Diagnostic Checking for Weibull Autoregressive Conditional Duration Models

Abstract

We derive the asymptotic distribution of residual autocorrelations for the Weibull autoregressive conditional duration (ACD) model, and this leads to a portmanteau test for the adequacy of the fitted Weibull ACD model. The finite-sample performance of this test is evaluated by simulation experiments and a real data example is also reported.

1 Introduction

First proposed by Engle and Russell [3], the autoregressive conditional duration (ACD) model has become very popular in the modeling of high-frequency financial data. ACD models are applied to describe the duration between trades for a frequently traded stock such as IBM and it provides useful information on the intraday market activity. Note that the ACD model for durations is analogous to the commonly used generalized autoregressive conditional heteroscedastic (GARCH) model [1, 2] for stock returns. Driven by the strong similarity between the ACD and GARCH models, various extensions to the original ACD model of Engle and Russell [3] have been suggested. However, despite the great variety of ACD specifications, the question of model diagnostic checking has received less attention.

The approach used by Engle and Russell [3] and widely adopted by subsequent authors to assess the adequacy of the estimated ACD model consists of applying the Ljung–Box Q-statistic [7] to the residuals from the fitted time series model and to its squared sequence. The latter case is commonly known as the McLeod–Li test [8]. As pointed out by Li and Mak [5] in the context of GARCH models, this approach is questionable, because this test statistic does not have the usual asymptotic chi-square distribution under the null hypothesis when it is applied to residuals of an estimated GARCH model. Following Li and Mak [5], Li and Yu [6] derived a portmanteau test for the goodness-of-fit of the fitted ACD model when the errors follow the exponential distribution.

In this paper, we consider a portmanteau test for checking the adequacy of the fitted ACD model when the errors have a Weibull distribution. This paper has similarities to [6] since the two papers both follow the approach by Li and Mak [5] to construct the portmanteau test statistic. Besides the difference in the distribution of the error term, the functional form of the ACD model in the present paper is more general than that of [6], because the latter only discusses the ACD model with an ARCH-like form of the conditional mean duration.

The remainder of this paper is organized as follows. Section 2 presents the portmanteau test for the Weibull ACD model estimated by the maximum likelihood method. In Sect. 3, two Monte Carlo simulations are performed to study the finite-sample performance of the diagnostic tool and an illustrative example is reported to demonstrate its usefulness.

2 A Portmanteau Test

2.1 Basic Definitions and the ML Estimation

Consider the autoregressive conditional duration (ACD) model,

$$\begin{aligned} x_i=\psi _i\varepsilon _i, \qquad \psi _i=\omega +\sum _{j=1}^p\alpha _jx_{i-j}+\sum _{j=1}^q\beta _j \psi _{i-j}, \end{aligned}$$

(1)

where $t_0<t_1<\cdots<t_n<\cdots$ are arrival times, $x_i=t_i-t_{i-1}$ is an interval, $\omega >0$, $\alpha _j\ge 0$, $\beta _j\ge 0$, and the innovations $\{\varepsilon _t\}$ are identically and independently distributed (i.i.d.) nonnegative random variables with mean one [3].

For ACD model at (1), we assume that the innovation $\varepsilon _i$ has the density of a standardized Weibull distribution,

$$ f_{\gamma }(x)=\gamma c_{\gamma } x^{\gamma -1}\exp \{-c_{\gamma }x^{\gamma }\}, \qquad x\ge 0, $$

where $c_{\gamma }=[\varGamma (1+\gamma ^{-1})]^{\gamma }$, $\varGamma (\cdot )$ is the Gamma function, and $E(\varepsilon _i)=1$. The Weibull distribution has a decreasing (increasing) hazard function if $\gamma <1$ ($\gamma >1$) and reduces to the standard exponential distribution if $\gamma = 1$. We denote this model by WACD(p, q) in this paper.

