Generalized Autoregressive Conditional Intensity Models with Long Range Dependence

Transcription

1 Generalized Autoregressive Conditional Intensity Models with Long Range Dependence Nikolaus Hautsch Klausurtagung SFB 649 Ökonomisches Risiko Motzen 15. Juni 2007 Nikolaus Hautsch (HU Berlin) GLMACI Models 1 / 61

2 1. Introduction 1. Introduction Modelling of financial data at the transaction level is an ongoing topic in the area of financial econometrics. Seminal papers: Hasbrouck (1991), Engle and Russell (1998), Engle (2000), Russell and Engle (2005) New area of research: High-Frequency Econometrics Issues in high-frequency econometrics: Market microstructure investigations Volatility estimation Liquidity estimation Trading models Nikolaus Hautsch (HU Berlin) GLMACI Models 2 / 61

3 1. Introduction Statistical Properties of Financial Transaction Data Irregular Spacing in Time Discreteness of price changes Bid ask bounce Strong intraday seasonalities Clustering of the trading intensity Nikolaus Hautsch (HU Berlin) GLMACI Models 3 / 61

4 1. Introduction The Irregularity in Trade Arrivals Nikolaus Hautsch (HU Berlin) GLMACI Models 4 / 61

5 1. Introduction The Irregularity in Quote Arrivals Nikolaus Hautsch (HU Berlin) GLMACI Models 5 / 61

6 1. Introduction A multivariate point process Nikolaus Hautsch (HU Berlin) GLMACI Models 6 / 61

7 1. Introduction Using the time information... to explicitly account for the irregular spacing of the data to analyze market microstructure relationships and order book dynamics: trade/quote intensities, buy/sell intensities, order aggressiveness for volatility measurement: time between cumulative absolute price changes for liquidity measurement: time until a certain (buy or sell) buy volume is traded time until a certain net order flow is observed to study financial markets at different time scales ( intrinsic time, Dacorogna et. al, 2001) Nikolaus Hautsch (HU Berlin) GLMACI Models 7 / 61

8 1. Introduction Financial durations - statistical properties positive serial dependence high persistence duration distributions imply non-monotonic hazard functions price durations reveal overdispersion and volume durations reveal underdispersion strong impact of intraday seasonality Nikolaus Hautsch (HU Berlin) GLMACI Models 8 / 61

9 1. Introduction Two Ways to Model Financial Durations Modelling inter-event waiting times (durations) x i = t i t i 1 Autoregressive Conditional Duration (ACD) (Engle/Russell, 1998) Arises from GARCH literature Accelerated Failure Time (AFT) model not suitable for multivariate specifications and time-varying covariates Modelling the (stochastic) intensity 1 λ(t; F t ) = lim 0 E[N(t + ) N(t) F t] Autoregressive Conditional Intensity (ACI) (Russell, 1999) Arises from point process literature or duration literature resp. Proportional Intensity (PI) model Continuous-time specification Nikolaus Hautsch (HU Berlin) GLMACI Models 9 / 61

10 Focus of this Paper 1. Introduction Bringing together ACD and ACI models in a unified generalized ACI framework Relaxing the assumption of a proportional intensity model and allowing for generalized accelerated failure time specifications. Allowing for long range dependence in the intensity process Allowing for more distributional flexibility by proposing a semiparametric specification of the baseline intensity component. The resulting model is called Generalized Long Memory Autoregressive Conditional Intensity (GLMACI) Model. Nikolaus Hautsch (HU Berlin) GLMACI Models 10 / 61

11 Outline of the Talk 1. Introduction 1. Introduction 2. Long Range Dependence in Financial Durations 3. Autoregressive Conditional Duration 4. Univariate Generalized Long Memory ACI Models 5. Multivariate Generalized Long Memory ACI Models 6. Statistical Inference 7. Empirical Evidence 8. Conclusions Nikolaus Hautsch (HU Berlin) GLMACI Models 11 / 61

