Quantiled conditional variance, skewness, and kurtosis by Cornish-Fisher expansion

Learning the conditional variance, skewness, and kurtosis of a univariate time series is the core issue in many financial and economic applications. The classical tools to study the conditional variance are the generalized autoregressive conditional heterosecdasticity (GARCH) model and its variants. See Engle (1982), Bollerslev (1986), Francq and Zakoïan (2019), and references therein. However, except for some theoretical works on parameter estimation in Escanciano (2009) and Francq and Thieu (2019), the GARCH-type models commonly assume independent and identically distributed (i.i.d.) innovations, resulting in the constant conditional skewness and kurtosis. As argued by Samuelson (1970) and Rubinstein (1973), the higher moments like skewness and kurtosis are nonnegligible, since they are not only exemplary evidence of the non-normal returns but also relevant to the investor’s optimal decision. Along this way, a large body of literature has demonstrated the importance of conditional skewness and kurtosis in portfolio selection (Chunhachinda et al., 1997), asset pricing (Harvey and Siddique, 2000), risk management (Bali et al., 2008), return predictability (Jondeau et al., 2019), and many others. These empirical successes indicate the necessity to learn the dynamic structures of the conditional skewness and kurtosis simultaneously with the conditional variance.

Although there is a large number of studies on the conditional variance, only a few of them have taken account of the conditional skewness and kurtosis. The pioneer works towards this goal are Hansen (1994) and Harvey and Siddique (1999), followed by Jondeau and Rockinger (2003), Brooks et al. (2005), León et al. (2005), Grigoletto and Lisi (2009), and León and N̄íguez (2020). All of these works assume a peculiar conditional distribution on the innovations of GARCH-type model, where the conditional skewness or kurtosis either directly has an analogous GARCH-type dynamic structure rooting in re-scaled shocks or indirectly depends on dynamic structure of distribution parameters. See also Francq and Sucarrat (2022) and Sucarrat and Grønneberg (2022) for a different investigation of conditional skewness and kurtosis via the GARCH-type model with a time-varying probability of zero returns. However, the aforementioned parametric methods have two major shortcomings: First, they inevitably have the risk of using wrongly specified parametric models or innovation distributions; Second, they usually have unstable model estimation results in the presence of dynamic structure of skewness and kurtosis.

This paper proposes a new novel method to simultaneously learn the conditional variance, skewness, and kurtosis by the so-called quantiled conditional variance, skewness, and kurtosis, respectively. Our three quantiled conditional moments (QCMs) are formed based on the spirit of the Cornish-Fisher expansion (Cornish and Fisher, 1938), which exhibits a fundamental relationship between the conditional quantiles and CMs. By replacing the unknown conditional quantiles with their estimators at $n$ quantile levels, the QCMs (with respect to variance, skewness, and kurtosis) are simply computed at each fixed timepoint by using the ordinary least squares (OLS) estimator of a linear regression model, which stems naturally from the Cornish-Fisher expansion. Surprisingly, our way to compute the QCMs does not require any estimator of the conditional mean. The precision of the QCMs is controlled by the errors of the proposed linear regression model, which comprises two components: First, the expansion error encompasses higher-order conditional moments in the Cornish-Fisher expansion that are not taken as the regressors in the linear regression model; Second, the approximation error arises from the use of estimated conditional quantiles (ECQs). Under certain conditions on the regression model error, we show that the QCMs are consistent estimators of the corresponding CMs with the convergence rate $n^{-1/2}$. Simulation studies reveal that, when considering various scenarios of approximation errors caused by the biased ECQs from the use of contaminated or mis-specified conditional quantile models, (i) the quantiled conditional variance and skewness exhibit robust and satisfactory performance, regardless of the non-negligibility of expansion error; (ii) the quantiled conditional kurtosis has a larger dispersion for heavier-tailed data with non-negligible expansion error, which is unavoidable due to the less accuracy of CF expansion on the heavier-tailed distributions (Lee and Lin, 1992).

In the application, we study the QCMs of return series for three exchange rates. During the COVID-19 pandemic outbreak in March 2020, we find that the values of quantiled conditional variance and kurtosis increased rapidly, and the values of quantiled conditional skewness decreased sharply before March 19 or 20 in all of examined exchange rates, shedding light on the worldwide perilous financial crisis at that time. After March 19 or 20, we find that the values of quantiled conditional variance, skewness, and kurtosis exhibited totally opposite trends, so demonstrating the effectiveness of financial rescue plans issued by governments. Moreover, since the existing parametric forms of “news impact curve” (NIC) functions for the CMs are chosen in an ad-hoc way (Engle and Ng, 1993; Harvey and Siddique, 1999; León et al., 2005), we give a data-driven method to scrutinize the parametric forms of the NIC functions by using the QCMs. Our findings suggest that the parametric forms of NIC functions for conditional variance are appropriate, while those for conditional skewness and kurtosis may be unsuitable.

