Forecasting using high-frequency data: a comparison of asymmetric financial duration models

Copyright © 2008 John Wiley & Sons, Ltd.

Forecasting Using High-Frequency Data: A Comparison of Asymmetric Financial Duration Models

QI ZHANG, CHARLIE X. CAI* AND KEVIN KEASEYLeeds University Business School, UK

ABSTRACT

The fi rst purpose of this paper is to assess the short-run forecasting capabilities of two competing fi nancial duration models. The forecast performance of the Autoregressive Conditional Multinomial–Autoregressive Conditional Duration (ACM-ACD) model is better than the Asymmetric Autoregressive Conditional Duration (AACD) model. However, the ACM-ACD model is more complex in terms of the computational setting and is more sensitive to starting values. The second purpose is to examine the effects of market microstructure on the forecasting performance of the two models. The results indicate that the fore-cast performance of the models generally decreases as the liquidity of the stock increases, with the exception of the most liquid stocks. Furthermore, a simple fi lter of the raw data improves the performance of both models. Finally, the results suggest that both models capture the characteristics of the micro data very well with a minimum sample length of 20 days. Copyright © 2008 John Wiley & Sons, Ltd.

key words Autoregressive Duration Model (ACD); forecasting; high- frequency data; market microstructure

INTRODUCTION

Building on recent developments in modelling irregularly spaced time series data, this study com-pares the performance of two models concerned with the joint dynamics of the duration and price change of trades. The Autoregressive Conditional Duration (ACD) model proposed by Engle and Russell (1998) opened up a new way of fi nancial modelling and there have been numerous develop-ments and extensions following their initial work on the subject. Bauwens and Giot (2000) propose a Log ACD model which ensures the expectation of duration is positive. Bauwens and Giot (2003) also propose an Asymmetric ACD model (AACD) for the quote revision process in which the dura-tion of uptick and downtick processes are allowed to be different. Russell and Engle (2005) further propose an Autoregressive Conditional Multinomial–Autoregressive Conditional Duration model (ACM-ACD) to study the joint density of the duration and price change of trades.

Journal of ForecastingJ. Forecast. 28, 371–386 (2009)Published online 29 October 2008 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/for.1100

* Correspondence to: Charlie X. Cai, Leeds University Business School, Maurice Keyworth Building, The University of Leeds, Leeds LS2 9JT, UK. E-mail: [email protected]

372 Q. Zhang, C. X. Cai and K. Keasey

Copyright © 2008 John Wiley & Sons, Ltd. J. Forecast. 28, 371–386 (2009) DOI: 10.1002/for

The fi rst purpose of this study is to compare the forecast accuracy of the ACM-ACD and AACD models. More specifi cally, the specifi cation and estimation, the out-of-sample forecasts and the computational effi ciency of the models are directly compared.

The second purpose is to examine the impact of the market microstructure on the forecasting ability of the models. This study examines the forecasting performance of both models on stocks with different quotation update frequency to test the effects of liquidity on the forecastability of stock returns. Furthermore, this paper studies the effects of fi ltering the raw data with relevant eco-nomic indicators on the forecasting performance. Finally, this study examines the length of sampling period needed to provide a reasonable characterization of the micro data.

The empirical fi ndings offer a number of results. In terms of the fi rst purpose, both models are found to have better forecasting power than the naïve model.1 The ACM-ACD has a slight better ‘in-sample’ and ‘out-of-sample’ performance than AACD. However, the difference of the forecast-ing accuracy is not very large and in terms of the estimation effi ciency AACD model seems to be more robust.

In terms of the second purpose, the forecasting performance of the stock price change is found to be inversely related to the liquidity of a stock, with the exception of the most liquid stocks. This suggests that illiquidity slows the price adjustment to a new market equilibrium and, therefore, increases the predictability. Second, using higher levels of price change as a fi lter to generate data points improves the forecasting power. This suggests that fi ltering high-frequency data using eco-nomic variables can reduce the noise in the data and improve the information content of the data. However, the choice of such variables and the appropriate fi ltering criteria are questions for further theoretical and empirical study. Finally, the forecasting power of both models is robust to the choice of sampling period and length, and both models capture the characteristics of the micro data very well with a minimum sample length of 20 days. This supports the fact that most of the existing market microstructure research has used suffi cient data (normally 3–6 months of data) to character-ize the structure of the data and to test hypotheses.

The rest of the paper is organized as follows. The next section gives a brief literature review of the development of the ACD model and its variants, with a focus on the details of the ACM-ACD and AACD models. The third section presents the data used in the study. The estimation and com-parison of the models are given in the fourth section. The paper concludes and offers future directions for research in the fi nal section.

THE MODELS

Over the past two decades numerous studies have been undertaken to test the predictability of stock returns (e.g., Campbell, 1987; Fama and French, 1988; Huang and Stoll, 1994). Most of these tests have focused on yearly, monthly, weekly and daily data. An exception is Huang and Stoll (1994), who tested the predictability of stock return on 5-minute interval data. They found that microstruc-ture theory and associated variables helped to predict intraday stock returns. They did not, however, include the durations between trades and quote revisions in their model.

Theoretical studies (Easley and O’Hara, 1987) in market microstructure show that time between intraday market events, like trades and quote updates, may contain information about the market

1 The naïve model refers to a simple unconditional forecast model with equal probability (50%) of price up and price down.