12 2. Long Range Dependence in Financial Durations 2. Long Range Dependence in Financial Durations Formal Definition Definition: Let X t be a (weakly) stationary process with autocorrelation function of order k, ρ(k). Then, X t implies long range dependence if with α (0, 1) and c > 0. lim k ρ(k)/[ck α ] = 1 in terms of the Hurst (self-similarity) parameter: H = 1 α/2 for α (0, 1) resp. H (0.5, 1): k= ρ(k) = V[ x n ] cn α with c > 0 Nikolaus Hautsch (HU Berlin) GLMACI Models 12 / 61

13 2. Long Range Dependence in Financial Durations log ρ(k) against log k, intertrade durations, AOL, NYSE Nikolaus Hautsch (HU Berlin) GLMACI Models 13 / 61

14 2. Long Range Dependence in Financial Durations log ρ(k) against log k, intertrade durations, Allianz, XETRA Nikolaus Hautsch (HU Berlin) GLMACI Models 14 / 61

15 2. Long Range Dependence in Financial Durations log ρ(k) against log k, intertrade durations, BHP, ASX Nikolaus Hautsch (HU Berlin) GLMACI Models 15 / 61

16 2. Long Range Dependence in Financial Durations log ρ(k) against log k, $0.1 price durations, IBM, NYSE Nikolaus Hautsch (HU Berlin) GLMACI Models 16 / 61

17 2. Long Range Dependence in Financial Durations V[ X n ] against n, intertrade durations, AOL, NYSE Nikolaus Hautsch (HU Berlin) GLMACI Models 17 / 61

18 2. Long Range Dependence in Financial Durations V[ X n ] against n, intertrade durations, Allianz, XETRA Nikolaus Hautsch (HU Berlin) GLMACI Models 18 / 61

19 2. Long Range Dependence in Financial Durations V[ X n ] against n, intertrade durations, BHP, ASX Nikolaus Hautsch (HU Berlin) GLMACI Models 19 / 61

20 2. Long Range Dependence in Financial Durations V[ X n ] against n, $0.1 price durations, IBM, NYSE Nikolaus Hautsch (HU Berlin) GLMACI Models 20 / 61

21 3. Autoregressive Conditional Duration 3. Autoregressive Conditional Duration Inter-event duration: x i := t i t i 1. Principle of the ACD model (Engle/Russell, 1998) where Ψ i := Ψ i (θ) = E[x i F ti 1 ; θ] ε i = x i Ψ i i.i.d., Basic specification: Ψ i = ω + P α j x i j + Q β j Ψ i j, where ω > 0, α 0, β 0. j=1 j=1 Exponential ACD (EACD) model: ε i Exp(1) Nikolaus Hautsch (HU Berlin) GLMACI Models 21 / 61

22 3. Autoregressive Conditional Duration ARMA representation: max(p,q) x i = ω + (α j + β j )x i j where η i := x i Ψ i. Intensity representation: j=1 λ(t; F t ) = λ ε ( x(t) ΨÑ(t)+1 ) Q β j η i j + η i, j=1 1 ΨÑ(t)+1, where λ ε (s) denotes the hazard function of ε i, Ñ(t) denotes the left-continuous counting function and x(t) := t tñ(t). Accelerated Failure Time Model. Nikolaus Hautsch (HU Berlin) GLMACI Models 22 / 61

23 3. Autoregressive Conditional Duration Specifications of Ψ i Linear Model: Ψ i = ω + P α j x i j + Q β j Ψ i j, j=1 j=1 Log ACD based on ACD innovations (Bauwens/Giot, 01): lnψ i = ω + P α j ε i j + Q β j lnψ i j, j=1 j=1 Alternative: ε i = Λ(t i 1, t i ) := ti t i 1 λ(s)ds = 1 Ψ i ti t i 1 λε ( ) x(s) Ψ i Brown/Nair (88): Λ(t i 1, t i ) := t i t i 1 λ(s)ds corresponds to the increment of a Poisson process. Nikolaus Hautsch (HU Berlin) GLMACI Models 23 / 61

