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Long-Horizon Return Predictability from Realized Volatility in Pure-Jump Point Processes
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Research in return predictability has been an active area for decades. Fama and French (1988), Campbell and Schiller (1987, 1988), Mishkin (1990, 1992), Boudoukh and Richardson (1993) found return predictability based on long-run aggregated financial variables such as the dividend yield, price-dividend ratio, and functions of interest rates. In these studies, a predictive regression with the overlapped aggregated return as the response and the similarly overlapped aggregated or non-aggregated financial variable as the predictor, was implemented. As the aggregation horizon increases, long-run return predictability was found to increase.
It has also been well documented in the literature that, even without aggregation, the strong autocorrelation of the predictors employed in these empirical studies induces bias in the OLS estimator of the predictive coefficient. Stambaugh (1999) considered a predictive regression model with an AR(1) predictor. He obtained an expression for the bias of the OLS estimator of the predictive coefficient. Amihud and Hurvich (2004) also considered an AR(1) predictor and introduced the augmented regression method to reduce the bias and studied a bias-corrected hypothesis test for return predictability. Chen, Deo, and Yi (2013) again considered an AR(1) regressor and proposed the quasi restricted likelihood ratio test (QRLRT) for inference on the predictive coefficient. They proposed a test that maintained the nominal size uniformly as the AR(1) coefficient approaches $1$ while delivering higher power than the competing procedures. Boudoukh, Richardson, and Whitelaw (2008) considered a sequence of predictive regressions where overlapped aggregated returns at multiple horizons were regressed on an autocorrelated predictor without aggregation. They showed that the OLS estimators of the predictive coefficients were highly correlated across horizons under the assumption of no return predictability, inflating the size of a joint test that assumes no correlation. They conducted a joint Wald test for the OLS estimators of AR(1) predictors and found much weaker evidence of return predictability.
Furthermore, the overlapping aggregation of an originally-autocorrelated predictor strengthens the degree of autocorrelation and increases the likelihood of spurious findings of long-run predictability of aggregated returns. Valkanov (2003) demonstrated that the ordinary $t$-test in an OLS regression with overlapped aggregated autocorrelated regressors tends to over-reject the null hypothesis of no return predictability.
Kostakis, Magdalinos, and Stamatogiannis (2018) considered a predictive regression on short-memory, integrated, and local-to-unity predictors. They proved that the long-horizon OLS estimator in a predictive regression is inconsistent when the predictor is integrated or is studied in a local-to-unity framework. Using a procedure that robustified against the unknown persistence of the predictor, they found far weaker evidence of return predictability with increasing aggregation horizon compared with most previous empirical literature.
The literature described above considers regressors that either have short memory, are integrated, or are viewed in a local-to-unity framework. This literature does not consider regressors that are fractionally integrated. Nevertheless, it has been established that predictors related to volatility, such as realized variance or VIX, are indeed fractionally integrated. For example, Andersen et al (2001) found that realized stock return volatility is well described by a long-memory process. Bandi and Perron (2006) found a fractionally-cointegrated relationship between realized and implied volatility, suggesting that both series are long-memory processes. Sizova (2013) considered a long-memory predictor in a predictive regression and proved that under certain assumptions as the level of aggregation increases, the population correlation between future returns and lagged realized variance converges to a constant which is a function of the long-memory parameter of the predictor. When the value of the long-memory parameter is zero, this population correlation converges to zero. Hence her theorem establishes that, under certain assumptions, a short-memory regressor does not lead to long-run return predictability as the level of aggregation increases.