Forecasting Using High-Frequency Data 373


states. Moreover, in the fi nancial econometrics fi eld, Engle and Russell (1998) construct an ACD model and provide a framework for modelling fi nancial duration. This development led onto a series of ACD type models. These studies include modifying the distribution assumption of residual terms (examples include Weibull, Gamma, and Burr distributions) or the specifi cation of expected duration formula (examples include Log-ACD by Bauwens and Giot, 2000, and Threshold-ACD by Zhang et al., 2001).

Besides focusing on the dynamics of duration, ACM-ACD and AACD model the direction of price change in conjunction with the duration. Both models treat the price movement as a discrete variable and model the joint density of duration and price movement. After estimation of the models, they can be used for forecasting owing to their autoregressive structure. These two models provide a framework to predict future price movements considering the irregular space of the durations and potentially incorporating other explanatory variables.

The two models have been chosen for this study because of their similarity in terms of modelling objective—that is, they both jointly model price change and duration—and the differences in their modelling approach. The detail of the two model settings is discussed below.

Following Russell and Engle’s (2005) notation, let ti denote the time that the tth market event (e.g., transaction, quote update) occurs. In the ACM-ACD model of Russell and Engle (2005) the market event is a transaction and in the AACD model of Bauwens and Giot (2003) it is the direction change of the mid quote prices. Although these two models initially focus on different marked point processes, it is easy and straightforward to extend to other marked point processes for each model.

Let ti denote the duration between two market events occurring at times ti−1 and ti.Let yi denote the price change about this market event. In Russell and Engle’s (2005) model,

yi is the transaction price change, which is a discrete variable with fi ve dimensions. In Bauwens and Giot’s (2003) model, this is the direction of change of mid quote price, which has two dimensions.

Choices for the construction of the yi and ti series can be various. For yi, it can be two dimensions including only up and down tick price movements, or three dimensions including up, down and zero tick movements, or fi ve dimensions including two up ticks, one up tick, zero tick, one down tick and two down ticks, as in Russell and Engle (2005). For ti, it can be trade duration or quote revision duration. Futhermore, the construction of the price movement yi and duration ti series can be conditional on thresholds of other variables. The most obvious choice of threshold variable is the price change itself. For example, yi only contains those points with price movements larger than two ticks.2

In order to compare the performance of the two models, we have to make a choice on which point processes to use. We choose to model the mid quote change and duration in our study for two reasons. First, use of the mid quote change removes the microstructure effect of bid–ask bounce and, there-fore, reduces the noise in the data. Second, forecasting mid quote change has a practical advantage over forecasting trade price change in that the forecast mid point forms a useful reference point to investors regardless of whether they are buyers or sellers.

For the mid quote price change,3 yi, although both models can be extended to higher dimensions, we set a two-dimension framework for simplicity. We let yi denote the direction of change of mid quote price, and yi has two dimensions:

2 Tick is the minimum price change for a security traded. In the period of our study, the minimum tick size is 1/16 in the NYSE.3 For simplicity, in the rest of this paper we use ‘mid quote price change’ and ‘price change’ interchangeably.



yi = 1 if the mid price increased over duration ti; andyi = −1 if the mid price decreased over duration t i

4.

Both ACM-ACD and AACD model the joint density of yi and ti, as

f y yi ii i, ,τ τ−( ) −( )( )1 1 (1)

where y(i−1) = (yi−1, yi−2, . . . , y1), and t (i−1) = (ti−1, ti−2, . . . , t1), and the difference between these two models lies in the way this joint density is specifi ed and determined by past information. In general, the autoregressive conditional form is used in both models. The specifi cation and estimation of both models are briefl y discussed in the following subsections.

ACM-ACD modelThe ACM-ACD model decomposes the joint conditional density of yi and ti into the product of the conditional density of the mark and the marginal density of the arrival times, both conditional on the past information set:

f y y g y y q yi ii i

ii i

ii i, , , ,τ τ τ τ τ−( ) −( ) −( ) ( ) −( ) −( )( ) = ( ) ( )1 1 1 1 1 (2)

where g(·) denotes the probability density function of the price changes, yi, conditional on t (i) and y(i−1), and q(·) denotes the density function of the ith quote duration conditional on t (i−1) and y(i−1).

Russell and Engle (2005) use the Autoregressive Conditional Multinomial (ACM) specifi cation to model the g(·). In the two-dimension case, let x̃i be a 2 × 1 vector indicating the possible states of the discrete mid quote price change, yi. x̃i takes the jth column of the 2 × 2 identity matrix if the jth state occurred. Let p̃i denote a 2 × 1 vector of conditional probabilities associated with the states. The jth element of p̃i corresponds to the probability that the jth element of x̃i takes the value 1. p̃i must satisfy the following two conditions:

• all elements of p̃i are non-negative; and• all columns must sum to unity.

Russell and Engle (2005) propose using an inverse logistic transformation to impose such condi-tions directly. Let the pi denote the probability of an up movement of mid quote price, and defi ne the log of the probability ratios as h(pi) = ln(pi/(1 − pi)). The ACM model assumes h(pi) follows an autoregressive conditional specifi cation:

h c a x b hi j i j i jj

p

j i j i ij

π π π χ τ χ τ( ) = + −( ) + ( ) + ( ) + ( )− −=

− −=

∑1

1 2 1ln ln11

q

∑ (3)

The conditional probability of an up movement of mid quote price is easily recovered from the above inverse logistic transformation using the following equation:

ππ π χ τ χ τ

i

j i j i jj

pj i j i ij

c a x b h=

+ −( ) + ( ) + ( ) + ( )− −= − −=∑exp ln ln1 1 2 111

1 1 21

q

j i j i jj

pj i j ic a x b h

∑∑

+ + −( ) + ( ) + ( ) +− −= −exp ln lnπ π χ τ χ τ iij

q−=

( ) ∑ 11

(4)

4 yi only contains quote update which changes the mid price. Quote change which does not affect mid price is not recognized as an event point.



and the probability of a down movement of mid quote price is calculated as 1 − pi. Given the initial condition, the entire path of p̃i can be constructed, and the log-likelihood is expressed as

L x xij ijj

K

i

N

i ii

N

= ( )( ) = ′ ( )== =

∑∑ ∑� � � �ln lnπ π11 1

(5)

In the above equation, x̃i are the realized observations and p̃i contains the variables needed to be estimated.

