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DYNAMIC ECONOMETRIC MODELS 
Vol. 11 – Nicolaus Copernicus University – Toruń – 2011 

Katarzyna Bień-Barkowska   
Warsaw School of Economics, National Bank of Poland 

Distribution Choice for the Asymmetric ACD Models† 

 
A b s t r a c t. In the paper, I generalize the Asymmetric Autoregressive Conditional Duration  
(AACD) model proposed by Bauwens and Giot (2003) with respect to the generalized gamma and 
the Burr distribution for an error term. I derive the log likelihood functions for the augmented 
models and show how to check the goodness-of-fit of the distributional assumptions with the 
application of the probability integral transforms proposed by Diebold, Gunther and Tay (1998). 
Moreover, I present an exemplary empirical application of the Asymmetric ACD model for the 
durations between submissions of market or best limit orders on the interbank trading platform for 
the Polish zloty. I test the impact of selected market microstructure factors (i.e. the bid-ask spread, 
volatility) on the time of order submissions.  

K e y w o r d s: Asymmetric ACD model, financial durations, probability integral transforms, 
market microstructure.  

Introduction  

 Econometric models for financial durations (i.e. time spells between 
selected events of the trading process) have gained an extreme popularity over 
the last decade. The standard Autoregressive Conditional Duration (ACD) 
model of Engle and Russell (1998) and its numerous extensions have become 
a standard tool in modeling irregularly-spaced financial data. The vast survey of 
different ACD specifications has been recently presented in the study of Pecurar 
(2008). The models have been used in order to test selected market 
microstructure hypotheses (Bauwens, Giot, 2001; Zhang et al., 2001; Hautsch, 
2004; Nolte, 2008, among others), as well as in order to augment the volatility 
measures (Ghysels, Jasiak, 1998; Giot, 2000; Kalimipalli, Warga, 2002; 
Grammig, Wellner, 2002; Giot, 2005). The outline of the ACD models has been 

                                                 
† This work has been financed from the Polish science budget resources in year 2011 as the 

part of the research project 03/S/0007/11. 



Katarzyna Bień-Barkowska 56

covered in many econometric textbooks (Bauwens, Giot, 2001; Gouriéroux, 
Jasiak, 2001; Tsay, 2002; Osińska, 2006). 

 In this paper I focus on the asymmetric ACD model (henceforth AACD 
model) of Bauwens and Giot (2003). It is a flexible model for the conditional 
density of financial durations that elapse if one of two possible end states (i.e. 
events pointed on the micro-scale by appropriate “thinning” the data) occurs. As 
an exemplary application of the model, the authors suggest a dynamic 
description of mid-price durations for selected stocks traded on the NYSE. They 
categorize time intervals between subsequent price movements into (1) “a price 
increase” duration (if price of a stock increases during the duration) and (2) 
“a price decrease” duration (if a price of a stock decreases during the duration). 
The AACD model entwines a logarithmic ACD (LACD) specification for the 
conditional expectations of durations, hence the model accounts for the well-
known clustering that characterizes such processes (see Bauwens, Giot, 2000). 
On the other hand, the AACD model has a major advantage over the standard 
ACD specification, as it discriminates between end states of the time spells 
under study. The model allows for a separate description of expected durations 
ending as a “price increase” and expected durations ending as a “price 
decrease”. Accordingly, it accounts for the different dynamics characterizing 
each of the two processes. It also allows for an easy inclusion of explanatory 
factors that – in this particular setup – may exert a totally different impact on 
the pace according to which one or the other event (end state) occurs.  

 More recently, the AACD model has been used by Lo and Sapp (2008) to 
test the impact of different microstructure factors on the choice and timing of 
market/limit1 order placement in the Reuters Dealing 2000-2 Spot Matching 
system2. The authors account for various explanatory variables that reflect the 
state of the order book and some other market conditions (i.e. volatility, time-
of-day) and check their influence on the expected time to a market order or 
a limit order arrival on the ask and bid side of the market. The application of the 
AACD specification allows for the joint modelling of both: (1) the order choice 
and (2) the time that elapses between subsequent order submissions.  

 Both of the aforementioned studies apply the AACD model with the 
Weibull-distributed error term. Such a setup allows for a parsimonous and 
tractable specification. Nevertheless, the studies (Bauwens, Giot, 2001; 
Bauwens et al., 2004) proved the superiority of some more general and flexible 
distributions (the Burr or generalized gamma distribution) over the exponential 
or Weibull ones, as far as the goodness-of-fit of the ACD models is concerned. 
Moreover, a simulation study presented in (Grammig, Mauer, 2000) proved that 
                                                 

1 A market order is an order to immediately buy or sell an asset at the best prevailing bid or 
ask prices. A limit order is an order to buy or sell an asset at a specified price (or better) and it is 
executed if it can be matched with an upcoming order on the opposite market side.  

