Conditions for the Propagation of Memory Parameter from Durations to Counts and Realized Volatility

We establish sufficient conditions on durations that are stationary with finite variance and memory parameter d 2 [0; 1=2) to ensure that the corresponding counting process N(t) satisfies VarN(t) » Ct2d+1 (C > 0) as t ! 1, with the same memory parameter d 2 [0; 1=2) that was assumed for the durations. Thus, these conditions ensure that the memory parameter in durations propagates to the same memory parameter in the counts. We then show that any Autoregressive Conditional Duration ACD(1,1) model with a sufficient number of finite moments yields short memory in counts, while any Long Memory Stochastic Duration model with and all finite moments yields long memory in counts, with the same d. Next, we present a result implying that the only way for a series of counts aggregated over a long time period to have nontrivial autocorrelation is for the counts to have long memory. In other words, aggregation ultimately destroys all autocorrelation in counts, if and only if the counts have short memory. Finally, under assumptions on the pure-jump price process, we show that the memory parameter in durations propagates all the way to the realized volatility, under both calendar-time sampling and transaction-time sampling.Statistics Working Papers Serie

TESTING FOR LONG MEMORY IN VOLATILITY

We consider the asymptotic behavior of log-periodogram regression estimators of the memory parameter in long-memory stochastic volatility models, under the null hypothesis of short memory in volatility. We show that in this situation, if the periodogram is computed from the log squared returns, then the estimator is asymptotically normal, with the same asymptotic mean and variance that would hold if the series were Gaussian. In particular, for the widely used GPH estimator dGPH under the null hypothesis, the asymptotic mean of mýdGPH is zero and the asymptotic variance is piò/24 where m is the number of Fourier frequencies used in the regression. This justifies an ordinary Wald test for long memory in volatility based on the log periodogram of the log squared returns.Statistics Working Papers Serie

Estimating long memory in volatility

We consider semiparametric estimation of the memory parameter in a model which includes as special cases both the long-memory stochastic volatility (LMSV) and fractionally integrated exponential GARCH (FIEGARCH) models. Under our general model the logarithms of the squared returns can be decomposed into the sum of a long-memory signal and a white noise. We consider periodogram-based estimators which explicitly account for the noise term in a local Whittle criterion function. We allow the optional inclusion of an additional term to allow for a correlation between the signal and noise processes, as would occur in the FIEGARCH model. We also allow for potential nonstationarity in volatility, by allowing the signal process to have a memory parameter d 1=2. We show that the local Whittle estimator is consistent for d 2 (0; 1). We also show that a modi ed version of the local Whittle estimator is asymptotically normal for d 2 (0; 3=4), and essentially recovers the optimal semiparametric rate of convergence for this problem. In particular if the spectral density of the short memory component of the signal is suficiently smooth, a convergence rate of n2=5-δ for d 2 (0; 3=4) can be attained, where n is the sample size and δ > 0 is arbitrarily small. This represents a strong improvement over the performance of existing semiparametric estimators of persistence in volatility. We also prove that the standard Gaussian semiparametric estimator is asymptotically normal if d = 0. This yields a test for long memory in volatility.Statistics Working Papers Serie

Propagation of Memory Parameter from Durations to Counts

We establish sufficient conditions on durations that are stationary with finite variance and memory parameter $d \in [0,1/2)$ to ensure that the corresponding counting process $N(t)$ satisfies $\textmd{Var} N(t) \sim C t^{2d+1}$ ($C>0$) as $t \to \infty$, with the same memory parameter $d \in [0,1/2)$ that was assumed for the durations. Thus, these conditions ensure that the memory in durations propagates to the same memory parameter in counts and therefore in realized volatility. We then show that any utoregressive Conditional Duration ACD(1,1) model with a sufficient number of finite moments yields short memory in counts, while any Long Memory Stochastic Duration model with $d>0$ and all finite moments yields long memory in counts, with the same $d$

Conditions for the Propagation of Memory Parameter from Durations to Counts and Realized Volatility

We establish sufficient conditions on durations that are stationary with finite variance and memory parameter $d \in [0,1/2)$ to ensure that the corresponding counting process $N(t)$ satisfies $Var N(t) \sim C t^{2d+1}$ ($C>0$) as $t \rightarrow \infty$, with the same memory parameter $d \in [0,1/2)$ that was assumed for the durations. Thus, these conditions ensure that the memory parameter in durations propagates to the same memory parameter in the counts. We then show that any Autoregressive Conditional Duration ACD(1,1) model with a sufficient number of finite moments yields short memory in counts, while any Long Memory Stochastic Duration model with $d>0$ and all finite moments yields long memory in counts, with the same $d$. Finally, we provide some results about the propagation of long memory to the empirically-relevant case of realized variance estimates affected by market microstructure noise contamination.New York University, Stern School of Business, CeDER, Universite Paris

Estimating long memory in volatility

We consider semiparametric estimation of the memory parameter in a model which includes as special cases both the long-memory stochastic volatility (LMSV) and fractionally integrated exponential GARCH (FIEGARCH) models. Under our general model the logarithms of the squared returns can be decomposed into the sum of a long-memory signal and a white noise. We consider periodogram-based estimators which explicitly account for the noise term in a local Whittle criterion function. We allow the optional inclusion of an additional term to allow for a correlation between the signal and noise processes, as would occur in the FIEGARCH model. We also allow for potential nonstationarity in volatility, by allowing the signal process to have a memory parameter d 1=2. We show that the local Whittle estimator is consistent for d 2 (0; 1). We also show that a modi ed version of the local Whittle estimator is asymptotically normal for d 2 (0; 3=4), and essentially recovers the optimal semiparametric rate of convergence for this problem. In particular if the spectral density of the short memory component of the signal is suficiently smooth, a convergence rate of n2=5-δ for d 2 (0; 3=4) can be attained, where n is the sample size and δ > 0 is arbitrarily small. This represents a strong improvement over the performance of existing semiparametric estimators of persistence in volatility. We also prove that the standard Gaussian semiparametric estimator is asymptotically normal if d = 0. This yields a test for long memory in volatility.Statistics Working Papers Serie

Asymptotics for Duration-Driven Long Range Dependent Processes

We consider processes with second order long range dependence resulting from heavy tailed durations. We refer to this phenomenon as duration-driven long range dependence (DDLRD), as opposed to the more widely studied linear long range dependence based on fractional differencing of an $iid$ process. We consider in detail two specific processes having DDLRD, originally presented in Taqqu and Levy (1986), and Parke (1999). For these processes, we obtain the limiting distribution of suitably standardized discrete Fourier transforms (DFTs) and sample autocovariances. At low frequencies, the standardized DFTs converge to a stable law, as do the standardized sample autocovariances at fixed lags. Finite collections of standardized sample autocovariances at a fixed set of lags converge to a degenerate distribution. The standardized DFTs at high frequencies converge to a Gaussian law. Our asymptotic results are strikingly similar for the two DDLRD processes studied. We calibrate our asymptotic results with a simulation study which also investigates the properties of the semiparametric log periodogram regression estimator of the memory parameter