Permutation-based tests for discontinuities in event studies^††thanks: We thank the Coeditor and two anonymous referees for comments and suggestions that have greatly improved the manuscript.

We first prove a new result that is broadly useful for establishing the asymptotic validity of permutation tests. Because of its independent theoretical interest, we develop the theory under high-level conditions. In Section 2.2, below, we shall specialize this general result in event-study applications under a more specific infill time-series setting, for which the existing theory on permutation tests is not applicable.

Consider an array $(Y_{n,i})_{i\in\mathcal{I}_{n}}$ of $\mathbb{R}$-valued observed variables defined on a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$, which may be either “raw” data or preliminary estimators. Our econometric goal is to decide whether two subsamples $(Y_{n,i})_{i\in\mathcal{I}_{1,n}}$ and $\left(Y_{n,i}\right)_{i\in\mathcal{I}_{2,n}}$ have “significantly” different distributions, where $(\mathcal{I}_{1,n},\mathcal{I}_{2,n})$ is a partition of $\mathcal{I}_{n}\subseteq\mathbb{Z}$. For ease of exposition, we assume that $\mathcal{I}_{1,n}$ and $\mathcal{I}_{2,n}$ contain the same number of observations, denoted by $k_{n}$.⁴⁴4 All of our results can be easily extended to the case when $\mathcal{I}_{1,n}$ and $\mathcal{I}_{2,n}$ have different sizes, but with the same order of magnitude. We stress from the outset that $k_{n}$ may either be fixed or grow to infinity in the subsequent analysis. As such, our analysis speaks to not only the classical finite-sample analysis of permutation tests, but also the large-sample analysis routinely used in econometrics.

To implement the test, we first estimate the empirical cumulative distribution functions (CDF) for the two subsamples using

$$\widehat{F}_{j,n}\left(x\right)\equiv\frac{1}{k_{n}}\sum_{i\in\mathcal{I}_{j,n}}1{\left\{Y_{n,i}\leq x\right\}},\quad j\in\{1,2\}.$$

We then measure their difference via the Cramér–von Mises statistic, given by

$$\widehat{T}_{n}\equiv\frac{1}{2k_{n}}\sum_{i\in\mathcal{I}_{n}}\left(\widehat{F}_{1,n}\left(Y_{n,i}\right)-\widehat{F}_{2,n}\left(Y_{n,i}\right)\right)^{2}.$$

(1)

For a significance level $\alpha\in\left(0,1\right)$, we compute the critical value via a standard permutation algorithm as in Lehmann and Romano (2005, page 633), which we specify in Algorithm 1 below. We use $\pi$ to denote a permutation of the elements of $\mathcal{I}_{n}$, that is, a bijective mapping from $\mathcal{I}_{n}$ to itself. Let $G_{n}$ denote the collection of all possible permutations of $\mathcal{I}_{n}$, with $M_{n}$ being its cardinality.

If the data $(Y_{n,i})_{i\in\mathcal{I}_{n}}$ are i.i.d., then the null hypothesis of the classical two-sample problem holds, and Lehmann and Romano (2005, Theorem 15.2.1) implies that the aforementioned permutation test has exact size control in finite samples. This is a remarkable property of the permutation test, as it holds without requiring any specific distributional assumptions on the data. In contrast to the classical two-sample problem, however, we shall not assume that the data are independent, or even “weakly” dependent (e.g., mixing). As mentioned in the Introduction, the main goal of this paper is to study the permutation test for time-series data observed within a short event window (say, a few days or hours), which can be serially highly dependent in practice. Our key theoretical insight is that the permutation test is still asymptotically valid if the data $(Y_{n,i})_{i\in\mathcal{I}_{n}}$ can be approximated, or “coupled,” by another collection of variables that are conditionally independent, as formalized by the following assumption.

Assumption 2.1 lays out the high-level structure for bridging our analysis with the classical theory on permutation tests, which we carry out in Theorem 2.1 below. Condition (i) sets up the “coupling” problem, which corresponds to a conditional version of the classical two-sample problem, treating the $(U_{n,i})_{i\in\mathcal{I}_{1,n}}$ and $(U_{n,i})_{i\in\mathcal{I}_{2,n}}$ variables as “data”. In part (a) of Theorem 2.1, we consider the situation in which both subsamples have the same conditional distribution. In this case, our coupling variables $(U_{n,i})_{i\in\mathcal{I}_{n}}$ give rise to an infeasible permutation test that can be analyzed as a classical two-sample problem. In particular, this infeasible permutation test attains the exact finite-sample size under our conditions.