2 Reuers Dealing 2000-2 Spot Matching System is a fully automated, order-driven, interbank 
market for currency trading.    



Distribution Choice for the Asymmetric ACD Models 57

the distributional misspecification may lead to serious inference problems: loss 
of efficiency and bias in the estimated parameters of duration expectations.   

 The aim of this paper is to generalize the standard AACD model of 
Bauwens and Giot (2003) with respect to its distributional assumptions. I derive 
the likelihood function for the AACD model with the Burr distribution 
(henceforth the B-AACD model) and with the generalized gamma distribution 
(henceforth the GG-AACD model). Moreover, I show how to derive the 
probability integral transforms (PIT) of Diebold, Gunther and Tay (1998) for 
the generalized AACD specifications in order to verify their dynamic and 
distributional goodness-of-fit. The theoretical issues have been supported with 
the empirical example. As in the study of Lo and Sapp (2008), I apply the 
AACD model to durations between the order submissions in the Reuters 
Dealing 3000 Spot Matching System which is a leading interbank trading 
platform for the EUR/PLN currency pair.  

1. Econometric Models 

1.1. Outline of the Asymmetric ACD Model  

The Asymmetric ACD model of Bauwens and Giot (2003) describes the 
marked point process { , }i ix y , where 1=i i ix t t   is a duration between 

moments in which certain events occur: it  and 1it  , and iy  is a qualitative 

variable indicating a type of an event:  ,iy a b . In the primary study of 

Bauwens and Giot (2003), the states a  and b  correspond, respectively, to 
“a price increase” and “a price decrease”; whereas in the study of Lo and Sapp 
(2008) they refer to  “submission of a market order” and “submission of a limit 
order”. Therefore, at the end of each duration one of two events (states, risks) 
can be observed. This modeling setup can be perceived as if the two potential 
risks were competing with each other and only one of them: a  or b  could be 
realized at it . Accordingly, the model belongs to a class of competing risks 

models. The duration ix  is treated as an outcome variable of a function 

, ,= min( , )i i a i bx x x , where ,i ax   and ,i bx  are durations that would end up in state 

a  and b , respectively. In this modeling framework, only one duration, i.e. ,i ax  

(or bix , ), is realized (i.e. the shorter one), which depends on whether an event 

a  or b  occurs first. The duration that is not realized is treated as truncated. The 
conditional bivariate density for a pair ix , iy  can be described as:  

1 1 1

1 1

( , | ) = ( | ) ( | )

( | ) ( | ),

a

a a

b

b b

I
i i i x i i x i i

I
x i i x i i

f x y F h x F S x F

h x F S x F

  

 
  (1)  



Katarzyna Bień-Barkowska 58

where ( )
axh   ( ( )

bxh  ) and ( )
axS   ( ( )

bxS  ) denote the hazard and the survival 

functions for the aix ,  ( bix , ) variable, aI  and bI  are dummy indicators, i.e. 

= 1aI  ( =1bI ) if state =iy a  ( =iy b ) is observed and, analogously, 0=aI   

( 0=bI ) if state =iy b  ( =iy a ) is observed at the end of duration ix . 1iF   

denotes a conditioning information set up to time 1t   which contains past 
realizations of ix  and iy .  

For example, if an event a  happens at it , aix ,  is observed and bix ,  
is 

truncated. Accordingly, the realized duration iai xx ,  contributes to the 

conditional density function given by equation (1) via its density function 

1( | )
ax i if x F   and the unrealized (truncated) duration ibi xx ,  via its survival 

function 1( | )
bx i iS x F  :  

1 1 1 1

1 1

( , | ) = ( | ) ( |, ) ( | )

( | ) ( | ).
a a b

a b

i i i x i i x i i x i i

x i i x i i

f x y a F h x F S x F S x F

f x F S x F

   

 




  (2) 

 As proposed by Bauwens and Giot (2000) or Lo and Sapp (2008) we 
parameterize the expectations of the conditional density functions of  aix ,  and 

bix ,  with the application of the Logarithmic ACD model. The conditional 

duration expectations, , , 1( | )i a i a iE x F    and , , 1( | )i b i b iE x F   , are specified 

in a dynamic fashion such as both factors (the previously observed states and 
the past realized durations) can exert an influence on the expected time to an 
event a  or b :  

, , , 1 1

, , 1 1 1,

= ( ln )

( ln ) ,

a
i a a a a a i i

b
b a b a i i a i a

x I

x I

  

   
 

  



  
  (3) 

, , , 1 1

, , 1 1 1,

= ( ln )

( ln ) ,

b
i b a b a b i i

b
b b b b i i b i b

x I

x I

  

   
 

  



  
  (4) 

where , ,ln( )i a i a   .  

