Econometr ic Model of Tr ading Volume
1. Intr oductionEquation Section 1
A standard mixture of normal distribution model (e.g. Tauchen and Pitts (1983)) can be summarized as follows.
| ~ (0, )
p n N bn
∆ , (1.1)
V n| ~ N cn dn( , ) , (1.2)
Cov(∆p V n, | )=0 , (1.3) where ∆p is the change in security price, V is the trading volume and n (representing
“news”) is the mixing variable. It is certainly possible to consider a more general specification in (1.1) in which the variance of price change is a function of mixing variable n, i.e. ∆p~ N( , ( ))0 f n . For example, if we let ∆p= 1+bnη , η1 1 σ1
2
0 ~ N( , ), and let n be Bernoulli distributed (e.g. Ball and Torous’s (1983) Bernoulli jump process),
∆p obeys the widely used two component normal mixture model. Similarly, we may
write V= +(1 cn)µ2+ 1+dnη2, 2
2 ~ N(0, 2)
η σ . When n is Bernoulli distributed, this is
equivalent to a switching regression model for trading volume.
Straightforward computations lead to the following moment equations:
E[∆p2]=bE n( ) (1.4) Cov p V[∆ 2, ]=bcvar( )n (1.5) E V( )=cE n( ) (1.6) var( )V =dE n( )+c2var( )n (1.7) I use ∆p2 to denote the square of price changes, not the difference in square prices.
Result (1.5) is a confirmed empirical regularity that price variability and trading volume is positively correlated. Owing to the fact that the first two moments completely characterize the normal distribution, there exist only four independent moment equations, but there are five parameters in the system, b, c, d, E(n) and var(n). Hence, as in any
model with a latent structure, the mixture of normal system is not identified. Some sort of normalization is absolutely required. The normalization used in Tauchen and Pitts (1983) is equivalent to setting E(n)=1. Generally, parameter restrictions must be imposed upon
the probability model of the mixing variable.
The Poisson model for the mixing variable proves to be quite convenient since the mean and variance are identical in Poisson distribution. Poisson model has its own appeal, yet it is perhaps too restrictive in the sense that the numbers of occurrence (the number of intra-day equilibria, in Tauchen and Pitts’ terminology) are independent over non-overlapping time intervals. If the news arrival process is correlated, then Poisson model is inappropriate. Other specifications for the mixing variable documented in the literature
include the log-normal distribution in Clark (1973) and Tauchen and Pitts (1983), the inverted gamma distribution in Praetz (1972) and Blattberg and Gonedes (1974). It is an empirical question regarding which distribution fits the data better.
The contribution of the mixture distribution hypothesis is that it explains the leptokurtic nature of many financial time series while preserving the property of finite second moment, and this is why MDH is replacing its main competitor, the stable distribution hypothesis in the 70’s. The assumption of serially independent sequence {(∆p V, )} seems to me the main problem of MDH. Yet researcher has devoted little effort to this problem.
2. ARCH in Mean Equation Section 2
In this article I propose a new model, focusing on the most important implication from the MDH, equation (1.5). No independence assumption is required in my methodology, nor is the distributional assumption on the mixing variable. Rewrite (1.1) and (1.2) as 1, 1 ~ (0,1) t t t t p cn Z Z N ∆ = (2.1) 2, 2 ~ (0,1) t t t t t V =cn + dn Z Z N (2.2)
Obviously, there is no way to eliminate the mixing variable n, hence an explicit
functional relationship between price change and trading volume is not available. However, if the price change and trading volume are indeed driven by the same mixing variable as proposed in the MDH, a linear model for the price variability exists since ∆p2
and trading volume must be positively correlated as in (1.5).
Begin with equation (2.1). If n is serially independent, then it cannot account for t
the ARCH effect in price change series ∆pt . However, the ARCH effect in many
individual ∆pt series is visible. Let the mixing variable be consistent with the conditional
heteroskedasticity in {∆pt} series. Through the construction of the variance equation, we can eliminate the mixing variable and derive a certain structure for trading volume and price variability. One possibility is to let
2 0 1 1,
t t
n =α α+ ∆p− (2.3)
Obviously, GRACH model (Bollerslev (1986)), nt =α α0+ ∆1 pt2−1+β1nt−1 , is a
straightforward extension. EGARCH can also be tried. Another extension that requires
more estimation effort is to consider 2
0 1 1
t t t
n =α α+ ∆p− +ε , where ε is the random error t term. This is precisely the stochastic volatility model in the literature. Specification (2.4) also has intuitive interpretation. The market tends to be volatile if it is volatile at time t-1
Having given nt =α α0+ ∆1 pt2−1, it follows that t t it p h Z ∆ = , 2 0 1 1 t t h =a + ∆a p− (2.4) 0 0 , 1 1 ,| 1| 1 a =α b a =αb a <
Straightforward computation shows that
2 2 1 2 2 0 1 2 (1 ) var( ) (1 3 ) t b a n a α − = − , hence MDH structure
further imposes the restriction |a1| 1 3< .
