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The realized empirical distribution function of stochastic variance with application to goodness-of-fit testing
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Stochastic volatility is a central concept in financial economics with numerous implications for capital allocation, asset- and derivatives pricing, or risk management. It is therefore essential to select a model for stochastic volatility that is able to reproduce the main features observed in data from financial markets, such as the implied volatility surface, as closely as possible. There is, however, no consensus about which model is “best,” as the extensive list of papers on stochastic volatility suggests \citep*[the literature is too big to enumerate, but a partial and necessarily incomplete set of proposed specifications can be found in][and the references therein]{andersen-benzoni-lund:02a, barndorff-nielsen-shephard:02a, chernov-gallant-ghysels-tauchen:03a, christoffersen-jacobs-mimouni:10a, comte-renault:98a, gatheral-jaisson-rosenbaum:18a, heston:93a, hull-white:87a}.
The goodness-of-fit of a stochastic volatility model can be assessed in several ways. If the parameters are estimated with a method of moment-based estimator, a standard diagnostic check is available via a test of overidentifying restrictions \citep*[see also, e.g.,][]{gallant-hsieh-tauchen:97a}. This approach can be highly inefficient, however, as it depends heavily on the selection of moment conditions and the associated estimation of the weighting matrix, which may produce significant size distortions in finite samples \citep*{andersen-sorensen:96a}.
Another idea is to compare the model-implied distribution of volatility with a nonparametric empirical measure of it \citep*[e.g.,][]{ait-sahalia:96a}. In a continuous-time setting, for instance, a model is often formulated via a stochastic differential equation for the variance process (or a transformation thereof, such as the square root or natural logarithm). This, in turn, implies that the marginal distribution of spot volatility (at least in the stationary case) belongs to a particular class of distributions. So if the underlying volatility was observed, a specification test based on the distance---with respect to a suitable norm---between the empirical distribution function (EDF) and the marginal distribution imposed by the model would be feasible. However, as spot volatility is not directly observed, this approach is not immediately applicable.
The recent access to financial high-frequency data has alleviated these concerns, as it enables the computation of an error-free measure of realized volatility, allowing for a more direct evaluation of the goodness-of-fit of stochastic volatility models. Inspired by the above, a common approach is to compare (conditional) moments of the integrated variance of a parametric model with a nonparametric estimator hereof \citep*[see, e.g.,][]{bollerslev-zhou:02a, corradi-distaso:06a, dette-podolskij:08a, dette-podolskij-vetter:06a, todorov:09a, todorov-tauchen-grynkiv:11a, vetter-dette:12a}. \citet*{zu:15a} proposes a test based on a de-convolution kernel density estimator of the distribution of the integrated variance, \citet*{lin-lee-guo:13a, lin-lee-guo:16a} resort to a characteristic function approach, while \citet*{bull:17a} proposes a wavelet-based test. As pointed out by todorov-tauchen:12a, however, the mapping between the probability distribution of the spot and integrated variance is in general not one-to-one. Thus, goodness-of-fit tests based on the latter suffer from lack of power against some alternatives, due to the smoothing entailed by integrating spot volatility over a discrete time interval.
In this article, we therefore construct goodness-of-fit tests for stochastic volatility models that are based on a realized EDF (REDF hereafter) of spot volatility. We build on \citet*{li-todorov-tauchen:13a, li-todorov-tauchen:16a}, who show that inference about the EDF can be based on the volatility occupation measure \citep*[e.g.,][]{geman-horowitz:80a}. In their framework, the latent spot variance at any point in time is retrieved by employing a \citet*{foster-nelson:96a} rolling window-type estimator computed on small non-overlapping blocks of high-frequency data. In contrast, we employ overlapping blocks, which is theoretically more efficient, but it also induces further technical problems in the part of the proofs, where a uniform bound on the estimation errors of spot variance is derived.
In addition, we allow the asset price to be recorded at the tick-by-tick level, so that the instantaneous variance can be recovered as accurately as possible. This is important if the whole distribution, including the tails, is of interest and not only measures of central tendency. At that sampling frequency, however, the data are distorted by microstructure noise \citep*[see, e.g.][]{hansen-lunde:06b}, and this complicates the issue. The derived problem of estimating the integrated---or cumulative---variance in the presence of noise has received a lot of attention in the literature \citep*[see, e.g.][]{barndorff-nielsen-hansen-lunde-shephard:08a, jacod-li-mykland-podolskij-vetter:09a, zhang-mykland-ait-sahalia:05a}. We here adapt the pre-averaging approach of \citet*{jacod-li-mykland-podolskij-vetter:09a} and \citet*{podolskij-vetter:09a, podolskij-vetter:09b} to design a consistent spot volatility estimator from contaminated high-frequency data \citep*[][suggest an alternative procedure]{zu-boswijk:14a}. The REDF is then constructed from the recovered sample path of spot variance.
We show that under weak regularity conditions the REDF is a consistent noise- and jump-robust estimator of the EDF over any fixed time interval in the infill limit, thus extending \citet*{li-todorov-tauchen:13a, li-todorov-tauchen:16a} to a setting with overlapping blocks and microstructure noise. This is merely a stepping stone in our context, however, as the main goal is to do goodness-of-fit tests for stochastic volatility models. In the second part, we therefore prove a functional CLT for the REDF in a long-span asymptotic framework by letting time increase to infinity. This amounts to establish convergence of what \citet*{vaart-wellner:07a} call an empirical process indexed by an estimated function, but here time is continuous and the observations are not i.i.d.
The theoretical results are refined into two new goodness-of-fit tests for measuring the discrepancy between the marginal density implied by a candidate stochastic volatility model and the nonparametric gauge at the EDF represented by the REDF. The first is reminiscent to a Kolmogorov-Smirnov statistic for an observed process, while the second is based on a weighted $L^{2}$ norm. Both are trivial to compute once the REDF has been constructed, but there are a number of subtleties that render the asymptotic distribution hard to evaluate, making it difficult to set critical regions and determine p-values. This is to some extent related to the complexities encountered in a classical setup, when the parameters of the model under the null are estimated. We resolve the issue via a parametric bootstrap \citep*[based on, e.g.,][and described in Appendix (ref)]{bull:17a}, which leads to fast evaluation of the $t$-statistic.
The paper is organized as follows. In Section (ref), we cover the setting, assumptions, and introduce the EDF of volatility. In Section (ref), the properties of the REDF are analyzed. The asymptotic theory for our goodness-of-fit tests is also developed here. Section (ref) is devoted to a Monte Carlo study of the REDF and an assessment of the proposed goodness-of-fit tests. An empirical application is conducted in Section 5, where we recover the REDF from stock market high-frequency data and test goodness-of-fit of several marginal distributions that are induced by mainstream stochastic volatility models. We conclude in Section 6. The proofs appear in Appendix (ref).