Let $\varvec{\alpha }=(\alpha _1,\ldots ,\alpha _p)^{\prime }$, $\varvec{\beta }=(\beta _1,\ldots ,\beta _q)^{\prime }$ and $\varvec{\theta }=(\omega ,\varvec{\alpha }^{\prime },\varvec{\beta }^{\prime })^{\prime }$. Denote by $\varvec{\lambda }=(\gamma ,\varvec{\theta }^{\prime })^{\prime }$ the parameter vector of the Weibull ACD model, and its true value $\varvec{\lambda }_0=(\gamma _0,\varvec{\theta }_0^{\prime })^{\prime }$ is an interior point of a compact set $\varLambda \subset \mathbb {R}^{p+q+2}$. The following assumption gives some constraints on the parameter space $\varLambda$.

Assumption 1

$\omega >0$, $\alpha _j>0$ for $1\le j\le p$, $\beta _j>0$ for $1\le j\le q$, $\sum _{j=1}^p\alpha _j+\sum _{j=1}^q\beta _j<1$, and Polynomials $\sum _{j=1}^p\alpha _jx^j$ and $1-\sum _{j=1}^q\beta _jx^j$ have no common root.

Given nonnegative observations $x_1,\ldots ,x_n$, the log-likelihood function of the Weibull ACD model is

$$\begin{aligned} L_n(\varvec{\lambda })&=\sum _{i=1}^n\left\{ \log f_{\gamma }\left( \frac{x_i}{\psi _i(\varvec{\theta })}\right) -\log \psi _i(\varvec{\theta })\right\} \\&=\sum _{i=1}^n\left\{ -\gamma \log [\psi _i(\varvec{\theta })] -c_{\gamma }\left[ \frac{x_i}{\psi _i(\varvec{\theta })}\right] ^{\gamma }\right\} +(\gamma -1)\sum _{i=1}^n\log (x_i) +n\log (\gamma \cdot c_{\gamma }). \end{aligned}$$

Note that the above functions all depend on unobservable values of $x_i$ with $i\le 0$, and some initial values are hence needed for $x_0, x_{-1},\ldots ,x_{1-p}$ and $\psi _0(\varvec{\theta }),\psi _{-1}(\varvec{\theta }),\ldots ,\psi _{1-q}(\varvec{\theta })$. We simply set them to be $\bar{x}=n^{-1}\sum _{i=1}^nx_i$, and denote the corresponding functions respectively by $\widetilde{\psi }_i(\varvec{\theta })$ and $\widetilde{L}_n(\varvec{\lambda })$. Thus, the MLE can be defined as

$$ \widetilde{\varvec{\lambda }}_n=(\widetilde{\gamma }_n,\widetilde{\varvec{\theta }}_n^{\prime })^{\prime }=\displaystyle \mathop {{{\mathrm{argmax}}}}_{\varvec{\lambda }\in \varLambda }\widetilde{L}_n(\varvec{\lambda }). $$

Let

$$ c_1(x,\gamma )=-\frac{\partial \log f_{\gamma }(x)}{\partial x}x-1=-\gamma (1-c_{\gamma }x^{\gamma }) $$

and

$$ c_2(x,\gamma )=\frac{\partial \log f_{\gamma }(x)}{\partial \gamma } =-c_{\gamma }x^{\gamma }\log (x)+\log (x)-c_{\gamma }^{\prime }x^{\gamma }+\gamma ^{-1}+c_{\gamma }^{\prime }/c_{\gamma }, $$

where $c_{\gamma }^{\prime }=\partial c_{\gamma }/\partial \gamma$. It can be verified that $E[c_1(\varepsilon _i,\gamma _0)]=0$ and $E[c_2(\varepsilon _i,\gamma _0)]=0$. Denote $\kappa _1=\text {var}[c_1(\varepsilon _i,\gamma _0)]$, $\kappa _2=\text {var}[c_2(\varepsilon _i,\gamma _0)]$, $\kappa _3=\text {cov}[c_1(\varepsilon _i,\gamma _0),c_2(\varepsilon _i,\gamma _0)]$ and