In Russell and Engle (2005), the dynamics of the duration are specifi ed by an exponential-ACD model. This means that the hazard function is constant.

The conditional density function is generalized to a Weibull-Log-ACD model, which means the hazard function could be the increasing or decreasing function of the duration. The log-likelihood associated with the ith duration is

ln , ln lnq yii i

i

i

i

τ τ γτ

γ γ τψ

−( ) −( )( )( ) =

+ +( )

−1 1 1 1 1Γ Γ ++( )

1 γ τψ

γi

i

(6)

where

ln lnψ ω α ε β ψυ

i j i jj

u

j i jj

( ) = + + ( )−=

−=

∑ ∑1 1

(7)

The estimation of the ACM-ACD model can be performed by maximizing the sum of the ACM log-likelihood function and the ACD log-likelihood function or separately maximizing the two log-likelihood functions.

AACD modelBauwens and Giot (2003) constructed an asymmetric ACD model by letting the conditional hazard function not only depend on the previous duration, but also on the previous state of the price move-ment. The model is defi ned via the following equations. If the end state of duration ti is yi = 1, the hazard for ti, assuming a Weibull distribution, is given by

h y yi ii i

i

i

i

τ τ γ τ γ

=( ) =

−( ) −( )+

+ +

−+

1 1 11

, ,Ψ Ψ

(8)

with Ψ+i = ey +

i e +i , the autoregressive process on y +

i is specifi ed as

ψ ω α ε ω α ε β ψi i i i i iI I+−+

−+

−+

−− +

−+= +( ) + +( ) +1 1 1 1 2 2 1 1 1 (9)

where xi = ey +i e +

i , which means that the model follows a Log-ACD framework.If the end state of duration ti is yi = −1, the hazard for ti is given by

h y yi ii i

i

i

i

τ τ γ τ γ

= −( ) =

−( ) −( )+

+ +

−+

1 1 11

, ,Ψ Ψ

(10)



where Ψ−i = ey −

i e −i , and

ψ ω α ε ω α ε β ψi i i i i iI I−−−

−+

−−

−− −

−−= +( ) + +( ) +3 3 1 1 4 4 1 1 1 (11)

where xi = ey −i e−

i .The joint density function of yi and ti, conditional on the past information, is given by

f y y ei ii i

i

i

i

Ii i

iτ τ γ τ γ τ

, ,−( ) −( )+

+ +

− −( ) =

++

+1 11

Ψ ΨΨ

−

− −

− −

+−

−

−

−

γ γ

γ τ τ

Ψ ΨΨ

i

i

i

r Iii

ie1

(12)

The parameters can be obtained by maximum likelihood estimation. Bauwens and Giot (2003) point out the following:

f y yf y y

f yi

i i i ii i

ii i

−( ) ( )−( ) −( )

−( ) −( )( ) = ( )( )

11 1

1 1,

, ,,

τ τ ττ τ

(13)

and

f y yii i i

i

i

I

i

i

i

i

−( ) ( )

+

+ +

−

− −

( ) =

−

++

1

1

, τ

γ τ γ τγ

Ψ Ψ Ψ Ψ

γγ

γ γγ τ γ τ

−−

+ −

−

+

+ +

− −

− −

−

+

1

1 1

I

i

i

i i

i

i

i

Ψ Ψ Ψ Ψ

(14)

Using the above equations, the conditional probability of the direction of mid quote price change is calculated. This is the same measure as pi in the ACM-ACD model.

In summary, both the ACM-ACD and AACD models have similar objectives in modelling the joint probabilities of price change and duration, and both are based on the ACD framework. Fur-thermore, the information and variables used in both models are identical, which are price change and duration. However, what is different between these two models is their modelling strategy. As we have seen in the above discussion, ACM-ACD models the price change conditional on duration while AACD focuses on modelling the duration of the up-tick and down-tick price changes separately.

Given the similarity and difference in the two models, the next sections examine and compare how well each model can forecast future prices.

SAMPLE AND DATA

The dataset is based on 50 stocks from the New York Stock Exchange (NYSE) and, in order to give a broad representation of the stocks trading on the NYSE, the sample is constructed by randomly selecting one stock from each of the 50 volume-sorted groups. One exception is the IBM stock, which is chosen because it has been used in previous research.

Data were abstracted from the Trades and Quotes (TAQ) dataset distributed by the NYSE. The sample period is from 3 January to 31 March 2000, giving a total of 63 trading days. Following



Engle and Russell (1998), the fi rst half hour of every day of data is deleted to reduce the effect of overnight information and market opening pressure. The data-generating process in the fi rst half hour is different and warrants a separate research focus (see, for example, Webb and Smith, 1994; Gallo and Pacini, 1998).