Let $X = (X_{t})_{t\geq0}$ denote the efficient log-price process of a financial asset, which is defined on a filtered probability space $\big( \Omega,\mathcal{F}, ( \mathcal{F}_{t} )_{t \geq 0}, \mathbb{P} \big)$. As consistent with no-arbitrage \citep*[e.g.,][]{delbaen-schachermayer:94a}, we assume $X$ is an It\^{o} semimartingale:
where $X_{0}$ is $\mathcal{F}_{0}$-measurable, $(b_{t})_{t \geq 0}$ is a locally bounded, predictable drift, $( \sigma_{t})_{t \geq 0}$ is an adapted, c\`{a}dl\`{a}g volatility, $(W_{t})_{t \geq 0}$ a Brownian motion, and $(J_{t})_{t \geq 0}$ is a jump process:
where $\mu$ is a Poisson random measure on $(\mathbb{R}_+,\mathbb{R})$, $\nu(d(s,z)) = ds \otimes \lambda(dz)$ is a compensator, $\lambda$ is a $\sigma$-finite measure on $\mathbb{R}$, and $\delta: \Omega \times \mathbb{R}_{+} \times \mathbb{R} \rightarrow \mathbb{R}$ is a predictable function. Furthermore, we assume that there exists a function $\Gamma$ such that $| \delta(\omega, s,z)|\leq \Gamma(z)$ for all $(\omega, s, z)$ and
for some $r \in [0,2]$.
The constant $r$ in (ref) controls the activity level of the price jumps and is an upper bound on the Blumenthal-Getoor index. When $r=0$, the jumps are of finite activity, whereas $r>0$ corresponds to the infinite-activity setting.
We collect assumptions about the drift, volatility, and jumps of $X$ below.
This places some moment conditions on $b_{t}$, $\sigma_{t}$, and the large jumps. While it is general and encompasses many volatility models applied in practice, it may feel a bit restrictive. The assumption can be relaxed, but this entails that a constant $\iota > 0$ (restricting the rate at which $T \rightarrow \infty$ and $\Delta_{n} \rightarrow 0$, cf. Section (ref)) has to be bounded from below, while it can be made arbitrarily small within the above setup.
Assumption 1(i) implies $\sigma$ is continuous. This means it can be approximated by a locally constant process over short time intervals. The condition covers both “rough” and long-memory volatility processes \citep*[e.g.,][]{comte-renault:98a, gatheral-jaisson-rosenbaum:18a} with Hurst exponent different from one-half.
The conditions in Assumption 1(ii), which says $\sigma$ is an It\^{o} semimartingale, are common in the literature. It allows volatility to exhibit fairly unrestricted jump dynamics (at the expense of excluding fractional Brownian motion as in 1(i)) and also enables a leverage effect, as it does not restrict the correlation structure between the increments of $X$ and $\sigma$.
Next, we define the EDF of volatility, which in the continuous-time setting is given by the random function:
where $V_{t} \equiv \sigma_{t}^{2}$.\footnote{In probability theory, $F_{T}(x) = \int_{0}^{T} 1_{ \{ Y_{t} \leq x \}} dt$ is called the occupation---or local---time of the stochastic process $Y$ \citep*[e.g.,][]{geman-horowitz:80a}. This convention was adopted by \citet*{li-todorov-tauchen:13a, li-todorov-tauchen:16a}, who applied it to high-frequency volatility estimation. We normalize $F_{T}$ by $T$ here, as we are heading toward a setting with stationary volatility and an asymptotic theory with $T \rightarrow \infty$.}
This condition is fulfilled by most stochastic volatility models. It does not impose the existence of a density nor does it require $\sigma$ itself to be continuous.
The right-hand side of (ref) can be written $T^{-1} \int_{0}^{T} g(V_{t}) dt$, where $g(v) = 1_{\{ v \leq x\}}$.\footnote{Much work in the high-frequency literature studies integrals of the form $\int_{0}^{T} g( V_{t}) dt$, where $g$ is smooth \citep*[e.g.,][]{barndorff-nielsen-graversen-jacod-podolskij-shephard:06a, jacod-rosenbaum:13a}. Here, in contrast, $g$ is discontinuous, which makes the theory a lot more inaccessible.} Then, if $F_{T}$ is absolutely continuous with respect to the Lebesgue measure---which is stronger than Assumption (ref)---and $g$ is a bounded or non-negative Borel function:
where $f_{T}$ is the density of $F_T$ with respect to the Lebesgue measure. The EDF therefore encapsulates all the information about $(V_{t})_{t \geq 0}$ available in $[0,T]$ and can be viewed as a pathwise version of the distribution function of volatility.
The main difficulty is that $F_{T}$ is latent, because $\sigma$ is not observable. The aim of this paper is therefore to construct a consistent estimator of $F_{T}$, while making the above minimal assumptions about $X$.
Of course, in an ideal world with no markets frictions and continuous trading---i.e. if the entire trajectory of $X$ is available---we can recover the volatility process perfectly and also the exact time and size of jumps in $X$ and $\sigma$ \citep*[in constrast to the drift, which cannot be consistently estimated in finite time, not even if it is constant, see, e.g.,][]{merton:80a, foster-nelson:96a}. This setup is not realistic, however. In practice, we operate with discrete high-frequency data, which we assume are available at times $i \Delta_{n}$, for $i = 0, 1, \ldots, n$, where $\Delta_{n}$ is the time gap between consecutive observations and $n = \lfloor T / \Delta_{n} \rfloor$ is the sample size (equidistant sampling is enforced here, but it can be weakened). In a near-ideal world, the database then constitutes a high-frequency record of $X$, i.e. $(X_{i \Delta_{n}})_{i=0}^{n}$, and the estimator proposed by \citet*{li-todorov-tauchen:13a, li-todorov-tauchen:16a} can be applied without further ado.
The microstructure of financial markets adds measurement error, however, which implies that we do not observe $X$ directly (e.g., due to bid-ask bounce or price discreteness). Instead, we record a contaminated log-price $Z$, which we assume is related to $X$ as follows:
This setting is comparable to Assumption (K) in \citet*{jacod-li-mykland-podolskij-vetter:09a} or Assumption (4) in the Appendix of \citet*{li-todorov-tauchen:17a} \citep*[see, e.g.,][for a related approach]{bibinger-winkelmann:18a}. The noise is permitted to be both heteroscedastic and serially dependent via $(\omega_{t})_{t \geq 0}$, so the model is broad enough to replicate realistic structures in the microstructure of financial markets \citep*[e.g.,][]{diebold-strasser:13a, hansen-lunde:06b}. The moment condition is not standard, however, but it is merely made for technical convenience. In the proofs, we only require moments of $\varepsilon_{i\Delta_{n}}$ up to some order (for instance, in Lemma (ref) the 4th moment should exist).
Later, we impose a stronger smoothness assumption on $( \omega_{t})_{t \geq 0}$:
We define the noisy log-return:
To deal with the noise, we pre-average the return series \citep*[see, e.g.,][]{jacod-li-mykland-podolskij-vetter:09a, podolskij-vetter:09a, podolskij-vetter:09b}:
where $k_{n}$ is a positive integer, while $g$ is a kernel.