The association between long-run future returns and current and past variance has been extensively studied in the financial economics literature. Bansal and Yaron (2004), Bollerslev, Tauchen and Zhou (2009), and Drechsler and Yaron (2011) considered the impact of long-run economic uncertainty on consumption fluctuations, and thus the effect on the expected long-run returns. They proved that the variance premium, defined as the difference between the variance of returns measured under the risk-neutral and physical probability measures, was a good proxy for the latent economic shocks and was correlated with the expected excess returns. Hence these models imply that the variance premium predicts long-run returns. In an empirical study, Bandi and Perron (2008) found that the long-run excess market return had a stronger correlation with past realized variance than it had with the classical dividend yield or the consumption-to-wealth ratio proposed by Lettau and Ludvigson (2001). Hence following Bandi and Perron (2008), we will use realized variance as a predictor in our predictive regressions.
In spite of the long-memory property of the realized variance, Bandi and Perron (2008) did not model market variance as a fractionally integrated process. As they pointed out, if stock returns have short memory, then a predictive regression with a long-memory regressor would be unbalanced. Indeed, rudimentary analysis of stock returns suggests that they have short memory although a weak long-memory component would be hard to detect. Researchers have adopted two frameworks to reconcile the apparently unbalanced nature of this predictive regression. One is to impose assumptions on the functional relationship between the time series of returns and regressors as done by Sizova (2013), who assumed that the predictor is in the domain of attraction of a fractional Brownian motion, and that the error term is additive and has shorter memory than the predictor, so that both sides of the predictive regression equation have long memory with the same memory parameter. The other is to directly model the continuous-time log price process, which then determines the returns and regressor at any level of aggregation. Our work falls into the second framework. Under our model, the long-memory parameter propagates unchanged from the transaction level drift to the calendar-time returns and the realized variance with the same memory parameter, leading endogenously to a balanced predictive regression equation.
In the world of ultra-high-frequency data, the actual transaction‐level stock price is naturally viewed as a pure-jump process, that is, a marked point process where the points are the transaction times and the marks are the prices. The stock price observed in continuous time is a step function as opposed to a diffusion process. Engle and Russell (1998) studied the time intervals, i.e. durations, between successive transactions and proposed the ACD (Autoregressive Conditional Duration) model for the durations. Since their seminal work, several studies have been conducted to investigate the propagation of transaction-level properties of pure-jump processes under aggregation to lower-frequency time series in discrete time. Deo, Hsieh, and Hurvich (2010) studied the intertrade durations, counts, (i.e. the number of trade), squared returns, and realized variance of 10 NYSE stocks. They found the presence of long memory in all of theses series. In light of this stylized fact, they proposed the LMSD (Long Memory Stochastic Duration) model for the durations. Deo et al (2009) provided sufficient conditions for the propagation of the long-memory parameter of durations to the corresponding counts and realized variance.
Cao, Hurvich, and Soulier (2017) studied the effect of drift in pure-jump transaction-level models for asset prices in continuous time, driven by a point process. Under their model, the drift is proportional to the driving point process itself, i.e. the cumulative number of transactions. This link reveals a mechanism by which long memory of the intertrade durations leads to long memory in the returns, with the same memory parameter. Our proposed price model follows Cao, Hurvich, and Soulier's (2017) framework. The calendar-time return derived by our model has a component that is proportional to the counts. Under this model, both calendar-time returns and realized variance are long-memory processes, which leads to a balanced predictive regression. \footnote{Note that Bollerslev, Sizova, and Tauchen (2012) also considered generalizing their framework by modeling the volatility of the consumption growth rate as a long-memory process in the continuous-time framework of Comte and Renault (1996). Though their generalized model also leads to a balanced predictive regression (since the components of their variance premium regressor are fractionally-cointegrated and the response is a short memory return series), it would follow from Theorem 3 of Sizova (2013) if its assumptions held that the model of Bollerslev, Sizova, and Tauchen (2012) would not lead to long-run return predictability.}
Our paper makes the following contributions to the existing literature. We propose a parametric transaction-level model for the log price. Our model for the log price implies properties of the calendar-time returns and realized variance that are consistent with the stylized facts. We propose an estimation procedure for the model parameters that is easy to implement. For assessing return predictability, we propose to aggregate $\tilde{H} = \tilde{T}^{\kappa}$ ($\kappa \in (0,1)$) calendar-time returns in contrast to Sizova's choice of $\tilde{H}=\theta \tilde{T}$ ($\theta \in (0,1)$), where $\tilde{T}$ is the number of available calendar-time returns. Sizova (2013) used the hypothesis test of Valkanov (2003) for long-run return predictability under the linear aggregation framework $\tilde{H}=\theta \tilde{T}$, but did not establish the consistency of this test. Within the power-law framework $\tilde{H} = \tilde{T}^{\kappa}$, we propose a hypothesis test for long-run return predictability based on the sample correlation between the future aggregated returns and the past realized variance. We establish a central limit theorem for the test statistic under the null hypothesis of no long-run return predictability. Our test is consistent, unlike the one used by Sizova (2013). We also provide simulations on a parametric bootstrap approach to testing for long-run return predictability under the power-law framework, and find that the test has a high power while maintaining the nominal size. We discuss the applicability of Sizova's (2013) assumptions for certain models in Section (ref).