This infeasible test, however, only plays an auxiliary role in our analysis, because our interest is on the feasible test $\hat{\phi}_{n}$ formed using the original $(Y_{n,i})_{i\in\mathcal{I}_{n}}$ data. Therefore, a key component of our theoretical argument in Theorem 2.1 is to show that the feasible test for the original data inherits asymptotically the same rejection properties from the infeasible test. Conditions (ii) and (iii) in Assumption 2.1 are introduced for this purpose. Specifically, condition (ii) requires the variable $U_{n,i}$ to be non-degenerate, in the sense that its conditional probability mass within any small $\left[x-\eta,x+\eta\right]$ interval is of order $O\left(\eta\right)$ in probability.⁵⁵5Condition (ii) is satisfied if $U_{n,i}$ has a conditional probability density that is uniformly bounded in probability. Condition (iii) specifies the requisite approximation accuracy of the coupling variables. We note that this condition is easier to hold when $k_{n}$ is smaller, because the joint coupling requirement would involve a smaller number of variables and the $o_{p}(k_{n}^{-2})$ error bound is easier to attain. This condition can be verified under more primitive conditions pertaining to the smoothness of underlying processes and an upper bound on the growth rate of $k_{n}$, as detailed in Section 2.2.⁶⁶6In our applications, we can often verify condition (iii) with $\widetilde{Y}_{n,i}=Y_{n,i}$. Nonetheless, allowing $\widetilde{Y}_{n,i}\neq Y_{n,i}$ is useful when $Y_{n,i}$ is itself an estimator. For example, if $(Y_{n,i})_{i\in\mathcal{I}_{n}}$ is a finite collection of estimators that converge jointly in distribution, then the coupling required in Assumption 2.1(iii) can be obtained via Skorokhod representation. As such, the theory developed in Canay and Kamat (2017) can be cast in our framework with $\mathcal{G}_{n}$ being the trivial information set and $k_{n}$ being a fixed constant, although our anti-concentration condition in Assumption 2.1(ii) is slightly stronger than the continuity condition in Canay and Kamat (2017, Assumption 4.2).

Theorem 2.1 characterizes the asymptotic rejection probabilities of the feasible test $\hat{\phi}_{n}$ under the null and alternative hypotheses of the two-sample problem for the coupling variables. Part (a) pertains to the situation in which the two subsamples of coupling variables, $(U_{n,i})_{i\in\mathcal{I}_{1,n}}$ and $(U_{n,i})_{i\in\mathcal{I}_{2n}}$, have the same conditional distribution, which corresponds to the null hypothesis. In this case, the theorem shows that the asymptotic rejection probability of the feasible test is equal to the nominal level $\alpha$. It is relevant to note that this result holds whether $k_{n}$ is fixed or divergent. This property is clearly reminiscent of the permutation test’s finite-sample exactness in the classical setting.

Part (b) of Theorem 2.1 concerns the power of the feasible test $\hat{\phi}_{n}$. It shows that the feasible test rejects with probability approaching one when the conditional distributions of the two coupling subsamples, $Q_{1,n}$ and $Q_{2,n}$, are different, in the sense that their “distance” measured by $\int\left(Q_{1,n}(x)-Q_{2,n}(x)\right)^{2}d\overline{Q}_{n}\left(x\right)$ is asymptotically non-degenerate, where the mixture distribution $\overline{Q}_{n}$ captures approximately the distribution of the permuted data.⁷⁷7When the two subsamples have different sizes, the same result goes through with $\overline{Q}_{n}$ defined as the sample-size weighted average between $Q_{1,n}$ and $Q_{2,n}$, given by $\overline{Q}_{n}=(Q_{1,n}|\mathcal{I}_{1,n}|+Q_{2,n}|\mathcal{I}_{2,n}|)/(|\mathcal{I}_{1,n}|+|\mathcal{I}_{2,n}|)$. This consistency-type result requires that the information available from each subsample grows with the sample size, i.e., $k_{n}\rightarrow\infty$. This result appears to be new in the context of permutation-based tests under a fixed alternative for the coupling variables. In particular, we note that an analogous result is unavailable in Canay and Kamat (2017), as they restrict attention to an asymptotic framework with a fixed $k_{n}$, which makes a consistency-type result unavailable. Our proof relies on applying Lehmann and Romano (2005, Theorem 15.2.3) to the infeasible test, for which we use the coupling construction developed by Chung and Romano (2013) to show that the so-called Hoeffding (1952) condition is satisfied. We note that this argument is used to establish the consistency of the permutation test rather than its asymptotic size property.

