## Abstract

In this paper we develop a dynamic model for integer counts to capture fundamental properties of financial prices at the transaction level. Our model relies on an autoregressive multinomial component for the direction of the price change and a dynamic count data component for the size of the price changes. Since the model is capable of capturing a wide range of discrete price movements it is particularly suited for financial markets where the trading intensity is moderate or low. We present the model at work by applying it to transaction data of two shares traded at the NYSE traded over a period of one trading month. We show that the model is well suited to test some theoretical implications of the market microstructure theory on the relationship between price movements and other marks of the trading process. Based on density forecast methods modified for the case of discrete random variables we show that our model is capable to explain large parts of the observed distribution of price changes at the transaction level.

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## Notes

The data used stems from the NYSE Trade and Quote database. We have removed all trades outside the regular trading hours and each day's first trade, to circumvent contamination due to the opening call auction at the NYSE. Besides, all trades are treated as split transactions, if they exhibited exactly the same timestamp. In this case we have aggregated their volume to one transaction and we have assigned the last price in the sequence to the aggregated transaction.

See, for example, Campbell et al. (1997), Chap. 3.2.

Alternatively, one could specify the p.d.f. of the transformed count

*Y*_{ i }−1 conditional on*Y*_{ i }>0 using a standard count data approach. This approach was adopted by Rydberg and Shephard (2003) in their decomposition model.According to the classification by Cox (1981), our ACM model belongs to the class of observationally driven models where time dependence arises from a recursion on lagged endogenous variables. Alternatively, our model could be based on a parameter driven specification, in which the log–odds ratios Λ

_{ i }are determined by a dynamic latent process. However, the estimation and the diagnostics of the latter approach results in a substantially higher computational burden than for the ACM model. On the other hand, models driven by latent processes are usually more parsimonious than comparable dynamic models based on lagged dependent variables. A comparison of the two alternatives should be the subject of future research.See Russel and Engle (2002) for a more detailed discussion of the stochastic properties of the ACMARMA(p,q) model.

See, for example, Lütkepohl (1993).

Computed as variance over mean.

See, for example, Cameron and Trivedi (1998) (Ch. 4.2.2.).

See, for instance, Nelson (1991).

Computed as \(z_{{1,2}} = {{_{1} }} {2} {1} {2}{{^{2}_{1} + 4_{2} } }.\)

See O'Hara (1995) for a comprehensive survey on the theoretical literature on market microstructure.

The LR-test clearly rejects the null hypothesis of symmetric price reactions.

## References

Andersen TG (1996) Return volatility and trading volume: an information flow interpretation of stochastic volatility. J Financ 51:169–204, 169–231

Ball C (1988) Estimation bias induced by discrete security prices. J Financ 43:841–865

Bauwens L, Giot P (2003) Asymmetric ACD models: introducing price information in ACD models with a two-state transition model. Empir Econ 28:709–731

Berndt EK, Hall BH, Hall RE, Hausman JA (1974) Estimation and inference in nonlinear structural models. Ann Econ Soc Meas 3/4:653–665

Bollerslev T, Melvin M (1994) Bid-ask spreads and volatility in the foreign exchange market – an empirical analysis. J Int Econ 36:355–372

Cameron AC, Trivedi PK (1998) Regression analysis of count data. Cambridge University Press, Cambridge

Campbell JY, Lo AW, MacKinlay AC (1997) The econometrics of financial markets. Princeton University Press

Cho DC, Frees EW (1988) Estimating the volatility of discrete stock prices. J Financ 43:451–466

Clark PK (1973) A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41:135–155

Cox D (1981) Statistical analysis of time series: some recent developments. Scand J Statist 8:93–115

Davis R, Dunsmuir W, Streett S (2003) Observation driven models for poisson counts. Biometrika 90:777–790

Denuit M, Lambert P (2005) Constraints on concordance measures in bivariate discrete data. J Multivar Anal 93:40–57

Diamond DW, Verrecchia RE (1987) Constraints on short-selling and asset price adjustment to private information. J Financ Econ 18:277–311

Diebold FX, Gunther TA, Tay AS (1998) Evaluating density forecasts, with applications to financial risk management. Int Econ Rev 39:863–883

Easley D, O'Hara M (1987) Price, trade size, and information in securities markets. J Financ Econ 19:69–90

Easley D, O'Hara M (1992) Time and the process of security price adjustment. J Financ 47:577–607