The paper is organized as follows. In Section (ref), we present our proposed price model. In Section (ref), we demonstrate return predictability by evaluating the correlation between the aggregated return and the lagged aggregated variance. In Section (ref), we propose a theoretically-based hypothesis test for long-run return predictability and use simulations to evaluate the size and the power of the bootstrap test. We compare our work with Sizova (2013) in Section (ref) and conclude the paper in Section (ref). In Appendix Section (ref), we propose formulas for estimating the model parameters. In Section (ref), we describe a procedure for simulating our model. The proofs of our theorems are provided in Section (ref).
In this section, we propose our transaction-level model based on a pure-jump point process. We then obtain the corresponding calendar-time return series and its properties.
We start with the Cox process which is a key ingredient in our model. The points of any point process $N$ consist of the sequence $\{T_{i},i\in\mathbb{Z}\}$ (here, the transaction arrival times) such that $T_i < T_{i+1}$ for all $i\in\mathbb{Z}$ and $T_0 \leq 0 < T_1$. For all measurable sets $A\subset \mathbb{R}$, we define the point process $N$ by
Let $\lambda(t), t \in \mathbb{R}$ be a non-negative stochastic intensity function, and define the random measure $\Lambda$ by
for all measurable sets $A$. A Cox process, $N$, with mean measure $\Lambda$, is defined as follows: conditionally on $\Lambda$, $N$ is a Poisson process with mean measure $\Lambda$.
We now make further assumptions on the stochastic intensity function which guarantee long memory in returns and volatility when $d > 0$, where $d$ is the long-memory parameter. Let
where $Z_H$ is a Gaussian stationary process with mean zero, variance one and Hurst index $1/2 \le H <1$ (where $H=\frac{1}{2}+d$), which implies that the lag-$r$ autocovariance function $\gamma_Z$ of $Z_H$ satisfies
Equation ((ref)) implies that the autocovariance of $Z_H$ is positive for large lags. The unconditional mean of $N(A)$ is equal to
An example of such a process $Z_H$ satisfying ((ref)) is fractional Gaussian noise defined as the increment process
where $B_H(t)$ is fractional Brownian motion and $c > 0$ is a time-scaling constant to ensure that our model is invariant to the choice of the time unit. Under Equation ((ref)), which we assume for the remainder of the paper, $Z_H$ has autocovariance function given by
$\forall r \in \mathbb{R}$, $\gamma_Z(r) \ge 0$ (see Lemma (ref)).
For the short-memory case of $H=\frac{1}{2}$ ($d=0$), if $|r| \ge \frac{1}{c}$, $\gamma_Z(r)=0$; if $|r| < \frac{1}{c}$, $\gamma_Z(r)= 1-c|r|$. (see Lemma (ref)). We show in Theorem (ref) that when $H=\frac{1}{2}$ there is no long-run predictability of aggregated future returns.