Theorem 2.1 establishes the relation between the rejection probability of the feasible test $\hat{\phi}_{n}$ and the homogeneity (or the lack of it) across the two coupling subsamples $(U_{n,i})_{i\in\mathcal{I}_{1,n}}$ and $(U_{n,i})_{i\in\mathcal{I}_{2,n}}$. This result does not speak directly to hypotheses formulated in terms of the original $\left(Y_{n,i}\right)_{i\in\mathcal{I}_{n}}$ observations. Rather, its theoretical significance is to “absorb” all generic technicalities stemming from the (feasible) permutation test, which in turn considerably simplifies our overall analysis. The residual issue for any specific application is to explicitly construct the coupling variables and translate their homogeneity in terms of the primitive structures of the original empirical problem, which can be done using domain-specific techniques. We provide general results along this line in the infill time-series context, as detailed in Section 2.2 below.

To help anticipate the general discussion, it is instructive to sketch the scheme in a basic running example. Let $Y_{n,i}=\Delta_{n}^{-1/2}(P_{\left(i+1\right)\Delta_{n}}-P_{i\Delta_{n}})$ be the scaled increment of the asset price process $P_{t}$ over the $i$th sampling interval. Let $\tau$ be a “cutoff” time point of interest that is known to the researcher (e.g., the announcement time of a news release) and $i^{\ast}$ be the unique integer such that $\tau\in[i^{\ast}\Delta_{n},(i^{\ast}+1)\Delta_{n})$.⁸⁸8The integer $i^{\ast}$ depends on $n$. We suppress this dependence in our notation for simplicity and to avoid having nested subscripts. We consider two index sets $\mathcal{I}_{1,n}=\{i^{\ast}-k_{n},\ldots,i^{\ast}-1\}$ and $\mathcal{I}_{2,n}=\{i^{\ast}+1,\ldots,i^{\ast}+k_{n}\}$, which collect observations before and after the cutoff, respectively. We consider an asymptotic setting in which these subsamples are “local” in calendar time, that is, $k_{n}\Delta_{n}\rightarrow 0$. Note that this implies that $\Delta_{n}\rightarrow 0$, which means that we are considering an infill asymptotic setting. If $P_{t}$ is an Itô process with respect to an information filtration $(\mathcal{F}_{t})_{t\geq 0}$, we may represent $Y_{n,i}$ as

$$Y_{n,i}=\Delta_{n}^{-1/2}\int_{i\Delta_{n}}^{\left(i+1\right)\Delta_{n}}b_{s}ds+\Delta_{n}^{-1/2}\int_{i\Delta_{n}}^{\left(i+1\right)\Delta_{n}}\sigma_{s}dW_{s},\quad\text{for}\quad i\in\mathcal{I}_{n},$$

(2)

where $b_{t}$ is the drift process, $\sigma_{t}$ is the stochastic volatility process, and $W_{t}$ is a standard Brownian motion.⁹⁹9Note that $\mathcal{I}_{n}$ does not include the $i^{\ast}$th return observation. Therefore, although the returns in (2) do not contain price jumps, an event-induced price jump is allowed to occur at time $\tau$. If the $\sigma_{t}$ process is smooth (e.g., Hölder continuous) in a local neighborhood before $\tau$, then the volatility throughout the pre-event subsample $\mathcal{I}_{1,n}$ is approximately $\sigma_{(i^{\ast}-k_{n})\Delta_{n}}$. Further recognizing that the drift term is negligible relative to the Brownian component, we can approximate $Y_{n,i}$ for each $i\in\mathcal{I}_{1,n}$ using the coupling variables

$$U_{n,i}=\sigma_{(i^{\ast}-k_{n})\Delta_{n}}\Delta_{n}^{-1/2}(W_{\left(i+1\right)\Delta_{n}}-W_{i\Delta_{n}})\sim\mathcal{MN}\left(0,\sigma_{(i^{\ast}-k_{n})\Delta_{n}}^{2}\right),$$

