On diagnostic checking the autoregressive conditional intensity model
Simon Kwok and Wai Keung Li
Cornell University

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Introduction

Autoregressive Conditional Duration (ACD) Model (Engle, Russell, 1998) is useful for modeling duration clustering in high-frequency …nancial time series. Autoregressive Conditional Intensity (ACI) Model (Russell; 1998) describes the arrival times from the intensity perspective. This paper solves two problems regarding ACI model:
Justify the independence of the generalized residuals. Develop a residual autocorrelations test that is more accurate and powerful than standard statistical tests for time series models.

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Why duration models are important?
They describe and capture the stylised facts of durations in high-frequency …nancial data set (e.g. clustering of transactions, leptokurtosis of duration). Trade durations (i.e. interarrival time between transactions) contain valuable information (Easley and O’ hara, 1992). Duration models are able to extract such information (e.g. probability of informed trading (Easley, Kiefer, O’ hara, Paperman, 1996), volatilities (Engle, 2000)). They o¤er a platform on which market microstructure theories are tested (Tay, et. al., 2004; Kwok, et. al., 2009). For instance, Easley and O’ hara (1992) concluded that low trading intensity indicates the absence of private information, while Diamond and Verrecchia (1997) claimed that low trading intensity means bad news were held by informed traders.

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ACD(p,q) model Dynamics of duration
xi = ψi εi ψi = E [xi jti
1 , . . . , t0 ] = ω +

xi = ti

ti

1

:
p j

j =1

∑ αj xi

q

j

+ ∑ βj ψi
j =1

εi are i.i.d. random variables with non-negative support.

Di¢ culties:
hard to model multivariate marked point processes. hard to incorporate into the model new information that arrives in between two transactions.

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Unmarked Point process

Data: arrival times fti gn=1 i Counting process: N(t) = Assumptions:

i =1

∑ 1(ti

n

t)

Orderliness: P(dN(t) > 1) = 0. Integrability: E (N(t)) < ∞ for all t

0.

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Marked point process

Data: fti , wi gn=1 , ti is the i th arrival time, with associated mark i wi 2 f1, . . . , K g. Counting process (pooled): N(t) = where N k (t) =
n k =1

∑ N k (t),

K

i =1

∑ 1(tik

t),

and tik is the i th arrival time with mark k, k = 1, . . . , K . Assumptions of orderliness and integrability are imposed on the pooled counting process N(t).

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Marked point process
2 marks: 1 & 2
pooled arrival times

time t0
t1

t2

t3

t4

arrival times for k=1

k=1

time t0

t

1 1

t

1 2

arrival times for k=2

time
k=2 t0

t12

2 t2

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One-mark ACI(1,1) model

The arrival times fTi g admit the following dynamics: R Ti λ(t)dt (generalized residual) εi = 1 Ti 1 where ( expfω + φN (t ) g ACI basic model λ(t) = (t ti 1 )γ expfω + φN (t ) g ACI extended model Dynamics of (log) intensity is described by an ARMA(1,1) structure: φN (t ) = αεN (t ) + βφN (t ) 1 for all t 2 [Ti 1 , Ti ). N(t) =
i =1

∑ 1(Ti

∞

t) is the counting process.

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K-mark ACI(1,1) model
The arrival times fTik g associated with mark k (k = 1, . . . , K ) admit the following dynamics: R T ik k λ (t)dt, εk = 1 i Tk
i1

where
k

λ (t) =

8 > < > :

j =1

∏ U j (t)γ

K

expfω k + φk (t ) g N
k j

ACI basic model

expfω k + φk (t ) g ACI extended model N

Dynamics of (log) intensity is described by an ARMA(1,1) structure: k φN (t ) = αk εk k (t ) + B k φN (t ) 1 , U k (t) = t TN k (t ) for all ˘ N t 2 [TN (t ) , Tik ), where φN (t ) = [φ1 (t ) , . . . , φK (t ) ]0 . N N N k (t) = N(t) =
i =1 K

∑ 1(Tik

∞

t) is the unpooled counting process.

k =1

∑ N k (t) is the pooled counting process.
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Kwok and Li (Cornell University)

Interpretation of intensity function

If λ(t) is right continuous and bounded by an integrable random variable for all t, then λ(s)ds = limt #s E [ N(t) N(s)j Fs ] = P[ dN(s) = 1j Fs ]

Rt M(t)h= N(t) λ(r )dr is a martingale. 0i Rt ) E s λ(r )dr Fs = E [ N(t) N(s)j Fs ]

(t > s

0)

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Generalized residuals

R Ti λ(r )dr =1 Ti 1 εi is a martingale di¤erence sequence: E ( εi j FT i a ) = E ( M(Ti ) M(Ti 1 )j FT i a ) = 0 for all d = 1, 2, . . .
εi := M(Ti ) M(Ti
1)