Let $N(t) = N((0,t])$ denote the number of transactions in $(0,t]$. For all $t \ge 0$, the log price process is defined as
where
$\mu > 0$ is the drift term and $\{e_k\}$ are the efficient shocks, which are assumed to be $i.i.d$ random variables with mean zero and variance $\sigma^2_e$, independent of $N$. The efficient shock $e_k$ is assumed to reflect the true value of a stock and thus causes a permanent change to the $\log$ price. Under ((ref)), Equation ((ref)) can be expressed as
This together with Equation ((ref)) imply that $\mathbb{E}[\log P(t)] = \mu\mathbb{E}[N(t)] = \mu (\lambda\mathrm{e}^{\frac{1}{2}}t) > 0$. Hence the expected $\log$ price is a linear function of $t$, which reflects the long-term growth of stock price as observed in the empirical finance literature. For example, the average real annual returns of the Standard and Poor $500$ Index over the 90-year period from 1889 to 1978 is about $7\%$ (see Mehra and Prescott(1985)).
Though we will not pursue it in this paper, the model in Equation ((ref)) could be generalized to multiple drift terms and point processes, for example
where $N_1(t)$ and $N_2(t)$ are mutually independent counting processes; $\{e_{1,k}\}$ and $\{e_{2,k}\}$ are i.i.d. efficient shocks independent of $N_1(t)$ and $N_2(t)$, both with means zero and the same standard deviation $\sigma^2_e$, $\mu_1 > 0$, $\mu_2 < 0$, and $\mu_1 + \mu_2 > 0$. Here $N_1(t)$ and $N_2(t)$ can be thought of representing the numbers of buy and sell transactions. As a result, the observed price reflects the net effect of buy and sell transactions. An active period for $N_1$ puts upward pressure on the price, while an active period for $N_2$ exerts downward pressure on the price. The assumption $\mu_1 + \mu_2 > 0$ reflects the long-term upward trend in the log prices. The positive drift term in Equation ((ref)) could represent the overall effect of $\mu_1$ and $\mu_2$ from ((ref)). Thus, we consider model ((ref)) as a simplified version of model ((ref)). As we show in Lemma (ref) and Corollary (ref), both models exhibit properties of long-term return predictability. Henceforth, unless specified explicitly, we will focus on this simplified model ((ref)).
The calendar-time return series $\{r_t\}^{\infty}_{t=1}$ measured at fixed clock-time intervals of width $\Delta t$ is $r_t = \log P(t\Delta t) - \log P((t-1)\Delta t)$. Note that $\Delta t$, for example, can be one minute, five minutes, 30 minutes, or one day. Here we take $\Delta t=1$ (one day) for notational simplicity. Thus the return series becomes $\{r_t\}^{\infty}_{t=1}$, which is given by
where $\Delta N(t) = N(t) - N(t-1)$ is the number of transactions within the interval $(t-1,t]$. We refer to the series $\{\Delta N(t)\}^{\infty}_{t=1}$ as the counts.
Next we present properties of the calendar-time returns $\{r_t\}$.
Lemma (ref) shows that the returns $\{r_t\}$ have memory parameter $d$. In this paper, we consider returns which can be obtained directly from prices rather than excess returns, which cannot. The long memory of returns may appear to contradict empirical analysis, but the long memory may be hard to detect when embedded in noise. In Table (ref) and Figure (ref), we present the estimated long memory parameter as well as the ACF and the log-log periodogram plots of returns simulated from the log price model with $d=0.35$. The estimated long memory parameters based on different bandwidths are all very close to zero and the lags in the ACF plot appear to be statistically insignificant \footnote{Indeed, $excess$ returns appear to have short memory in spite of the fact that in theory they would have memory parameter equal to $one$ if the risk free rate has a unit root as is found empirically.}.
Theorem (ref) implies that the realized variance is a long-memory process. Therefore, the calendar-time return and the realized variance derived from our model are long memory processes with the same memory parameter, which leads endogenously to a balanced predictive regression equation.
