Modeling dynamic diurnal patterns in high frequency financial data Ryoko Ito 1 1 Faculty of Economics, Cambridge University Tinbergen Institute Amsterdam, January 2013 Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 1/27
Introduction We want to build a model for high frequency observations of financial data Not easy: need to capture stylized features – Concentration of zero-observations – Diurnal patterns – Skewness, heavy tail – Seasonality (intra-weekly, monthly, quarterly...) – Highly persistent dynamics (long-memory?) An extension of DCS model works very well! Several advantages over other existing methods. Methods: – Distribution decomposition at zero – Unobserved components – Dynamic cubic spline (Harvey and Koopman (1993)) Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 2/27
Data Trade volume of IBM stock traded on the NYSE. The number of shares traded. Period: 5 consequtive trading weeks in February - March 2000 Sampling frequency: 30 seconds Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 3/27
Empirical features Diurnal U-shaped patterns 400,000 4,000 350,000 3,500 300,000 3,000 250,000 2,500 200,000 2,000 150,000 1,500 100,000 1,000 50,000 500 0 0 Mon ‐ 20 ‐ Mar Tue ‐ 21 ‐ Mar Wed ‐ 22 ‐ Mar Thu ‐ 23 ‐ Mar Fri ‐ 24 ‐ Mar Mon ‐ 20 ‐ Mar Tue ‐ 21 ‐ Mar Wed ‐ 22 ‐ Mar Thu ‐ 23 ‐ Mar Fri ‐ 24 ‐ Mar Figure: IBM trade volume (left column) and the same series smoothed by the simple moving average (right column). Time on the x-axis. Monday 20 - Friday 24 March 2000. Each day covers trading hours between 9.30am-4pm (in the New York local time). Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 4/27
Empirical features Trade volume bottoms out at around 1pm. 100,000 2,000 90,000 1,800 80,000 1,600 70,000 1,400 60,000 1,200 50,000 1,000 40,000 800 30,000 600 20,000 400 10,000 200 0 0 09 11 13 15 09 11 13 15 Figure: IBM trade volume (left column) and the same series smoothed by the simple moving average (right column). Time on the x-axis. Wednesday 22 March 2000, covering 9.30am-4pm (in the New York local time). Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 5/27
Empirical features Sample autocorrelation. Highly persistent. 0.3 95% confidence interval 0.25 Sample Autocorrelation of y 0.2 0.15 0.1 0.05 0 −0.05 −0.1 0 20 40 60 80 100 120 140 160 180 200 Lag Figure: Sample autocorrelation of IBM trade volume. Sampling period: 28 February - 31 March 2000. The 200th lag corresponds approximately to 1.5 hours prior. Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 6/27
Empirical features Skewness, long upper-tail. 1.5 100 90 80 Empirical CDF % 70 Frequency % 1 60 50 40 0.5 30 20 10 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 All volume All volume 4 4 x 10 x 10 Figure: Frequency distribution (left) and empirical cdf (right) of IBM trade volume. Sampling period: 28 February - 31 March 2000. Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 7/27
Empirical features Sample statistics. Observations (total) 19,500 Mean 10,539 Median 6,100 Maximum 1,652,100 Minimum 0 Std. Dev. 26,071 Skewness 29 99.9% sample quantile 293,654 Max - 99.9% quantile 1,358,446 Frequency of zero-obs. 0.47% (92 obs.) Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 8/27
Intra-day DCS with dynamic cubic spline Our model: intra-day DCS with dynamic cubic spline Time index : intra-day time τ on t -th trading day as · t ,τ τ = 0 , . . . , I and t = 1 , . . . , T . τ = 0 and τ = I : the moments of market open and close. Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 9/27
Distribution decomposition at zero Define CDF F : R ≥ 0 → [0 , 1] of a standard random variable X ∼ F – the origin has a discrete mass of probability – strictly positive support is captured by a conventional continuous distribution Formally, P F ( X = 0) = p ∈ [0 , 1] p P F ( X > 0) = 1 − p (1) F ∗ ( x ) P F ( X ≤ x | X > 0) = x > 0 F ∗ : R > 0 → [0 , 1] is the cdf of a conventional standard continuous random variable with constant parameter vector θ ∗ . Decomposition technique: Amemiya (1973), Heckman (1974), McCulloch and Tsay (2001) Rydberg and Shephard (2003), Hautsch, Malec, and Schienle (2010) Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 10/27