Any $g:[0,1] \mapsto \mathbb{R}$ with $g$ continuous and piecewise $C^{1}$ with Lipschitz derivative $g'$, such that $g(0) = g(1) = 0$ and $\int_{0}^{1} g(x)^{2}dx > 0$ is permitted. We follow the default choice in the literature by setting $g(x) = \min(x,1-x)$.
The selection of $k_{n}$ entails a trade-off. The intuition is that while pre-averaging lessens the noise, it also smooths out the underlying volatility of $X$. This can render it hard to construct estimates of spot volatility that fit the tails of the distribution if a too wide pre-averaging window is applied. As shown by \citet*{jacod-li-mykland-podolskij-vetter:09a}, an optimal $k_{n}$ is achieved via:
where $\theta > 0$ is a tuning parameter, which controls the balance struck between the above forces in small samples. As consistent with prior work, we base our analysis on $\theta = 1/3$ and $k_{n} = [\theta/ \sqrt{ \Delta_{n}}]$.
We now exploit the pre-averaged high-frequency data to form local estimates of $V_{t}$. The estimator we propose is both jump- and noise-robust and can therefore be plugged into a “realized” version of the EDF (or REDF, as defined in (ref)). It is computed on small blocks of pre-averaged returns. We denote the number of increments in a block by a sequence of positive integers $h_{n}$ with $h_{n} \Delta_{n} \rightarrow 0$ and $h_{n}/k_{n} \rightarrow \infty$, as $\Delta_{n} \rightarrow 0$, so the time span of the block is decreasing, but there is an increasing amount of data within it.
Then, we set:
where $v_{n} = \alpha \Delta_{n}^{ \bar{ \omega}}$, for some $\bar{ \omega} \in (0, 1/4)$ and $\alpha>0$.
As the notation suggests, $\tilde{V}_{i \Delta_{n}}$ is an estimator of the spot variance at time $i \Delta_{n}$, i.e. $V_{i \Delta_{n}}$. It is convenient to extend this definition to $t \in [0, T]$. We do this by setting $\tilde{V}_{t} = \tilde{V}_{i \Delta_{n}}$, for $t \in [i \Delta_{n}, (i+1) \Delta_{n})$, while for $t > (Tn - h_{n} - k_{n} + 1) \Delta_{n}$: $\tilde{V}_{t} = \tilde{V}_{(Tn - h_{n} - k_{n} + 1) \Delta_{n}}$, thus holding the final spot variance estimate fixed up to time $T$.
Finally, we introduce the following constants, which depend on the weight function:
With this notation in hand, we have the following result for our preliminary estimator of the spot variance.
As an aside, we should point out it is also possible to show that $\tilde{V}_{i \Delta_{n}}$ retains its consistency even in absence of truncation. However, in order to derive rates of convergence, which play an important role in the subsequent theory, truncation is required when $X$ is discontinuous. We therefore do not pursue this idea further.\footnote{The corresponding analysis of rates of convergence with the non-truncated spot variance estimator and continuous $X$ appeared in an earlier working paper version of this article (available at request).}
To strip out the residual noise variation, we need an estimator of $\omega_{t}^{2}$. There are several options around \citep*[e.g.,][]{bandi-russell:06a, hansen-lunde:06b, oomen:06a} and each one has its own merits and disadvantages \citep*[see, e.g.,][for a comparison]{gatheral-oomen:10a}. In this paper, we adopt the following:
which is a consistent estimator of $\omega_{i \Delta_n}^{2}$ (as demonstrated in the proof of Theorem (ref)). If we define $\hat{ \omega}_{t}^{2} = \hat{ \omega}_{i \Delta_{n}}^2$ for $t \in [i \Delta_{n}, (i+1) \Delta_{n})$, it then follows that
Subtracting the bias implies $\hat{V}_{t}$ can be negative in finite samples. Nevertheless, not a single point estimate fell below zero neither in our simulations nor empirical work.
We then construct the REDF:
In this section, we cover the asymptotic theory of the REDF. In Section (ref), we deal first with consistency for the EDF in the infill setting, where we assume that the noisy log-price is recorded over ever shorter intervals but the total time elapsed is constant, as formalized by $\Delta_{n} \rightarrow 0$ with $T$ fixed. In the following Section (ref), we then further assume high-frequency data are collected on an expanding time window by also letting $T \rightarrow \infty$ and deduce convergence toward the marginal distribution function. We exploit these theoretical insights to develop some new goodness-of-fit tests for stochastic volatility models in Section (ref).
We start this section with showing that $F_{n,T}$ is a consistent estimator of the EDF.
The consistency of $F_{n,T}$ is derived directly from that of the spot volatility estimator and the continuity of the EDF. It thus inherits the jump robustness of the estimator in (ref), regardless of the activity level of jumps in $X$.
We define $Q_{n,T}( \alpha) = \inf \big\{x:F_{n,T}(x) \geq \alpha \big\}$ and $Q_{T}( \alpha) = \inf \big\{x:F_{T}(x) \geq \alpha \big\}$ as the uniquely determined $\alpha$-quantile of $F_{n,T}$ and $F_{T}$, for $\alpha \in(0,1)$. Then, using the uniform convergence in probability of $F_{n,T}$, we deduce $Q_{n,T}( \alpha)$ is a consistent estimator of $Q_{T}( \alpha)$. As it is an implication of Lemma 21.2 in \citet*{vaart:98a}, we omit a proof.
The goal here is to extend the above analysis (with $T$ fixed) to estimation of the stationary distribution function, $F$. To do so, we study an asymptotic framework, where the time span $T \rightarrow \infty$ jointly with $\Delta_{n} \rightarrow 0$. The key result is a feasible CLT for the REDF, which enables us to compute the accuracy with which the marginal distribution can be recovered from the data. To facilitate the derivation of this theory we require some additional assumptions on the volatility process.
The stationarity condition in Assumption (ref) is required to make the target of inference well-defined. We then restrict the memory of the volatility, so that $(V_{t})_{t \geq 0}$ is not “too” strongly dependent. This implies that $F_{T}$ (as an estimator of the marginal distribution, $F$) is consistent, as $T \rightarrow \infty$. We exploit this to deduce that $F_{n,T}$ also converges in probability to $F$, once we show that the discretization error embedded in the recovery of the volatility path is asymptotically negligible (as $\Delta_{n} \to 0$ fast enough, because the errors accumulate with $T$). Moreover, it is an essential part in showing the weak convergence of the empirical processes that we construct in the derivation of our goodness-of-fit test statistics in Section (ref).
