A Modified Fractionally Co-integrated VAR for Predicting Returns

Transcription

1 A Modified Fractionally Co-integrated VAR for Predicting Returns Xingzhi Yao Marwan Izzeldin Department of Economics, Lancaster University 13 December 215 Yao & Izzeldin (Lancaster University) CFE (215) 12/215 1 / 21

2 Outline 1 Introduction and Motivation 2 Contribution 3 The Model 4 Monte Carlo Simulation 5 Empirical Examples 6 Concluding Remarks Yao & Izzeldin (Lancaster University) CFE (215) 12/215 2 / 21

3 Introduction Consider a fractionally co-integrated VAR (FCVAR) model of Johansen (28) and Johansen and Nielsen (212) X t I (d) k d X t = αβ d b L b X t + Γ c d L c b }{{} X t + ε t c=1 long-run equilibrium }{{} short-run dynamics d = (1 L) d, L d = 1 d, ε t is i.i.d.(, Ω) α: speed of adjustment, β : the cointegrating vector, Γ c : lag matrix β X t I (d b), 1 > d b > A wide range of applications: stock price and volatility forecasting, market return predictability, economic voting hypothesis... Yao & Izzeldin (Lancaster University) CFE (215) 12/215 3 / 21

4 Motivation Consider a FCVAR d X t = αβ d b L b X t + k c=1 Γ c d L c b X t + ε t X t = (x t, y t, z t ), where x t, y t I (d), z t I () fractional co-integration between I (d) variables is affected by adding z t, which gives rise to a biased estimate of β the model is mis-specified when d > b: error correction terms β d b X t are over-differenced Yao & Izzeldin (Lancaster University) CFE (215) 12/215 4 / 21

5 Contribution We propose a modified FCVAR (M-FCVAR) to accommodate a mixture of I (d) and I () variables, which restricts shocks arising from the I () to only have transitory effects on long-memory variables, an extension of Fisher et al. (215) to fractional models allows for a commonly encountered case where d b Compared with FCVAR, M-FCVAR results in lower MSEs of co-fractional estimates higher degree of predictability of the I () variable We verify our claims using a monte carlo simulation an empirical application using high frequency financial data (SP5, SPY and VIX index) from 23 to 213 Yao & Izzeldin (Lancaster University) CFE (215) 12/215 5 / 21

6 FCVAR vs M-FCVAR FCVAR M-FCVAR System X t = (RV t, VIXt 2, r t ) Y t = ( d b RV t, d b VIXt 2, r t ) Set-up d X t = αβ d b L b X t + k c=1 Γc d L c ( ) b Xt + εt b Y t = αβ L b Y t + k c=1 Γc b L c byt + εt ( ) β 1 β1 1 β1 1 1 Error correction term β d b X t β Y t α 11 α 12 α α 21 α 22 α 21 α 22 α 31 α 32 α 31 α 32 C g h i j k l a d RV t, VIXt 2 : CI (d, b), r t I () C denotes the long-run responses of each variable to shocks in ε t and β C = 2 3 ε t = (ε p 1t, εt 2t, εt 3t ) : one permanent shock and two transitory shocks ideally, C = a d To illustrate the restrictions on α, we introduce the Impulse-Response Functions (IRFs). Yao & Izzeldin (Lancaster University) CFE (215) 12/215 6 / 21

7 M-FCVAR: IRFs Consider IRFs to investigate the model-implied dynamic dependencies d = (1 L) d = i= θ i (d)l i with θ i (d) = ( 1) i ( d i ); L d = 1 d = i=1 θ i (d)l i The model can be written as Y t = = i=1 ( I θ i (b) αβ θ i (b) + Ξ i L i Y t + ε t i=1 k c=1 ( 1) c Γ c K c,i )L i Y t + ε t where K c,i = b L c b = θ l (b)k c 1,i l (l = 1, 2, 3, ) Invert the AR polynomial Ξ i to obtain the IRF coeffi cient Φ j Φ = I, Φ j = j 1 i= Ξ j i Φ i Y t = j= Φ j ε t j Yao & Izzeldin (Lancaster University) CFE (215) 12/215 7 / 21

8 M-FCVAR: transitory shocks from the I() Fractional co-integration is unaffected if the shocks arising from the I () are transitory Let RVt = d b RV t, which is the only variable containing permanent shocks within the model RVt = L b RVt = + α 11 L b ξ t + α 12 L b r t + (Γ Γ 1 β 12)L b b RVt + 1 Γ 1 1 β 12L b b ξ t 1 +Γ 1 13L b b r t + ε t θ i (b)l i RV t α 11 i=1 i=1 + 1 Γ 1 12 β K 1,i L i ξ t + Γ i=1 i=1 θ i (b)l i ξ t α 12 K 1,i L i r t + ε t1 where ξ t = d b (RV t + β 1 VIX 2 t ) I () and r t I (). θ i (b)l i r t + (Γ Γ 1 i=1 β 12) 1 i=1 K 1,i L i RV t Shocks associated with ξ t and r t are expected to have no long-run effects on RV t α 11 = α 12 = no restrictions on Γ 1 12, Γ1 13 Yao & Izzeldin (Lancaster University) CFE (215) 12/215 8 / 21

9 M-FCVAR: restrictions on Alpha -.1 sum of i (b) sum of K 1,i Long-run effects Parameter b Long-run effects contributed by the lag components are negligible or even zero once b exceeds.6. Suffi cient to set α 11 = α 12 = to ensure that shocks from I () error correction terms produce no permanent effects on RV t. Yao & Izzeldin (Lancaster University) CFE (215) 12/215 9 / 21

