+1 and σ t+1 are the predicted conditional mean and standard deviation, respectively, and b τ is the τ th sample quantile of the residuals { η 1 , . . . , η t }. Then we move the window forward by one and repeat the above procedure until all data are used. Finally, we obtain 187 one-month-ahead negative VaRs for each τ . For illustration, the rolling forecasts at τ = 5% are displayed in Figure 3, which indicates that the negative VaRs change accordingly to the volatility of the data.
To compare the forecasting performance of the proposed model with other counterparts, we also perform the rolling forecasting procedure using a linear DAR model of order three (LDAR) and an AR(3) model with the threshold GARCH(1,1) errors (AR-TGARCH). Note that the threshold GARCH(1,1) model is widely used to depict asymmetric effects in volatilities, while the LDAR model ignores the asymmetric effect. For comparison, the LDAR and AR-TGARCH models are also fitted by the QMLE, and their VaR forecasts are computed in the same way as for the ALDAR model. To evaluate the performance of three models in forecasting VaRs, we calculate the empirical coverage rate (ECR), and perform VaR backtests for the VaR forecasts at τ = 5%, 10%, 90% and 95%. Specifically, ECR is calculated as the proportion of observations that fall below the corresponding conditional quantile forecast for the last 187 data points. Two VaR backtests, i.e. the likelihood ratio test for correct conditional coverage (CC) in Christoffersen (1998) and the dynamic quantile (DQ) test in Engle and Manganelli (2004) are employed. Denote the hit by H t = I(Y t < Q Yt (τ | F t−1 )). The null hypothesis of CC test is that, conditional on F t−1 , {H t } are i.i.d. Bernoulli random variables with success probability being τ . For the DQ test, following Engle and Manganelli (2004), we regress H t on regressors including a constant, four lagged hits H t−i , i = 1, 2, 3, 4, and the contemporaneous VaR forecast. The null hypothesis of DQ test is that all regression coefficients are zero and the intercept equals to the quantile level τ . Table 6 reports ECRs and p-values of two VaR backtests for the one-step-ahead forecasts by the fitted ALDAR, LDAR and AR-TGARCH models at the lower and upper 5% and 10% conditional quantiles, i.e. 5% and 10% VaRs for long and short positions. In terms of backtests, the proposed ALDAR model and the LDAR model perform satisfactorily at all the four quantile levels with p-values not less than 0.1, while the AR-TGARCH model performs poorly at the upper 5% and 10% conditional quantiles. With respect to ECRs, it can be seen that those of the ALDAR model are closest to the nominal quantile level τ . Therefore, we conclude that the proposed ALDAR model dominates the other two competitors in forecasting VaRs for NASDAQ Composite Index.

Conclusion and discussion
This paper proposes the asymmetric linear double AR model which takes into account asymmetric effects for conditional heteroscedastic time series in the presence of a conditional mean structure. The strict stationarity of the new model is derived, and inference tools, including a Gaussian QMLE and a mixed portmanteau test, are constructed without any moment condition on the data. Based on the QMLE, a BIC and its modified version are proposed for order selection, and simulation results suggest that the modified BIC performs better in small and moderate samples. The Wald, Lagrange multiplier and quasi-likelihood ratio test statistics are constructed to detect asymmetric effects, and it is shown that, the Wald and Lagrange multiplier tests are asymptotically equivalent in the size and power, while the asymptotics of the quasi-likelihood ratio test become non-standard. The usefulness of the new model is confirmed by our empirical evidence.
Our model can be extended to allow for asymmetric effects in both the conditional mean and the standard deviation, then the proposed asymmetry tests could adapt to detect the asymmetry from the conditional location and scale separately or jointly. Moreover, since financial time series can be heavy-tailed such that E(η 4 t ) = ∞, it is also of interest to consider more robust estimation methods than Gaussian QMLE, for example, the quasi-maximum exponential likelihood estimation of Zhu and Ling (2011).

Appendix: Technical proofs
This appendix includes technical details for Theorems 1-7. To show Theorems 2 and 5, Lemmas 1-6 are introduced with proofs. Throughout the appendix, for a vector x = (x 1 , . . . , x p ) , the n-norm is defined as x n = ( p i=1 |x i | n ) 1/n ; for a matrix or column vector A, we define A = tr(AA ), where tr(·) denotes the trace of a square matrix.
Consider the test function g(x) = 1 + |x 1 | κ + p−1 i=1 r i |x i+1 | κ as for Case (i). Define ρ as in (A.5). Note that g(x) = O(1 + ||x|| κ κ ), this together with (A.8), implies that It can be shown that (A.2) and (A.3) hold, i.e. Tweedie's drift criterion is verified. Similar to the proof of Case (i), we can show that, there exists a strictly stationary solution {y t } to model (2.1), and this solution is unique and geometrically ergodic with E (|y t | κ ) < ∞. This accomplishes the second part.