 The specifications for ,i a  and ,i b  change with the previously realized 

state of iy . Thus, the expected time to a given event varies with the type of 
previously observed state and with the length of preceding duration. In the 
AACD framework, the observed duration ix  stems from a mixing process:  

 
, 1 , , 1 ,

, , , ,

= [ ( | ) ] [ ( | ) ]

= [ ] [ ] ,

a b
i i a i i a i i b i i b i

a b
i a i a i i b i b i

x E x F I E x F I

I I

 

 
 

  
 (5)  



Distribution Choice for the Asymmetric ACD Models 59

where ,i a  ( bi, ) is an independent and identically distributed error term with 

,( ) =1i aE  , ( ,( ) 1i bE   ). In this setup ,i a  and ,i b  can both have the 

generalized gamma or the Burr distribution (see Appendix 1 for the theoretical 
outline of these distributions).  

 In order to parameterize the conditional bivariate density for { , }i ix y  
outlined in equation (1), it is crucial to derive the conditional hazard and the 
conditional survival functions for ix  under given assumptions about the 

distribution of ai,  
and ,i b . If we denote hazard and survival function of an 

error term ,i a  ( bi, ) as ( )
a

h   ( ( )
b

h  ) and ( )
a

S   ( ( )
b

S  ), respectively, under 

the necessary assumption that ,( ) =1i aE   ( ,( ) 1i bE   ), the conditional hazard 

and the conditional survival function of , ,i a i a ax    (a similar result holds for 

, ,i b i b bx   ) can be given as: 

1
, ,

1
( | ) ,

a a

i
x i i

i a i a

x
h x F h

 
     

 (6)  

1
,

( | ) ,
a a

i
x i i

i a

x
S x F S

 
    

 (7)  

where , , /i a i a a    and a  is the expectation of a Burr-distributed  

(or a generalized gamma-distributed) random variable (see Appendix 1).  

1.2. AACD Model with the Burr Distribution 

The Burr distribution has two shape parameters  and 2 .  Lancaster (1990) 
proves that it can be derived as a gamma mixture of Weibull distributions.  
It contains exponential (if 1  , 2 0  ), Weibull (if 2 0  ) and log-logistic 
(if 2 1  ) distributions as its limiting cases. In contrast to a Weibull 
distribution, the Burr distribution allows for a non-monotonic hazard functions, 
which can extensively improve the goodness-of-fit of the ACD models  
(see Grammig, Maurer, 2000; Bauwens et al., 2004).  

I apply the formulas for the hazard and survival functions of a Burr-distributed 
random variable (see Appendix 1) in equations (6) and (7) in order to derive the 
hazard and the survival functions ( )

axh   ( ( )
bxh  ) and ( )

axS   ( ( )
bxS  ). Then, it is 

straightforward to rewrite the conditional bivariate density outlined in (1) as: 



Katarzyna Bień-Barkowska 60

   

   

2

2

1 1
,( ) 2

1 ,2

1 1
, 2

,2

( , | ) = 1
1

1 .
1

a
i

a a

a a
a

a a

b
i

b b

b b
b

b b

I

a i i aB
i i i a i i a

a i i

I

b i i b
b i i b

b i i

x
f x y F x

x

x
x

x

 
  

 

 
  

 











 


 

 




 
   
   

 
   
   

  (8) 

The marginal distribution of ix  in the B-AACD model can be derived as: 

   

   2 2

( ) ( ) ( )
1 1 1

1 1

2 2
, ,

1 1
2 2

, ,

( | ) = ( , | ) ( , | )

1 1

1 1 .

a b

a a b b b

a a b b
a b

B B B
i i i i i i i i

a i b i

i a a i i i b b i i

a i i a b i i b

f x F f x y a F f x y b F

x x

x x

x x

 

    

    

 
 

 

  

 

 

  

  

 
  
       

     

  (9) 

 Bauwens and Giot derive the conditional (with respect to a current duration 
and the past filtration 1iF  ) transition probabilities between state a  and b  for 
the AACD model with the Weibull distribution (see Bauwens, Giot, 2003).  
In the case of the B-AACD model, such conditional transition probabilities can 
be given as: 

   

   

( )
( ) 1

1 ( )
1

1 12 1 2 1
, ,

1
1 1

2 2
, ,

( , | )
( | , ) =

( | )

( ) ( )

.
1 1

a b
i i

a a a b b b

a b

a a b b b

B
B i i i

i i i B
i i

I I

a i i a a i b i i b b i

a i b i

i a a i i i b b i i

f x y F
f y x F

f x F

x x x x

x x

x x

     

 

    

   

 
 






  


 

 

    

 
  
       

  (10) 

As ( )
1( | , )B

i i if y x F   depends on ix , ix and iy are not independent. 