A summary of the data is given in Table I. The mean average daily number of trades for the 50 stocks is 687.36, and the mean average daily trading volume is US$93.74 million. After deleting the fi rst half hour of each trading day, we select the point at which the mid quote price has changed, and then we calculate the time between any two continuous points. By this method, we construct the duration sequence of the mid quote revision for each stock. The mean of the total number of the mid quote changes is 20,610, which implies that, on average, each day has 327 mid quote changes. The mean average duration betweens quotes is 100.74 seconds.

Furthermore, Engle and Russell (1998) suggest that before estimating the ACD models the dura-tion sequence data need to be diurnally adjusted for intraday patterns. For both the ACM-ACD and AACD models, a quadratic function with indicator variables from Tsay (2002, Ch5, pp. 194–206) is used to adjust the diurnal pattern and remove the intraday patterns. After the diurnal adjustment, the intraday patterns are removed.

EMPIRICAL RESULTS

For simplicity and consistency of comparison, both models are set to only include the fi rst lag order. For the ACM-ACD model, the initial values of y in equation (7) and p in equation (3) at the begin-ning of each day are set to the unconditional mean of duration and probability of price transition. For the AACD model, the initial values of both of y + in equation (9) and y − in equation (11) are set to the unconditional mean of duration at the beginning of each day.

Estimation of the ACM-ACD model is performed by separately maximizing the functions given in equations (5) and (6). Estimation of the AACD model is achieved by maximizing the function in

Table I. Summary statistics

Mean Median Max. Min.

Panel A. Trading activitiesAverage daily number of trades 687.36 510.50 2,404.00 84.00Average daily dollar volume (m) 93.74 34.92 722.57 1.80Average price level 50.40 41.74 196.33 7.63Total return −0.04 −0.04 0.62 −0.75

Panel B. DurationTotal number of quotes 20,610.14 16,619.00 90,794.00 3161.00Average duration between quotes 100.74 82.05 429.19 15.02Median duration between quotes 43.78 40.00 156.00 9.00Minimum duration between quotes 1.00 1.00 1.00 1.00Maximum duration between quotes 4,187.10 2,999.50 13,936.00 591.00

Note: This table reports the summary statistics for the trading activities and duration measures for 50 NYSE stocks. Panel A reports statistics on trading activities, which include average daily number of trades, average daily dollar volume in mil-lions, average price level, and total return during the three month (January–March 2000) period. Panel B reports the summary statistics on quotes and durations. Duration is measured in seconds. The mean, median, maximum and minimum of the 50 stocks are reported for each measure.



equation (12). We then set g + = g − and estimate the model again and test the hypothesis (g + = g −) via the likelihood ratio test. Test results suggest acceptance of the hypothesis (g + = g −) for all of the 50 stocks. Therefore, we use equation (14) to calculate the conditional probability of the direction of mid quote price changes.

The models are estimated with the data from the fi rst 32 trading days and the rest of the 31 trading days are used to analyse forecast accuracy. The two models differ in terms of forecast process. For the ACM-ACD model, the initial values of y and p at the beginning of each day are set to the unconditional mean of duration and the probability of price transition of the previous day, respec-tively. The forecast yi of the duration in equation (6) is used to replace ti in equation (3). For the AACD model, the initial values of the both of the y + and y − at the beginning of each day are set to the unconditional mean of the durations of the previous day. With the estimated parameters, initial values are set as above and the forecasting sequences for the probabilities of mid quote price change are constructed for the 31 remaining days. In determining the directions of the forecast price move, a forecast probability of an up movement greater than 0.5 is treated as a forecast of an up movement and vice versa.

Estimation and forecastingTable II summarizes the out-of-sample forecasting performance of the two models (ACM-ACD and AACD). Panel A reports the mean, median, standard deviation, maximum and minimum of the out-of-sample forecasting accuracy for the 50 NYSE stocks. The Shapiro–Wilk tests for normality are rejected at the 1% level for the results of both models. Therefore, non-parametric tests are used in examining the statistical properties of the forecasting results. The signed rank test is used to test whether the forecast accuracy is different from 50%, which is the expected accuracy for an uncon-ditional naïve strategy. Panel B reports the number of stocks for which each model outperforms the

Table II. Summary of out-of-sample forecasting results

ACM-ACD AACD

Panel A. Summary of forecasting performanceMean 0.5396 0.5368Median 0.5283 0.5240Std 0.0386 0.0348Max. 0.6891 0.6883Min. 0.4897 0.4876Normal 0.8922*** 0.8368***Signed rank test (m = 0.5) 604.5*** 628.5***

Panel B. Comparisons of forecasting performanceNumber of forecasts which outperform the other model 31 19Pairwise signed rank test 131.5

Note: This table reports and compares the forecasting performance of the two models (ACM-ACD and AACD). Panel A reports the mean, median, standard deviation, maximum and minimum of the out-of-sample forecasting accuracy for 50 NYSE stocks. The parameter of the model is estimated using the fi rst 32 days of the sample and the rest is used for out-of-sample forecasts. The Normal row reports the Shapiro–Wilk test for normality. The signed rank test is used to test whether the forecast accuracy is different from 50%, which is the expected accuracy for an unconditional naïve strategy. Panel B reports the number of stocks where each model outperforms the other and a pair-wise signed rank test for testing the difference in forecasting accuracy between the two models. Asterisks indicate signifi cance at the ***1%, **5% and *10% levels.



other and a pair-wise signed rank test for testing the difference in forecasting accuracy between the two models.

The means of the correct rate of forecasting of the 50 stocks for the ACM-ACD and AACD models are 0.5396 and 0.5368, respectively. The best forecast stocks in both models achieve 69% accuracy, while the worse forecast stocks have accuracies of less than 50%. A signed rank test shows that both of these forecast results are signifi cantly different from 0.5 and, therefore, both models outperform an unconditional naive strategy.

