1. On the components of the transaction process
1.1. Models of the transaction process. This chapter concentrates on the analysis of the relationship between the two components of the process of price changes and the time between transactions. Other variables such as the volume associated with a certain transaction or the transaction costs, i.e. the bid-ask spread prevailing at this point in time are left for a later treatment as these can be assessed in a straightforward extension of the econometric model suggested in this work.
As this work concentrates on the analysis of transaction data with some well-known properties which have been outlined in chapter 2, approaches which are not adapted to the properties of the data will not be discussed in detail. Hasbrouck (1991) e.g. in his work on price changes and quote revisions plainly applies standard VAR techniques, which can be justified in some sense, as he concentrates on the effects of explanatory variables for the conditional mean function. This work is extended by Dufour and Engle (2000). They build on the setup by Hasbrouck in a way which allows the parameters of the process of price changes to depend on the trade intensity modelled by an ACD specification. Ghysels and Jasiak (1998) and Grammig and Wellner (1999) modify their estimators of volatility to account for the irregular spacing of observations. However, they apply a standard GARCH approach going back to the work of Drost and Nijman (1993) which allows the scaling of a modified GARCH model to account for a change in the aggregation interval.
In the literature there are mainly two approaches available which take the mi-crostructure properties of transaction data into account and which allow for a flexible time series structure to capture the salient features of the data. First, there is the already mentioned approach by Rydberg and Shephard (1998) which decomposes
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the price change dt observed at time t into three components. An indicator for the occurrence of a price change, an indicator for the sign of the price change and the size of the price change itself. This decomposition allows to apply dynamic exten-sions to count data models in the context of GLMs as they were already outlined in chapter 3. A feature of the modeling strategy pursued by Rydberg and Shephard worth emphasizing is the discrete nature of time. They consider each second of the trading day and characterize it by the above mentioned random variables.
Second, Russell and Engle (1998) pursue a somewhat different approach by specify-ing a dynamic model for the evolution of price changes in the form of a generalized Markov Chain which was also outlined in chapter 3. Their approach is to com-bine this model with an ACD model, which is described in chapter 4, to obtain a joint model of price changes and time between transactions in a continuous time framework.
The model proposed in this work combines both approaches of Rydberg and hard and Russell and Engle. The decomposition framework of Rydberg and Shep-hard is modified to allow for a joint model of the size of price changes and the time between transactions within a continuous time framework. This approach yields a parsimonious model compared to both alternatives, as will be pointed out in the course of this chapter.
1.2. On the relationship of components of the transaction process. All of these procedures share the property that the joint distribution of the potentially vector valued price process is decomposed into marginal and conditional models which permit the estimation of the involved parameters and in turn an economic interpretation of the observed relationships. Assume for the moment that the trans-action process yt consists of two components, the time between transactions τt and the price change dt, i.e. yt := h
τt dt i
, the conditional joint density is given by f(τt, dt|¯τt−1,d¯t−1,x¯t−1, θ) using θ as the parameter vector of the joint distribution.
Engle, Hendry, and Richard (1983) have defined the notions of weak, strong, and super exogeneity to describe the properties of the joint distribution concisely. The
concepts of exogeneity are always related to a set of parameters which are of interest to the researcher. The different concepts indicate which conclusions might be drawn from particular distributional assumptions or rather how results of inference are to be interpreted economically.
If one definesfd|τ as the conditional density with parametersθ1 of the price changes dtgiven the standard information set and the contemporaneous time between trans-actions τt and fτ as the marginal density of the time between transactions with parametersθ2, then weak exogeneity as given in Engle, Hendry, and Richard (1983, def. 2.5) expresses whether inference on θ1 is independent from inference on θ2. Consider the factorization of the joint density
f(τt, dt|¯τt−1,d¯t−1,x¯t−1, θ) = fd|τ(dt|τt,τ¯t−1,d¯t−1,x¯t−1, θ1)·
fτ(τt|¯τt−1,d¯t−1,x¯t−1, θ2). (5.1) Two prerequisites have to be fulfilled for weak exogeneity. The factorization as given in (5.1) has to be possible under the condition that the parameters θ1 and θ2 are variation free. This means that there should be no parameter restrictions involving both θ1 and θ2. If this is the case, then the estimation is greatly simplified as the likelihood based on fτ can be maximized independently with respect to θ2 without loss of information.