(3)

where $\mathcal{MN}$ denotes the mixed normal distribution. Since the Brownian motion has independent and stationary increments, it is easy to see that the coupling variables $(U_{n,i})_{i\in\mathcal{I}_{1,n}}$ are $\mathcal{F}_{(i^{\ast}-k_{n})\Delta_{n}}$-conditionally i.i.d. Moreover, if the volatility process $\sigma_{t}$ does not jump at the cutoff time $\tau$, we may follow the same logic to extend the approximation in (3) further to $i\in\mathcal{I}_{2,n}$. In other words, if the volatility process process does not jump then the coupling variables $(U_{n,i})_{i\in\mathcal{I}_{n}}$ are conditionally i.i.d., which corresponds to the situation in part (a) of Theorem 2.1. On the other hand, if the volatility process jumps at time $\tau$, say by a constant $c\neq 0$, then the coupling variables for the $\mathcal{I}_{2,n}$ subsample will instead take the form $U_{n,i}=\left(\sigma_{(i^{\ast}-k_{n})\Delta_{n}}+c\right)(W_{\left(i+1\right)\Delta_{n}}-W_{i\Delta_{n}})$. In this case, the two subsamples of $U_{n,i}$’s have distinct conditional distributions (i.e., mixed normal with different conditional variances), corresponding to the scenario in part (b) of Theorem 2.1.

Within the context of this illustrative example, we can further clarify a key feature of the proposed test that holds more generally. It is not aimed at detecting “small” time-variations in the distribution of the observed data. In fact, by allowing the drift $b_{t}$ and the volatility $\sigma_{t}$ to be time-varying, a smooth form of heterogeneity is always built in. The test instead detects abrupt changes, or discontinuities, in the evolution of the distribution, which can be more plausibly associated with the “lumpy” information carried by the underlying economic announcement, as emphasized by Nakamura and Steinsson (2018b). Specifically in this example, the asset returns are locally centered Gaussian (due to the assumption that the price is an Itô process), and hence, the temporal discontinuity in the return distribution manifests itself as a volatility jump. The empirical scope of our permutation test, however, is far beyond volatility-jump testing depicted in this illustration, as we shall demonstrate in the remainder of the paper.

We now specialize the general Theorem 2.1 into an infill asymptotic time-series setting that is particularly suitable for event studies. By introducing a mild additional econometric structure, we shall establish the asymptotic validity of the permutation test under more primitive conditions that are easy to verify in a variety of concrete empirical settings. As in the running example above, we consider an event occurring at time $\tau\in[i^{\ast}\Delta_{n},(i^{\ast}+1)\Delta_{n})$, which separates two subsamples indexed by $\mathcal{I}_{1,n}=\{i^{\ast}-k_{n},\ldots,i^{\ast}-1\}$ and $\mathcal{I}_{2,n}=\{i^{\ast}+1,\ldots,i^{\ast}+k_{n}\}$, respectively. All limits in the sequel are obtained under the infill asymptotic setting with $\Delta_{n}\rightarrow 0$.

We suppose that the data are generated from an approximate state-space model of the form

$$Y_{n,i}=g\left(\zeta_{i\Delta_{n}},\epsilon_{n,i}\right)+R_{n,i},\quad i\in\mathcal{I}_{n},$$

(4)

where the state process $\zeta_{t}$ is càdlàg, adapted to a filtration $\mathcal{F}_{t}$, and takes values in an open set $\mathcal{Z}\subseteq\mathbb{R}^{\dim(\zeta)}$; $(\epsilon_{n,i})_{i\in\mathcal{I}_{n}}$ are i.i.d. random disturbances taking values in some (possibly abstract) space $\mathcal{E}$; $g\left(\cdot,\cdot\right)$ is a “smooth” transform; and $R_{n,i}$ is a residual term that is negligible relative to the leading term $g\left(\zeta_{i\Delta_{n}},\epsilon_{n,i}\right)$ in a proper sense detailed below. A simpler version of this state-space model without the $R_{n,i}$ residual term has been used by Li and Xiu (2016) and Bollerslev et al. (2018), among others, for modeling market variables such as trading volume and bid-ask spread. By introducing the $R_{n,i}$ residual term, we can use a unified framework to accommodate a broader class of models, which in particular include increments of an Itô semimartingale. We now revisit the model in (2) as the first illustration.