Our p is constant. OK as the number of zero-observations is small. – Possible extension: time-varying p t ,τ using logit link [Rydberg and Shephard (2003), Hautsch, Malec, and Schienle (2010)] Apply DCS filter only to positive observations. – irregular term u t ,τ : the score of F ∗ – λ t ,τ driven by u t ,τ − 1 1 l { y t ,τ − 1 > 0 } Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 11/27
Unobserved components Assumption 1 : periodicity and autocorrelation in data are due to the time-varying scale parameter α t ,τ = exp( λ t ,τ ). Standardized observations are iid and free of periodicity and autocorrelation. ⇒ Let λ t ,τ have a component structure to capture each feature of data. Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 12/27
Unobserved components Unobserved components : y t ,τ = ε t ,τ exp( λ t ,τ ) , ε t ,τ |F t ,τ − 1 ∼ iid F λ t ,τ = δ + µ t ,τ + η t ,τ + s t ,τ µ t ,τ : low-frequency component. Captures highly persistent nonstationary dynamics µ t ,τ = µ t ,τ − 1 + κ µ u t ,τ − 1 1 l { y t ,τ − 1 > 0 } η t ,τ : stationary (autoregressive) component. A mixture of AR components captures long-memory. � J η ( j ) η t ,τ = t ,τ j =1 η ( j ) t ,τ = φ ( j ) 1 η ( j ) t ,τ − 1 + φ ( j ) 2 η ( j ) m η ( j ) t ,τ − 2 · · · + φ ( j ) t ,τ − m ( j ) + κ ( j ) η u t ,τ − 1 1 l { y t ,τ − 1 > 0 } for some J ∈ N > 0 and m ( j ) ∈ N > 0 . s t ,τ : periodic component capturing diurnal patterns Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 13/27
Dynamic cubic spline s t ,τ : dynamic cubic spline (Harvey and Koopman (1993)) � k l { τ ∈ [ τ j − 1 ,τ j ] } z j ( τ ) · γ † s t ,τ = 1 j =1 k : number of knots τ 0 < τ 1 < · · · < τ k : coordinates of the knots along time-axis γ † = ( γ 1 , . . . , γ k ) ⊤ : y-coordinates (height) of the knots z j : [ τ j − 1 , τ j ] k → R k : k -dimensional vector of functions. Conveys information about (i) polynomial order, (ii) continuity, (iii) periodicity, and (iv) zero-sum conditions. Bowsher and Meeks (2008): “special type of dynamic factor model” Time-varying spline: let γ † → γ † t ,τ where t ,τ − 1 + κ ∗ · u t ,τ − 1 1 γ † t ,τ = γ † l { y t ,τ − 1 > 0 } Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 14/27
Why use this spline? Alternative options used by many: Fourier representation Sample moments for each intra-day bins Diurnal pattern = deterministic function of intra-day time (Andersen and Bollerslev (1998), Engle and Russell (1998), Shang et al. (2001), Campbell and Diebold (2005), Engle and Rangel (2008), Brownlees et al. (2011), Engle and Sokalska (2012).) So why use this spline? Allows for changing diurnal patterns No need for a two-step procedure to “diurnally adjust” data Formal test for the day-of-the-week effect. Compare shape of dirunal patterns. – Unlike the alternative: seasonal dummies. Test for level differences. Used by many (e.g. Andersen and Bollerslev (1998), Lo and Wang (2010)) Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 15/27
Estimation Apply spline-DCS model to IBM trade volume data Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 16/27
Estimation results Assumption 1: � ε t ,τ = y t ,τ / � α t ,τ has to be free of autocorrelation. Satisfied - no autocorrelation in � ε t ,τ . 0.3 0.3 95% confidence interval 95% confidence interval Sample Autocorrelation of res Sample Autocorrelation of y 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 −0.05 −0.05 −0.1 −0.1 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Lag Lag Figure: Sample autocorrelation of trade volume (top), of � ε t ,τ (left). The 95% confidence interval is computed at ± 2 standard errors. Ryoko Ito, Faculty of Economics, Cambridge University, UK Modeling dynamic diurnal patterns in high freq. fin. data 17/27
Recommend
More recommend