In Appendices (ref) and (ref), we provide method-of-moments estimators of the model parameters and a methodology for simulating realizations of the log price model. We show that the proposed estimators are consistent in Theorem (ref) and we evaluate the performance of these estimators in simulations.
{We use the aggregated past realized variance as the regressor to predict the aggregated future return. For concreteness, we assume that $\tilde{t}=1, 2, \cdots, \tilde{T} $ is measured in months, there are $m$ trading days per month, and $r_t$ is the return for the $t^{th}$ day. The aggregated return over the next $\tilde{H}$ months is given by
The realized variance over the past $\tilde{H}$ months is given by
To evaluate return predictability over the next $\tilde{H}$ months, one can compute the covariance between the future return and the past realized variance, $\mathrm{cov}(R_{\tilde{t}, \tilde{t}+\tilde{H}}, RV_{\tilde{t}-\tilde{H}, \tilde{t}})$ or the correlation
Suppose first that $d > 0$. In Lemma (ref), we show that $\mathrm{cov}(R_{\tilde{t}, \tilde{t}+\tilde{H}}\;, RV_{\tilde{t}-\tilde{H}, \tilde{t}})$ can be expressed as
and that $\mathrm{cov}(r_L, r^2_0) > 0$, for every positive integer L. Hence $\mathrm{cov}\left(R_{\tilde{t}, \tilde{t}+\tilde{H}}, RV_{\tilde{t}-\tilde{H}, \tilde{t}} \right)$ is positive and there is return predictability. As the horizon $\tilde{H}$ increases, the correlation between $R_{\tilde{t}, \tilde{t}+\tilde{H}}$ and $RV_{\tilde{t}-\tilde{H}, \tilde{t}}$ increases, and it converges to a function of the long-memory parameter $d$:
This theorem has the same form as Theorem 3 in Sizova (2013), though our assumptions differ from hers. On the other hand, if $d=0$, i.e. short memory, then there is no long-term return predictability as shown in the following theorem:
Hence as pointed out by Sizova (2013), it is the presence of the long memory component, strengthened by the aggregation of the returns and the squared returns that causes the long-term return predictability. From now on, we use the notation $\rho$ to represent $\mathrm{corr}\left(R_{\tilde{t},\tilde{t}+\tilde{H}}, RV_{\tilde{t}-\tilde{H},\tilde{t}}\right)$ and $\hat{\rho}$ to represent the sample correlation between these quantities as estimated by}
where $\overline{R_{\tilde{t},\tilde{t}+\tilde{H}}}$ and $\overline{RV_{\tilde{t}-\tilde{H},\tilde{t}}}$ are the sample means of $R_{\tilde{t}, \tilde{t}+\tilde{H}}$ and $RV_{\tilde{t}-\tilde{H}, \tilde{t}}$.
Theorem (ref) establishes that when $d=0$, $\rho \to 0$ as $\tilde{H} \to \infty$ and thus there is no long-run return predictability. Hence we wish to test the hypotheses of no long-run return predictability,
We consider two different asymptotic frameworks for the degree of aggregation $\tilde{H}$. The first framework is linear growth, $\tilde{H} = \theta \tilde{T}$, for $\theta \in (0,1)$, as used by Sizova (2013). The second framework is a power law, $\tilde{H} = \tilde{T}^{\kappa}$, for $\kappa \in (0,1)$. Under the linear aggregation framework, we will consider using $\hat{\rho}_{\tilde{H}, \tilde{T}}$ as the test statistic without further normalization. Under the power-law framework, we will consider the test statistic $\sqrt{\tilde{T}^{1-\kappa}}\hat{\rho}_{\tilde{H}, \tilde{T}}$, or $\sqrt{\tilde{T}^{1-\kappa}}\tilde{\rho}_{\tilde{H}, \tilde{T}}$, where $\tilde{\rho}_{\tilde{H}, \tilde{T}}$ is a slightly modified version of $\hat{\rho}_{\tilde{H}, \tilde{T}}$ given by ((ref)).