The notion of strong exogeneity relies on the additional condition that dt does not Granger cause τt, i.e. the conditional distribution of τt is independent of past dt given past values of τt. Then, the joint distribution can be reduced to
f(τt, dt|¯τt−1,d¯t−1,x¯t−1, θ) = fd|τ(dt|τt,τ¯t−1,d¯t−1,x¯t−1, θ1)·
fτ(τt|¯τt−1,x¯t−1, θ2).
This means that not only estimation can be carried out independently of the second distribution, but also forecasts ofτtdepend exclusively on the history ofτtand other exogenous variables and not on the history of dt. The same line of arguments can obviously be drawn for the weak and strong exogeneity of dt with respect to τt. If
strong exogeneity would hold in both directions one could consider both processes to be independent.
Super exogeneity is strictly related to the question whether it is possible to control the process by manipulating a particular variable. What is needed to ensure this property is that in addition to weak exogeneity the parameter of interest is either invariant for any change in the distribution of conditioning variables or least invari-ant with respect to a particular class of interventions. See definitions 2.7 and 2.8 of Engle, Hendry, and Richard (1983). However, this concept will not play a significant role in the empirical analysis pursued in this work.
2. A GLM model in discrete time
2.1. A decomposition in terms of conditional probabilities. Due to the fact that the approach suggested by Rydberg and Shephard (1998) employs a discrete time framework, the index t of observations is here identical with the clock time measured in seconds since the trading start. Thus an observation is available for each second of the trading day. To discriminate between the price change dt in a continuous time framework, wheredt is only observed if a trade occurs and the price change in a discrete time model, random variables are marked by a circle in the latter case. So d◦t stands for the price change in the discrete time context.
The decomposition of Rydberg and Shephard heavily utilizes the dynamic framework for GLMs outlined in chapter 3, to capture the dynamics of the three components, price activitya◦t, direction of price changess◦t and size of price changesz◦t. The price activity, a◦t, is a variable taking on a value of one, if a transaction was observed at time t and a price change has taken place and is zero otherwise. The price change itself is decomposed into its direction s◦t taking on values +1 and −1 and its size zt◦, which is integer valued, i.e.z◦t ∈ {0,1,2, . . .}. As already mentioned the discrete time framework leads to an observation in each second having the form of a product d◦t =a◦ts◦tzt◦. (5.2)
Using appropriate definitions of the three components, the distribution of d◦t can be formulated without loss of generality based on the conditional distribution of the components, where vi is one of J possible distinct price changes.
Prob [d◦t =vi|Ft−1] =
Note that this specification does not discriminate between the occurrence of a trade without a price change and the event that no trade occurs at time t. Both events are captured bya◦t = 0 and therebyd◦t = 0. Note thats◦t = 0 andzt◦ = 0 are defined to equal zero if a◦t = 0.
2.2. GLMs for individual components. Price activity is modeled as a GLM based on a logistic distribution including ARMA(p,q) type dynamics as outlined in chapter 3. The dynamic model for a◦t is thus specified as
µa,t = Prob [a◦t = 1|Ft−1] (5.4)
G(µa,t) = x0tβ+ga,t. (5.5)
The variable µa,t is the conditional expectation of the random variable a◦t given the relevant information Ft−1, i.e. Prob [a◦t = 1|Ft−1] = E [a◦t| Ft−1]. The link function G is motivated from the logistic distribution, thus defined as
G(µa,t) = exp(µa,t)
1 + exp(µa,t). (5.6)
The systematic component capturing the dynamics of a◦t is defined as in chapter 3
where ua,t is a martingale difference sequence, mimicking the error term in a con-ventional ARMA model. The model of the trade direction is quite similar to this specification. Actually, it is just an auto logistic model as the empirical results show that the direction of price changes seems to be driven mainly by the bid-ask bounce.
The size of price changes zt◦ is at least potentially a count variable. However, price changes larger than one are observed only rarely in liquid markets. To accommodate a count model Rydberg and Shephard employ again a GLM based on a negative binomial (NegBin) distribution with an ARMA dynamic to model the size of price changes minus one, zt◦ −1. The NegBin generalizes the Poisson distribution as it allows for overdispersion, i.e. conditional mean and variance need not necessarily be restricted to a constant ratio. Using Γ as the Gamma function andvz,i ∈ {1,2,3, . . .}
as the possible values zt◦ can take on, the model is specified as Prob [zt◦ =vz,i|Ft−1, a◦t = 1, s◦t] = The additional parameter α allows for some flexibility in the relationship of the conditional mean and the conditional variance as can be seen from the implied moments
E [z◦t| Ft−1, a◦t = 1, s◦t] = 1 +µz,t Var [z◦t| Ft−1, a◦t = 1, s◦t] =µz,t+ 1
αµ2z,t.