In comparing the performance of the two models in panel B, the ACM-ACD model outperforms the AACD model for 31 of the stocks and a pair-wise signed rank test offers the same conclusion.

However, the estimations of the AACD model converge more quickly and are less fi ckle as to the initial estimated parameter setting. The estimations of ACM-ACD are more time-consuming in terms of convergence and often fail in the progress of estimation because of matrix singularity. When failure occured other initial parameter values were used to rerun the estimation.

In summary, the average forecasting accuracy of the ACM-ACD model is higher than that of the AACD model, and the number of stocks, in which the ACM-ACD model outperforms the AACD model, is dominant in the sample. This evidence suggests a tentative conclusion that the ACM-ACD model is better than the AACD model. However, the difference of the forecasting accuracy is not very large and in terms of the estimation effi ciency the AACD model seems to be more robust. The ACM-ACD model is more complex in terms of the computational setting and takes longer to estimate and is more sensitive to starting values than the AACD model.

Effect of quotation frequency on forecasting performanceMarket microstructure has been found (see, for example, Huang and Stoll, 1994) to have important infl uences on the forcastability of high-frequency data. In the context of the current research, the liquidity of a stock may be relevant to the forecastability of the stock return because it affects the informativeness of the stock price. In essence, illiquidity may slow the adjustment to a new market equilibrium and as a consequence can increase the predictability of a stock (see Psaradakis et al., 2004). Chordia et al. (2007) study the predictability for NYSE stocks that traded every day from 1993 through 2002. Their fi ndings support the notion that liquidity stimulates arbitrage activity and reduces autocorrelations in price, which, in turn, enhances market effi ciency. Therefore, we would expect the predictability of the stock return to be inversely related to its liquidity.

We use the frequency of quotation updates to capture the liquidity of a stock. There are a number of other liquidity measures such as bid–ask spread, liquidity ratio, variance ratio and market impact. Given the objective of the current research, we choose the frequency of quote update as a measure of liquidity for two reasons. First, it is a measure highly correlated with other measures of trading activities such as trading volume and trading frequency. More importantly, it is a direct measure of the activity in quote price revisions.

Table III reports the forecasting performance of the two models in fi ve groups ranked by their average daily number of quotes. The fi rst group consists of 10 stocks with the lowest daily number of quotes and the fi fth group consists of 10 stocks with the highest daily number of quotes. It reports the mean of the out of sample forecasting accuracy for each group.

The DailyNumQuote column reports the average daily number of quotes for each group. The mean daily number of quotes ranges from 116 updates to 730. It is worth noting that the highest group is updated twice as often as the next group.

Except for the highest group, the predictability of the stock return does decrease as the frequency of quotation increases. The forecasting performance of the ACM-ACD model is better than the



AACD model in all these groups. However, the group with the highest quote update frequency singles itself out from the rest of the stocks. The predictability of the stock return is higher than the next group with a lower liquidity measure (quote update frequency). Furthermore, the forecasting performance of the ACM-ACD model is not as good as that of the AACD model. This result has two implications. First, it indicates that there are other factors affecting the predictability of stock returns. Second, in studying market microstructure effects such as liquidity, those stocks with highest trade/quote frequency may have to be examined separately from the rest. Those extremely liquid and most traded stocks are also those stocks with the largest market capitalization and with the most analysts. With the presence of a large and very diverse investor base (institutional and individual), trading in these stocks contains as much noise as information.

In summary, the ACM-ACD model outperforms the AACD model for those stocks with low and medium liquidity, whereas the AACD model outperforms the ACM-ACD model for the most liquid stocks. Furthermore, consistent with our expectations, the predictability of the stock return is found to decrease as the liquidity of a stock increases with the exception of the most liquid stocks. This supports the notion that illiquidity slows the price adjustment to new market equilibrium and, there-fore, increases the predictability of a stock price. However, the structural change in the performance of the two econometric models and the predictability of the stocks in the highest liquidity group warrant further research.

In theory, these models should work equally well with all stocks, no matter what the trading frequency. However, in empirical practice, one extra dimension needs to be considered—that is, the unit of measurement. In most studies of duration, and this is also the case here, the unit of measure-ment for duration is a second. For high-frequency stocks, there can be more than one trade at a given second. Furthermore, the quote can be updated nearly every second throughout the day, especially with the participation of public limit orders. In this context, the duration generated from these point processes contain no extra information as the irregular spaced high-frequency data effectively becomes regularly spaced high-frequency data. Therefore, with these highly liquid stocks, duration has little potential to add explanatory power.

Table III. Quotation frequency and forecasting performance

Quotegroup ACM-ACD AACD DailyNumQuote

Lowest 0.5523 0.5474 1162 0.5454 0.5405 1963 0.5423 0.5347 2724 0.5175 0.5166 376Highest 0.5403 0.5448 730

Note: This table reports the forecasting performance of the two models for fi ve groups ranked by their average daily number of quotes. The fi rst group consists of 10 stocks with the lowest daily number of quotes and the fi fth group consists of 10 stocks with the highest daily number of quotes. It reports the mean of the out-of-sample forecasting accuracy for each group. The parameter of the model is esti-mated using the fi rst 32 days of the sample and the rest is used for out-of-sample forecasts. The DailyNumQuote column reports the average daily number of quotes for each group.