Under the linear growth framework, Sizova (2013) proved that $\hat{\rho}_{\tilde{H}, \tilde{T}}$ converges in distribution to a functional of fractional Brownian Motion subject to several assumptions (e.g.\;the dynamics of (14) and (15) as well as Assumptions 1 and 2 of that paper for both $d=0$ and $d> 0$). Under these dynamics and assumptions, the resulting asymptotic distribution $F_\rho(A^d(\tau),B^d(\tau))$ for $d=0$ could be used to obtain critical values for a hypothesis test based on $\hat{\rho}_{\tilde{H}, \tilde{T}}$. Though such a test is asymptotically correctly sized, it is inconsistent as implied by Sizova's (2013) Theorem 4 since the test statistic converges in distribution under both the null hypothesis and the alternative hypothesis. For our model, Figures (ref) and (ref) show the sampling distribution of $\hat{\rho}_{\tilde{H}, \tilde{T}}$ under the linear growth framework for $\tilde{H}$ based on simulated returns with $d=0$ and $d > 0$. These results suggest that $\hat{\rho}_{\tilde{H}, \tilde{T}}$ converges in distribution when $d=0$ and $d > 0$, and thus that the power of a hypothesis test for $d=0$ based on $\hat{\rho}_{\tilde{H}, \tilde{T}}$ will not go to $1$ in this framework. Further support for this conclusion is provided by Tables (ref) and (ref), where we calculate the variance of $\hat{\rho}_{\tilde{H}, \tilde{T}}$ for several values of $\tilde{T}$ under the linear growth framework for our model with $d=0$ and $d > 0$. The variance of $\hat{\rho}_{\tilde{H}, \tilde{T}}$ remains essentially constant. Apparently, the power of a correctly-sized test based on $\hat{\rho}_{\tilde{H}, \tilde{T}}$ would not approach $1$ as $\tilde{T} \to \infty$ under our model in the linear growth framework. Next we obtain the critical values of the correctly-sized test based on the simulated sampling distribution of $\hat{\rho}_{\tilde{H}, \tilde{T}}$ from our model under $d=0$ \footnote{Here we do not consider using the critical values obtained from the asymptotic null distribution in Sizova's (2013) Theorem 4 for the test. This is because our return model implies an endogenous relationship between the return and the realized variance, whereas Sizova's (2013) Theorem 4 assumes dynamic (as in her Eq.(15)) under which the realized variance is an exogenous variable.}. Tables (ref) and (ref) show the size and the power of the hypothesis test based on $\hat{\rho}_{\tilde{H}, \tilde{T}}$ under the linear growth framework with nominal size of $5\%$. This test is approximately correctly sized but has low power even as the sample size increases, as explained above.
We next consider the power-law framework, which we show is more promising than the linear framework for testing for long-run return predictability. The density plots of $\hat{\rho}_{\tilde{H}, \tilde{T}}$ under the power-law framework shown in Figures (ref) and (ref) and Tables (ref) and (ref) indicate that as $\tilde{T}$ increases, the variance of $\hat{\rho}_{\tilde{H}, \tilde{T}}$ decreases. Furthermore, the mean of $\hat{\rho}_{\tilde{H}, \tilde{T}}$ increases with $\tilde{T}$ when $d > 0$, perhaps approaching $2^{2d}-1$, though this is not clear based on the sample sizes considered. This suggests the possibility to construct a consistent test for long-run return predictability based on a rescaled version of $\hat{\rho}_{\tilde{H}, \tilde{T}}$. To determine an appropriate rescaling factor, we study the behavior of the variance of $\hat{\rho}_{\tilde{H}, \tilde{T}}$ as $\tilde{T}$ increases under the null hypothesis, $d=0$. To study the convergence rate of the variance of $\hat{\rho}_{\tilde{H}, \tilde{T}}$, we created a scatter plot for the logged variance of $\hat{\rho}_{\tilde{H}, \tilde{T}}$ and $\log {\tilde{T}}$ presented in Figure (ref). For each value of $\kappa$, the scatter plot appears to be linear. To confirm this, we regress the logged variance of $\hat{\rho}_{\tilde{H}, \tilde{T}}$ on $\log \tilde{T}$ and present the fitted equations in Table (ref) . The fitted slope coefficients are very close to the values of $\kappa -1$. Hence we conjecture that the variance of $\hat{\rho}_{\tilde{H}, \tilde{T}}$ is proportional to $\tilde{T}^{\kappa-1}$. This leads to the rescaled test statistic $\sqrt{\tilde{T}^{1-\kappa}}\hat{\rho}_{\tilde{H}, \tilde{T}}$.