$$ \varSigma =\left( \begin{array}{cc} \kappa _2 &{} \kappa _3 E[\psi _i^{-1}(\varvec{\theta }_0){\partial \psi _i(\varvec{\theta }_0) }/{\partial \varvec{\theta }^{\prime }}]\\ \kappa _3 E[\psi _i^{-1}(\varvec{\theta }_0){\partial \psi _i(\varvec{\theta }_0) }/{\partial \varvec{\theta }}] &{}\kappa _1 E\{\psi _i^{-2}(\varvec{\theta }_0)[{\partial \psi _i(\varvec{\theta }_0) }/{\partial \varvec{\theta }}][{\partial \psi _i(\varvec{\theta }_0) }/{\partial \varvec{\theta }^{\prime }}]\} \end{array} \right) . $$

If Assumption 1 holds, then $\widetilde{\varvec{\lambda }}_n$ converges to ${\varvec{\lambda }}_0$ in almost surely sense as $n\rightarrow \infty$, and $\sqrt{n}(\widetilde{\varvec{\lambda }}_n-\varvec{\lambda }_0) \rightarrow _d N(0,\varSigma ^{-1})$ as $n\rightarrow \infty$; see Engle and Russell [3] and Francq and Zakoian [4].

Denote by $\{\widetilde{\varepsilon }_i\}$ the residual sequence from the fitted Weibull ACD model, where $\widetilde{\varepsilon }_i=x_i/\widetilde{\psi }_i(\widetilde{\varvec{\theta }}_n)$. For the quantities in the information matrix $\varSigma$, $\kappa _1$, $\kappa _2$, $\kappa _3$, $E[\psi _i^{-1}(\varvec{\theta }_0){\partial \psi _i(\varvec{\theta }_0) }/{\partial \varvec{\theta }}]$, and $E[\psi _i^{-2}(\varvec{\theta }_0)({\partial \psi _i(\varvec{\theta }_0) }/{\partial \varvec{\theta }})({\partial \psi _i(\varvec{\theta }_0) }/{\partial \varvec{\theta }^{\prime }})]$, we can estimate them respectively by

$$ \widetilde{\kappa }_1=\frac{1}{n}\sum _{i=1}^n[c_1(\widetilde{\varepsilon }_i,\widetilde{\gamma }_n)]^2, \quad \widetilde{\kappa }_2=\frac{1}{n}\sum _{i=1}^n[c_2(\widetilde{\varepsilon }_i,\widetilde{\gamma }_n)]^2, \quad \widetilde{\kappa }_3=\frac{1}{n}\sum _{i=1}^nc_1(\widetilde{\varepsilon }_i,\widetilde{\gamma }_n) c_2(\widetilde{\varepsilon }_i,\widetilde{\gamma }_n), $$

$$ \frac{1}{n}\sum _{i=1}^n \frac{1}{\widetilde{\psi }_i(\widetilde{\varvec{\theta }}_n)}\frac{\partial \widetilde{\psi }_i(\widetilde{\varvec{\theta }}_n) }{\partial \varvec{\theta }} \text {and} \quad \frac{1}{n}\sum _{i=1}^n\frac{1}{\widetilde{\psi }_i^2(\widetilde{\varvec{\theta }}_n)}\frac{\partial \widetilde{\psi }_i(\widetilde{\varvec{\theta }}_n) }{\partial \varvec{\theta }}\frac{\partial \widetilde{\psi }_i(\widetilde{\varvec{\theta }}_n) }{\partial \varvec{\theta }^{\prime }}. $$

The above estimators are all consistent, and hence a consistent estimator of the information matrix $\varSigma$. Moreover,

$$\begin{aligned} \sqrt{n}(\widetilde{\varvec{\theta }}_n-\varvec{\theta }_0) \rightarrow _d N(0,\varSigma _1^{-1}) \quad \text {as } n\rightarrow \infty , \end{aligned}$$