The log-likelihood function of the B-AACD model is obtained as a sum of N 
logarithms of conditional probabilities given in equation (8) and it can be 
decomposed into two parts: 

1 2 1 2( , ) ( ) ( ),a bL L L        (11) 

where: 

 
 

( 1)

,

2
1

1

2
2

( ) ln ln ln ln 1

1
ln 1 ,

a
a a a

i i a

a a

N
a

a i a a i i
i

a i i
a

L I x x

x

   

 

 




 





         


   




 (12) 



Distribution Choice for the Asymmetric ACD Models 61

 
 

,

( 1) 2
2

1

2
2

( ) ln ln ln ln 1

1
ln 1 ,

b b b b

i i b

b b

N
b

b i b b i i
i

b i i
b

L I x x

x

   

 

 




 





         


   




 (13) 

and 

   2 2
, , , , , , , ,, , , , , , , , , , , , , .a a a a a b a a a b a a b b b a b b b a b b b b                

The model can be easily estimated by maximising the joint likelihood given by 
equation (11). As the two components of the likelihood function ( 1L  and 2L ) 

depend on different parameters, they can also be maximized separately, as 
suggested by Bauwens, Giot (2003) in the case of the AACD model with the 
Weibull distribution.  

1.3. AACD with the Generalized Gamma Distribution 

 Generalized gamma distribution has two shape parameters   and  . As the 
Burr distribution, it allows for different, non-monotonic shapes of the hazard 
function. It nests a gamma distribution (if 1  ), a Weibull distribution  

(if 1  ) and an exponential distribution (if 1  , 1  ). As the Burr 
distribution, the generalized gamma distribution is often applied to the ACD 
models (see Bauwens, Giot, 2001; Bauwens et al., 2004, among others). 

 Substitution of the hazard and the survival of a generalized gamma 
distribution for ,i a  ( ,i b ) into equations (6) and (7) results in  the hazard and 

survival function for ,i ax  ( ,i bx ). Accordingly, the conditional bivariate density 

 of the pair { , }i ix y  is given as:  

 

( )
1

1
, ,

,
,

1
, ,

,

( , | ) =

exp( )
1 ( , )

( )(1 ( , )

exp( )
1 ( ,

( )(1 ( , )

a
i

a a a a a a

a a

a a

b
i

b b b b b b

b

b b

GG
i i i

I

a i i a i i a i
a i i ai

a a i i a

I

b i i b i i b i
b i ii

b b i i b

f x y F

x x
x

x

x x
x

x

     
 

 

     


 




 




 



  




  



   
        

   
         

 , ) ,b
b


  (14) 

where   denotes the gamma function and i  is the incomplete gamma function 
(see Appendix 1 for details).   

 The marginal distribution of ix  can be derived as: 



Katarzyna Bień-Barkowska 62

( ) ( ) ( )
1 1 1

, ,

,

, ,

,

( | ) = ( , | ) ( , | )

exp( )

( )(1 ( , )

exp( )

( )(1 ( , )

1 ( ,

a a a a a a

a a

b b b b b b

b b

GG GG GG
i i i i i i i i

a i i a i i a

i
i a a i i a

b i i b i i b

i
i b b i i b

i
a i

f x F f x y a F f x y b F

x x

x x

x x

x x

x

     

 

     

 




 


 



  

 



 



  

   
     

  
     

     , ,) 1 ( , ) ,a a b bi
i a b i i bx       

  (15) 

and the conditional transition probabilities between states a  and b  are: 
( )

( ) 1
1 ( )

1

1
, ,

,

1
, ,

,

( , | )
( | , ) =

( | )

exp( )

( )(1 ( , )

exp( )

( )(1 ( , )

a
i

a a a a a a

a a

b
i

b b b b b b

b b

GG
GG i i i

i i i GG
i i

I

a i i a i i a

i
a a i i a

I

b i i b i i b

i
b b i i b

a

f x y F
f y x F

f x F

x x

x

x x

x

     

 

     

 


 


 








  



  



   
       

   
       

 , ,

,

1

, ,

,

exp( )

( )(1 ( , )

exp( )
.

( )(1 ( , )

a a a a a a

a a

b b b b b b

b b

i i a i i a

i
i a a i i a

b i i b i i b

i
i b b i i b

x x

x x

x x

x x

     

 

     

 

 


 

 



 



   
    

  
     

  (16) 

From that it can be seen that ix  and iy  are not independent. 

 The log-likelihood of the GG-AACD model can be derived as a sum of 
N logarithms of conditional probabilities given in (14). In a close analogy to the 
B-AACD model, the log-likelihood can be decomposed into two components, 

1 2 1 2( , ) ( ) ( )a bL L L      , where: 




1
1 , ,

1

, ,

( ) ln ln ln( )

ln( ( )) ln(1 ( , ) ln(1 ( , ) ,

a a a a

i

a a a a

N
a

a i a a a i a i i a
i

i i
a a i i a a i i a

L I x x

x x

   

   

  

  

 



 

        

        


  (17) 




1
2 , ,

1

, ,

( ) ln ln ln( )

ln( ( )) ln(1 ( , ) ln(1 ( , ) ,

b b b b

b b b b

N
b

b i b i b b i b i i b
i

i i
b b i i b b i i b

L I x x

x x

   

   

  

  

 



 

        

        


  (18) 

and 



Distribution Choice for the Asymmetric ACD Models 63

   , , , , , , , ,, , , , , , , , , , , , , .a a a a a b a a a b a a b b b a b b b a b b b b                   

The estimation may be performed on the joint log-likelihood function or in 
two steps, separately for 1L and 2L .  