24 4. Univariate GLMACI Models 4. Univariate GLMACI Models The basic structure of the univariate GLMACI model is given by λ(t) = λ 0 (η(t))φ N(t)+1 s(t), [ δ η(t) := x(t) Φ N(t)+1 s(t)]. Φ N(t) is a function capturing the model dynamics as well as possible covariates. s(t) is a (deterministic) function of time capturing possible seasonality effects. λ 0 ( ) denotes a baseline intensity function in terms of η(t). η(t) corresponds to the backward recurrence time x(t) := t t N(t) scaled by Φ i and s(t). Nikolaus Hautsch (HU Berlin) GLMACI Models 24 / 61

25 4. Univariate GLMACI Models Parameterization of Φ i : A Fractionally Integrated Process ) Φ i = exp ( Φ i + z i 1 γ. Innovation term: with Λ(t i 1, t i ) i.i.d.exp(1). ε i := 1 Λ(t i 1, t i ) Infinite series representation (Koulikov 03, Hosking, 1981): Φ i = ω + α(1 βl) 1 (1 L) d ε i 1 with β < 1 α(1 βl) 1 (1 L) d ε i 1 = α θ j 1 ε i j, θ j := j αβ k θj k, θ j := k=0 j=1 Γ(d + j) Γ(d)Γ(1 + j) j 0 Nikolaus Hautsch (HU Berlin) GLMACI Models 25 / 61

26 4. Univariate GLMACI Models Stationarity Properties of Φ i Adapting the results by Koulikov (2003) for MD-ARCH( ) processes: Sufficient condition for covariance stationarity: Necessary conditions: α < 1, β < 1 d < 0.5. θj 2 < 1, j=0 Non-summable autocorrelation functions functions for d (0, 1). Nikolaus Hautsch (HU Berlin) GLMACI Models 26 / 61

27 4. Univariate GLMACI Models Special Cases d = 0 and δ = 0 Basic ACI(1,1) specification (Russell, 99): Φ i = ω + α(1 βl) 1 ε i 1 = ω + αε i 1 + β( Φ i 1 ω) Nikolaus Hautsch (HU Berlin) GLMACI Models 27 / 61

28 Special Cases - cntd. 4. Univariate GLMACI Models If λ 0 ( ) is non-specified, and δ = 1, ω = α = β = 0: AFT model (Kalbfleisch/Prentice, 1980) with λ(t) = λ 0 [ x(t)exp(z N(t) γ) ] exp(z N(t) γ). Then: lnx i = z i 1γ + ξ i If λ 0 ( ) is non-specified, and δ = ω = α = β = 0: Semiparametric PI model (Cox, 1972) with λ(t) = λ 0 [x(t)]exp(z N(t) γ). Nikolaus Hautsch (HU Berlin) GLMACI Models 28 / 61

29 Special Cases - cntd. 4. Univariate GLMACI Models If λ 0 ( ) = 1: Long memory Log-ACD model with Φ i = E[x i F ti 1 ] 1 := Υ 1 i and ) Υ i = exp( Υ i z i 1γ Υ i = ω + α(1 βl) 1 (1 L) d (x i 1 /Υ i 1 1), If λ 0 ( ) = 1 and d = 0: Log-ACD model with Υ i = ω + α(x i 1 /Υ i 1 1) + β( Υ i 1 ω) Nikolaus Hautsch (HU Berlin) GLMACI Models 29 / 61

30 4. Univariate GLMACI Models Special Cases - cntd. λ 0 (η(t)) = p η(t) p 1 (1 + κη(t) p ) 1 and δ = 1: Burr type (long memory) ACD model with [ ] Φ i = E[x i F ti 1 ] 1 κ 1+1/p Γ(1 + 1/κ) Γ(1 + 1/p)Γ(κ 1 := Υ i, 1/a) where Υ i = ω α(1 βl) 1 (1 L) d ε i 1. If d = 0: Burr-ACD model with Υ i = ω αε i 1 + β( Υ i 1 ω), where ε i = 1 Λ(t i 1, t i ) equals the integrated intensity. Nikolaus Hautsch (HU Berlin) GLMACI Models 30 / 61