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Independence of generalized residuals

Theorem

Marked compensators Rt Λk (t) := 0 λk (r )dr

for all k = 1, . . . , K

Suppose a multivariate point process fN 1 (t), . . . , N K (t)g is formed from arrival times fTi1 g, . . . , fTiK g and has absolutely continuous compensator fΛ1 (t), . . . , Λk (t)g such that Λk (∞) = ∞ for all k = 1, . . . , , K . Then the point processes fΛ1 (Ti1 )g, . . . , fΛK (TiK )g are independent standard Poisson processes. (Meyer, 1971)

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Independence of generalized residuals
Corollary
Assume that, for all k = 1, . . . , K , the marked compensator Λk (t) satisfying Λk (∞) = ∞ is absolutely continuous with respect to the Lebesgue measure and admits the predictable process λk (t) as its density. De…ne the generalized residuals εk of N k (t) by i εk i

=1

Z Tk i

T ik 1

λk (t)dt

for all k = 1, . . . , K . Then 0 (i) εk and εk0 are independent for i 6= i 0 , or k 6= k 0 , or both; i i (ii) εk has a translated exponential distribution with mean 0 for i i = 1, 2, . . . and k = 1, . . . , K .

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Diagnostic checking
H0 : The sequence fεk g is independent over all i and k. i 0 H1 : εk and εk0 are dependent for some i 6= i 0 or k 6= k 0 . i i We concentrate on autocorrelations test. We can reduce the diagnostic checking problem of a multiple-marked ACI model to that of a single-marked ACI model.
Reason: Under H0 , the marked generalized residuals εk are the i durations of independent Poisson processes Λk (t) all with intensity 1 (by theorem), so the combined unmarked residuals are the durations of the Poisson process Λ(t) = 2007)
k =1

∑ Λk (t) with intensity K .

K

(Bowsher,

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Autocorrelations test of 1-mark ACI models

Data: arrival times fti gn=1 i

ˆ Generalized residual: εi = 1

ˆ The corollary implies that εi follows an iid translated exponential distribution with mean 0 and variance 1. Lag-m autocorrelation " : rm = ˜
1 nm i =m+1

R ti
ti

1

λ(r )dr

∑

n

ˆ εi n.
m.

ˆ ε

ˆ εi

m

ˆ ε

#, "

1 n

i =1

∑

n

ˆ εi

ˆ ε

2

#

,

m = 1, . . . , M, M Let rm = ˆ
1 n

i =m+1

∑

n

ˆˆ εi εi

By WLLN, rm ˜

rm ! 0. ˆ

P

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Autocorrelations test of 1-mark ACI models
Let r = (ˆ1 , r2 , . . . , rM )0 be the residual autocorrelations with MLE ˆ rˆ ˆ ˆ parameters θ. Consider the Taylor series expansion around true parameters θ: ∂r ˆ r = r + ∂θ (θ θ) + Op (n 1 ). ˆ

Theorem p0 p 1 nˆ pVar ( nˆ ) r r r ! χ2 , where ˆ M Var ( nˆ ) = IM + XG 1 X 0 XG 1 Y Y 0 G 1 X 0 +ho(1), where r i ∂r 1 ∂2 L X = plim ∂θ , Y = plim ∂L r 0 and G = plim E n ∂θ∂θ 0 . ∂θ
L = L(θ) =
k =1

function. X , Y and G can be estimated consistently. Explicit forms are derived for ACI(1,1) extended and basic models.
Kwok and Li (Cornell University) ACI model diagnostics 16 / 26

∑ 4Λk (T )

K

2

N k (T ) i =1

∑

log λk (tik )5 is the log likelihood

3

Test statistics

Our test statistic: Box-Pierce: Ljung-Box: Modi…ed Ljung-Box:

p Q1 (M) = nˆ 0 [Var ( nˆ )] r r
Q2 (M) = nˆ 0 r = n rˆ Q3 (M) = n(n + 2) Q4 (M) = Q3 (M) +
m=1 M

1r ˆ

∑

M

rm ˆ2
rm ˆ2 nm

m=1 M (M +1) 2n

∑

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Simulation experiments: Size
Exp S1a S1b S1c S1d S2a S2b S2c S2d DGP ACI(1,1) xtd ACI(1,1) xtd ACI(1,1) xtd ACI(1,1) xtd ACI(1,1) basic ACI(1,1) basic ACI(1,1) basic ACI(1,1) basic Fitted Model ACI(1,1) xtd ACI(1,1) xtd ACI(1,1) xtd ACI(1,1) xtd ACI(1,1) basic ACI(1,1) basic ACI(1,1) basic ACI(1,1) basic α 0.15 0.05 0.05 0.02 0.15 0.05 0.05 0.02 θ in β 0.6 0.8 0.9 0.95 0.6 0.8 0.9 0.95 DGP ω -0.07 -0.07 -0.07 -0.07 -0.07 -0.07 -0.07 -0.07

γ -0.3 -0.3 -0.3 -0.3

Rt φi = αεi + βφi 1 where εi = ti i 1 λ(t)dt ACI(1,1) basic model: λ(t) = expfω + φi 1 g for t 2 [ti 1 , ti ). ACI(1,1) extended model: λ(t) = (t ti 1 )γ expfω + φi 1 g for t 2 [ti 1 , ti ).