The mixing condition is slightly weaker than the comparable assumptions made in related work with joint infill and long-span asymptotics \citep*[see, e.g.,][]{todorov-tauchen:12a, andersen-thyrsgaard-todorov:19a}. It encompasses several stochastic volatility models, for instance a large class of processes driven by Brownian motion \citep*[e.g.,][]{heston:93a} or the L\'{e}vy-driven Ornstein-Uhlenbeck model of \citet*{barndorff-nielsen-shephard:01a}, where volatility is governed by general (positive) processes.\footnote{Assume that $V$ is a solution of the stochastic differential equation: $dV_{t} = \tilde{b}(t, V_{t})dt + \tilde{ \sigma}(t, V_{t})dW_{t}$, where $\tilde{ \sigma}(t, V_{t}) \leq K \big(1 + | V_{t}|^{1/2} \big)$, for some $K > 0$. If there exists an $S$ such that for all $|s|\geq S$ and $t\geq 0$, $b(t,s) \leq -\gamma$, for some $\gamma>0$, Theorem 2 in \citet*{veretennikov:88a} implies that $\alpha(t) = O(t^{-\gamma})$.}\footnote{If $V$ is of the form $d V_{t} = -\kappa V_{t} dt + dZ_{ \kappa t}$, where $\kappa > 0$ and $Z$ is a positive L\'{e}vy process (e.g., a subordinator) with L\'{e}vy measure $\nu$, then it follows from \citet*{jongbloed-meulen-vaart:05a} that $V$ is stationary if $\int_{2}^{\infty} \ln(x) \nu(dx) < \infty$. If further $\mathbb{E} \big[| V_{1} |^{p} \big] < \infty$, for some $p>0$, there exists $a > 0$ such that $\alpha(t) = O (e^{-at})$. The decay of the mixing coefficient can even be deduced if the driving process in volatility is a fractional Brownian motion \citep*[see, e.g.,][]{magdziarz-weron:11a}.} As a result, the setup is not restrictive in practice as it can capture a wide variety of marginal distributions.
Next, we replace the pathwise smoothness imposed on the EDF of volatility from Assumption (ref) with the following condition.
In Theorem (ref), the block length $h_{n}$ is selected so that the discrepancy between the empirical process associated with the REDF and EDF vanishes at the fastest rate. To control this error, we need to restrict the relative speed at which $T \rightarrow \infty$ and $\Delta_{n} \rightarrow0$.
The first restriction is due to the untangling of the spot variance and Brownian component. The least binding is to take $h_{n}$ as indicated. Note that $h_{n}$ is an increasing function of $H$. In particular, if $\sigma$ is a continuous semimartingale (i.e., $H = 1/2$), the contribution of this error to $\sqrt{T} \big(F_{n,T}(x) - F_T(x) \big)$ has order $T^{1/2+ \iota} \Delta_{n}^{1/8- \iota}$. With a smoother volatility ($H \rightarrow 1$), the rate of convergence improves to $T^{1/2 + \iota} \Delta_{n}^{1/6- \iota}$, whereas it gets arbitrarily slow as $H \rightarrow 0$. The intuition is that it is virtually impossible to retrieve spot variance from a “rough” path. On the other hand, if $H = 1/2$ the optimal choice of $h_{n}$ and the associated speed condition is unchanged, irrespective of whether volatility jumps or not.
The second is due to the presence of jumps in $X$. It gets more restrictive, the higher $r$ is. Indeed, if $r > \frac{2}{H+1}$ this is the leading term, but for that to happen, the price jumps must not only be of infinite activity, but also infinite variation (i.e., $r>1$).
The noise-robustness of $F_{n,T}$ plays a critical role in determining the above condition. To compare, if there was no error in our measurement of $X$, a standard non-noise-robust realized measure suffices for estimation of spot volatility \citep*[see, e.g.,][]{jacod-protter:12a}. Here, the rate condition improves to $T^{1/2 + \iota} \Delta_{n}^{1/4-\iota} \rightarrow 0$ with $H = 1/2$, as is consistent with the fixed $T$ asymptotic theory in \citet*{li-todorov-tauchen:13a}. The deterioration of the speed condition is due to the added complexity from retrieving volatility robustly, which cuts the rate of convergence in half. This problem is well-known in the literature on estimation of integrated variance \citep*[see, e.g.,][]{zhang-mykland-ait-sahalia:05a, barndorff-nielsen-hansen-lunde-shephard:08a, jacod-li-mykland-podolskij-vetter:09a}. Moreover, while a convergence rate of (almost) $\Delta_{n}^{1/8}$ may appear slow, it is optimal for extraction of spot variance in noisy diffusion models \citep*[e.g.,][p. 296]{ait-sahalia-jacod:14a}.
In addition, note that the error of the spot noise variance estimator in (ref) is always negligible. For example, in Theorem (ref)(i) $| \hat{ \omega}_{t}^{2} - \omega_{t}^{2}| = O_{p} \big( h_{n}^{-1/2} \vee h_{n} \Delta_{n} \big)$, where $h_{n} \asymp \Delta_{n}^{- \frac{4H+1}{4H+2}}$, which is smaller than the spot variance estimation error $| \hat{V}_{t} - V_{t}|$ of at least order $\Delta_{n}^{ \frac{H}{4H+2}}$ (the latter is bigger than $\Delta_{n}^{1/6}$, as seen by letting $H \rightarrow 1$).
Notice that the asymptotic variance in Theorem (ref), $\Sigma(x)$, is non-stochastic, so that the usual machinery of stable convergence from high-frequency analysis is not required to conclude that $\sqrt{T} \big( F_{n,T}(x) - F(x) \big) / \sqrt{ \Sigma(x)} \xrightarrow{d} N(0,1)$.
$\Sigma(x)$ depends crucially on the memory of the volatility process. If a series is highly dependent, which is true empirically for volatility, it becomes harder to recover its marginal distribution, which results in larger values of $\Sigma(x)$ and wider confidence intervals for $F(x)$. The only way to compensate for this effect is to collect a larger sample.
$\Sigma(x)$ can be computed either analytically or, for example, by numerical integration. As shown by \citet*{dehay:05a}, however, for finite $T$:
The reduction by $(1-t/T)$ in the true variance of $F_{T}$ is an edge effect. We compute (ref) in the simulation section, as it provides an infeasible assessment for the accuracy of the approximation in (ref) for finite $T$. On the other hand, if $\Delta_{n}$ is not small enough to ignore the pathwise discretization error, there is an extra source of randomness in $F_{n,T}$ induced by the estimation of spot volatility. As such, (ref) may actually understate the variation of $F_{n,T}$.
To estimate $\Sigma(x)$, we fix $\xi \in (0, 1/3)$ and set:
where
Then, as shown in the proof of Theorem (ref), $\Sigma_{n,T}(x) \xrightarrow{ \mathbb{P}} \Sigma(x)$.
The estimation of $\Sigma(x)$ is related to the problem of estimating the long-run variance in a stationary time series. In that context, the lag length has to grow slower than $\sqrt{T}$ to ensure consistency, and the optimal bandwidth is usually $O(T^{1/3})$, see, e.g., \citet*{newey-west:94a}. The condition imposed on $\xi$ is identical to that.
In practice, we may also be interested in determining how accurate the associated quantile(s) of $F$ are estimated (e.g., for risk management reporting). These questions are, of course, intertwined. Indeed, from the assumed existence of the density $f$ and Corollary 21.5 in vaart:98a, we deduce the following result.