31 Extensions of the Basic Univariate GLMACI Model Extensions of the Basic Univariate GLMACI Model Component-Specific Acceleration Effects Motivation: Accounting for the multiplicative nature of the model. Acceleration effects might be driven by individual components. We allow for component-specific acceleration effects by re-formulating the basic model as η(t) := x(t) Φ δ Φ N(t)+1 s(t)δs, where δ Φ and δ s are specific acceleration parameters affecting the individual components separately. Acceleration/decelaration effects might be driven individually by intraday dynamics and seasonality effects. Basic model for δ Φ = δ s = δ. Nikolaus Hautsch (HU Berlin) GLMACI Models 31 / 61

32 Extensions of the Basic Univariate GLMACI Model Flexible Baseline Intensities Motivation: Burr distribution is good starting point for financial duration processes (Grammig/Maurer, 00) but often not sufficient (Bauwens et al, 04, Bauwens/Hautsch, 06). Define ν(t) := 1 exp( x(t)) with ν(t) [0; 1]. Idea: Augmentation of a Burr parameterization by a flexible Fourier form. Then, λ 0 (η(t)) = p η(t)p κη(t) p [ M ] exp p m,s sin(2mπν(t)) + p m,c cos(2mπν(t)), m=1 where M denotes the order of the process and p m,s and p m,c are coefficients. Nikolaus Hautsch (HU Berlin) GLMACI Models 32 / 61

33 Extensions of the Basic Univariate GLMACI Model Asymmetric News Response Motivation: Recent literature on financial duration models illustrates presence of nonlinear news impact functions (Dufour/Engle, 01, Fernandes/Grammig, 05, or Hautsch, 06.) Following Nelson (91) we can re-specify the process Φ i as Φ i = ω + α(1 βl) 1 (1 L) d ε i 1 + ς(1 βl) 1 (1 L) d ε i 1, where ε i := ε i E[ ε i ] and ς denotes the asymmetry parameter. Nikolaus Hautsch (HU Berlin) GLMACI Models 33 / 61

34 5. Multivariate GLMACI Models 5. Multivariate GLMACI Models K-variate marked point process with arrival times {ti k } i {1,...,n k }, and N k (t) := i 1 1l {ti k t}, k = 1,...K. k-type intensity: E[N k (s) N k (t) F t ] = E [ s ] t λk (u)du F t with compensator Λ k (t) := t 0 λk (u)du. ( Λ1 The processes of compensators (t), Λ 2 (t),..., Λ (t)) K are independent Poisson processes with unit intensity (Brown/Nair, 1988). The k-type integrated intensities Λ k (ti 1 k, tk i ) := ti k λ k (s)ds ti 1 k k = 1,..., K are independently standard exponentially distributed. Nikolaus Hautsch (HU Berlin) GLMACI Models 34 / 61

35 Basic Model 5. Multivariate GLMACI Models The multivariate GLMACI model is given by [ ] λ k (t) = λ k 0 η 1 (t),...,η K (t) Φ k N(t)+1 s k (t), where η k (t) := { η k (t N(t) ) + x(t) Ξ(t) if t N(t) was of type r k x(t) Ξ(t) if t N(t) was of type k and Ξ(t) := K r=1 ( Φ r N(t)+1) δ k r,φ (s r (t)) δk r,s. Hence, the acceleration/deceleration effects act piecewise on the backward recurrence times. Nikolaus Hautsch (HU Berlin) GLMACI Models 35 / 61

36 5. Multivariate GLMACI Models Specification of Φ k i Φ k i = exp ( Φ k i + z i 1γ k) Φ k i = ω k + ε i = K k=1 K j=1 Then: Φ k i = ω k + θ k s,i := {α k j (1 β k j L) 1 (1 L) dk ε i 1 } y j i 1, ( ) 1 Λ k (ti 1, k ti k ) yi k { θ k θ k s { K j=1 α k j } θs 1,iε k i s y j i 1, s=1 s if s = 1 + s m 1 K m=1 θk s m r=0 j=1 βk j yj i r if s > 1. Nikolaus Hautsch (HU Berlin) GLMACI Models 36 / 61

37 5. Multivariate GLMACI Models Stationarity Properties of Φ k i Sufficient conditions for covariance stationarity: j=0 ( θ k j,i) 2 < 1, i = 1, 2,... Necessary conditions: α k < 1, d k 0.5 Sufficient condition for β k j : βk j < 1 Nikolaus Hautsch (HU Berlin) GLMACI Models 37 / 61