Now the volume process can be derived. Using (2.2) and plug in n specification, t
2 2 0 1 1 0 1 1 2 ( ) ( ) t t t t V =cα α+ ∆p− + d α α+ ∆p− Z = ( 0 1 t21) 0 1 t21 2t t t 2t c c d d p d d p Z g g Z d α +α ∆ − + α +α ∆ − =d + . It follows that 2 t t t t V =δg + g Z , (2.5) 2 0 1 1 t t g = + ∆γ γ p− 0 0 1 1 / , , c d d d δ = γ =α γ =α .
Substitute the trading volume model in (2.6) into (2.3), a particular model for the price variability can be derived.
2 0 1 0 1( 2 ) t t t t t t t p ω ωV ε ω ω δg g Z ε ∆ = + + = + + + = ω ω0+ 1*gt+( g Zt 2t+εt) (2.6)
Suitably reparameterizing, (2.7) suggests an ARCH-in-mean model for the price variability. Model (2.6) and (2.7) can be estimated across a spectrum of individual stocks.
Estimating model (2.6) was carried out, yet the empirical results were somewhat disappointing. Among all the sample stocks I tried, achieving estimates convergence proved to be a challenging task. It is my conjecture that the ARCH-in-mean representation may not be an adequate model for trading volume. However, as the next section shows, modeling conditional heteroskedasticity in volume series is much more successful.
3. Conditional Heter oskedasticity in Tr ading Volume
Another strategy is to model the news arrival process to be consistent with the conditional heteroskedasticity in volume series. Specifically,
1 | ~ ( , ) t t t t V ψ− N k n , 2 0 1 1, 0 1( 1 0 1 2) t t t t t k = +φ φV− n =π +π V− − −φ φV− (2.7)
This is the ARCH(1,1) model in Tsay’s (1987) term, which can be easily extended to ARCH(p,q). Clearly, we have let the news arrival process n depend on the surprised t
trading volume at t-1.
As shown in Tsay (1987), an ARCH(1,1) model is second-order equivalent to the autoregressive model with random AR coefficient. It follows that the price variability regression in (2.3) is equivalent to a regression against lagged trading volumes with time varying parameters. 2 0 ( 1 1) 1 ( 2 2 ) 2 t t t t t t p m m υ V− m υ V− ξ ∆ = + + + + + , (2.8) 1 2 ~ (0, ) t t niid υ υ Σ
, ξ is a white noise, and t
1 2 t t υ υ
and ξ are mutually independent. The t
advantage of this model is that all regressors are ex-ante, its forecast performance can be used as an objective criterion to evaluate the adequacy of specification of news arrival process.
Two major market indices were estimated. Results were summarized in Table I. Standard errors are included in parentheses. LM test rejects the null of homoskedasticity with nil prob-value. Furthermore, the both coefficients for lagged trading volume are highly significant.
Table I
Market Index LM test for HSK GLS Coeff of Vt−1 GLS Coeff of Vt−2
NASDAQ (T=4463) 119.61 92.96 (14.49) 57.78 (14.02)
DOW (T=4958) 54.54 133.97 (58.25) 422.24 (55.74)
Table II summarizes empirical results for individual stocks, with t statistics included in
the parentheses. LM test rejects the null of homoskedasticity for the following 17 stocks. There have been some sample stocks in which the null was not rejected. The ratio rejecting versus accepting the null for all sample stocks is about six to four. Hence model (2.8) is a worthy empirical model. Table II also shows that almost all coefficients of trading volume are highly significant, except the case of IBM. Moreover, these coefficients were mostly positive, indicating that the relation between price variability and lagged trading volume is positive.
Table II
Stock Symbol LM test for HSK GLS Coeff of Vt−1 GLS Coeff of Vt−2
DELL 16.77 0.034 (20.29) 0.016 ( 9.82) ORCL 26.24 0.175 (14.83) 0.006 ( 5.33) C 8.26 0.105 (38.47) -0.183 (-6.18) AA 61.19 0.161 (27.24) 0.026 ( 4.51) ADBE 14.57 0.617 (21.37) 0.256 (10.30) AMAT 18.21 0.045 (18.16) 0.028 (10.71)
AMTD 13.70 0.377 (14.73) 0.389 (12.05) CSCO 59.49 -0.007 (-0.64) 0.033 (27.59) CTXS 541.69 1.104 (23.75) -0.415 (-16.99) AMZN 18.91 0.436 (12.47) 0.226 ( 6.29) GLW 8.74 0.120 (4.77) 0.255 ( 9.76) GE 107.56 0.075 (55.33) -0.229 (-18.51) IBM 9.95 0.579 (23.41) 0.023 ( 0.98) MEL 39.09 0.296 (25.39) 0.039 ( 4.79) MRK 12.38 0.139 (16.23) 0.100 (10.81) MU 34.96 0.296 (22.32) 0.077 (6.41) MWD 35.71 0.604 (18.64) 0.135 (4.27) 4 Tentative Conclusion
In this research we propose a new model of conditional heteroskedasticity for trading volume. Empirical results were mostly in favor of this new model. The regression of price variability on lagged trading volumes is particularly attractive due to its forecasting ability. Comparison of the forecasting performance is reported in the full-versioned draft of this research.
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