Note that the mean is shifted by one, due to the fact that the model describes the size of price changes minus one. Note that this shift cancels out in the conditional variance. If the parameter α goes to infinity the model specializes to the Poisson
model and the conditional variance is equal to the conditional mean minus the de-terministic shift. The GLM specification is quite straightforward asµz,t is restricted to be positive we have
µz,t = exp(x0tβ+gz,t), (5.9) where gz,t is specified similar to ga,t in (5.7) withuz,t appropriately defined as
uz,t= zt◦−1−µz,t
q
µz,t+ α1µ2z,t
. (5.10)
The modeling framework proposed by Rydberg and Shephard has the considerable advantage that the decomposition of the price process allows to build flexible models for each component and allows to consider and interpret each component in turn.
On the other hand, the individual processes can be combined via (5.2) or a similar definition of a composed random variable of interest, to obtain empirical evidence on the trading process as a whole.
Yet, the main problem associated with the discrete time framework chosen by Ryd-berg and Shephard is the identity of the observation indextwith the clock time. The problem becomes apparent if one considers a very active trading phase, e.g. trading after the opening of an exchange, and a very slow trading phase, e.g. during lunch time. Assume further, to have an easy example, yet without invalidating the argu-ment in a more complex setting, that the true DGP is such that the current time between transactions depends via an ACD(1,0) model only on the preceding time between transactions and the time of day, governing the intraday seasonals. In the log ACD context outlined in chapter 4, the difference between the slow and fast trading could be captured by an additive component, e.g. a trigonometric seasonal, in the mean function. For the model in discrete time however, a time varying coef-ficients model would be necessary to capture the seasonality effects. This is easy to understand: in a fast trading phase lags of the dependent variable with a lower order should have the largest weight. In a slow trading phase, on the other hand, lags with a higher order should have a greater weight. A model for this kind of seasonally varying coefficients was suggested in the GARCH context by Bollerslev and Ghysels
(1996) in the form of a periodic GARCH model. Their work was also motivated by an intraday analysis on the basis of equally spaced observations. They, however, used time aggregates which of course suffer from the same effect, if the aggregation interval is chosen small enough. A modification of the given GLM models in the sense of Bollerslev and Ghysels would however complicate matters considerably, if it is possible at all. These effects might be the reason for the large number of lags Rydberg and Shephard (1998) need to model the price activity a◦t in a satisfactory fashion. This is also the reason why a continuous time framework employing an ACD type dynamics is used in this work.
3. A GLM model in continuous time
3.1. A decomposition in terms of the conditional hazard rate. Engle and Russell (1998) in their work on ACD models propose to decompose the price process into price changes and transaction intensity, i.e. time between transactions
y0t:=
h dt τt
i
. (5.11)
The joint density of yt can be written as the product of the conditional density of the price changesfd|τ and the marginal density of the time between transactionsfτ without any further implications as
fy(yt|d¯t−1,τ¯t−1) = fd|τ(dt|d¯t−1,τ¯t)·fτ(τt|¯τt−1,d¯t−1). (5.12) The autoregressive conditional multinomial model (ACM) which is used to model fd|τ has already been outlined in chapter 3 as a generalization of a Markov chain.
Compared to the latter it allows for a more complex dynamic. The marginal density of the durationfτ is modeled using the ACD model which was already described in chapter 4.
Russell and Engle (1998) specify their joint model as a special competing risks model in which the hazard rate is split into intensities of price changes of size vj or more concisely, the hazard rateλACM(τ|Ft−1) is aJ-dimensional vector, assuming thatdt
can take on J different values. It has the form
Note that here a joint probability is used as a basis to formulate the hazard rate as opposed to the standard definition of the hazard rate in (4.2). The joint probability in (5.13) is then factored by Russell and Engle into a product of the conditional density to observe a certain price change and the standard ACD hazard rate λACD, as it is implied by a regular ACD model, e.g. in(4.17),
λACMt (s|Ft−1) =λACDt (s|d¯t−1,τ¯t−1)µACMt ( ¯dt−1,τ¯t). (5.14) Thus, it remains to specify a process for the conditional probabilities µACMt .