Forecast performance with fi ltered dataOne criticism to using high-frequency data is that the raw data contain excessive noise. This is especially the case for the most frequently traded stocks. The aim of applying econometric and other computational methods to high-frequency fi nance data is to fi lter noise and extract information from it. By doing so, it makes high-frequency data, which is otherwise limited because it contains too much noise, useful for decisions regarding stock choice. Both the ACM-ACD and AACD models impose a parametric structure to the data-generating process of the high-frequency return series. The root to these analyses is that quote-by-quote duration contains useful information in assessing the true value of the stock and can be used to predict the next price movement.

As already noted, in the case of highly frequently traded stocks, the duration generated by every quote update may not contain information as they have been updated very often for liquidity reasons other than information—i.e., noise. The optimal duration measure with the most information is the duration between the points in a process where each event point relates to changes in information about the process. Whether this point process is the raw process recorded or a fi ltered process is an empirical question which we examine here.

In the microstructure literature, it has been shown (see, for example, Psaradakis et al., 2004) that in the presence of transaction costs a rational informed trader will only trade when the current price deviates from his/her estimate of the ‘true price’ by more than the transaction costs to be incurred. Due to the presence of transaction costs, it follows, therefore, that small price changes contain less information than larger price changes in the market. On the other hand, if only large price changes are considered, the informativeness of the process will be reduced as some valuable data points are dropped from the system. Therefore, we expect there to be an optimal threshold for fi ltering the data.

The analyses in previous sections are based on the point process of the mid quote price change. We select all points at which the mid quote price has changed regardless of its magnitude. Since the tick size of NYSE in the year 2000 is 1/16, the minimum change of the mid quote price is 1/32. The raw point processes can be seen as being constructed by setting the threshold of the price change greater or equal to 1/32. To assess the impact of noise within the data, we increase the price change threshold to 1/16, 1/8, and 1/4. Consequently, we construct three new point processes. The difference of the new processes with the original process is that they delete the information about the small price changes, which may be considered to be noise. Therefore, the estimation and forecasting of the new processes can tell us about the performance of the models after reducing noise.

However, when applying fi lters to the raw data the number of data points in the series is reduced. This is especially a problem for those stocks with a low number of quote updates in a given day. For comparability of model estimation and forecast, after applying the fi lters we drop those stocks which have one or more days with a zero number of quote updates. The effects of the fi ltering on the number of stocks and average number of quotes are reported in Table IV. Panel A shows that for the low quote update frequency groups (1 and 2) fi ltering the data with a threshold higher than 1/16 will reduce the number of stocks in the groups signifi cantly. For the medium quote update frequency groups (3 and 4), fi ltering the data with a threshold higher than 1/8 will reduce the number of stocks in the groups signifi cantly; while for the highest quote update frequency group, fi ltering up to 1/4 does not affect the number of stocks in the estimations. Panel B shows that, as the thresh-old doubles, the average daily number of quotes reduces approximately by half for all quote update frequency groups.

Table V summarizes the forecasting performance of the two models, with different thresholds (1/32, 1/16, 1/8 and 1/4) being used to fi lter the raw data. Panel A reports the mean, median, standard



Table IV. Data fi ltering and number of quotes

Threshold Lowest 2 3 4 Highest All

Panel A. Number of stocks1/32 10 10 10 10 10 501/16 10 9 10 10 10 491/8 5 8 10 10 10 431/4 1 2 6 8 10 27

Panel B. Average number of quotes per day1/32 116 196 272 376 730 3271/16 45 83 115 139 353 1481/8 23 33 42 45 153 651/4 11 20 18 15 59 32

Note: This table reports the effect of data fi ltering with different thresholds (1/32, 1/16, 1/8 and 1/4) on the raw data. Panel A reports the number of stocks in each group after applying the fi lter. Panel B reports the average number of quotes per day in each of fi ltering threshold and quote update frequency group. The fi rst group consist of 10 stocks with the lowest daily number of quotes and the fi fth group consist of 10 stocks with the highest daily number of quotes.

Table V. Data fi ltering and forecasting performance

Panel A. All stocks

Threshold Mean Median Std Max. Min. Signrank

ACM-ACD 1/32 0.5396 0.5283 0.0386 0.6891 0.4897 −153.5** 1/16 0.5441 0.5340 0.0421 0.6569 0.4767 −144** 1/8 0.5484 0.5388 0.0565 0.6802 0.4533 140*** 1/4 0.5354 0.5156 0.0563 0.6525 0.4573AACD 1/32 0.5368 0.5240 0.0348 0.6883 0.4876 −392.5*** 1/16 0.5456 0.5372 0.0384 0.7000 0.4988 −71** 1/8 0.5467 0.5335 0.0474 0.6421 0.4589 131*** 1/4 0.5241 0.5165 0.0335 0.5958 0.4759

Panel B. Stocks by quote frequency groups

Quotegroup Lowest 2 3 4 Highest

ACM-ACD 1/32 0.5523 0.5454 0.5423 0.5175 0.5403 1/16 0.5429 0.5358 0.5567 0.5249 0.5595 1/8 0.5262 0.5391 0.5732 0.5278 0.5627 1/4 0.4944 0.6013 0.5430 0.5040 0.5469AACD 1/32 0.5474 0.5405 0.5347 0.5166 0.5448 1/16 0.5495 0.5420 0.5564 0.5282 0.5518 1/8 0.5491 0.5442 0.5665 0.5297 0.5446 1/4 0.4916 0.4944 0.5284 0.5168 0.5365

Note: This table reports the forecasting performance of the two models with different thresholds (1/32, 1/16, 1/8 and 1/4) used to fi lter the raw data. Panel A reports the mean, median, standard deviation, maximum and minimum of the forecasting accuracy for 50 NYSE stocks. The Signrank column reports the signed rank test on the difference between the current and the next row. Panel B reports the mean forecasting accuracy for each threshold in the fi ve quotation frequency groups. The fi rst group consists of 10 stocks with the lowest daily number of quotes and the fi fth group consists of 10 stocks with the highest daily number of quotes. Asterisks indicate signifi cance at the ***1%, **5% and *10% levels.



deviation, maximum and minimum of the forecasting accuracy for the 50 NYSE stocks. The Sign-rank column reports the signed rank test on the difference between the current and the next row. Panel B reports the mean forecasting accuracy for each threshold in the fi ve quote update frequency groups.