A.2 Proof of Theorem 2
To show Theorem 2, we introduce the following lemmas.
Lemma 2. If Assumptions 2 and 4 hold, then Proof. These follow from Lemma 1 and Theorem 3.1 in Ling and McAleer (2003).
Lemma 3. If Assumptions 2 and 4 hold, then E t (θ) has a unique maximum at θ 0 .
Proof of Theorem 2. By Lemma 2(i) and Lemma 3, we have established all the conditions for consistency in Theorem 4.1.1 in Amemiya (1985), and hence θ n → p θ 0 as n → ∞.

A.3 Technical details for model selection
In this section, notations Θ p , θ p , θ p n and Σ p are employed to emphasize their dependence on the order p. We first derive the proposed BICs in Section 3.2, and then establish their consistency in order selection.

A.3.1 Derivation of BICs
Denote Y as the observed data and m(Y ) as its marginal distribution. Let π(p) (p ∈ {1, 2, . . . , p max }) be a discrete prior over the order set {1, 2, . . . , p max }, and g(θ p | p) be a prior on θ p given the order p. Moreover, denote L(θ p | Y ) as the likelihood of Y under the model with order p, then we have ln L(θ p | Y ) = L n (θ p ). By Bayes's Theorem, the joint posterior of p and θ p can be written as Then the posterior probability of p is given by This together with (A.17) and ln L( θ p n | Y ) = L n ( θ p n ), implies that −2 ln P (p | Y ) ≈ −2L n ( θ p n ) + (3p + 1) ln n − p 2π + ln det Σ p = BIC 2 (p).
As n → ∞, ignoring the O(1) terms in the above approximation, we obtain that Motivated by the above approximations, the BICs are defined as in (3.3) and (3.4).

A.3.2 Proof of Theorem 3
Proof. Denote the true order of model (2.1) as p 0 . Note that BIC 2 (p) = BIC 1 (p) − (3p + 1) ln(2π) + ln(det( Σ p )), and −(3p + 1) ln(2π) + ln(det( Σ p )) is bounded as n → ∞. It suffices to show that, for any p = p 0 , We first consider the case with p > p 0 , i.e. the model is overfitted. Note that the model with order p corresponds to a bigger model, and it holds that . Similar to the proof of Theorem 3, we can show that ν = O p (1). By Taylor's expansion and Slutsky's theorem, it can be shown that Similarly, it can be verified that L n ( θ p n ) − L n (θ p 0 ) = O p (1). As a result, Hence, we have as n → ∞. Therefore, (A.18) holds for p > p 0 .
We next consider the case with p < p 0 , i.e. the model is underfitted. Let θ p 0 = arg max θ∈Θ p E[ t (θ p )]. Similar to the proof of Theorem 3 and (A.19), we can verify that √ n( θ p n − θ p 0 ) = O p (1) and Since the model with order p corresponds to a smaller model, we have E t (θ p 0 0 ) ≥ E t (θ p 0 ) + for some positive constant . By ergodic theorem, we have n −1 L n (θ p 0 ) = E t (θ p 0 ) + o p (1). Thus it holds that Therefore, we have This together with (3p + 1) ln(n − p) − (3p 0 + 1) ln(n − p 0 ) = O(ln n), implies that We show the null distributions of the Wald, LM and QLR test statistics, respectively.
Then under H 0 , by Lemma 4, it follows that where Ξ = Σ −1 ΩΣ −1 . Then by Slutsky's theorem, under H 0 , it holds that This completes the first part.
(ii) Lagrange Multiplier (LM) test Let k ∈ R p be a vector. Under H 0 , the lagrangian can be formulated as Denote ( θ n , k) = arg inf θ,k L(θ, k), where θ n is the restricted QMLE under H 0 , and k is the lagrangian multiplier. Taking the first derivatives of L(θ, k) with respect to θ and k at ( θ n , k) respectively, we have Under H 0 , by Taylor's expansion and (A.20), it holds that This together with (A.16) and (A.21), implies that Under H 0 , note that R θ n = 0 p by (A.21) and Rθ 0 = 0 p , then by Theorem 2, we have Recall that ∆ = RΣ −1 R . Multiplying both sides of (A.22) by R, we have This together with (A.22) and Slutsky's theorem, implies that, under H 0 , This completes the second part.
e j x j as n → ∞, where Ψ = ∆ −1/2 RΞR ∆ −1/2 , e j is the j-th eigenvalue of Ψ, and x j 's are the i.i.d. random variables following the χ 2 1 distribution. This completes the third part.
To prove Theorem 5, the following two lemmas are required.
(ii) By Taylor's expansion, we have where θ * is between θ n,h and θ n . Moreover, by the Martingale Central Limit Theorem and the Cramér-Wold device, together with Lemma 6(ii), we can show that, as n → ∞, Recall that θ n = θ 0 + h/ √ n. Then by (i) and Lemma 6(i), we have The proof of this theorem is accomplished.
N (0, 1) t 5 st 5,−1.2 n Bias ESD ASD Bias ESD ASD Bias ESD ASD α 500 -0.024 0.515 0.520  Table 3: Empirical sizes of three tests W n , L n and Q n at 5% significance level, where the innovations follow the standard normal, standardized Student t 5 or standardized skewed Student st 5,−1.2 distribution.
N (0, 1) t 5 st 5,−1.2 n W n L n Q n W n L n Q n W n L n Q n 500 0.