1.4. Testing the Distribution Choice with the PIT 

 The goodness-of-fit of the ACD models can be checked with the probability 
integral transforms (PIT) proposed by Diebold, Gunther and Tay (1998). This 
testing procedure has been used to check the adequacy of the distribution choice 
and the quality of the conditional mean specification in numerous studies on 
financial durations (e.g. Bauwens et al., 2004; Grammig, Mauer, 2000; Hautsch, 
2004; Bień, 2006, among others). In a shortcut, this approach  can be presented 

as following. If  1 1
( | )

m

i i if x F  denotes a sequence of one-step-ahead density 

forecasts of the ACD model and  1 1
( | )

m

i i ip x F  is a sequence of conditional 

densities for the data generating process of financial durations, the ACD model 
will be correctly specified if the following equation holds: 

   1 11 1
( | ) ( | )

m m

i i i i i if x F p x F    (19) 

 Although the sequence  1 1
( | )

m

i i ip x F   cannot be observed, Diebold, 

Gunther and Tay (1998) show that if equation (19) holds true, the sequence of 
density transforms  iz  for durations  ix  should be i.i.d. uniformly distributed 

on (0,1): 

( ) , ~ . . . (0,1).
ix

i i iz f t dt z i i d U


    (20) 

 It can be seen from formula (20) that in order to compute the sequence  iz

, we need the cumulative distribution function (CDF) for ix  under given ACD 

specification. The marginal densities of ix  under the B-AACD data generating 
process (DGP) or under the GG-AACD DGP  have been derived in equations 
(9) and (15), respectively.  

 The sequence of integrated density transforms for the B-AACD model can 
be calculated as: 

   2 2

1 1
ˆ ˆ ˆ( ) 2 2ˆ ˆ

, ,
ˆ ˆˆ ˆˆ 1 1 1 ,a a b ba b

B
i a i i a b i i bz x x     

           (21) 

which can be seen from the Proof 1 or the Proof 2: 

 

 

 



Katarzyna Bień-Barkowska 64

Proof 1: 

   

   

 

2 2

2 2

1 1( ) ˆ 1ˆ ˆ ˆ ˆ1 2 2ˆ ˆ
, ,

,

1 1
ˆ 1ˆ ˆ ˆ ˆ1 2 2ˆ ˆ

, ,
,

ˆ 1

ˆ
ˆ ˆ2

,
,

ˆ ˆ ˆ ˆˆ ˆ ˆ1 1

ˆ ˆ ˆˆ ˆ ˆ1 1

ˆ ˆ

ˆ ˆˆ1

a
a a a b ba b

b
b b b b bb a

a

a
a a

B
i

a i a i i a b i i b
i a

i

b i b i i b a i i a
i b

a i b

a i i a
i a

dz
x x x

dx

x x x

x

x

     

     



  

  

  

 



     

     



 

     

     

 
    

   2 2

ˆ 1

ˆ
ˆ ˆ2

,
,

1 1
ˆ ˆ2 2ˆ ˆ

, , 1

ˆ ˆˆ1

ˆˆ ˆˆ ˆ1 1 ( | )

b

b
b b

b b b ba b

i

b i i b
i b

a i i a b i i b B i i

x

x

x x f x F



  

    



 



 

  


 
 
 

   
 

     

    

Proof 2: 

If min( , )a b
i i ix x x , iz can be computed as: 

1 1 1 1
ˆ ˆ ˆˆˆ ( | ) 1 ( | ) 1 ( | ) ( | )i i i i i a i i b i iz CDF x F S x F S x F S x F           

 Analogously, from Proof 2, iz  for the GG-AACD model can be estimated 

as:  

   ˆ ˆ ˆ ˆ( )
, ,

ˆ ˆˆ ˆˆ 1 1 ( , ) 1 ( , ) ,a a b bGG i i
i a i i a b i i bz x x                (22) 

 Application of the PIT diagnostic procedures often boils down to checking 
whether ˆiz  is i.i.d. and uniformly distributed. Diebold, Gunther and Tay (1998) 
and Bauwens et al. (2004) emphasize visual inspection of graphs which depict 
dynamics and distributional properties of ˆiz .  

