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Modelling financial transaction price movements: a dynamic integer count data model


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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  1. 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.

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

  3. 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.

  4. 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.

  5. See Russel and Engle (2002) for a more detailed discussion of the stochastic properties of the ACMARMA(p,q) model.

  6. See, for example, Lütkepohl (1993).

  7. Computed as variance over mean.

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

  9. Similar to the alternative specification discussed in the context of the ACM model one could also specify a dynamic latent process for ω i . See Zeger (1988) and Jung and Liesenfeld (2001) for examples.

  10. See, for instance, Nelson (1991).

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

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

  13. See Andersen (1996) and Liesenfeld (1998, 2001) for extensions of the mixture models.

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

  15. This technique of continuization is widely used to describe the properties of the p.d.f. of discrete random variables, see e.g. Stevens (1950) and Denuit and Lambert (2005).


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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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Correspondence to Winfried Pohlmeier.



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

$$u \sim {\user1{\mathcal{U}}}{\left( {u^{l}_{y} ,u^{u}_{y} } \right)},$$

where the boundaries are u y l=Pr(Y≤y−1), u y u=Pr(Yy) (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

$$\Pr {\left( {u \leqslant c} \right)} = {\sum\limits_{y \in \Delta } {\Pr {\left( {\left. {u \leqslant c} \right|Y = y} \right)}\Pr {\left( {Y = y} \right)},\quad c \in {\left[ {0,1} \right]},} }$$


$$\Pr {\left( {\left. {u \leqslant c} \right|Y = y} \right)} = \frac{{c - u^{l}_{y} }} {{u^{u}_{y} - u^{l}_{y} }}{\user1{\mathcal{I}}}_{{\left[ {{\user1{}}u^{l}_{y} ,u^{u}_{y} } \right)}} {\left( c \right)} + {\user1{\mathcal{I}}}_{{{\left[ {u^{u}_{y} ,1} \right]}}} {\left( c \right)}$$
$$\Pr {\left( {Y = y} \right)} = u^{u}_{y} - u^{l}_{y} ,$$

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

$${\Pr {\left( {u \le c} \right)} = {\sum\limits_{y \in \Delta } {{\left\{ {{\left( {c - u^{l}_{y} } \right)}{\user1{{\cal I}}}_{{\left[ {{\user1{}}u^{l}_{y} ,u^{u}_{y} } \right)}} {\left( c \right)} + {\left( {u^{u}_{y} - u^{l}_{y} } \right)}{\user1{{\cal I}}}_{{{\left[ {u^{u}_{y} ,1} \right]}}} {\left( c \right)}} \right\}}} }.}$$

Assuming that c∈[u j l, u j u], j∈Δ, we find

$$\begin{array}{*{20}c} {\Pr {\left( {u \leqslant c} \right)} = c{\user1{\mathcal{I}}}_{{{\left[ {0,1} \right]}}} {\left( c \right)} - \Pr {\left( {Y \leqslant j - 1} \right)} + \ldots } \\{ + \Pr {\left( {Y \leqslant j - 3} \right)} - \Pr {\left( {Y \leqslant j - 4} \right)}} \\{ + \Pr {\left( {Y \leqslant j - 2} \right)} - \Pr {\left( {Y \leqslant j - 3} \right)}} \\{ + \Pr {\left( {Y \leqslant j - 1} \right)} - \Pr {\left( {Y \leqslant j - 2} \right)}} \\{ = c{\user1{\mathcal{I}}}_{{{\left[ {0,1} \right]}}} {\left( c \right)},} \\\end{array} $$

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).

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


  • C22
  • C25
  • G10