3.2. A GLM the process of price changes. The conditional probability µACMt is derived from a modified GLM based on a multinomial Logit with an ARMA dynamic and some additional terms included to account for the time between trans-actions.
where ψt is the conditional expectation of the time between transactionsτt implied by the ACD model. The matrix Vt is a scaling factor which possibly compensates the variation of the “residual” Dt−µACMt . In the modeling framework of Rydberg and Shephard this factor is defined as the conditional variance of this difference term. Russell and Engle leave the definition of the factor more or less open.
Russell and Engle claim that τt could not be considered weakly exogenous in this model because τt enters the conditional likelihood of the transaction price changes’
model. Yet, simultaneity of the of the time between transactions τt and the price changes dt would involve a dependence between the martingale difference sequence ut and the error term of the ACD model t in (4.15). If however both error terms were correlated the estimates of the coefficients of the functions of the durations τt
in (5.16) would be rendered inconsistent by a simultaneous equations bias. Because of this reason a slightly different approach to build a multivariate model is chosen in this work. Here, the hazard rate is not specified state dependent as in (5.13) but an assumption is made on the joint distribution of the error terms of the dynamic for the durations τt and price changes dt.
A comparison of the solutions of Russell and Engle and Rydberg and Shephard yields not a really clear cut picture. The advantage of Russell and Engle (1998) is the use of an ACD model to obtain a parsimonious duration model in continuous time, which allows e.g. a straightforward inclusion of seasonalities. The advantage of Rydberg and Shephard’s approach is the stringent and parsimonious inclusion of dynamics in the specifications of the components of the process of price changes and the decomposition of the transaction process into components of interest, e.g. the size of price changes. Both models unites however the drawback that the simultaneity of individual components of the transaction process cannot be assessed from the estimation results but needs to be imposed by assumption. A model which allows to assess simultaneity issues between the individual variables and which is based on a decomposition of the price process while using a continuous time framework is proposed in the next section.
4. A latent linear model of price changes in continuous time
4.1. The decomposition of the price process in sign and size. As the work of Rydberg and Shephard has shown, some flexibility in modeling can be gained by decomposing the price process. A significant drawback of their approach lies however in the discrete nature of the price frequency model employed. Therefore
a modified approach is pursued in this context, where the time between transactions is taken as a continuous variable which can be modeled by an ACD type model. How-ever, before the bivariate model of trade frequency and price changes is proposed, the decomposition of the price process in different components is discussed, leav-ing open the duration for later treatment. By conditionleav-ing all probabilities on the time between transactionsτt, these components can then be easily employed in the next section to obtain a model of the joint distribution of the transaction process yt consisting in this context of durations τt and individual components of the price changes dt.
The price change dt which is in this context only observed if a trade occurs, is decomposed into the size of price changes zt and the sign of price changes st. This deviates substantially from the price changed◦t employed by Rydberg and Shephard which is observed at every period (second) of the trading day. Here again, the sign takes on values st ∈ {−1,0,1} and the size zt ∈ {0,1,2, . . .}. The definition of the decomposition is thus
dt=zt·st. (5.17)
Contrary to Rydberg and Shephard there is no particular component which accounts for the occurrence of a price change, like a◦t, this is captured in this model by the random variablezt. Finally, the occurrence of a transaction is explicitly modelled by the time between transactions random variableτt. Trade occurrence is a component of this model and not only price changes.
The probability to observe a certain price change dt of size vi at timet Prob [dt =vi|τt,Ft−1] = Prob [zt= abs (vi), st = sgn (vi)|τt,Ft−1], can thus be decomposed into
Prob [zt = abs (vi), st= sgn (vi)|τt,Ft−1] =
Prob [st= sgn (vi)|zt=abs(vi), τt,Ft−1]·Prob [zt=abs(vi)|τt,Ft−1]. The two processes of size ztand sign stas defined above can not be independent, as it is quite obvious that the size of the price change is informative for the direction
of the price change with respect to the zero price change event, i.e.
Prob [st= 0|zt=abs(vi) = 0, τt,Ft−1] = 1 (5.18) and Prob [st= 0|zt=abs(vi)>0, τt,Ft−1] = 0. (5.19) However, it can be assumed that the size of price changes beyond the zero event carries no additional information for the sign of price changes, or more concisely
Prob [st = sgn (vi)|zt=abs(vi)>0, τt,Ft−1] =
Prob [st= sgn (vi)|zt>0, τt,Ft−1], (5.20) which reflects the fact that the explicit sizeabs(vi) of the price changes, in addition
Prob [st= sgn (vi)|zt>0, τt,Ft−1], (5.20) which reflects the fact that the explicit sizeabs(vi) of the price changes, in addition