2. Empirical Example 

 As in the study of Lo and Sapp (2008), I apply the AACD model to analyze 
the order submission process in the Reuters Dealing 3000 Spot Matching 
System3 (RDSM). The RDSM system is a fully automated4 order-driven market 
where the interbank currency trading takes place. On this trading platform 
currency dealers can submit two major order types, i.e. market orders or limit 
orders, to buy or sell a given amount of the base currency5. From the viewpoint 
of the market microstructure, market orders are perceived as liquidity 
consuming  – they are immediately executed against limit orders listed on the 

                                                 
3 The Reuters Dealing 3000 Spot Matching System is an updated version of the Reuters 

Dealing 2000-2 Matching System described by Lo and Sapp (2008).  
4 Orders are automatically matched if they arrive to opposite market sides and if their prices 

agree. 
5 In case of the EUR/PLN currency pair, euro is the base (transaction) currency and zloty is 

the counter (quote) currency.  



Distribution Choice for the Asymmetric ACD Models 65

opposite side of an order book, hence they exhaust liquidity measured as market 
depth. Limit orders are liquidity supplying – they wait for a possible execution 
in future, hence they replenish depth on the ask or the bid side of a market.  

 I use data on order submissions on the EUR/PLN market during four days, 
from 2nd to 5th January 2007. The vast majority of Polish zloty trading takes 
place in the offshore market (between London banks) and in Poland. Therefore, 
in order to account for periods when trading is high, I consider orders placed 
after 8:00 CET and before 18:00 CET6. In the sample there are 10 515 orders, 
i.e. 4 848 market orders and 5 667 limit orders7. Order durations are defined as 
time intervals between subsequent moments of order submissions.  

 In the first step I deseasonalized durations as suggested in the literature on 
ACD models. I assume a multiplicative intraday seasonality factor is , such as 

i i ix s x . As suggested by Bauwens and Veredas (2004), the intraday 

seasonality factor is  has been estimated with the application of the kernel 

regression of ix  on a time-of-day variable8. Estimation9 of the AACD models 

has been conducted on diurnally adjusted durations ix .  

 For the sake of completeness of my study I estimated four specifications of 
AACD models, the E-AACD model (with the exponential distribution of an 
error term), the W-AACD model (with the Weibull distribution of an error 
term), the B-AACD model and the GG-AACD model. The specifications of 
conditional expectations were always the same – as in equations (3)–(4). The 
Bayesian Information Criterion (BIC = 2.6197) favorises the B-AACD model 
over the GG-AACD (BIC = 2.6540), the W-AACD model (BIC = 2.7296) and 
the E-AACD model (BIC = 2.9146 ).  

 Histograms and autocorrelation functions for the probability integral 
transforms ˆiz  are depicted in Figure 1 and 2. As can be seen in Figure 1, neither 
the exponential, nor the Weibull distribution are proper for the AACD model. 
Long (but not very long) durations are underrepresented in both specifications 
and the distribution of the probability integral transforms is far from being 
uniform. 

 Additionally, the Weibull distribution demands more observations of 
a small value than registered in the duration series. Both parsimonious 
distributions are not flexible enough to reflect the shape of the true data 
generating process. The generalized gamma distribution does not fit the data as 
                                                 

6 Similar truncation was performed by Lo and Sapp (2008). 
7 As in (Lo, Sapp 2008) I accounted for the best limit orders only, i.e. orders that are placed 

within the best ask and bid prices in the order book. 
8 Quartic kernel is used, with the bandwidth computed as 2.78sN -1/5, where s is the standard 

deviation of the data.. For details of the estimation procedure please refer to (Bauwens, Veredas, 
2004).  

9 The whole empirical study has been performed in Gauss 8.0.  



Katarzyna Bień-Barkowska 66

well. Just as the Weibull distribution, it gives too much probability mass to 
small durations and too little probability mass to a medium-sized durations 
(from the third to fifth quantile of the IPF distribution). Better distribution 
choice would require less observations in the lower tail of the distribution and 
more observations of a middle-sized value. Although the visual inspection 
suggests that the B-AACD has won the competition among the models, the 
choice of the Burr distribution is not optimal as well. The standard Pearson’s 
goodness-of-fit statistic10 2̂  for the uniformity of the ( )ˆ B

iz  distribution equals 

158.53. Because 2*ˆ 30.14   (5% significance level), so the null of uniformity 
should be rejected.    

 The dynamic properties of the model are not perfect, as the ACF function 
for the PIT depicts significant autocorrelation of the first order. Nevertheless, 
once more B-AACD model seems to provide the best fit among selected 
specifications. The AACD models were parsimoniously parameterized in terms 
of the conditional mean functions in order to avoid a burdensome estimation, 
but the literature on the ACD models shows that it is very difficult to find 
a satisfying model as far as its dynamic features are concerned (see Bauwens 
et al., 2004).    
                E-AACD                W-AACD 

 
GG-AACD                     B-AACD 

  
Figure 1.  Histogram of the probability integral transforms for the AACD models. 