(2)

where

$$ \varSigma _1=\kappa _1\cdot E\left[ \frac{1}{\psi _i^2(\varvec{\theta }_0)}\frac{\partial \psi _i(\varvec{\theta }_0) }{\partial \varvec{\theta }}\frac{\partial \psi _i(\varvec{\theta }_0) }{\partial \varvec{\theta }^{\prime }}\right] -\frac{\kappa _3^2}{\kappa _2}\cdot E\left[ \frac{1}{\psi _i(\varvec{\theta }_0)}\frac{\partial \psi _i(\varvec{\theta }_0) }{\partial \varvec{\theta }}\right] E\left[ \frac{1}{\psi _i(\varvec{\theta }_0)}\frac{\partial \psi _i(\varvec{\theta }_0) }{\partial \varvec{\theta }^{\prime }}\right] . $$

2.2 The Main Result

This subsection derives asymptotic distributions of the residual autocorrelations from the estimated Weibull ACD model, and hence a portmanteau test for checking the adequacy of this model. Note that the residuals are nonnegative, and the residual autocorrelations here are also the absolute residual autocorrelations.

Without confusion, we denote $\widetilde{\psi }_i(\widetilde{\varvec{\theta }}_n)$ and $\psi _i(\varvec{\theta }_0)$ respectively by $\widetilde{\psi }_i$ and $\psi _i$ for simplicity. Consider the residual sequence $\{\widetilde{\varepsilon }_i\}$ with $\widetilde{\varepsilon }_i=x_i/\widetilde{\psi }_i$. Note that $n^{-1}\sum _{i=1}^n\widetilde{\varepsilon }_i=1+o_p(1)$ and then, for a positive integer k, the lag-k residual autocorrelation can be defined as

$$ \widetilde{r}_k=\frac{\sum _{i=k+1}^n(\widetilde{\varepsilon }_i-1)(\widetilde{\varepsilon }_{i-k}-1)}{\sum _{i=1}^n(\widetilde{\varepsilon }_i-1)^2}. $$

We next consider the asymptotic distributions of the first K residual autocorrelations, $\widetilde{R}=(\widetilde{r}_1,\ldots ,\widetilde{r}_K)^{\prime }$, where K is a predetermined positive integer.

Denote $\widetilde{\psi }_i(\widetilde{\varvec{\theta }}_n)$ and $\psi _i(\varvec{\theta }_0)$ respectively by $\widetilde{\psi }_i$ and $\psi _i$, and let $\widetilde{\varepsilon }_i=x_i/\widetilde{\psi }_i$. Let $\widetilde{C}=(\widetilde{C}_1,\ldots ,\widetilde{C}_K)^{\prime }$ and $C=(C_1,\ldots ,C_K)^{\prime }$, where

$$ \widetilde{C}_k=\frac{1}{n}\sum _{i=k+1}^n(\widetilde{\varepsilon }_i-1)(\widetilde{\varepsilon }_{i-k}-1) \quad \text {and}\quad C_k=\frac{1}{n}\sum _{i=k+1}^n(\varepsilon _i-1)(\varepsilon _{i-k}-1). $$

By the $\sqrt{n}$-consistency of $\widetilde{\varvec{\theta }}_n$ at (2) and the ergodic theorem, it follows that $n^{-1}\sum _{i=1}^n(\widetilde{\varepsilon }_i-1)^2=\sigma _{\gamma _0}^2+o_p(1),$ where $\sigma _{\gamma _0}^2=\text {var}(\varepsilon _i)$, and thus it suffices to derive the asymptotic distribution of $\widetilde{C}$.