Panel A shows that, for both models, as the thresholds increase forecasting performance initially improves and then declines. This means that deleting noise can improve the performance for both models but there is a limit to this improvement.

Further examination on the effect of fi ltering on quotation frequency groups is given in panel B. It shows that the effect of fi ltering is different across the different quotation frequency groups. For the two groups with the lowest quote update frequency, the fi lters applied to the raw data decrease the forecasting performance in general. Together with the results in Table IV, this suggests the raw duration data for these stocks is optimal. For the other groups with higher quotation frequency, the threshold of 1/8 is found to be the optimal fi lter for the raw data.

Furthermore, in comparing the performance of the two models, the ACM-ACD model is shown to be more sensitive to noise in the data than the AACD model as the forecasting performance in the former changes in a wider range than the latter as the threshold level changes.

In summary, fi ltering data by raising the threshold of counting price changes can reduce noise and improve forecasting performance for both models in the whole sample. The effects of fi ltering on the fi ve quotation frequency groups are different. Filtering data only affects the three highest liquid-ity groups. The raw duration process of the two lowest liquidity groups is optimal in terms of infor-mation contents. However, there are various choices for the fi ltering criteria. Beside price change, it could be the duration, duration and price change together, or other microstructure variables such as bid–ask spread, trading intensity, etc. Optimal choices of such variables and the fi ltering criteria for the forecasting performance are questions for future studies.

Sampling length and forecasting performanceThere is always a trade-off between sample size and computational resource in fi nancial modelling. This becomes an even more important issue in modelling with high-frequency data. The choice of sample length in statistical analysis is mainly determined by the representativeness of the sample as compared to its population. Econometricians have to ensure the sample used represents and captures the data-generating process of the population while ensuring the computational aspect of the analy-sis is feasible. On the other hand, in practising fi nancial modelling, market timing is a very important factor for implementing program trades. For example, a model which has superior predicting power for a return series for the next 30 minutes is not useful to practitioners if its update of parameters takes 1 hour to complete.

In market microstructure research, researchers often choose 3 months of high-frequency data to test hypotheses. In the context of the current research, we examine the effect of sample length on forecasting performance. To our knowledge, we are the fi rst study to examine the effect of sample length in using high-frequency data in market microstructure research.

Table VI reports the forecasting performance of the two models using different sampling lengths. It reports the mean, median, standard deviation, maximum and minimum of the out-of-sample fore-casting accuracy for the 50 NYSE stocks. The parameter for the forecasting model is estimated using 32, 20, and 10 days immediately before the out-of-sample period, which consists of 31 trading days.

Table VI shows that forecasting performance marginally decreases as the sample period shortens. The ACM-ACD model is shown to be more sensitive to the sampling length. The signed rank test



shows that the forecasting performance of the ACM-ACD model is signifi cantly worse when the sample is reduced from 20 days to 10 days. For the AACD model, the medians of the forecasting accuracy are indifferent across the choice of the three sample lengths. In general, the ACM-ACD model has better forecast performance than the AACD model across all sample lengths.

In summary, in most cases, reducing sample length does not signifi cantly worsen the forecasting performance of both models. It suggests that the forecasting power of both models is robust to the choice of sample length. Both models are found to be able to capture the characteristics of the micro data with a minimum sample length of 20 days. This supports the fact that most of the existing market microstructure research has used suffi cient data (normally 3–6 months of data) to character-ize the structure of the data and to test hypotheses.

CONCLUSIONS

Models and forecasts of short-run security returns with high-frequency data are assessed using two competing econometric models: namely, the ACM-ACD and AACD models. In general, the forecast performance of the ACM-ACD model is found to be better than the AACD model. The ACM-ACD model has a higher forecasting accuracy than the AACD model for the whole sample and outper-forms the AACD model for most of the stocks. However, the difference of the forecasting accuracy is not very large and the AACD model is more convenient to estimate.

The effect of market microstructure on the forecasting performance of the two time series model is also examined. The forecast performance of the models generally decreases as the liquidity of the stock (which is measured by quote update frequency) increases, with the exception of the most liquid

Table VI. Sampling length and forecasting performance

Mean Median Std Max. Min.