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Simulation results - ACI(1,1) basic model
Length Q1 (20) 5000 10000 15000 5000 10000 15000 5000 10000 15000 5000 10000 15000 0.0567 0.0544 0.0439 0.0280 0.0327 0.0253 0.0314 0.0363 0.0307 0.0552 0.0608 0.0554 Size Q2 (20) Q3 (20) Q4 (20) Q1 (5) Q2 (5) Exp S2a: ACI(1,1) basic (α = 0.15, β = 0.6) 0.0250 0.0253 0.0253 0.0454 0.0116 0.0276 0.0276 0.0276 0.0403 0.0111 0.0231 0.0231 0.0231 0.0341 0.0097 Exp S2b: ACI(1,1) basic (α = 0.05, β = 0.8) 0.0270 0.0277 0.0274 0.0277 0.0181 0.0313 0.0313 0.0313 0.0257 0.0173 0.0213 0.0217 0.0213 0.0283 0.0183 Exp S2c: ACI(1,1) basic (α = 0.05, β = 0.9) 0.0290 0.0294 0.0290 0.0320 0.0273 0.0340 0.0340 0.0340 0.0297 0.0240 0.0263 0.0267 0.0267 0.0330 0.0273 Exp S2d: ACI(1,1) basic (α = 0.02, β = 0.95) 0.0411 0.0411 0.0411 0.0535 0.0324 0.0471 0.0471 0.0471 0.0537 0.0337 0.0364 0.0374 0.0367 0.0587 0.0373 Q3 (5) 0.0116 0.0111 0.0097 0.0181 0.0177 0.0183 0.0273 0.0240 0.0273 0.0324 0.0340 0.0373 Q4 (5) 0.0116 0.0111 0.0097 0.0181 0.0173 0.0183 0.0273 0.0240 0.0273 0.0324 0.0340 0.0373

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Simulation results - ACI(1,1) extended model
Length Q1 (20) 5000 10000 15000 5000 10000 15000 5000 10000 15000 5000 10000 15000 0.0442 0.0452 0.0413 0.0441 0.0491 0.0423 0.0513 0.0471 0.0397 0.0536 0.0551 0.0467 Size Q2 (20) Q3 (20) Q4 (20) Q1 (5) Q2 (5) Exp S1a: ACI(1,1) extended (α = 0.15, β = 0.6) 0.0268 0.0279 0.0272 0.0685 0.0118 0.0275 0.0275 0.0275 0.0728 0.0121 0.0239 0.0239 0.0239 0.0785 0.0102 Exp S1b: ACI(1,1) extended (α = 0.05, β = 0.8) 0.0256 0.0259 0.0259 0.0341 0.0203 0.0296 0.0299 0.0296 0.0289 0.0164 0.0245 0.0249 0.0249 0.0321 0.0177 Exp S1c: ACI(1,1) extended (α = 0.05, β = 0.9) 0.0282 0.0296 0.0292 0.0333 0.0257 0.0324 0.0324 0.0324 0.0287 0.0227 0.0267 0.0267 0.0267 0.0320 0.0247 Exp S1d: ACI(1,1) extended (α = 0.02, β = 0.95) 0.0335 0.0342 0.0338 0.0377 0.0293 0.0374 0.0384 0.0381 0.0370 0.0300 0.0307 0.0307 0.0307 0.0390 0.0310
ACI model diagnostics

Q3 (5) 0.0118 0.0121 0.0102 0.0203 0.0164 0.0177 0.0257 0.0227 0.0247 0.0297 0.0300 0.0310

Q4 (5) 0.0118 0.0121 0.0102 0.0203 0.0164 0.0177 0.0257 0.0227 0.0247 0.0297 0.0300 0.0310

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Simulation experiments: Power

Exp P1a P1b P2a P2b P2c P2d

DGP ACI(2,1) xtd ACI(1,2) xtd ACI(2,1) b ACI(3,1) b ACI(1,2) b ACI(1,3) b

Fitted Model ACI(1,1) xtd ACI(1,1) xtd ACI(1,1) b ACI(1,1) b ACI(1,1) b ACI(1,1) b α1 0.1 0.1 0.14 0.1 0.1 0.05