By the Taylor expansion, it holds that

$$\begin{aligned} \widetilde{C}=C+H^{\prime }(\widetilde{\varvec{\theta }}_n-\varvec{\theta }_0)+o_p(n^{-1/2}), \end{aligned}$$

(3)

where $H=(H_1,\ldots ,H_K)$ with $H_k=-E [\psi _i^{-1}(\varepsilon _{i-k}-1)\partial \psi _i/\partial \varvec{\theta }]$. Moreover,

$$\begin{aligned} \sqrt{n}(\widetilde{\varvec{\theta }}_n-\varvec{\theta }_0)=A \varSigma ^{-1}\cdot \frac{1}{\sqrt{n}}\sum _{i=1}^n \left[ c_2(\varepsilon _i,\gamma _0),\frac{c_1(\varepsilon _i,\gamma _0)}{\psi _i} \frac{\partial \psi _i}{\partial \varvec{\theta }^{\prime }}\right] ^{\prime }+o_p(1), \end{aligned}$$

(4)

where the $c_j(\varepsilon _i,\gamma _0)$ is as defined in Sect. 2.1, and the matrix $A=(0,\mathbf {I})$ with $\mathbf {I}$ being the $(p+q+1)$-dimensional identity matrix. Note that $E[\varepsilon _ic_2(\varepsilon _i,\gamma _0)]=0$ and $E[\varepsilon _ic_1(\varepsilon _i,\gamma _0)]=1$. By (3), (4), the central limit theorem and the Cramér-Wold device, it follows that

$$ \sqrt{n}\widetilde{R} \rightarrow _d N(0,\varOmega ) \quad \text {as } n\rightarrow \infty , $$

where $\varOmega =\mathbf {I}-\sigma _{\gamma _0}^{-4}H^{\prime }\varSigma _1^{-1}H$, $\sigma _{\gamma _0}^2=\text {var}(\varepsilon _i)$, $H=(H_1,\ldots ,H_K)$ with $H_k=-E [\psi _i^{-1}(\varepsilon _{i-k}-1)\partial \psi _i/\partial \varvec{\theta }]$, and $\varSigma _1$ is as defined in Sect. 2.1.

Let $\widetilde{\sigma }_{\gamma _0}^2=n^{-1}\sum _{i=1}^n(\widetilde{\varepsilon }_i-1)^2$, $\widetilde{H}_k=-n^{-1}\sum _{i=1}^n \widetilde{\psi }_i^{-1}(\widetilde{\varepsilon }_{i-k}-1)\partial \widetilde{\psi }_i/\partial \varvec{\theta }$ and $\widetilde{H}=(\widetilde{H}_1,\ldots ,\widetilde{H}_K)$. Then we have $\widetilde{H}=H+o_p(1)$ and hence a consistent estimator of $\varOmega$ can be constructed, denoted by $\widetilde{\varOmega }$. Let $\widetilde{\varOmega }_{kk}$ be the diagonal elements of $\widetilde{\varOmega }$, for $1\le k\le K$. We therefore can check the significance of $\widetilde{r}_k$ by comparing its absolute value with $1.96\sqrt{\widetilde{\varOmega }_{kk}/n}$, where the significance level is 5 %.

To check the significance of $\widetilde{R}=(\widetilde{r}_1,\ldots ,\widetilde{r}_K)^{\prime }$ jointly, we can construct a portmanteau test statistic,

$$ Q(K)=n\widetilde{R}^{\prime }\widetilde{\varOmega }^{-1}\widetilde{R}, $$

and it will be asymptotically distributed as $\chi _K^2$, the chi-square distribution with K degrees of freedom.

3 Numerical Studies

3.1 Simulation Experiments

This subsection conducts two Monte Carlo simulation experiments to check the finite-sample performance of the proposed portmanteau test in the previous section.

The first experiment evaluates the sample approximation for the asymptotic variance of residual autocorrelations $\varOmega$, and the data generating process is

$$ x_i=\psi _i\varepsilon _i, \qquad \psi _i=0.1+\alpha x_{i-1}+\beta \psi _{i-1}, $$

where $\varepsilon _i$ follows the standardized Weibull distribution with the parameter of $\gamma$. We consider $\gamma =0.8$ and 1.2, corresponding to a heavy-tailed distribution and a light-tailed one, and $(\alpha ,\beta )^{\prime }=(0.2,0.6)^{\prime }$ and $(0.4,0.5)^{\prime }$. The sample size is set to $n=200$, 500 or 1000, and there are 1000 replications for each sample size. As shown in Table 1, the asymptotic standard deviations (ASDs) of the residual autocorrelations at lags 2, 4 and 6 are close to their corresponding empirical standard deviations (ESDs) when the sample size is as small as $n=500$.