It is worth noting that our QCM method essentially transforms the problem of CM estimation to that of conditional quantile estimation. This brings us two major advantages over the aforementioned parametric methods, although we need do the conditional quantile estimation $n$ different times to implement the QCM method, and could face the risk of getting inaccurate estimate of conditional kurtosis for the very heavy-tailed data.

First, the QCM method can largely reduce the risk of model mis-specification, since the QCMs are simultaneously computed without any prior estimation of the conditional mean, and their consistency holds even when the specifications of conditional quantiles are mis-specified. This advantage is attractive and unexpected, since we usually have to estimate conditional mean, variance, skewness (or kurtosis) successively via some correctly specified parametric models. The reason of this advantage is that the QCM method is regression-based. Specifically, the conditional mean formally becomes one part of the intercept parameter, so it has no impact on the QCMs that are computed only from the OLS estimator of all non-intercept parameters; meanwhile, the impact of biased ECQs from the use of wrongly specified conditional quantile models can be aggregately offset by another part of the intercept parameter, ensuring the consistency of QCMs to a large extent. In a sense, without specifying any parametric forms of CMs, this important feature allows us to view the QCMs as the “observed” CMs, and consequently, many intriguing but hardly implemented empirical studies could become tractable based on the QCMs (see, e.g., our empirical studies on NICs for the CMs).

Second, the QCM method can numerically deliver more stable estimators of the CMs than the parametric methods. As shown in Jondeau and Rockinger (2003), there exists a moment issue that places a necessary nonlinear constraint on the conditional skewness and kurtosis, leading to a complex restriction on the admission region of model parameters. This restriction not only raises the computational burden of parameter estimation but also makes the estimation result unstable, so it has been rarely considered in the existing parametric methods. In contrast, the QCM method directly computes the QCMs at each fixed timepoint, and this interesting feature ensures that the nonlinear constraint on the conditional skewness and kurtosis can be simply examined using the computed QCMs at each timepoint. Particularly, if this nonlinear constraint is violated at some timepoints, it is straightforward to replace the OLS estimator with a constrained least squares estimator to propose the QCMs which then satisfy the constraint automatically.

The remaining paper is organized as follows. Section 2 proposes the QCMs based on the linear regression, and discusses the issues of conditional mean and moment constraints. Section 3 establishes the asymptotics of the QCMs. Section 4 provides the practical implementations of the QCMs. Simulation studies are given in Section 5. An application to study the QCMs for three exchange rates and their related NICs is offered in Section 6. Concluding remarks are presented in Section 7. Proofs and some additional simulation results are deferred into the supplementary materials.

This section studies the asymptotics of $\widehat{h}_{t}$, $\widehat{s}_{t}$, and $\widehat{k}_{t}$ at a fixed timepoint $t$. Let $\overset{p}{\longrightarrow}$ denote convergence in probability. To derive the consistency of $\widehat{h}_{t}$, $\widehat{s}_{t}$, and $\widehat{k}_{t}$ in (2.8), the following assumptions are needed.

We offer some remarks on the aforementioned assumptions. Assumption 3.1 is regular for linear regression models, and it holds as long as $\bm{Z}$ is fully ranked (i.e., Rank($\bm{Z}$)=4). Because $\varepsilon_{t,i}=\varepsilon_{t,i}^{\ast}+\varepsilon_{t,i}^{\circ}-\gamma_{t}$, Assumption 3.2 is equivalent to

where $C_{t,\ast}=n^{-1}\sum_{i=1}^{n}\bm{Z}_{i}\varepsilon_{t,i}^{\ast}$, $C_{t,\circ}=n^{-1}\sum_{i=1}^{n}\bm{Z}_{i}\varepsilon_{t,i}^{\circ}$, and $C_{t,\gamma}=\big{(}n^{-1}\sum_{i=1}^{n}\bm{Z}_{i}\big{)}\gamma_{t}$. By law of large numbers for dependent and heteroscedastic data sequence (Andrews, 1988), it is reasonable to assert that $C_{t,\ast}=n^{-1}\sum_{i=1}^{n}\bm{Z}_{i}E(\varepsilon_{t,i}^{\ast})+o_{p}(1)$ and $C_{t,\circ}=n^{-1}\sum_{i=1}^{n}\bm{Z}_{i}E(\varepsilon_{t,i}^{\circ})+o_{p}(1)$. Then, since $\varepsilon_{t,i}^{\bullet}=\varepsilon_{t,i}^{\ast}+\varepsilon_{t,i}^{\circ}$, condition (3.1) holds if