Thus, inference about any given quantile is feasible, so long as an estimator of the density function of the marginal distribution evaluated at that quantile is available.
The preceding theory shows that $F_{n,T}$ converges to the stationary distribution function of volatility, as $\Delta_{n} \to 0$ and $T \rightarrow \infty$. In this section, we build upon this analysis to construct realized extensions of some classical goodness-of-fit tests for the volatility. To this end, we first derive a functional central limit theorem for the process $G_{n,T} = \Big\{ \sqrt{T} \big (F_{n,T}(x) - F(x) \big), x \in \mathbb{R}_+ \Big\}$.
Note that Theorem (ref) can also be extended to the empirical process associated with the quantile estimator: $G_{n,T}^{Q} = \Big\{ \sqrt{T} \big(Q_{n,T}( \alpha) - Q( \alpha) \big), \alpha \in (0,1) \Big\}$.
As shown in Lemma A.3 in Appendix (ref), $\Sigma(x,y)$ can be consistently estimated by
for $\xi \in (0,1/3)$, where
Appealing to Theorem (ref), we can now construct the test statistics for our goodness-of-fit tests, which evaluate the fit of an assumed stochastic volatility model:
where $w: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$ is a continuous weight function.
$rKS$ resembles a classical Kolmogorov-Smirnov test for an observed process, whereas $rL^{2}$ is a realized version of the goodness-of-fit test based on a weighted $L^{2}$ norm. The latter nests several existing tests, such as the Cramer-von-Mises (i.e., $w(x) = f(x)$) or Anderson-Darling (i.e., $w(x) = f(x)/[F(x)(1-F(x))]$). $rKS$ has the advantage that it is rather trivial to implement, once the REDF is constructed. On the other hand, the literature suggests that an $L^{2}$ statistic, at least in the i.i.d. setting, may be more powerful in finite samples, but it can also be tedious to compute, if a closed-form expression for $w$ is unavailable. This is, for example, the case for a Cramer-von-Mises test if the marginal density is unknown or hard to evaluate.
Now, suppose we are interested in testing the fit of a stochastic volatility model, indexed by a known parameter vector $\upsilon$. Later in this section, we return to the question of how parameter estimation affects the proposed tests. We let $F_{ \upsilon}(x) = \mathbb{P}( V_{t} \leq x; \upsilon)$ describe the marginal distribution of the model.\footnote{As $\upsilon \mapsto F_{ \upsilon}$ is not one-to-one, in general, the model itself is not fully identified by $F_{ \upsilon}$, unless further constraints are imposed on the drift and volatility \citep*[e.g.,][]{ait-sahalia:96a,bibby-skovgaard-sorensen:05a}.} The null---and associated alternative---hypothesis is then written as:
It follows that---under $\mathcal{H}_{0}$---$rKS(F_{ \upsilon}) \xrightarrow{d} \sup_{x \in \mathbb{R}_+} |G_{F_{ \upsilon}}(x)|$, whereas $rKS(F_{ \upsilon}) \overset{ \mathbb{P}}{ \rightarrow} \infty$ under $\mathcal{H}_{a}$, because the alternative means there is at least one $x$, such that $F_{n,T}(x) \overset{ \mathbb{P}}{ \nrightarrow} F_{ \upsilon}(x)$. An equivalent result holds for $rL^{2}(F)$. As large values of the $t$-statistics discredit that $F = F_{ \upsilon}$, the tests are therefore one-sided.
A drawback of this approach is that the limiting distribution of $rKS(F)$ and $rL^{2}(F)$ depends on the model under $\mathcal{H}_{0}$. This differs from the classical theory, for example the Kolmogorov-Smirnov statistic in the i.i.d. case with $F$ continuous and known \citep*[see, e.g.,][]{vaart:98a}. Even there, however, the asymptotic distribution is not readily available if the parameters describing $F_{ \upsilon}$ are estimated, as we do below, although in that setting it often suffices to prepare a single table of family-wise critical values in advance---e.g., as in \citet*{lilliefors:67a,lilliefors:69a}---if $F_{ \upsilon}$ belongs to a location-scale family. This is due to the fact that the limiting distribution, in that case, does not depend on nuisance parameters of $F$, only the functional form of the probability distribution \citep*[e.g.,][]{david-johnson:48a}. This is not true here, and we are therefore forced to retrieve critical values through simulation on a case-by-case basis. In Appendix B, we offer a detailed recipe of how this procedure works both for Corollary (ref) and (ref).
In principle, we can construct a $t$-statistic that can be evaluated independently of $F$. To explain how, define $||f||_w = \sqrt{ \int_{ \mathbb{R}_{+}}w(x)f(x)^{2}dx}$, such that $\mathbb{E} \big[ ||G_{F}||_{w}^{2} \big] < \infty$.
The other ingredient is a normalizing constant:
with $\tilde{ \Sigma} < \infty$. Then, we set:
We can then define a test statistic that rejects $\mathcal{H}_{0}$, if we observe that $T(F) > \chi_{1 - \alpha}^{2}$, where $\chi_{1 - \alpha}^{2}$ is the $(1 - \alpha)$-quantile of a chi-squared distribution with one degree of freedom. Let $\alpha_{T(F)}$ be the significance level achieved by such a test. The following proposition summarizes the details.
We can thus construct a goodness-of-fit test for which critical values are readily available, however it achieves a significance level of at most $\alpha$ and may therefore be overly conservative for some $F$. Moreover, while the constant $\tilde{ \Sigma}$ can be calculated under $\mathcal{H}_{0}$, so that no estimation is required, it is complicated to derive for many common stochastic volatility models. We therefore do not explore this test further.
In practice, the parameter vector $\upsilon$ is typically unknown and not fixed a priori. It therefore has to be estimated, which invalidates the previous analysis (assuming $F_{ \upsilon}$ known). The intuition is that the estimation moves the model-implied marginal distribution closer to the EDF, making inference based on the preceding theory too conservative, as is well-known from the traditional literature on goodness-of-fit testing.
The complication brought about by parameter estimation can be viewed as a different testing problem that applies if we are interested in testing an entire class of stochastic volatility processes, as opposed to a particular member of that class.
The null is thus a composite hypothesis:
where $\Upsilon$ is the admissible parameter space (i.e., $\Upsilon = \{ ( \kappa, v_{0}, \xi) \in \mathbb{R}^{3}_{+} : 2 \kappa v_{0} \geq \xi^{2} \}$ for the \citet*{heston:93a} model due to the Feller condition).
