Limit Laws in Transaction-Level Asset Price Models

Aue, Alexander; Horváth, Lajos; Hurvich, Clifford M.; Soulier, Philippe

doi:10.1017/s0266466613000406

Cited by 4 publications

(7 citation statements)

References 28 publications

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“…For t ⩾0, define

x_{i, n} (t) = n^{- 1 / 2} \sum_{k = 1}^{N_{i} (n t)} ζ_{i, k}

. The present assumptions imply assumptions 2.1, 2.2, 2.3, and 2.4 of Aue et al (); thus, Theorem therein implies that 0

x_{i, n} \overset{f i . d i .}{\to} \sqrt{λ} σ_{i} B_{i}

, where B i are mutually independent standard Brownian motions. Since the sequences { e i , k } are independent of the point processes N i , the sequences of processes x i , n and { n − γ ( N i ( n t )− n λ i t ), t ⩾0}, i =1,2 converge jointly.…”

Section: Proofsmentioning

confidence: 74%

“…This allows for a leverage effect. See Aue et al (2014). Assumption 2.2 below implies that the microstructure noise becomes negligible after aggregation.…”

Section: A Univariate Model For Log Pricementioning

confidence: 99%

“…Next, we discuss the use of time deformation. This discussion is taken from Aue et al (2014). Let f be a deterministic or random function such that f is nondecreasing and has càdlàg paths with probability one.…”

Section: −→mentioning

confidence: 99%

“…This allows for a leverage effect. See Aue et al (2014). The leverage is then defined as the correlation between current calendar-time return r k and absolute value of the next calendar-time return |r k+1 | leverage = corr(r k , |r k+1 |) .…”

Section: Leverage Effectmentioning

confidence: 99%

“…Recent interest in the point process approach to modeling transactionlevel data was spurred by the seminal paper of Engle and Russell (1998), who proposed a model for inter-trade durations. Other work on modeling transactionlevel data as point processes and/or constructing duration models includes that of Prigent (2001), Bowsher (2007), Billingsley (1968), Hautsch (2012), Bacry et al (2011), Deo et al (2009), Deo et al (2010), Hurvich and Wang (2010), Aue et al (2014), Shenai (2012), Chen et al (2012).…”

Section: Introductionmentioning

confidence: 99%

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Drift in Transaction‐Level Asset Price Models

Cao

Hurvich

Soulier

2017

Journal Time Series Analysis

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We study the effect of drift in pure-jump transaction-level models for asset prices in continuous time, driven by point processes. The drift is assumed to arise from a non-zero mean in the efficient shock series. It follows that the drift is proportional to the driving point process itself, that is, the cumulative number of transactions. This link reveals a mechanism by which properties of intertrade durations (such as heavy tails and long memory) can have a strong impact on properties of average returns, thereby potentially making it extremely difficult to determine long-term growth rates or to reliably detect an equity premium. We focus on a basic univariate model for log price, coupled with general assumptions on the point process that are satisfied by several existing flexible models, allowing for both long memory and heavy tails in durations. Under our pure-jump model, we obtain the limiting distribution for the suitably normalized log price. This limiting distribution need not be Gaussian and may have either finite variance or infinite variance. We show that the drift can affect not only the limiting distribution for the normalized log price but also the rate in the corresponding normalization. Therefore, the drift (or equivalently, the properties of durations) affects the rate of convergence of estimators of the growth rate and can invalidate standard hypothesis tests for that growth rate. As a remedy to these problems, we propose a new ratio statistic that behaves more robustly and employ subsampling methods to carry out inference for the growth rate. Our analysis also sheds some new light on two long-standing debates as to whether stock returns have long memory or infinite variance. .Nevertheless, it must be recognized that time series of asset returns in discrete (say, equally spaced) time are still in widespread use and indeed may be the only recorded form of the data that encompass many decades. Such long historical series are of importance for understanding long-term trends (a prime focus of this article) and, arguably, for a realistic assessment of risk. So given the ubiquitous nature of the time series data and also keeping in mind the underlying price-generating process that occurred at the level of individual transactions, it is important to make sure that transaction-level models obey the stylized facts, not only for the intertrade durations but also for the lower-frequency time series.It has been observed empirically that time series of financial returns are weakly autocorrelated (although perhaps not completely uncorrelated), while squared returns or other proxies for volatility show strong autocorrelations that decay very slowly with increasing lag, possibly suggesting long memory (see Andersen et al. (2001)). It is also generally accepted that such time series show asymmetries, such as a correlation between the current return and the next period's squared return, and this effect (often referred to traditionally as the 'leverage effect') is addressed, for example, by the exponential generaliz...

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“…For t ⩾0, define

x_{i, n} (t) = n^{- 1 / 2} \sum_{k = 1}^{N_{i} (n t)} ζ_{i, k}

. The present assumptions imply assumptions 2.1, 2.2, 2.3, and 2.4 of Aue et al (); thus, Theorem therein implies that 0

x_{i, n} \overset{f i . d i .}{\to} \sqrt{λ} σ_{i} B_{i}

Section: Proofsmentioning

confidence: 74%

“…This allows for a leverage effect. See Aue et al (2014). Assumption 2.2 below implies that the microstructure noise becomes negligible after aggregation.…”

Section: A Univariate Model For Log Pricementioning

confidence: 99%

Section: −→mentioning

confidence: 99%

Section: Leverage Effectmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Drift in Transaction‐Level Asset Price Models

Cao

Hurvich

Soulier

2017

Journal Time Series Analysis

Self Cite

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show abstract

Estimation ofα,βand portfolio weights in a pure-jump model with long memory in volatility

Zhang

Hurvich

2022

Stochastic Processes and their Applications

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A Pure-Jump Transaction-Level Price Model Yielding Cointegration

Hurvich

Wang

2010

Journal of Business & Economic Statistics

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We propose a new transaction-level bivariate log-price model that 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, {τ k }, between trades and a pair of stationary noise processes, ({e k } and {η k }), 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 t. We prove that the cointegrating parameter can be consistently estimated by the ordinary least squares estimator, and we 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.

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Limit Laws in Transaction-Level Asset Price Models

Cited by 4 publications

References 28 publications

Drift in Transaction‐Level Asset Price Models

Drift in Transaction‐Level Asset Price Models

Estimation ofα,βand portfolio weights in a pure-jump model with long memory in volatility

A Pure-Jump Transaction-Level Price Model Yielding Cointegration

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