Condition (3.2) reveals an important fact that the role of $\gamma_{t}$ is to offset the possible non-identification effect caused by the non-zero mean of $\varepsilon_{t,i}^{\bullet}$. In other words, to achieve the identification, $\gamma_{t}$ should automatically tend to minimize the absolute difference

for large $n$. Clearly, if $d_{n,t}\approx 0$ for large $n$, Condition (3.2) holds automatically, so then Assumption 3.2 most likely holds.

Next, we study the behavior of $d_{n,t}$ in different cases. For the first case that $E(\varepsilon_{t,i}^{\bullet})\approx c_{t}$ for all $i$, we have $d_{n,t}\approx 0$ with $\gamma_{t}=c_{t}$. For the second case that $E(\varepsilon_{t,i}^{\bullet})\approx 0$ for most of $i$, we also have $d_{n,t}\approx 0$ with $\gamma_{t}=0$. For the third case that $n^{-1}\sum_{i=1}^{n}\bm{Z}_{i}\big{[}E(\varepsilon_{t,i}^{\bullet})\big{]}\approx\boldsymbol{\tau}_{t}\approx\boldsymbol{z}f_{t}$ and $n^{-1}\sum_{i=1}^{n}\bm{Z}_{i}\approx\boldsymbol{z}$ for large $n$, we again have $d_{n,t}\approx 0$ with $\gamma_{t}=f_{t}$. For other cases, we still have the chance to ensure $d_{n,t}\approx 0$, depending on the behavior of $E(\varepsilon_{t,i}^{\bullet})$ across $i$. In summary, the condition that $E(\varepsilon_{t,i}^{\bullet})\approx 0$ for all $i$ is not necessary for the validity of Assumption 3.2. This implies that the QCMs are able to have robust performances across diverse error scenarios, including situations where $E(\varepsilon_{t,i}^{\circ})$ has the large non-zero absolute values across $i$ (that is, the large biases of ECQs caused by the use of mis-specified conditional quantile models).

As expected, the condition that $d_{n,t}\approx 0$ for large $n$ may not always hold, and therefore, under certain circumstances, Assumption 3.2 may fail. For example, when the tail of $y_{t}$ becomes heavier, the impact of higher-order conditional moments in the Cornish-Fisher expansion becomes larger. In this case, $E(\varepsilon_{t,i}^{*})$ tends to have a more non-negligible exotic behavior across $i$, so that it is harder to offset the non-identification effect via $\gamma_{t}$, with the value of $d_{n,t}$ farther away from zero. Indeed, our simulation studies in Section 5 below indicate that the presence of non-negligible $\varepsilon_{t,i}^{*}$ has a larger impact on the consistency of $\widehat{k}_{t}$ than $\widehat{h}_{t}$ and $\widehat{s}_{t}$, which perform robustly in terms of the heavy-tailedness of $y_{t}$.

The following theorem establishes the consistency of $\widehat{h}_{t}$, $\widehat{s}_{t}$, and $\widehat{k}_{t}$.

Let $\overset{d}{\longrightarrow}$ denote convergence in distribution. We raise the following higher order assumption to replace Assumption 3.2:

Assumption 3.3 is regular for proving the asymptotic normality of OLS estimator (see White (2001)). The theorem below shows that $\widehat{h}_{t}$, $\widehat{s}_{t}$, and $\widehat{k}_{t}$ are $\sqrt{n}$-consistent but not asymptotically normal.

As shown before, the asymptotics of QCMs in Theorems 3.1–3.2 hold with no need to consider the specification of conditional mean and choose the specifications of conditional quantiles correctly. This important feature guarantees that the QCM method can largely reduce the risk of model mis-specification. The reason of this feature is that the QCM method is regression-based. Specifically, the conditional mean $\mu_{t}$ is absorbed into the intercept parameter $\beta_{0,t}$, so that it has no impact on the QCMs; meanwhile, the biases of ECQs from the use of wrongly specified conditional quantile models can be aggregately offset by the term $\gamma_{t}$, which is also nested in $\beta_{0,t}$.