We assume a consistent estimator of $\upsilon$, say $\hat{ \upsilon}$, has been found and introduce a pseudo empirical process $\tilde{G}_{n,T} = \Big\{ \sqrt{T} \big( F_{n,T}(x) - F_{ \hat{ \upsilon}}(x) \big), x \in \mathbb{R}_{+} \Big\}$. At a theoretical level, the problem then arises because the parameters of the underlying distribution typically cannot be recovered at a convergence rate faster than $T^{-1/2}$, i.e $\hat{ \upsilon} - \upsilon = O_{p} (T^{-1/2})$. This cancels with the $\sqrt{T}$ scaling in the construction of $\tilde{G}_{n,T}$, so that the estimation error inevitably affects the asymptotic distribution.\footnote{A standard rate of convergence is required in the following. Although we are not aware about the presence of so-called “super” consistent estimators in the stochastic volatility literature---at least not in the stationary setting---we point out that if $\hat{ \upsilon} - \upsilon = o_{p}(T^{-1/2})$ the sampling error is asymptotically negligible and the problem reverts back to Corollary (ref). On the other hand, a slower rate of convergence implies the error is blown up by the $\sqrt{T}$ scaling and then our approach just goes out the window.}
The requirement that $\hat{ \upsilon}$ should be asymptotically linear is fulfilled by most $\sqrt{T}$-consistent estimators proposed in the literature \citep*[e.g.,][]{bickel-klaassen-ritov-wellner:98a}.
We now appraise, via Monte Carlo simulation, our estimator of the EDF of spot variance introduced in Section (ref). We also evaluate the size and power of the goodness-of-fit tests for the marginal distribution of volatility proposed in Section (ref).
Starting at $X_{0} = 0$, and recalling that $\sigma_{t} = \sqrt{V_{t}}$, the efficient log-price is a convolution of a \citet*{heston:93a}-type stochastic volatility model and a pure-jump process:
where $W$ and $B$ are Brownian motions with $\text{corr}[dW_{t}dB_{t}] = \rho = -\sqrt{0.5}$. The other parameters are $\kappa = 0.05$, $v_{0} = 1$, and $\xi = 0.2$, which is calibrated to our empirical data and broadly aligns with previous studies \citep*[e.g,][]{ait-sahalia-kimmel:07a}.
In this framework, the law of $V_{t}$ (conditional on $V_{s}$, for $s < t$) is---up to a scale factor---non-central chi-square with $d = 4 v_{0} \kappa \xi^{-2}$ degrees of freedom and non-centrality parameter $\displaystyle \lambda = \frac{4 \kappa e^{- \kappa(t-s)}}{\xi^{2}(1-e^{- \kappa(t-s)})} V_{s}$ \citep*[e.g.,][]{cox-ingersoll-ross:85a}. As $t \to \infty$, the conditional distribution converges to a $\text{Gamma}(2 \kappa v_{0} \xi^{-2}, 2\kappa \xi^{-2})$, which is also the unconditional law of $V_{t}$ for any finite $t$, if $V_{0}$ is drawn from this distribution. We do that here. The variance of $F_{T}(x)$ in (ref) is then computed from these expressions by numerical integration of the joint density function.
$J_{t}$ is a symmetric tempered stable process with L\'{e}vy measure \citep*[e.g.,][]{rosinski:07a}:
where $c > 0$, $\lambda > 0$ and $r \in [0,2)$. The degree of price jump activity is measured by $r$. We choose $r = 0.5$, which produces an infinite-activity, finite-variation process with many small jumps and a few large ones. We assume $\lambda = 3$ and calibrate $c$ such that $J_{t}$ accounts for 20% of the quadratic variation. This complies with previous work in the field \citep*[e.g.,][]{ait-sahalia-jacod-li:12a, ait-sahalia-xiu:16a}.
We generate 10,000 “continuous-time” paths with $23,400$ updates per unit of time $T$. As described in \citet*{todorov-tauchen-grynkiv:14a}, $J_{t}$ is simulated as the difference between two spectrally positive tempered stable processes using the acceptance-rejection algorithm of \citet*{baeumer-meerschaert:10a}. The rest of the system is discretized by an Euler approach. To gauge how close the REDF is to the population counterpart, we extract a coarser grid of $n = 2,340T$ log-price increments. This emulates a financial market, where new transactions arrive regularly every tenth second over a 6.5 hour trading day, which is aligned to the sample sizes in our empirical work.
We add heteroscedastic noise $Z_{i \Delta_{n}} = X_{i \Delta_{n}} + U_{i \Delta_{n}}$ with $U_{i \Delta_{n}} \sim N(0, \omega_{i \Delta_{n}}^{2})$, where $\omega_{i \Delta_{n}}^{2}$ is set by fixing the noise-to-signal ratio \citep*[e.g.,][]{oomen:06a}: $\gamma = \sqrt{ \displaystyle \frac{1}{\Delta_{n}} \frac{\omega_{i \Delta_{n}}^{2}}{ V_{i \Delta_{n}}}}$. As consistent with \citet*{christensen-oomen-podolskij:14a}, we take $\gamma = 0.5$.\footnote{We made a robustness check with $\gamma = 2.0$ as in \citet*{ait-sahalia-jacod-li:12a}. The pre-averaging estimator was hardly affected by this change, while the sparse estimator was substantially more biased, due to the amplification of the noise.} Thus, $(Z_{i \Delta_{n}})_{i=0}^{n}$ comprises our discretely observed sample used to estimate the spot variances, which we recover with $\tilde{V}_{i \Delta_{n}}$ from (ref) combined with (ref). We pre-average with $g(x) = \min(x,1-x)$, $k_{n} = [\theta / \sqrt{ \Delta_{n}}]$ and $\theta = 1/3$. We also truncate using a standard approach. In particular, a pre-averaged increment is reset to zero if $|\bar{Z}_{i}| \geq q_{1-\alpha} \sqrt{ \widehat{IV}} \Delta_{n}^{ \bar{ \omega}}$, where $q_{1- \alpha}$ is the $1- \alpha$ quantile of the standard normal distribution, $\widehat{IV}$ is the pre-averaged bipower variation, and $\bar{ \omega} \in (0,0.25)$ controls how fast the cutoff goes to zero with $\Delta_{n}$ (note $\bar{Z}_{i} = O_{p} \big( \Delta_{n}^{1/4} \big)$ if $X$ is continuous). Here, $\alpha = 0.001$ and $\bar{ \omega} = 0.20$. The debiasing in $\hat{V}_{i \Delta_{n}}$ is done with $\hat{ \omega}_{i \Delta_{n}}^{2}$ from (ref), which we measure locally on a block that covers all the noisy high-frequency data that were included in the calculation of the pre-averaged increments that are part of the computation of $\tilde{V}_{i \Delta_{n}}$.
In Figure (ref), we examine how $F_{n,T}$ recovers $F_{T}$. We set $T = 1$ and freeze volatility across replica to keep the estimation target fixed. The sample path of $V_{t}$ is shown in Panel A, while the associated EDF appears in Panel B. We report our estimator and compare it to \citet*{li-todorov-tauchen:13a}. We construct the latter statistic from 1- and 5-minute low-frequency returns. As recommended in their work and to facilitate comparison, $h_{n}$ is set throughout so that it spans about 1.5 hours worth of data.