Horizontal lines depict the 99% confidence interval 

 

                                                 
10 






m

i

i

Np

Npn

1

2
2 )( , where m=20 (number of histogram bins), p=1/m. 



Distribution Choice for the Asymmetric ACD Models 67

                 E-AACD        W-AACD 

  
GG-AACD                       B-AACD 

  
Figure 2.  The ACF function of the probability integral transforms for the AACD 

models. Horizontal lines depict the 99% confidence interval 

 In the last step of this study, I introduce two explanatory variables into the 
B-AACD specification. I check the impact of the bid-ask spread and the 
EUR/PLN volatility – separately – on the expected durations to a market order 
versus an expected duration to a limit order. There is a large body of theoretical 
and empirical studies proving that these both factors matter as far as order 
choice decisions are considered (Foucault, 1999; Hautsch, 2004; Lo, Sapp, 
2008; among others). The bid-ask (difference between the best ask and bid price 
at the moment of order submission) and the volatility (the realized volatility 
estimate during 10 minutes interval before the order submission) were cleared 
form the intraday seasonality in the same way as the order durations.  

 In Table 1, I report the ML estimates and the corresponding p-values (for 
the robust standard errors) of the B-AACD model. The estimated shape 
parameter proves that the assumption of the Weibull distribution for the error 
terms is not proper. As 2ˆ 0.9425a   and 2ˆ 1.0777b  , the obtained Burr 
distribution seems to be rather “closer” to a log-logistic distribution (where 

2 1)  , than to a Weibull one ( 2 0)  . As far as the dynamic properties of 
the AACD model are concerned, obtained results agree with the results of Lo 
and Sapp (2008). For bk  , that is for durations ending with a market order I 
have ,ˆa a  ,ˆa b  and ,ˆa a  ,ˆa b , hence in result of a previously observed 

market order, the expected duration to another market order shrinks stronger 
than directly after a limit order. Such a result seems to agree with a study of 
Biais et al. (1995) performed for selected stocks traded on the Paris Bourse. It is 



Katarzyna Bień-Barkowska 68

predicted there that market orders placed on the same side of a market cluster 
together as traders: (1) may split large orders into small ones to avoid a huge 
price impact, (2) mimic each other or (3) react similarly to given events. In this 
study I do not differentiate between ask and bid side of a market, but I suspect 
that overall clustering of market orders may be caused by similar behaviour 
patterns. On the other hand, the complicated dynamic structure of the AACD 
model makes it extremely difficult to capture all information given by distinct 
parameters and to interpret  them accordingly. As far as expected durations that 
end with a limit orders are concerned ( k b ), the obtained relations are more 
unequivocal and easy to interpret. As ,ˆb a  ,ˆb b

 
and ,ˆb a  ,ˆb b , the expected 

duration to a limit order shrinks more considerably after a market order than 
after a limit order. Market order always “erodes” depth on the ask or bid side of 
a market which may result in a wider bid-ask spread. Therefore it is more 
profitable to submit a limit order than a market order, large bid-ask spread 
makes liquidity consumption more costly.  The complete dynamics of the model 
can be captured in a more detailed way with a simulation of the whole process, 
as proposed by Bauwens and Giot, (2003). 

 The bid-ask spread has a significant positive impact on the expected time to 
a market order and it has a negative impact on the expected time to best limit 
orders. This result agrees with findings of several studies (Ranaldo, 2004; 
Verhouven et al., 2003; Ellul et al., 2007; Lo, Sapp, 2008). If the bid-ask spread 
is large, it becomes more costly for a trader to cross the difference between the 
best bid and ask prices in order to buy or sell the currency in an immediate way. 
The bid-ask spread constitutes the cost of such quick transaction. On the other 
hand, the traders opt for limit orders. As the differences between best bid and 
ask prices are large, it is much easier for them to compete for the transaction 
priority by offering a price at least one tick better than the current one (“the tick 
rule”).   

 An increase in volatility prompts market orders, as it has a negative 
significant impact on the expected time to a market order submission. A rise in 
uncertainty about the future movement of the EUR/PLN rate encourages traders 
to close their open currency positions or to realize their gains quickly. The 
positive impact of volatility on the expected time to a limit order can be easily 
understood with the nature of a limit order. The submission of a limit order is 
the same as writing of the option – if the FX rate moves in the undesirable 
direction, it can be executed at an unfavourable price (so called “free option 
risk” of a limit order). As prices in limit orders are frozen during their lifetime, 
an increased volatility increases the risk of incurring potential losses. 