38 5. Multivariate GLMACI Models An Alternative Specification for Φ k i Idea: Exploiting the fact that N(t) t 0 λ(s)ds is a MD. Then: (N k (t i ) N k (t i 1 )) t i t i 1 λ k (s)ds is also a MD Alternative specification: Φ k i = ω k + K αj k (1 βj k L) 1 (1 L) dk ε j i 1, j=1 ε j i := (N j (t i ) N j (t i 1 )) Λ j (t i 1, t i ) Nikolaus Hautsch (HU Berlin) GLMACI Models 38 / 61

39 5. Multivariate GLMACI Models Specification of the Baseline Intensity Baseline intensity function λ k [ 0 η 1 (t),...,η K (t) ] : [ K λ k pr k η r (t) pk r 1 M ] 0 [ ] = exp p 1 + κ r=1 k r η r (t) pk m,r,s k sin(2 mπν r (t)) r m=1 [ M ] exp pm,r,c k cos(2 mπν r (t)), pr k > 0, κ k r 0, m=1 where ν k (t) := 1 exp( x k (t)). Nikolaus Hautsch (HU Berlin) GLMACI Models 39 / 61

40 6. Statistical Inference 6. Statistical Inference Log Likelihood (Karr, 1991): ln L(θ; {N(t)} t (0,T] ) [ K T = (1 λ k (s))ds + = k=1 n 0 i=1 k=1 (0,T] lnλ k (s)dn k (s) K [ ] ( Λ k (t i 1, t i )) + ln λ k (t i ) yi k + KT. Diagnostics based on three different types of model residuals: ei,1 k := ˆΛ k (ti 1, k ti k ), K ( ) e i,2 := 1 ˆΛ k (ti 1, k ti k ) yi k k=1 e i,3 := 1 ˆΛ(t i 1, t i ) = 1 K ˆΛ k (t i 1, t i ). k=1 Nikolaus Hautsch (HU Berlin) GLMACI Models 40 / 61 ],

41 7. Empirical Evidence 7. Empirical Evidence Transaction data of the GE stock traded at the NYSE, 01-02/2003. Financial durations: Trade durations: Time between consecutive transactions. Price durations: Time between absolute cumulative midquote changes. Volume durations: Time until a cumlative volume is traded. Net volume durations: Time untile a cumulative net volume is traded. Overnight spells as well as all trades before 9:45 and after 16:00 are removed. Seasonality function: s(t) = j=1 ν j(t τ j ) 1l {t>τj }, where τ j, j = 1...,6, denote the 6 nodes within a trading day and ν j the corresponding parameters. Nikolaus Hautsch (HU Berlin) GLMACI Models 41 / 61

42 7. Empirical Evidence Estimates of GLMACI models for price durations, GE stock, dp {0.01, 0.05} dp = 0.01 dp = 0.05 Est. p-v. Est. p-v. Est. p-v. Est. p-v. ω p κ p 1,s p 2,s p 3,s p 1,c p 2,c p 3,c δ α ς β d n LL BIC Mean e i S.D. e i LB(20) e i Exc. Disp. e i χ 2 20 PIT LB(20) PIT Nikolaus Hautsch (HU Berlin) GLMACI Models 42 / 61

43 7. Empirical Evidence QQ-plot and baseline intensity, ACI model, price durations, dp = Nikolaus Hautsch (HU Berlin) GLMACI Models 43 / 61

44 7. Empirical Evidence QQ-plot and baseline intensity, GLMACI model, price durations, dp = Nikolaus Hautsch (HU Berlin) GLMACI Models 44 / 61

45 7. Empirical Evidence QQ-plot and baseline intensity, ACI model, price durations, dp = Nikolaus Hautsch (HU Berlin) GLMACI Models 45 / 61

46 7. Empirical Evidence QQ-plots and baseline intensity, GLMACI model, price durations, dp = 0.05 Nikolaus Hautsch (HU Berlin) GLMACI Models 46 / 61