This running example further illustrates the distinct roles played by $\zeta_{t}$, $\epsilon_{n,i}$, and $R_{n,i}$ in our state-space model (4). The leading term $g\left(\zeta_{i\Delta_{n}},\epsilon_{n,i}\right)$ captures the “main feature” of the observed data; in addition, since the $\epsilon_{n,i}$ disturbance terms are i.i.d., any “large” change in the empirical distribution across the two subsamples must be attributed to the time-$\tau$ discontinuity in the state process $\zeta_{t}$. From this description, it follows that the hypothesis test for the continuity of the distribution of the main feature of the observed data can be formulated as

$$H_{0}:\Delta\zeta_{\tau}=0\quad\text{versus}\quad H_{a}:\Delta\zeta_{\tau}\neq 0,$$

(6)

where $\Delta\zeta_{\tau}\equiv\zeta_{\tau}-\zeta_{\tau-}\equiv\zeta_{\tau}-\lim_{s\uparrow\tau}\zeta_{s}$ denotes the jump of the state process at time $\tau$.

With the state-space model (4) in place, we can design more primitive sufficient conditions for establishing the asymptotic validity of the permutation test under the hypotheses in (6). We need some additional notation to describe these conditions. For each fixed $z\in\mathcal{Z}$, let $f_{z}\left(\cdot\right)$ and $F_{z}\left(\cdot\right)$ denote the probability density function (PDF) and the CDF of the random variable $g\left(z,\varepsilon_{n,i}\right)$, respectively. It is also convenient to introduce a “shifted” version of $\zeta_{t}$ defined as $\tilde{\zeta}_{t}\equiv\zeta_{t}-\Delta\zeta_{\tau}1\{t\geq\tau\}$, which has the same increments as $\zeta_{t}$ over time intervals not containing $\tau$.

Assumption 2.2 entails regularity conditions pertaining to the random disturbance terms, which are often easy to verify in concrete examples as demonstrated later in this subsection. Assumption 2.3 imposes a set of smoothness conditions that permits the approximation of the observed data using properly constructed coupling variables.¹⁰¹⁰10Note that the assumption is framed in a localized fashion using the stopping times $(T_{m})_{m\geq 1}$, which is a standard technique for weakening the regularity condition in the infill asymptotic setting. See Jacod and Protter (2012, Section 4.4.1) for a comprehensive discussion on the localization technique. Specifically, condition (i) requires that the random function $z\mapsto g\left(z,\epsilon_{n,i}\right)$ is Lipschitz in $z$ over compact sets under the $L_{2}$ distance. The $a_{n}$ sequence captures the scale of the Lipschitz coefficient. In many applications, we can verify this condition simply with $a_{n}\equiv 1$, but allowing $a_{n}$ to diverge to infinity is sometimes necessary (see Example 2 below). Condition (ii) states that the $\zeta_{t}$ process is locally compact (up to each stopping time $T_{m}$) and, upon removing the fixed-time discontinuity at $\tau$, it is $(1/2)$-Hölder continuous under the $L_{2}$ norm. This Hölder-continuity requirement can be easily verified using well-known results provided that the $\tilde{\zeta}$ process is an Itô semimartingle or a long-memory process (see Jacod and Protter (2012, Chapter 2) and Li and Liu (2020)). Condition (iii) imposes the requisite assumptions on the residual terms. In some applications, this condition holds trivially with $R_{n,i}\equiv 0$, but, more generally, it needs to be verified on a case-by-case basis using (relatively standard) infill asymptotic techniques. Theorem 2.2, below, establishes the size and power properties of the permutation test under the hypotheses described in (6).