Density plots in Figure (ref) as well as Shapiro-Wilk normality test results in Table $\ref{tab:k-s_normality_test}$ for $d=0$ indicate that $\sqrt{\tilde{T}^{1-\kappa}}\hat{\rho}_{\tilde{H}, \tilde{T}}$ may be asymptotically normal under the null hypothesis. Therefore, from now on we will focus on the power-law aggregation framework. We will also restrict attention to hypothesis testing under our model. We consider two options for hypothesis testing under the power law framework: (i) bootstrap method and (ii) asymptotic method.
In the bootstrap approach, given one realization of $\log P(t)$, $t \in (0,T]$, we compute the observed value of $\sqrt{\tilde{T}^{1-\kappa}}\hat{\rho}_{\tilde{H}, \tilde{T}}$, $\sqrt{\tilde{T}^{1-\kappa}}\hat{\rho}_{\tilde{H}, \tilde{T}}^{obs}$, and estimate the model parameters using the formulas in Section (ref). We then set $d=0$ and use the other estimated parameters to simulate 1000 replications of $\sqrt{\tilde{T}^{1-\kappa}}\hat{\rho}_{\tilde{H}, \tilde{T}}$. We use the $95^{th}$ percentile among the 1000 replications, $\sqrt{\tilde{T}^{1-\kappa}}{\hat{\rho}}_{95\%}$, as the critical value of the test. Thus, we compare $\sqrt{\tilde{T}^{1-\kappa}}{\hat{\rho}_{\tilde{H}, \tilde{T}}}^{obs}$ with ${\hat{\rho}}_{95\%}$ and reject the null hypothesis if $\sqrt{\tilde{T}^{1-\kappa}}\hat{\rho}_{\tilde{H}, \tilde{T}}^{obs} > \sqrt{\tilde{T}^{1-\kappa}}\hat{\rho}_{95\%}$.
Davidson and MacKinnon (1999, 2006) showed that subject to some assumptions, if the model parameter estimator is consistent under the null hypothesis, the rejection rate of a parametric bootstrap test converges to the nominal size, where the convergence rate depends on the order of convergence of the consistent estimator. In Theorem (ref), we prove that our model estimators are consistent under $d=0$. We find empirically that the rejection rates of our bootstrap hypothesis test (reported in Table (ref)) are almost never significantly different from the nominal size. Furthermore, the power (see Table (ref)) increases with the sample size and the power is highest for smaller values of $\kappa$. Nevertheless, we do not have a theoretical justification for the bootstrap hypothesis test in our framework.