Table 1 Empirical standard deviations (ESD) and asymptotic standard deviations (ASD) of residual autocorrelations at lags 2, 4 and 6

In the second experiment, we check the size and power of the proposed portmanteau test Q(K) using the data generating process,

$$ x_i=\psi _i\varepsilon _i, \qquad \psi _i=0.1+0.3x_{i-1}+\alpha _2 x_{i-2}+0.3 \psi _{i-1}, $$

where $\alpha _2 = 0$, 0.15 or 0.3, and $\varepsilon _i$ follows the standardized Weibull distribution with $\gamma =0.8$ or 1.2. All the other settings are preserved from the previous experiment. We fit the model of orders (1, 1) to the generated data; hence, the case with $\alpha _2 = 0$ corresponds to the size and those with $\alpha _2 > 0$ to the power. The rejection rates of test statistic Q(K) with $K=6$ are given in Table 2. For comparison, the corresponding rejection rates of the Ljung–Box statistics for the residual series and its squared process are also reported, denoted by $Q^*_1(K)$ and $Q^*_2(K)$. The critical value is the upper 5th percentile of the $\chi ^2_6$ distribution for all these tests. As shown in the table, the test Q(K) is oversized when $n=1000$, while the other two tests are largely undersized for some $\gamma$. Furthermore, we found that increasing the sample size to 9000 could result in Q(K) having sizes of 0.058 and 0.053 for $\gamma =0.8$ and 1.2, while the sizes of the other two tests do not become closer to the nominal value even for very large n. For the power simulations, it can be seen clearly that Q(K) is the most powerful test among the three and $Q^*_2(K)$ is the least powerful one. Moreover, the powers are interestingly observed to have smaller values when the generated data are heavy-tailed ($\gamma =0.8$).

Table 2 Rejection rates of the test statistics Q(K), $Q^*_1(K)$ and $Q^*_2(K)$ with $K=6$ and $\gamma =0.8$ or 1.2

Table 3 Model diagnostic checking results for the adjusted durations for IBM stock traded in first fifteen trading days of November 1990: p values for Q(K), $Q^*_1(K)$ and $Q^*_2(K)$ with $K=6, 12$ and 18, at the 5 % significance level

3.2 An Empirical Example

As an illustrative example, this subsection considers the trade durations of the US IBM stock on fifteen consecutive trading days starting from November 1, 1990. The data are truncated from a larger data set which consists of the diurnally adjusted IBM trade durations data from November 1, 1990, to January 31, 1991, adjusted and analyzed by Tsay [9, Chap. 5]. Focusing on positive durations, we have 12,532 diurnally adjusted observations.

We consider the WACD(p, q) models with $p=1$ and $q=1, 2$ or 3. The major interest is on whether the models fit the data adequately. To this end, the p values for Q(K), $Q^*_1(K)$ and $Q^*_2(K)$ with $K=6$, 12 and 18 at the 5 % significance level are reported in Table 3. It can be seen that the WACD(1, 3) model fits the data adequately according to all the test statistics. The fitted WACD(1, 1) model is clearly rejected by both Q(K) and $Q^*_1(K)$ with $K=6$, 12 and 18. For the fitted WACD(1, 2) model, both Q(K) and $Q^*_1(K)$ suggest an adequate fit of the data with $K=6$ or 12, but not with $K=18$. While for the data, Q(K) and $Q^*_1(K)$ always lead to the same conclusions, the fact that the p value for Q(K) is always smaller than that for $Q^*_1(K)$ confirms that Q(K) is more powerful than $Q^*_1(K)$. In contrast, $Q^*_2(K)$ fails to detect any inadequacy of the fitted WACD models.