Engle R (2000) The econometrics of ultra-high-frequency data. Econometrica 68(1):1–22

Harris L (1990) Estimation of stock variances and serial covariances from discrete observations. J Financ Quant Anal 25:291–306

Hausman JA, Lo AW, MacKinlay AC (1992) An ordered probit analysis of transaction stock prices. J Financ Econ 31:319–379

Jung R, Liesenfeld R (2001) Estimating time series models for count data using efficient importance sampling. Allg Stat Arch 85:387–407

Liesenfeld R (1998) Dynamic bivariate mixture models: modelling the behavior of prices and trading volume. J Bus Econ Stat 16:101–109

Liesenfeld R (2001) A generalized bivariate mixture model for stock price volatility and trading volume. J Econ 104:141–178

Lütkepohl H (1993) Introduction to multiple time series analysis. Springer, Berlin Heidelberg New York

Mullahy J (1986) Specification and testing of some modified count data models. J Econ 33:341–365

Nelson D (1991) Conditional heteroskedasticity in asset returns: a new approach. J Econ 43:227–251

O'Hara M (1995) Market microstructure theory. Blackwell Publishers, Oxford

Pohlmeier W, Ulrich V (1995) An econometric model of the two-part decision process in the demand for health. J Hum Resour 30:339–361

Russel J, Engle R (2002) Econometric analysis of discrete-valued, irregularly-spaced financial transactions data using a new autoregressive conditional multinomial model. University of California, San Diego (revised version of Discussion Paper 98–10)

Russell JR, Engle RF (1998) Econometric analysis of discrete-valued, irregularly-spaced financial transactions data using a new autoregressive conditional multinomial model, presented at Second International Conference on High Frequency Data in Finance, Zurich, Switzerland

Rydberg T, Shephard N (2002) Dynamics of trade-by-trade price movements: decomposition and models. Discussion paper, Nuffiled College, Oxford University, will be published in Journal of Financial Econometrics

Rydberg T, Shephard N (2003) Dynamics of trade-by-trade price movements: decomposition and models. J Financ Econ 1:2–25

Stevens W (1950) Fiducial limits of the parameter of a discontinuous distribution. Biometrika 37:117–129

Tauchen GE, Pitts M (1983) The price variability-volume relationship on speculative markets. Econometrica 51:485–505

Zeger S (1988) A regression model for time series of counts. Biometrika 75:621–629

## Acknowledgements

For helpful comments and suggestions we like to thank Bernd Fitzenberger, Nikolaus Hautsch, Neil Shephard, Gerd Ronning, Timo Teräsvirta and Pravin Trivedi. The work of the second and third co-author is supported by the Friedrich Thyssen Foundation and the European Community's Human Potential Programme under contract HPRN-CT-2002-00232, Microstructure of Financial Markets in Europe (MICFINMA), respectively.

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## Appendix

### Appendix

This appendix shows that under a correctly specified model for *Y*
_{
i
}, the *u*
_{
i
}'s drawn from the uniform distributions (4.3) follow a uniform distribution on the interval [0, 1].^{Footnote 15} Consider a discrete random variable *Y* with support \(\Delta \subseteq \mathbb{Z}\), and let *u* be a continuous random variable with the following conditional uniform distribution

where the boundaries are *u*
_{
y
}
^{l}=Pr(*Y≤y*−1), *u*
_{
y
}
^{u}=Pr(*Y*≤*y*) (for ease of notation we ignore the index *i* for the variables *u* and *Y*). Then, the c.d.f. of the unconditional distribution of *u* is

with

where \({\user1{\mathcal{I}}}_{A} {\left( z \right)}\) is an indicator function which is 1 if *z*∈*A* and zero for *z*∉*A*. Inserting Eqs. (A.3) and (A.4) into Eq. (A.2), we obtain

Assuming that c∈[*u*
_{
j
}
^{l}, *u*
_{
j
}
^{u}], *j*∈Δ, we find

which represents the c.d.f. of a uniform distribution on the interval [0, 1].

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Liesenfeld, R., Nolte, I. & Pohlmeier, W. Modelling financial transaction price movements: a dynamic integer count data model.
*Empirical Economics* **30**, 795–825 (2006). https://doi.org/10.1007/s00181-005-0001-1

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DOI: https://doi.org/10.1007/s00181-005-0001-1

### Keywords

- Financial transaction prices
- Autoregressive conditional multinomial model
- GLARMA
- Count data
- Market microstructure effects

### JEL

- C22
- C25
- G10