A Pure-Jump Transaction-Level Price Model Yielding Cointegration, Leverage, and Nonsynchronous Trading Effects

We propose a new transaction-level bivariate log-price model, which yields fractional or standard cointegration. The model provides a link between market microstructure and lower-frequency observations. The two ingredients of our model are a Long Memory Stochastic Duration process for the waiting times between trades, and a pair of stationary noise processes which determine the jump sizes in the pure-jump log-price process. Our model includes feedback between the disturbances of the two log-price series at the transaction level, which induces standard or fractional cointegration for any fixed sampling interval. We prove that the cointegrating parameter can be consistently estimated by the ordinary least-squares estimator, and obtain a lower bound on the rate of convergence. We propose transaction-level method-of-moments estimators of the other parameters in our model and discuss the consistency of these estimators. We then use simulations to argue that suitably-modified versions of our model are able to capture a variety of additional properties and stylized facts, including leverage, and portfolio return autocorrelation due to nonsynchronous trading. The ability of the model to capture these effects stems in most cases from the fact that the model treats the (stochastic) intertrade durations in a fully endogenous way.Tick Time; Long Memory Stochastic Duration; Information Share

Semiparametric Estimation of Fractional Cointegrating Subspaces

We consider a common components model for multivariate fractional cointegration, in which the s>=1 components have different memory parameters. The cointegrating rank is allowed to exceed 1. The true cointegrating vectors can be decomposed into orthogonal fractional cointegrating subspaces such that vectors from distinct subspaces yield cointegrating errors with distinct memory parameters, denoted by d_k for k=1,...,s. We estimate each cointegrating subsspace separately using appropriate sets of eigenvectors of an averaged periodogram matrix of tapered, differenced observations. The averaging uses the first m Fourier frequencies, with m fixed. We will show that any vector in the k'th estimated coingetraging subspace is, with high probability, close to the k'th true cointegrating subspace, in the sense that the angle between the estimated cointegrating vector and the true cointegrating subspace converges in probability to zero. The angle is O_p(n^{- \alpha_k}), where n is the sample size and \alpha_k is the shortest distance between the memory parameters corresponding to the given and adjacent subspaces. We show that the cointegrating residuals corresponding to an estimated cointegrating vector can be used to obtain a consistent and asymptotically normal estimate of the memory parameter for the given cointegrating subspace, using a univariate Gaussian semiparametric estimator with a bandwidth that tends to \infty more slowly than n. We also show how these memory parameter estimates can be used to test for fractional cointegration and to consistently identify the cointegrating subspaces.Fractional Cointegration; Long Memory; Tapering; Periodogram

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

A Pure-Jump Transaction-Level Price Model Yielding Cointegration, Leverage, and Nonsynchronous Trading Effects

We propose a new transaction-level bivariate log-price model, which yields fractional or standard cointegration. To the best of our knowledge, all existing models for cointegration require the choice of a fixed sampling frequency Delta t. By contrast, our proposed model is constructed at the transaction level, thus determining the properties of returns at all sampling frequencies. The two ingredients of our model are a Long Memory Stochastic Duration process for the waiting times tau(k) between trades, and a pair of stationary noise processes ( e(k) and eta(k) ) which determine the jump sizes in the pure-jump log-price process. The e(k), assumed to be iid Gaussian, produce a Martingale component in log prices. We assume that the microstructure noise eta(k) obeys a certain model with memory parameter d(eta) in (-1/2,0) (fractional cointegration case) or d(eta) = -1 (standard cointegration case). Our log-price model includes feedback between the shocks of the two series. This feedback yields cointegration, in that there exists a linear combination of the two components that reduces the memory parameter from 1 to 1+d(eta) in (0.5,1) and (0). Returns at sampling frequency Delta t are asymptotically uncorrelated at any fixed lag as Delta t increases. We prove that the cointegrating parameter can be consistently estimated by the ordinary least-squares estimator, and obtain a lower bound on the rate of convergence. We propose transaction-level method-of-moments estimators of several of the other parameters in our model. We present a data analysis, which provides evidence of fractional cointegration. We then consider special cases and generalizations of our model, mostly in simulation studies, to argue that the suitably-modified model is able to capture a variety of additional properties and stylized facts, including leverage, portfolio return autocorrelation due to nonsynchronous trading, Granger causality, and volatility feedback. The ability of the model to capture these effects stems in most cases from the fact that the model treats the (stochastic) intertrade durations in a fully endogenous way.Tick Time; Long Memory Stochastic Duration; Information Share; Granger causality

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 autocovariances at fixed lags. Finite collections of standardized 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.Long Memory; Structural Change

Hypothesis Testing in Predictive Regressions

We propose a new hypothesis testing method for multi-predictor regressions with finite samples, where the dependent variable is regressed on lagged variables that are autoregressive. It is based on the augmented regressiom method (ARM; Amihud and Hurvich (2004)), which produces reduced-bias coefficients and is easy to implement. The method's usefulness is demonstrated by simulations and by an empirical example, where stock returns are predicted by dividend yield and by bond yield spread. For single-predictor regressions, we show that the ARM outperforms bootstrapping and that the ARM performs better than Lewellen's (2003) method in many situations.Augmented Regression Method (ARM); Bootstrapping; Hypothesis Testing

Tracing the Source of Long Memory in Volatility

We study the effects of trade duration properties on dependence in counts (number of transactions) and thus on dependence in volatility of returns. A return model is established to link counts and volatility. We present theorems as well as a conjecture relating properties of durations to long memory in counts and thus in volatility. We then apply several parametric duration models to empirical trade durations and discuss our findings in the light of the theorems and conjecture.

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 $t \rightarrow \infty$, with the same memory parameter $d \in [0,1/2)$ that was assumed for the durations. Thus, these conditions ensure that the memory in durations propagates to the same memory parameter in counts and therefore in realized volatility. 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 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 short-term counts to have long memory. In other words, aggregation ultimately destroys all autocorrelation in counts, if and only if the counts have short memory.Long Memory Stochastic Duration, Autoregressive Conditional Duration, Rosenthal-type Inequality.