This theorem is proved by verifying the high-level conditions in Theorem 2.1 with properly constructed coupling variables analogous to those in equation (3). The condition $a_{n}k_{n}^{3}\Delta_{n}^{1/2}=o(1)$ mainly requires that the window size $k_{n}$ does not grow too fast, which ensures the closeness between the coupling variables and the original data. In the typical case with $a_{n}=1$, it reduces to $k_{n}=o(\Delta_{n}^{-1/6})$.¹¹¹¹11This sufficient condition for the growth rate of $k_{n}$ is different from the conditions needed for conventional asymptotic-Gaussian-based spot inference, which requires $k_{n}\asymp\Delta_{n}^{-\iota}$ for some $\iota\in(0,1/2)$. For the permutation test, $k_{n}$ may be fixed or grow to infinity. However, in the latter case, our condition on the growth rate of $k_{n}$ is more stringent than what is needed for the conventional spot inference theory. In general, a larger $k_{n}$ allows one to utilize more data, but the associated longer event window may also lead to a larger nonparametric bias, and hence, a more severe size distortion. Part (a) shows that the permutation test attains the desired asymptotic level under the null hypothesis in (6). Again, we stress that the test has valid asymptotic size control even in the “small-sample” case with fixed $k_{n}$. As in Theorem 2.1, the “large-sample” condition $k_{n}\rightarrow\infty$ is needed for establishing the consistency of the test under the alternative, as shown in part (b).

In the remainder of this subsection, we use a few prototype examples to demonstrate how the proposed test may be used in various empirical settings. In particular, we show how to cast the specific problems into the approximate state-space model (4), and discuss how to verify our sufficient regularity conditions. We start by revisiting the running example.

Example 1 shows that the permutation test $\hat{\phi}_{n}$ is asymptotically valid for testing the presence of a volatility jump. This is a relatively familiar problem in the literature. It is therefore useful to contrast the proposed permutation test with the standard approach, which is based on nonparametric “spot” estimators of the asset price’s instantaneous variances before and after the event time given by, respectively,

$$\hat{\sigma}_{\tau-}^{2}=\frac{1}{k_{n}}\sum_{i\in\mathcal{I}_{1,n}}Y_{n,i}^{2},\quad\hat{\sigma}_{\tau}^{2}=\frac{1}{k_{n}}\sum_{i\in\mathcal{I}_{2,n}}Y_{n,i}^{2}.$$

(7)

Assuming $k_{n}\rightarrow\infty$ and $k_{n}^{2}\Delta_{n}\rightarrow 0$, it can be shown that (see Jacod and Protter (2012, Chapter 13))

$$\frac{k_{n}^{1/2}\left(\hat{\sigma}_{\tau}^{2}-\hat{\sigma}_{\tau-}^{2}-(\sigma_{\tau}^{2}-\sigma_{\tau-}^{2})\right)}{\sqrt{2\hat{\sigma}_{\tau}^{4}+2\hat{\sigma}_{\tau-}^{4}}}\overset{d}{\longrightarrow}\mathcal{N}\left(0,1\right).$$

(8)

Thus, we can test $H_{0}:\Delta\sigma_{\tau}=0$ by comparing the t-statistic $k_{n}^{1/2}\left(\hat{\sigma}_{\tau}^{2}-\hat{\sigma}_{\tau-}^{2}\right)/\sqrt{2\hat{\sigma}_{\tau}^{4}+2\hat{\sigma}_{\tau-}^{4}}$ with critical values based on the standard normal distribution.

Two remarks are in order. First, note that the asymptotic size control of the standard approach relies on the asymptotic normal approximation (8), which depends crucially on $k_{n}\rightarrow\infty$ (in addition to having $\Delta_{n}\to 0$) because the underlying central limit theorem is obtained by aggregating a “large” number of martingale differences. Hence, the t-test may suffer from severe size distortion when $k_{n}$ is relatively small. This issue is empirically relevant because an applied researcher may use a short time window to capture short-lived “impulse-like” dynamics and/or to minimize the impact of other confounding economic factors in the background. Moreover, for “real-time” applications, the researcher may have no choice but to use a small $k_{n}$ simply because of the limited amount of available data soon after the event time $\tau$. In sharp contrast, the permutation test controls asymptotic size even when $k_{n}$ is fixed. This remarkable property is inherited from the coupling two-sample problem, in which the permutation test controls size exactly regardless of whether $k_{n}$ is fixed or grows to infinity.