The aforementioned feature is accompanied by two limitations. The first limitation is that we need to estimate conditional quantiles $n$ different times. Fortunately, this limitation seems mild, since the quantile estimation commonly can be computed easily by the linear programming method and the resulting estimation biases can be tolerated by the QCM method to a large extent. The second limitation of the QCMs is that when the data are very heavy-tailed, the expansion error could have a non-negligible impact to cause the identification problem, particularly for the conditional kurtosis. This limitation seems an unavoidable consequence of the Cornish-Fisher expansion, and it could not be addressed by simply increasing the order of expansion (Lee and Lin, 1992).

To compute three QCMs in (2.8), we only need to input $n$ different ECQs $\widehat{Q}_{t}(\alpha_{i})$, which can be computed in many different ways; see, for example, McNeil and Frey (2000), Kuester et al. (2006), Xiao and Koenker (2009) and the references therein for earlier works, and Koenker et al. (2017) and Zheng at al. (2018) for more recent ones. Without assuming any parametric specifications of CMs, Engle and Manganelli (2004) propose a general class of CAViaR models, which can decently specify the dynamic specification of conditional quantiles. Hence, the CAViaR models are appropriate choices for us to compute $\widehat{Q}_{t}(\alpha_{i})$. Following Engle and Manganelli (2004), we consider four CAViaR models below:

1. (1)

   Symmetric Absolute Value (SAV) model: $Q_{t}(\alpha)=\psi_{1,0}+\psi_{2,0}Q_{t-1}(\alpha)+\psi_{3,0}|y_{t-1}|$;

2. (2)

   Asymmetric Slope (AS) model: $Q_{t}(\alpha)=\psi_{1,0}+\psi_{2,0}Q_{t-1}(\alpha)+\psi_{3,0}(y_{t-1})^{+}+\psi_{4,0}(y_{t-1})^{-}$, where $(y_{t-1})^{+}=\text{max}(y_{t-1},0)$ and $(y_{t-1})^{-}=\text{min}(y_{t-1},0)$;

3. (3)

   Indirect GARCH (IG) model: $Q_{t}(\alpha)=(\psi_{1,0}+\psi_{2,0}Q^{2}_{t-1}(\alpha)+\psi_{3,0}y^{2}_{t-1})^{1/2}$;

4. (4)

   Adaptive (ADAP) model: $Q_{t}(\alpha)=Q_{t-1}(\alpha)+\psi_{1,0}\{[1+\text{exp}(N[y_{t-1}-Q_{t-1}(\alpha)])]^{-1}-\alpha\}$, where $N$ is a positive finite number.

Each CAViaR model above can be estimated via the classical quantile estimation method (Koenker and Bassett, 1978). For simplicity, we take the SAV model as an illustrating example. Let $\bm{\psi}=(\psi_{1},\psi_{2},\psi_{3})^{\prime}$ be the unknown parameter of SAV model, and $\bm{\psi}_{0}=(\psi_{1,0},\psi_{2,0},\psi_{3,0})^{\prime}$ be its true value. As in Engle and Manganelli (2004), we estimate $\bm{\psi}_{0}$ by the quantile estimator $\widehat{\bm{\psi}}_{n}\equiv\arg\min_{\bm{\psi}}\rho_{\alpha}(y_{t}-Q_{t}(\alpha,\bm{\psi}))$, where $\rho_{\alpha}(x)=x[\alpha-\mbox{I}(x<0)]$ is the check function, and $Q_{t}(\alpha,\bm{\psi})$ is defined in the same way as $Q_{t}(\alpha)$ in the SAV model with $\bm{\psi}_{0}$ replaced by $\bm{\psi}$. Once $\widehat{\bm{\psi}}_{n}$ is obtained, we then take $\widehat{Q}_{t}(\alpha)\equiv Q_{t}(\alpha,\widehat{\bm{\psi}}_{n})$ as our ECQ.