At the 1-minute resolution, the \citet*{li-todorov-tauchen:13a} estimator is severely distorted by market frictions, which leads to a systematic upward bias. This was to be expected, as it is not resistant to noise. On the other hand, while the 5-minute statistic does not accumulate any discernible bias, it delivers imprecise estimates, resulting in a highly overdispersed REDF. In contrast, as our noise-robust estimator is able to capitalize on tick-by-tick information, it is more accurate. This suggests the inferior rate of convergence embedded in pre-averaging is more than offset by being able to employ all the data.
Next, we vary $V_{t}$ randomly in each iteration and estimate $F_{T}$ as above. In Figure (ref), we plot the relative error $Q_{n,T}( \alpha)/Q_{T}( \alpha) - 1$ in Panel A and absolute error $\big|Q_{n,T}( \alpha) - Q_{T}( \alpha) \big|$ in Panel B, which are averaged across Monte Carlo trials, for the vector of quantiles $\alpha = (0.05, 0.10, \ldots,0.95)'$. As evident---and consistent with Figure (ref)---our estimator is less biased, especially in the tails. The shape of the upward sloping bias, common among estimators, is explained by the fact that the sampling distribution of $\hat{V}_{t}$, say, is spread out (not necessarily centered) around $V_{t}$. Hence, when volatility is low, $\hat{V}_{t}$ sometimes underestimates $V_{t}$, which translates into a downward bias in the low quantiles of the REDF vis-\'{a}-vis the EDF, and vice versa. The bias is more pronounced when $V_{t}$ is high, because it is harder to estimate spot variance in a volatile market. Overall, the REDF proposed in this paper is nevertheless roughly unbiased over a large spectrum of the distribution with a relative error within a few percent. This is further corroborated by Panel B. In general, while all estimators do a relatively poorer job at fitting the tails of the distribution of spot variance, there are again notable gains by applying our estimator.
We now turn to estimation of the marginal distribution of volatility based on Theorem (ref). We work with $T = 250$, corresponding to the number of trading days in about a year. This relatively low value is meant to be conservative and illustrate the potential of our estimator. As a comparison, $T = 2,517$ in our empirical work in Section (ref). Note that $n$ is kept fixed relative to $T$, so that $\Delta_{n}$ is constant. In Figure (ref), we report point estimates---for the first 100 realizations---and sample averages across all 10,000 simulations of $F_{n,T}(x)$ and kernel densities of the asymptotic pivot $\sqrt{T} \big( F_{n,T}(x) - F(x) \big) / \sqrt{ \Sigma(x)} \xrightarrow{d} N(0,1)$ evaluated at five distinct points, which cover a variety of low-to-high volatility states, i.e. $F(x) = 0.10, 0.25, 0.50, 0.75$ and $0.90$. We see that in general the REDF is unbiased and correctly scaled. If we look far in the tails, in particular the right-hand one, the density estimate is slightly skewed toward the center, which is intuitive due to the natural bounds on $F_{n,T}$. Still, the limiting distribution provides a good description of the finite sample variation of $F_{n,T}$.
Last but not least, we explore the properties of the realized Kolmogorov-Smirnov and Cramer-von-Mises (i.e., a weighted $L^{2}$ norm with $w(x) = f(x)$) goodness-of-fit test. This piece of analysis is based on a shorter run of 1,000 Monte Carlo replica due to the increased computational cost of the resampling scheme. We draw 500 bootstrap samples in each simulation trial and inspect both the setting where $v$ is known and estimated. We follow \citet*{corradi-distaso:06a} and recover the parameters via $\hat{v}$ using GMM to match the sample mean, variance, plus the first and second autocovariance of the bias-corrected pre-averaged bipower variation of \citet*{podolskij-vetter:09a} to the corresponding model-based moments of integrated variance (the former is consistent for the latter).
To be able to define a measure of the testing power we simulate from another model, i.e. the log-normal---or exponential Ornstein-Uhlenbeck---process, which is the solution of the stochastic differential equation:
where $v = ( \kappa, v_{0}, \xi) = (0.08, -0.3, 0.45)$ is comparable to what we observe in the real data. We leave the leverage correlation unchanged at $\rho = -\sqrt{0.5}$.
In Table (ref), we report the size and power of the test for a nominal significance level of $\alpha = 5\%$ and $T = 250$, $500$, $1,000$, and $2,000$. As seen, the test is roughly unbiased and has a rejection rate close to $\alpha$. We next comment on the power, as reported in the second half of Table (ref). It is calculated as the rejection rate achieved by simulating log-normal volatility and testing the square-root process. $\mathcal{H}_{0}$ without estimation is here based on the configuration of the \citet*{heston:93a} model with the true parameters used to simulate volatility for the size analysis. As shown, the test has moderate power if the sample size is small, but the rejection rate increases steadily as $T$ grows larger. At $T = 2,000$, the power is in the 60--90% range. The largest choice of $T$ corresponds to about eight years worth of high-frequency data, which is widely available for many assets in practice. The impact of parameter estimation is to “flatten” the power curve adding an upward bias for small $T$ and dampening it for large $T$. We conjecture this effect is caused by the inherent sampling error in the pre-averaged bipower variation, which leads to systematic upward biases in the estimate of both the mean reversion and volatility-of-volatility coefficient $\kappa$ and $\xi$.\footnote{The \citet*{heston:93a} model has an ARMA(1,1) representation for the integrated variance. We could exploit this structure by casting the whole system into state-space representation and then use a Kalman filter to extract a filtered (or smoothed) volatility series before estimating the model, as discussed in \citet*{barndorff-nielsen-shephard:02a} \citep*[see also, e.g.,][for an alternative approach]{todorov-tauchen-grynkiv:11a}. This tends to deliver less biased parameter estimates. However, we do not pursue this idea here.} The misspecification of the dynamic properties of the system invariably yields simulated critical values that are slightly off target and this effect appears more pronounced under the alternative. Finally, we notice the realized Kolmogorov-Smirnov test has slightly higher power than our weighted $L^{2}$ statistic, which is a somewhat surprising finding compared to the classical results for i.i.d. data.
We here illustrate the application of our new nonparametric estimator of the EDF of spot variance and the associated goodness-of-fit tests to real high-frequency data. We construct the $F_{n,T}$ measure from tick data based on selected exchange-traded funds (ETFs) that cover different sectors of the U.S. stock market. In addition to a proxy for the market index (ticker symbol SPY), we add the nine industry portfolios of the S&P 500, yielding a total of ten securities.\footnote{There is also a 10th sector ETF (XLRE), which is a diversified portfolio of companies exposed to real estate. However, as the inception date of this fund is rather recent, we exclude it from consideration due to the limited availability of high-frequency data. Further information about the ETFs is available at http://www.sectorspdr.com/.} High-frequency data was acquired from the TAQ database and comprise a complete series of trades and quotes for each stock for the sample period July 2008 -- June 2018. The analysis here is based on transaction data, which were filtered for outliers as in \citet*{christensen-oomen-podolskij:14a} and subsequently pre-ticked to an equidistant 10-second grid from 9:30am to 4:00pm EST, resulting in $T = 2,517$ days of $n = 2,340$ intraday returns.\footnote{Trading at NYSE terminates early at 1:00pm on a few regularly scheduled business days each year in observance of upcoming holidays. We do not collect data after the exchange has officially closed, as there is typically very little liquidity, and therefore use a shorter sample of $n = 1,260$ log-price increments on these days.} A list of ticker symbols and descriptive statistics of the sample are presented in Table (ref).