10 Predictive R-square Implied by the M-FCVAR e3 (,, 1) and Φ j denotes the IRF coeffi cient r t = e3 i= Φ i ε t i Decompose return into the expected and unexpected part r h t = e3 h 1 j= i=j+1 } {{ } expected (A) Φ i ε t+j i + e3 h 1 j j= i= Φ i ε t+j i } {{ } unexpected (B) Return predictability over h horizons implied by the M-FCVAR model R 2 h = var (A) var (A)+var (B ) Yao & Izzeldin (Lancaster University) CFE (215) 12/215 1 / 21

11 Simulation Design and Settings To highlight the gains of M-FCVAR, we simulate a set of fractionally co-integrated systems containing both I (d) and I () variables ε t i.i.d.n(, 1) System 1: X t (x t, y t ) = j= Φ j ε t j, Φ i (α, β, Γ, b) X t = ( b d x t, b d y t ), X t CI (d, b) System 2: Y t Y t = j= Φ j ε t j, Φ i (α, β, Γ, b) z t = e3 Y t, z t I () System 3: Z t = (X t, z t ) Estimate Z t by FCVAR and M-FCVAR, respectively. The distribution of estimates is obtained by 1 replications of a moving block bootstrap. T = 2 ( α ) β k ( Γ ) System 1: X t (1 1.8) α β k Γ.5.1 ( ) System 2: Y t Yao & Izzeldin (Lancaster University) CFE (215) 12/ / 21

12 Empirical Distributions: d=b= FCVAR d=b=.3 3 b FCVAR M-FCVAR b M-FCVAR Histogram Distribution from histfit Distribution computed manually Underlying value Yao & Izzeldin (Lancaster University) CFE (215) 12/ / 21

13 Empirical Distributions: d=b= FCVAR d=b= b FCVAR M-FCVAR 25 2 b M-FCVAR Histogram Distribution from histfit Distribution computed manually Underlying value Yao & Izzeldin (Lancaster University) CFE (215) 12/ / 21

14 Empirical Distributions: d=.7 b= FCVAR d=.7 b=.5 3 b FCVAR M-FCVAR b M-FCVAR Histogram Distribution from histfit Distribution computed manually Underlying value Yao & Izzeldin (Lancaster University) CFE (215) 12/ / 21

15 Simulation Results: parameter b b FCVAR b M FCVAR MSE Reduction d b Bias Std MSE Bias Std MSE % % % % % % % % % % % % % % Yao & Izzeldin (Lancaster University) CFE (215) 12/ / 21

16 Simulation Results: Beta β 1_FCVAR β 1_M FCVAR MSE Reduction d b Bias Std MSE Bias Std MSE % % % % % % % % % % % % % % Yao & Izzeldin (Lancaster University) CFE (215) 12/ / 21

17 Simulation Results: predictive R-square R 2 FCVAR (%) R 2 M FCVAR (%) d b h=1 h=5 h=22 h=1 h=1 h=5 h=22 h= Note: shocks, taken from the residuals, have been orthogonalised. Yao & Izzeldin (Lancaster University) CFE (215) 12/ / 21

18 Empirical Application Daily CBOE VIX volatility index 5 min observations of aggregate S&P 5 composite index, SPDR S&P 5 ETF TRUST (SPY) index Sample period: (2623 obs) 5 min intraday return r t,j = 1(log p t,j log p t,j 1 ), daily return r t = M j=1 r t,j Daily realised variance M rv t = rt,j 2 j=1 One-month forward realised return variation ( ) 22 RV t = log rv t+i i=1 Risk-neutral return variation VIX 2 t = log ( 3 ( 365 VIX CBOE t ) 2 ) Yao & Izzeldin (Lancaster University) CFE (215) 12/ / 21

19 FCVAR vs M-FCVAR M-FCVAR: Y t = ( d b RV t, d b VIXt 2, r t ) d b k α β AIC BIC.. (.) (.) 1 SP (.) (.18) (.13) (.38) (.1) (.71) (.271) SPY.681 (.).646 (.15) 1. (.).118 (.12) -.55 (.86). (.).16 (.61) (.494) (.1) FCVAR: X t = (RV t, VIXt 2, r t ) d b k α β AIC BIC (.5) (.24) 1 SP (.) (.2) (.15) (.66) (.1) (.73) (.675) SPY.681 (.).635 (.21) (.5).128 (.17) (.96).5 (.34).13 (.94) (1.119) (.) Yao & Izzeldin (Lancaster University) CFE (215) 12/ / 21

20 Predictive R-square: different models R-square(%) Predictive R-square for Return of SP5 M-FCVAR FCVAR VAR(1) AR(1) Horizon (day) Predictive R-square for Return of SPY M-FCVAR FCVAR VAR(1) AR(1) R-square(%) Horizon (day) Yao & Izzeldin (Lancaster University) CFE (215) 12/215 2 / 21

21 Conclusion We provide modifications for the traditional FCVAR model of Johansen (28) and Johansen and Nielsen (212), Our modified FCVAR (M-FCVAR) is better suited for the estimation of fractionally co-integrated systems containing a mixture of I (d) and I () variables. Specifically, the M-FCVAR restricts shocks emanating from the I () variable to only have transitory effects on the I (d) variables. accounts for long memory in the co-integration residuals by applying a partial differencing procedure. provides co-fractional estimates with lower MSEs. yields higher degree of predictability of the I () variable. We demonstrate the gains from adopting the M-FCVAR via Monte Carlo simulations and an empirical application using high frequency data. Yao & Izzeldin (Lancaster University) CFE (215) 12/ / 21