 

 

 



Distribution Choice for the Asymmetric ACD Models 69

Table 1.  ML estimates for the B-AACD model 

 ak   (market order) bk   (limit order) 
 parameters   estimate   p – val   estimate   p - val 

 ak ,
 0.0905 0.0404 0.5388 0.0000 

bk ,
  0.3541 0.0000 0.8631 0.0000 

 k   0.7478 0.0000 0.6912 0.0000 

 ak ,
  0.1638 0.0000 0.0908 0.0000 

bk ,
  0.1647 0.0000 0.2195 0.0000 

 k   0.9104 0.0000 1.2104 0.0000 
2
k  0.5844 0.0000 0.9840 0.0000 

bid-ask 
spread 

1.6472 0.0000 -0.6609 0.0000 

volatility -0.3537 0.0000 0.2304 0.0000 
BIC 2.4122 

Conclusions 

 In the paper I have extended the asymmetric ACD model of Bauwens, Giot 
(2003) with respect to more general distribution families: the Burr and the 
generalized gamma. As in the study of Bauwens and Giot (2003), I present the 
basic properties of the generalized specifications. Additionally, we showed an 
easy way of testing the goodness-of-fit of such a competing risk models with 
the probability integral transforms. As Lo and Sapp (2008), I have also 
presented an exemplary application of the AACD model to the order submission 
process on the interbank order-driven market for a Polish zloty. The obtained 
results with regards to the impact of the bid-ask spread and the volatility on the 
order choice confirm the main results from the empirical literature. 

 There are many possible extentions to my study. A most natural one would 
be a more finance-oriented empirical application. Introducing more explanatory 
factors, reflecting the larger scope of information that can be deduced from the 
order book or the external market environment would give more insight into the 
process of market liquidity fluctuation as studied by Lo and Sapp (2008). 
Second, different functional form for duration expectations or even some more 
general distributions for the error term would possibly result in a better fit of the 
model. The PIT diagnostic tool points the Burr distribution as a best one, 
although this distribution choice does not seem to be the optimal one. The 
possible solution would be even more general distribution, such as a generalized 
F distribution advocated by Hautsch (2001) for the ACD models. Third, the 
assymetric ACD model could be easily extended to more than two competing 
risks. It could result in a more flexible specification that, in the context of 
a current empirical study, could also discriminate between orders listed on 
either bid or ask side of the market. 



Katarzyna Bień-Barkowska 70

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Distribution Choice for the Asymmetric ACD Models 71

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Appendix 1 

Burr distribution with parameters 0  , 2 0  and scale parameter is set to 
1: 

survival: ( )S   (1+
2

1
2 ) ,  


 

density:  2

1

2 1 1
( ) ,

(1 )
f



 


 







 

hazard: 
1

2
( ) ,

1
h






 






 

expectation:
 

1

1 1
2

2(1 )
2

1
1

1
1

 







 



     
 
   
 

   if 2 .    

Generalized gamma distribution with parameters 0 , 0 where the 
scale parameter is set to 1: 

survival: ( ) 1 ( , )iS      ,  

where i  is the lower incomplete gamma function: 
1

0
( , )

( )

t
i t e

dt
   



 

 
  and )( - gamma function, 

density: 
1 exp( )

( ) ,
( )

f
  



 



 

hazard: 
1 exp( )

( ) ,
( ) ( , )i

f
 



 
  

 


 
 

expectation:
 1

.
( )

 




 



 

  



Katarzyna Bień-Barkowska 72

Wybór rozkładu składnika losowego w asymetrycznych modelach ACD 

Z a r y s  t r e ś c i W artykule dokonano uogólnienia asymetrycznego modelu ACD, 
zaproponowanego w pracy (Bauwens, Giot, 2003) w odniesieniu do nowych rozkładów składnika 
losowego: rozkładu Burra i uogólnionego rozkładu gamma. Wyprowadzono funkcję  
wiarygodności dla rozszerzonych specyfikacji i przedstawiono procedurę testowania jakości 
dopasowania modeli za pomocą transformat gęstości (Diebold i in., 1998).  Dodatkowo, 
przedstawiono przykładową aplikację asymetrycznych modeli ACD w odniesieniu do odstępów 
czasu (tzw. czasów trwania) pomiędzy momentami, w których składane są zlecenia z limitem 
ceny lub zlecenia rynkowe na kierowanym zleceniami międzybankowym kasowym rynku 
złotego. Dokonano weryfikacji wpływu dwóch czynników mikrostruktury rynku (spreadu bid-ask 
i zmienności) na tempo składania wyróżnionych typów zleceń.  

S ł o w a  k l u c z o w e: asymetryczny model ACD, finansowe czasy trwania, transformaty 
gęstości, mikrostruktura rynku.  

Acknowledgements 

The author thanks the Thomson Reuters and the National Bank of Poland for providing 
the data from the Reuters Dealing 3000 Spot Matching System and Professor 
Małgorzata Doman for valuable comments and suggestions. The views and opinions 
presented in the paper are those of the author and do not reflect those of the National 
Bank of Poland.  