Next, we develop an asymptotically correctly-sized consistent test based on $\sqrt{\tilde{T}^{1-\kappa}}\tilde{\rho}_{\tilde{H}, \tilde{T}}$, where $\tilde{\rho}_{\tilde{H}, \tilde{T}}$ is a slightly modified version of $\hat{\rho}_{\tilde{H}, \tilde{T}}$ (defined in ((ref))). When $d=0$, our return model implies that the daily return $r_t$ is $h$-dependent, where the lag $h$ is determined by the time-scaling constant $c$ (See ((ref)) for the definition of $h$). To take advantage of this fact (see Remark (ref) below), we propose a modified sample correlation coefficient $\tilde{\rho}_{\tilde{H}, \tilde{T}}$, where in the aggregation of returns, we skip the first $h$ daily returns from the first day of month $\tilde{t}$ and aggregate the remaining daily returns over the $\tilde{H}$ months to construct the modified aggregated return $\tilde{R}_{\tilde{t}, \tilde{t}+\tilde{H}}$ and then compute the sample correlation between $\tilde{R}_{\tilde{t}, \tilde{t}+\tilde{H}}$ and $RV_{\tilde{t}-\tilde{H}, \tilde{t}}$.
Since our paper is not in a local-to-zero framework, we consider a test based on $\hat{\rho}_{\tilde{H}, \tilde{T}}$ with critical value obtained from Theorem 4 of Sizova (2013) under the assumption $\tilde H/ \tilde T \rightarrow \theta \in (0,1)$. This theorem is based on a further assumption (Assumption 2) that a suitably normalized version of the regressor converges weakly to a fractional Brownian motion. Sizova presented several examples under which Assumptions 1 and 2 in that paper hold. The example that is closest to the context of this paper is Example 3, in which the continuous-time long-memory stochastic volatility model of Comte and Renault (1998) is assumed for the price. In this example, the regresssor is taken to be the integrated latent variance, $\sigma^2(t)=\int^t_{0}\sigma^2(u)du$, where $\sigma^2(u)$ is an unobserved latent variance, assumed to obey a stationary long-memory Ornstein-Uhlenbeck model. Sizova pointed out that for this regressor, Assumption 2 would hold, by an argument similar to the proof of the main result of Taqqu (1976). In comparison with Example 3 of Sizova (2013), we point out that the regressor in the example could not be used in practice even if the model assumed there held since $\sigma^2(t)$ is a latent process. Clearly, one could try using the realized variance to estimate $\sigma^2(t)$, and indeed Comte and Renault (1998, Proposition 5.1, Page 305) provide a theoretical result on the $L^2$ convergence and $L^1$ convergence of a suitably normalized version of realized variance to $\sigma^2(t)$. However, neither Sizova (2013) nor Comte and Renault (1998) has claimed or proved that Assumption 2 of Sizova (2013) would hold for some function of the realized variance. Furthermore, a practical issue that would arise in trying to estimate $\sigma^2(t)$ in the model of Comte and Renault (1998) is that realized variance is not a useful quantity when based on very short time increments because the squared returns are mostly zero. This is often attributed to microstructure noise in the literature, but in our viewpoint the problem is deeper, and is most fundamentally attributable to the step-function nature of the log price process.
Much of the literature on long-horizon return predictability finds that the evidence for predictability improves as the aggregation horizon increases. Many of these studies, however, were subject to spurious findings due to the amplified correlations in the predictors and the returns caused by the overlapping aggregation.
Our work provides a theoretical framework for exploring the correlation between the long-horizon returns and the realized variance. We propose a parametric transaction-level model for the continuous-time log price based on a pure-jump process. The model determines returns and realized variance at any level of aggregation and reveals a channel through which the statistical properties of the transaction-level data are propagated to induce long-horizon return predictability. Specifically, the memory parameter of returns at the transaction level propagates unchanged to the returns and realized variance at any calendar-time frequency, which leads to a balanced predictive regression. We show that if the long-memory parameter is zero, then there is no long-horizon return predictability. This extends a result shown previously by Sizova (2013) based on discrete-time assumptions. In addition, we provide consistent estimators for the model parameters. To assess return predictability, we propose a hypothesis test based on a power-law aggregation of the returns and realized variance. We demonstrate that the proposed test is asymptotically correctly-sized and is consistent, whereas the related test of Sizova (2013), which used a linear aggregation framework, is inconsistent.