Using the above CAViaR models, we can obtain different $\widehat{Q}_{t}(\alpha)$. However, at some quantile levels $\alpha$, some of models may be inadequate to specify the dynamic structure of $Q_{t}(\alpha)$, resulting in invalid $\widehat{Q}_{t}(\alpha)$. To screen out those invalid $\widehat{Q}_{t}(\alpha)$ before computing the QCMs, we consider the in-sample dynamic quantile (DQ) test $\mbox{DQ}_{IS}(\alpha)$ in Section 6 of Engle and Manganelli (2004). This test $\mbox{DQ}_{IS}(\alpha)$ aims to detect the inadequacy of CAViaR models by examining whether $\bar{\bm{X}}_{t}(\alpha)\mbox{Hit}_{t}(\alpha)$ has mean zero, where $\mbox{Hit}_{t}(\alpha)\equiv\mbox{I}(y_{t}<Q_{t}(\alpha))-\alpha$ and $\bar{\bm{X}}_{t}(\alpha)\equiv(\mbox{Hit}_{t-1}(\alpha),...,\mbox{Hit}_{t-4}(\alpha))^{\prime}$. The testing idea of $\mbox{DQ}_{IS}(\alpha)$ relies on the fact that $\bar{\bm{X}}_{t}(\alpha)\mbox{Hit}_{t}(\alpha)$ has mean zero when the CAViaR model specifies the dynamic structure of $Q_{t}(\alpha)$ correctly. Based on the sequence $\{\widehat{Q}_{t}(\alpha)\}_{t=1}^{T}$ from a given CAViaR model, we can compute $\mbox{DQ}_{IS}(\alpha)$ and then its p-value $P(\xi>\mbox{DQ}_{IS}(\alpha))$, where $\xi\sim\chi_{4}^{2}$ (a Chi-squared distribution with $4$ degrees of freedom). If the p-value of $\mbox{DQ}_{IS}(\alpha)$ is less than $p^{*}$, the corresponding ECQs $\{\widehat{Q}_{t}(\alpha)\}$ are deemed to be invalid, so they are not included to compute the QCMs.

Below, we summarize our aforementioned procedure to compute the QCMs:

Procedure 4.1. (The steps to compute $\widehat{h}_{t}$, $\widehat{s}_{t}$, and $\widehat{k}_{t}$)

1. 1.

   Obtain $\{\widehat{Q}_{t}(\alpha)\}_{t=1}^{T}$ at any quantile level $\alpha$ in $\bm{\alpha}$ based on any CAViaR model in $\mathcal{\bm{M}}$, where $\bm{\alpha}\equiv[0.01:0.01:0.99]$ is a sequence of real numbers from $0.01$ to $0.99$ incrementing by $0.01$, and $\mathcal{\bm{M}}\equiv\{\mbox{SAV},\,\,\mbox{AS},\,\,\mbox{IG},\,\,\mbox{ADAP}\}$.

2. 2.

   Apply the DQ test $\mbox{DQ}_{IS}(\alpha)$ to each $\{\widehat{Q}_{t}(\alpha)\}_{t=1}^{T}$ from Step 1, and discard those $\{\widehat{Q}_{t}(\alpha)\}_{t=1}^{T}$ with p-values of $\mbox{DQ}_{IS}(\alpha)$ less than $p^{*}$.

3. 3.

   Group all remaining $\widehat{Q}_{t}(\alpha)$ to form a set $\bm{S}_{t}$ at each given $t$. Then, take the $i$-th entry of $\bm{S}_{t}$ to be $Y_{t,i}$ in (2.6), and use its corresponding quantile level to compute $\bm{X}_{i}$ in (2.6), where $i=1,...,n_{0}$, and $n_{0}$ is the size of $\bm{S}_{t}$.

4. 4.

   Based on $\{Y_{t,i},\bm{X}_{i}\}_{i=1}^{n_{0}}$ from Step 3, compute the OLS estimator $\widehat{\bm{\theta}}_{t}$ in (2.7) and then the three QCMs $\widehat{h}_{t}$, $\widehat{s}_{t}$, and $\widehat{k}_{t}$ in (2.8).

In Procedure 4.1, the value of $n_{0}$ decreases with that of $p^{*}$, and it achieves the upper bound $n=99\times 4$ when $p^{*}=0$ (i.e., no ECQs are discarded). Clearly, the choice of $p^{*}$ reveals a trade-off between estimation reliability and estimation efficiency in the QCM method, since a large value of $p^{*}$ enhances the reliability of $\widehat{Q}_{t}(\alpha)$, but meanwhile, it reduces the efficiency of $\widehat{\bm{\theta}}_{t}$ as the value of $n_{0}$ becomes small. So far, how to choose $p^{*}$ optimally is unclear. Our additional simulation results in the supplementary materials show that $p^{*}=0.1$ is a good choice to balance the bias and variance of the estimation error of the QCMs. Hence, we recommend to take $p^{*}=0.1$ for the practical use.

This section examines the finite sample performance of three QCMs $\widehat{h}_{t}$, $\widehat{s}_{t}$, and $\widehat{k}_{t}$. For saving the space, some additional simulation results on the selection of $p^{*}$ are reported in the supplementary materials.