47 7. Empirical Evidence Estimates of GLMACI models for volume durations, GE stock v {5000, 10000} v = 5, 000 v = 10, 000 Est. p-v. Est. p-v. Est. p-v. Est. p-v. ω p κ p 1,s p 2,s p 3,s p 1,c p 2,c p 3,c δ α ς β d n LL BIC Mean e i S.D. e i LB(20) e i Exc. Disp. e i χ 2 20 PIT LB(20) PIT Nikolaus Hautsch (HU Berlin) GLMACI Models 47 / 61

48 7. Empirical Evidence QQ-plot and baseline intensity, ACI model, volume durations, v = 5, 000. Nikolaus Hautsch (HU Berlin) GLMACI Models 48 / 61

49 7. Empirical Evidence QQ-plot and baseline intensity, GLMACI model, volume durations, v = 5, 000. Nikolaus Hautsch (HU Berlin) GLMACI Models 49 / 61

50 7. Empirical Evidence QQ-plot and baseline intensity, ACI model, volume durations, v = 10, 000. Nikolaus Hautsch (HU Berlin) GLMACI Models 50 / 61

51 7. Empirical Evidence QQ-plot and baseline intensity, GLMACI model, volume durations, v = 10, 000. Nikolaus Hautsch (HU Berlin) GLMACI Models 51 / 61

52 7. Empirical Evidence QQ-plots and baseline intensity, ACI model, net volume durations, v = 5, 000 Nikolaus Hautsch (HU Berlin) GLMACI Models 52 / 61

53 7. Empirical Evidence QQ-plots and baseline intensity, GLMACI model, net volume durations, v = 5, 000 Nikolaus Hautsch (HU Berlin) GLMACI Models 53 / 61

54 7. Empirical Evidence Simultaneous Modelling of Price Intensities and Net Volume Intensities Relationship between speed of absolute price changes and one-sided volume absorption. Studying the relationship between intraday volatility and one-sided liquidity demand. Testing for Granger causalities. Dynamic properties? Measuring realized market depth: λ(t) dp -λ(t) nv Net volume per price change. Extension of VNET measure by Engle and Lange (2001). Dynamic properties? Predictability? Nikolaus Hautsch (HU Berlin) GLMACI Models 54 / 61

55 7. Empirical Evidence Estimates of Bivariate GLMACI models for price durations, dp = 0.02, and net volume durations v = 10, 000 ACI GLMACI Est. p-v. Est. p-v. δ δ δ δ α α α α β β β β d d n LL BIC ei,1 1 ei,1 2 e i,2 e i,3 ei,1 1 ei,1 2 e i,2 e i,3 Mean S.D LB(20) e i Exc. Disp. e i χ 2 20 PIT LB(20) PIT Nikolaus Hautsch (HU Berlin) GLMACI Models 55 / 61

56 7. Empirical Evidence QQ-plots, bivariate basic ACI model Nikolaus Hautsch (HU Berlin) GLMACI Models 56 / 61

57 7. Empirical Evidence QQ-plots, bivariate GLMACI model Nikolaus Hautsch (HU Berlin) GLMACI Models 57 / 61

58 7. Empirical Evidence Estimated intraday seasonalities Nikolaus Hautsch (HU Berlin) GLMACI Models 58 / 61

59 7. Empirical Evidence QQ-plots, Estimated realized market depth, λ(t) dp -λ(t) nv Nikolaus Hautsch (HU Berlin) GLMACI Models 59 / 61

60 7. Empirical Evidence QQ-plots, Estimated realized market depth, λ(t) dp -λ(t) nv Nikolaus Hautsch (HU Berlin) GLMACI Models 60 / 61

61 8. Conclusions 8. Conclusions Basic ACI model is not flexible enough. Accounting for long range dependence significantly improves the dynamical properties. However, often d (0.5, 0.6) No square-summability of coefficients θ j. Semiparametric specification of baseline intensity significantly improves the goodness-of-fit. Clear evidence for acceleration effects. Cross-effects in bivariate model. Proportional intensity specification rejected. Nikolaus Hautsch (HU Berlin) GLMACI Models 61 / 61