The second and perhaps practically more important difference between the two tests is that the permutation test is more versatile. Under the spot-estimation-based approach, both the design of the spot estimators in (7) and the convergence in (8) depend heavily on the fact that the increments of the Brownian motion are not only i.i.d., but also Gaussian. Gaussianity is obviously essential for the conventional approach because, among other things, it ensures that the instantaneous variance of the normalized returns are well-defined.¹²¹²12Recall that many distributions used in continuous-time models do not have finite second moments. For example, within the class of stable distributions, the Gaussian distribution is the only one with a finite second moment. Moreover, Gaussianity also implies that the variance of $\Delta_{n}^{-1}(W_{i\Delta_{n}}-W_{(i-1)\Delta_{n}})^{2}$ is 2, which explains the “2” factor in the denominator of the t-statistic. The permutation test, on the other hand, only exploits the i.i.d. property of the Brownian shocks, without relying on their Gaussianity. Therefore, the permutation test readily accommodates a more general model for asset returns with Lévy shocks, as we demonstrate in the following example.

So far, we have illustrated the use of the permutation test for high-frequency asset returns data. Under the settings of Examples 1 and 2, the distributional change of asset returns is mainly driven by the time-$\tau$ discontinuity in volatility, and hence, the permutation test is effectively a test for volatility jumps. Example 2, in particular, highlights the versatility and robustness of the permutation test compared with the conventional approach based on spot estimation. Going one step further, we now illustrate how to apply the permutation test to other types of economic variables.

The location-scale structure in Example 3 is by no means essential in applications, because the permutation test is valid provided that the more general conditions in Assumptions 2.2 and 2.3 hold. This illustration is pedagogically convenient, in that it permits a straightforward verification of our high-level conditions. That being said, this example does reveal a limitation of our theory developed so far. That is, the data variable needs to be continuously distributed, as required in Assumption 2.2(ii) (which in turn is related to Assumption 2.1(ii)). Observed data in actual applications are invariably discrete, but this continuous-distribution assumption is often deemed as a reasonable approximation to reality. In some situations, however, the discreteness in the data is more salient. For example, the trading volume of a relatively illiquid asset may take values as small integer multiples of the lot size (e.g., 100 shares).¹³¹³13This issue has become less important in the equity market as retail investors can now trade a single share, or even a fractional share, of a stock. However, the lot size is still relevant for less liquid assets such as option contracts or for equity data from earlier sample periods. This motivates us to directly confront the discreteness in the data, as detailed in the next subsection.

The extension will be carried out in similar steps as the theory developed above. We start with modifying the general result in Theorem 2.1 to accommodate discretely valued observations; we then specialize the general theory to a high-frequency setting under more primitive conditions. Recall that $Q_{j,n}\left(\cdot\right)$ denotes the $\mathcal{G}_{n}$-conditional distribution function of the coupling variable $U_{n,i}$ for $i\in\mathcal{I}_{j,n}$ and $j\in\left\{1,2\right\}$, and $\overline{Q}_{n}=(Q_{1,n}+Q_{2,n})/2$.

Theorem 2.3 establishes exactly the same asymptotic properties for the permutation test as Theorem 2.1, but under different conditions: it does not impose the anti-concentration requirement for the coupling variable (i.e., Assumption 2.1(ii)), and the “distance” between the observed data and the coupling variable is measured by the probability mass of $\{\widetilde{Y}_{n,i}\neq U_{n,i}\}$. These modifications seem natural for the discrete-data setting.

Next, we specialize the general result in Theorem 2.3 to the state-space model (4), starting with some motivating examples. The first is an alternative model for the trading volume that explicitly features discretely valued data, which shows an interesting contrast to Example 3.

To further broaden the empirical scope, we consider another example concerning the bid-ask spread of asset quotes. This example is econometrically interesting because of its resemblance to the discrete-choice models (e.g., probit and logit) commonly used for modeling binary and multinomial data.

We now proceed to establish the asymptotic validity of the permutation test for the hypotheses described in (6) for discretely valued observations; see Theorem 2.4 below. Since the state-space representation (4) holds with the residual term $R_{n,i}=0$ in the examples above, it seems reasonable to avoid unnecessary redundancy by restricting our analysis to a simpler version given by

$$Y_{n,i}=g\left(\zeta_{i\Delta_{n}},\epsilon_{n,i}\right),\quad i\in\mathcal{I}_{n}.$$

(9)

We replace Assumption 2.3 with the following assumption, where we recall that for each $z\in\mathcal{Z}$, $F_{z}\left(\cdot\right)$ denotes the CDF of the random variable $g\left(z,\varepsilon_{n,i}\right)$ and $\tilde{\zeta}_{t}=\zeta_{t}-\Delta\zeta_{\tau}1\{t\geq\tau\}$.