ACM-ACD-32d 0.5396 0.5283 0.0386 0.6891 0.4897ACM-ACD-20d 0.5377 0.5275 0.0397 0.6891 0.4895ACM-ACD-10d 0.5344 0.5238 0.0393 0.6891 0.4868AACD-32d 0.5368 0.5240 0.0348 0.6883 0.4876AACD-20d 0.5364 0.5231 0.0356 0.6883 0.4900AACD-10d 0.5340 0.5260 0.0342 0.6907 0.4839

Signed rank test

Tests Signrank

ACM-ACD-32d– ACM-ACD-20d 45.5ACM-ACD-20d– ACM-ACD-10d 199**ACM-ACD-32d– ACM-ACD-10d 294***AACD-32d– AACD-20d 18.5AACD-20d– AACD-10d 96AACD-32d– AACD-10d 67

Note: This table reports the forecasting performance of the two models using different sampling lengths. It reports the mean, median, standard deviation, maximum and minimum of the out-of-sample forecasting accuracy for 50 NYSE stocks. The parameter for the forecasting model is estimated using 32, 20, and 10 days immediately before the out-of-sample period, which consist of 31 trading days. The signed rank test is used to test the difference between the models with different in-sample lengths. Asterisks indicate signifi cance at the ***1%, **5% and *10% levels.



stocks. The structural change in the performance of the two econometric models and the predict-ability of the stocks for the highest liquidity groups warrant further research. The results further show that fi ltering raw data points does reduce noise in the data and improves the forecasting per-formance of both models but only for the more liquid stocks. In addition, the results show that forecasting performance only deteriorates marginally and insignifi cantly when the length of the sample is reduced from 32 days to 20 days. This suggests that both models capture the characteris-tics of the micro data very well and the structure of the micro data is stable over time. This supports the common practice within market microstructure research of using 3–6 months of data. Finally, the ACM-ACD model is shown to be more sensitive to the noise in the data and to sampling length than the AACD model.

A couple of directions for future work arise from the current research. First, the model specifi ca-tion in the current paper has been deliberately kept simple to aid comparison. Specifi cation of models adding other microstructure variables may enhance performance. Second, given the predictability of the stock returns by the models, it would be of great interest to fi nd out whether this predictability can be translated into profi table trading strategies.

ACKNOWLEDGEMENTS

We thank the participants in the 2007 International Symposium on Financial Engineering and Risk Management (FERM 2007), Beijing, for their helpful comments.

REFERENCES

Bauwens L, Giot P. 2000. The logarithmic ACD model: an application to the bid–ask quote process of three NYSE stocks. Annales d’Economie et de Statistique 60: 117–149.

Bauwens L, Giot P. 2003. Asymmetric ACD models: introducing price information in ACD models. Empirical Economics 28: 709–731.

Campbell J. 1987. Stock return and the term structure. Journal of Financial Economics 18: 373–399.Chordia T, Roll R, Subrahmanyam A. 2007. Liquidity and market effi ciency. Working paper. Available: http://ssrn.

com/abstract=794264 [27 August 2008].Easley D, O’Hara M. 1987. Price, trade size, and information in securities markets. Journal of Financial Econom-

ics 19: 69–90.Engle R, Russell J. 1998. Autoregressive conditional duration: a new approach for irregularly spaced transaction

data. Econometrica 66: 1127–1162.Fama E, French K. 1988. Permanet and temporary components of stock prices. Journal of Political Economy 96:

246–273.Gallo G, Pacini B. 1998. Early news is good news: the effects of market opening on market volatility. Studies in

Nonlinear Dynamics and Econometrics 2: 115–132.Huang R, Stoll H. 1994. Market microstructure and stock return predictions. Review of Financial Studies 7:

179–215.Psaradakis Z, Spagnolo F, Sola M. 2004. On Markov error-correction models, with an application to stock prices

and dividends. Journal of Applied Econometrics 19: 69–88.Russell J, Engle R. 2005. A discrete-state continous-time model of fi nancial transactions prices and times: the

autoregressive conditional multinomial-autoregressive conditional duration model. Journal of Business and Economic Statistics 23: 166–180.

Tsay R. 2002. Analysis of Financial Time Series. Wiley: New York.Webb R, Smith D. 1994. The effect of market opening and closing on the volatility of eurodollar futures prices.

Journal of Futures Markets 14: 51–78.



Zhang M, Russell J, Tsay R. 2001. A nonlinear autoregressive conditional duration model with applications to fi nancial transaction data. Journal of Econometrics 104: 179–207.

Authors’ biographies:Qi Zhang is a PhD student within the Centre for Advanced Studies in Finance (CASIF) at Leeds University Busi-ness School. Qi’s previous academic achievements are: a bachelors degree and a masters degree in economics from the School of Economics and Management, Tsinghua University, China. Jacky’s research interests are in the area of fi nancial economics, with a primary focus on the microstructure of fi nancial markets and fi nancial econometrics.

Dr Charlie X. Cai holds the position of senior lecturer in Accounting and Finance Department, Leeds University Business School, the University of Leeds. He is also a member of staff within the Centre for Advanced Studies in Finance (CASIF). Charlie’s research interests are in the areas of fi nancial market microstructure, asset pricing, corporate governance, behaviour fi nance and emerging market.

Kevin Keasey is Professor of Financial Services, Director of the International Institute of Banking and Financial Services and Director of the Centre for Advanced Studies in Finance, Leeds University Business School, The University of Leeds. Professor Keasey is an author of 10 books, monographs and edited volumes on behavioural fi nance, small fi rm fi nance and fi nancial markets, and corporate governance, fi nancial regulation and fi nancial services, and is the author of over 95 refereed articles in leading international journals and a range of other pub-lications. He is an editor, deputy editor and on the editorial board of a number of academic journals. He has directed a number of research projects for funding bodies such as the ESRC, ICAEW, CIMA, the Nuffi eld Foun-dation, etc.

Authors’ addresses:Qi Zhang, Charlie X. Cai and Kevin Keasey, Leeds University Business School, Maurice Keyworth Building, The University of Leeds, Leeds LS2 9JT, UK.

Forecasting using high-frequency data: a comparison of asymmetric financial duration models

Documents

Transcript of Forecasting using high-frequency data: a comparison of asymmetric financial duration models