The calculation of $\hat{V}_{t}$ proceeds as in Section (ref) -- (ref) in terms of tuning parameters. We further account for the diurnal pattern in intraday spot volatility. We follow the standard approach in the literature, which is to compute the average value of $\hat{V}_{t}$ at each time point of the day over the sample. The diurnal factor is then estimated by normalizing the sum of these averages to one. We correct $\hat{V}_{t}$ by this quantity and use the adjusted time series to construct $F_{n,T}$. In Panel A of Figure (ref), we show a kernel-based estimate of the probability density function implied by the time series of variance estimates of SPY, as representative of our data.
We compute $rKS(F)$ and $rL^{2}(F)$ with three choices of $F$ under $\mathcal{H}_{0}$. The first are the gamma and log-normal distribution inherent in the square-root and log-normal diffusion models that were presented in the simulation section. Furthermore, we inspect the goodness-of-fit of a non-Gaussian Ornstein-Uhlenbeck process \citep*[e.g.,][]{barndorff-nielsen-shephard:01a}. The latter is a pure-jump specification with SDE:
where $L_{t}$ is a subordinator with L\'{e}vy measure $\nu_{L}(dx)$.
To maintain a tractable setting, we proceed as in \citet*{todorov-tauchen-grynkiv:11a} by assuming that the distribution of the increments of the volatility process corresponds to that of a tempered stable process. This means it has a L\'{e}vy density:
where $c > 0$, $\lambda > 0$ and $\beta < 1$. The L\'{e}vy measure of the background driving process $\nu_{L}(dx)$ can then be backed out from $\nu_{V}(dx)$, see e.g., \citet*{barndorff-nielsen-shephard:01a}. We fix the activity level of the volatility jumps at $\beta = 0.5$, so that the resulting marginal distribution of volatility is inverse Gaussian, i.e. $V_{t} \sim IG(\mu, \nu)$ with:
and
is the density function of $V_{t}$.
The procedure in Section (ref) is applied to report a p-value for the $t$-statistic. The exception is that we boost the number of bootstrap samples to 1,000 for higher precision. The non-Gaussian Ornstein-Uhlenbeck process is discretized and simulated as explained in the appendix of \citet*{todorov-tauchen-grynkiv:11a}.\footnote{As in meddahi:03a, we exploit the ARMA(1,1) structure for the integrated variance of this model to compute the relevant moments: $\displaystyle \mathbb{E}[IV_{t}] = c \sqrt{ \frac{ \pi}{ \lambda}}$, $\displaystyle \text{var}(IV_{t}) = \frac{c \sqrt{ \pi}}{ \lambda^{3/2} \kappa^{2}} ( e^{- \kappa}+\kappa-1)$, $\displaystyle \text{cov}(IV_{t}, IV_{t-1}) = \sqrt{ \frac{ \pi c^{2}}{4 \lambda^{3}}} \frac{(1 - e^{-\kappa})^{2}}{ \kappa^{2}}$, and $\text{cov}(IV_t,IV_{t+j}) = e^{-\kappa} \text{cov}(IV_{t},IV_{t+j-1})$ for $j \geq 2$. We also calculated these expressions based on the ARMA coefficients reported in the appendix of \citet*{todorov-tauchen-grynkiv:11a}, but the results differ and we suspect there are some typos in that paper.}
In Table (ref), we present the outcome of these efforts, while Panel B of Figure (ref) shows a QQ-style plot, where the quantiles of $F_{n,T}$ (on the $x$-axis) are compared to the quantiles of the best fitting distribution function $F$ in each class (on the $y$-axis). The graph should align with the 45$^\circ$-line if the model is correct. As readily seen, the gamma distribution does not capture stochastic equity variance at all. The main problem with the square-root process is that it cannot generate a sufficient level of volatility-of-volatility (via $\xi$) without violating the Feller restriction, which is highly binding in practice. The p-values of the log-normal model are higher on average, but it also does not deliver an acceptable fit of the data across sectors. Although there are a few close calls, for example XLK, XLU and SPY, the model is generally rejected at standard levels of significance. In contrast, the p-values of the inverse Gaussian are typically much higher. This is in line with \citet*{todorov-tauchen:12a}, and it appears relatively consistent across the ETF space. Nevertheless, the latter also struggles (depending a bit on which $t$-statistic we ask), for instance it does not capture return variation exhibited by the highly volatile financial sector ETF (XLF), but it is generally the best fitting model of the ones estimated in this paper.
We construct a nonparametric jump- and noise-robust realized measure of the EDF of the latent volatility of a general It\^{o} semimartingale sampled at high-frequency on a fixed time interval. We extend previous work of \citet*{li-todorov-tauchen:13a, li-todorov-tauchen:16a} in the noise-free setting and prove that in the presence of microstructure noise the pre-averaged version of their estimator is consistent in the infill asymptotic limit. The resistance to market frictions enables our statistic to fully exploit information about volatility available in tick-by-tick high-frequency data, which improves its accuracy in a simulation study.
In a subsequent analysis, we let the time span tend to infinity. We show that our estimator then converges to the marginal distribution function of volatility. This result is subject to some rate conditions, which are more restrictive in the presence of jumps or roughness in volatility. We establish a functional CLT, from which we prove the validity of a family of goodness-of-fit tests for stochastic volatility models. The limiting distribution of our $t$-statistic depends on the true model under the null hypothesis, but critical values can be found via simulation. We show how the procedure can be adapted to the case with parameter estimation.
We apply the theory to high-frequency data from several ETFs that track different sectors of the U.S. stock market. The marginal distribution embedded in several classes of popular stochastic volatility models is tested against our nonparametric measure. There is strong evidence against the gamma or log-normal distribution, while the inverse Gaussian---implied by the class of non-Gaussian Ornstein-Uhlenbeck processes of \citet*{barndorff-nielsen-shephard:01a} with tempered stable increments---has more support. These findings are consistent with \citet*{todorov-tauchen-grynkiv:11a}, but in disagreement with \citet*{corradi-distaso:06a}.
Our paper can serve as a starting point for building more refined models of stochastic volatility. It appears natural, for example, to inspect the properties of the generalized inverse Gaussian (GIG) as the marginal distribution of volatility. It has the PDF:
where $K_{p}$ is the modified Bessel function of the second kind, while $a > 0, b > 0$ and $p$ a real number are the parameters. The GIG is self-decomposable (necessary and sufficient for stationary solutions of (ref) to exist) and nests the gamma distribution as $b \to 0$ (with $a > 0$) and the inverse Gaussian if $p = -1/2$. The extra degree of freedom may just be enough for a complete description of equity variance. We leave this assessment for future research.