θ in DGP (all ω = α2 α3 β1 0.55 0.05 0.8 0.55 0.4 0.05 0.8 0.03 0.01 0.8

0.07) β2 β3 0.4 0.3 0.2

γ -0.3 -0.3

0.1

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Simulation results - ACI(1,1) basic model
Length Q1 (20) 5000 10000 15000 5000 10000 15000 5000 10000 15000 5000 10000 15000 0.1973 0.4650 0.7160 0.1067 0.2558 0.4246 0.2907 0.6587 0.8777 0.1383 0.3097 0.5277 Power Q2 (20) Q3 (20) Q4 (20) Q1 (5) Exp P2a: ACI(1,1) basic vs ACI(2,1) 0.1880 0.1910 0.1890 0.3830 0.4587 0.4597 0.4593 0.7630 0.7110 0.7113 0.7110 0.9287 Exp P2b: ACI(1,1) basic vs ACI(3,1) 0.0953 0.0956 0.0953 0.1629 0.2398 0.2408 0.2398 0.4148 0.4073 0.4106 0.4096 0.6583 Exp P2c: ACI(1,1) basic vs ACI(1,2) 0.2820 0.2863 0.2840 0.5533 0.6527 0.6537 0.6533 0.8920 0.8727 0.8737 0.8737 0.9797 Exp P2d: ACI(1,1) basic vs ACI(1,3) 0.1283 0.1307 0.1297 0.2863 0.2980 0.2980 0.2980 0.5873 0.5160 0.5177 0.5170 0.7970
ACI model diagnostics

Q2 (5) basic 0.3523 0.7380 0.9187 basic 0.1283 0.3207 0.5560 basic 0.5087 0.8687 0.9737 basic 0.2357 0.5240 0.7507

Q3 (5) 0.3527 0.7380 0.9187 0.1283 0.3207 0.5563 0.5090 0.8687 0.9737 0.2363 0.5243 0.7510

Q4 (5) 0.3523 0.7380 0.9187 0.1283 0.3207 0.5560 0.5087 0.8687 0.9737 0.2360 0.5240 0.7507
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Simulation results - ACI(1,1) extended model

Length Q1 (20) 5000 10000 15000 5000 10000 15000 0.1977 0.3919 0.5923 0.3048 0.6616 0.8664

Power Q2 (20) Q3 (20) Q4 (20) Q1 (5) Exp P1a: ACI(1,1) extended vs ACI(2,1) 0.1676 0.1699 0.1686 0.3260 0.3646 0.3656 0.3656 0.6467 0.5772 0.5779 0.5779 0.8617 Exp P1b: ACI(1,1) extended vs ACI(1,2) 0.2803 0.2826 0.2816 0.5543 0.6582 0.6596 0.6589 0.8953 0.8741 0.8741 0.8741 0.9797

Q2 (5) extended 0.3100 0.6347 0.8563 extended 0.5080 0.8690 0.9730

Q3 (5) 0.3100 0.6350 0.8563 0.5080 0.8693 0.9730

Q4 (5) 0.3100 0.6347 0.8563 0.5080 0.8693 0.9730

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Application

Data: HSBC in HKSE, …rst 5000 stock transactions since January 2, 2003. Data cleansing following Engle and Russell (1998):
drop those transactions before the opening and after the close of stock exchange, and also those in the …rst 20 minutes on each trading day to mitigate opening auction e¤ect. apply diurnal adjustment to remove season e¤ect.

Choose small M = 4, justi…ed as the downward adjustment to the asymptotic variance of the residual autocorrelations is the most signi…cant in the …rst few lags (Ansley and Newbold, 1979).

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Result
ACI(1,1) basic 0.1435 (0.005218) 0.9551 (0.004009) -3.1007 (0.054568) 5000 -20438.58 15.5862 14.4745 14.4856 14.4775 ACI(1,1) xtd 0.1523 (0.006184) 0.9540 (0.004647) -2.8819 (0.056944) -0.0848 (0.008419) 5000 -20398.55 11.0347 9.2856 9.2931 9.2876

α β ω γ n L Q1 (4) Q2 (4) Q3 (4) Q4 (4)

χ2 (0.95) = 9.48 4

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Conclusions and future directions Conclusions
Proved the independence of the generalized residuals. Constructed a residual autocorrelations test that outperforms other o¤-the-shelf statistical tests in terms of size and power.

Future Directions
Consider even more ‡exible intensity functions (or ACI semiparametric models?). Apply ACI models to test empirically the market microstructure theories (ACI models with exogenous marks). Think about the reciprocal relationship between ACD models and ACI models.

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