Modeling the Persistence of Expected Returns Dooruj Rambaccussing - PowerPoint PPT Presentation

Modeling the Persistence of Expected Returns Dooruj Rambaccussing University of Exeter 21st January 2012 (University of Exeter) 21st January 2012 1 / 11

Introduction Research question: Does Expected Returns exhibit high persistence, typical of long memory structures ? Motivation: Koijen and Van Binsbergen (2010): µ t + 1 − δ 0 = δ 1 ( µ t − δ 0 ) + ε µ t + 1 ; δ 1 = 0 . 932 Geometric decay (best modeled by an ARMA( p , q )) or hyperbolic decay (ARFIMA( p , d , q ). Idea is simple: Can we model expected returns through an ARFIMA( p , d , q ) in a state space (Kalman Filter) model ? Long memory theorists (like Robinson or Davidson may not like it !). However to be in tune with Real-Time, a Kalman Filter is preferable. (University of Exeter) 21st January 2012 2 / 11

Fractional Integration and Long Memory Granger (1980) aggregation: Long memory properties stems from aggregation of micro-processes. µ j , t + 1 = a j µ j , t + b j w t .... j = 1 , .. N ... a j ∈ [ 0 , 1 ] � � N b j E ( µ t + 1 ) = µ t + 1 = 1 ∑ w t N 1 − a j L j = 1 µ t + 1 � E ( b ) � 1 1 − aL dF ( a ) w t where F ( a ) represents a Beta ( p , q ) distribution. Solving the above, and setting d = 1 − q , we have the interesting result that: µ t + 1 ∼ I ( d ) d is known as the fractional difference parameter or at times a measure of long memory. (University of Exeter) 21st January 2012 3 / 11

ARFIMA(p,d,q) An ARFIMA( p , d , q ) process, x t for t =1 .... T may be written as: ϕ ( L ) x t = θ ( L )( 1 − L ) − d η t (1) η t is white noise: ( η t ∼ N ( 0 , σ 2 η ) ϕ ( L ) and θ ( L ) are respectively equal to ( 1 − ϕ L − ϕ 2 L 2 − · · · − ϕ p L p ) and ( 1 − θ L − θ 2 L 2 − · · · − θ p L q ) . Persistence is measured by d . The following definitions apply: 0 < d < 0 . 5 (The process is long memory and exhibits persistence) (University of Exeter) 21st January 2012 4 / 11

ARFIMA(p,d,q) An ARFIMA( p , d , q ) process, x t for t =1 .... T may be written as: ϕ ( L ) x t = θ ( L )( 1 − L ) − d η t (1) η t is white noise: ( η t ∼ N ( 0 , σ 2 η ) ϕ ( L ) and θ ( L ) are respectively equal to ( 1 − ϕ L − ϕ 2 L 2 − · · · − ϕ p L p ) and ( 1 − θ L − θ 2 L 2 − · · · − θ p L q ) . Persistence is measured by d . The following definitions apply: 0 < d < 0 . 5 (The process is long memory and exhibits persistence) − 0 . 5 < d < 0 (The process is antipersistent) (University of Exeter) 21st January 2012 4 / 11

ARFIMA(p,d,q) An ARFIMA( p , d , q ) process, x t for t =1 .... T may be written as: ϕ ( L ) x t = θ ( L )( 1 − L ) − d η t (1) η t is white noise: ( η t ∼ N ( 0 , σ 2 η ) ϕ ( L ) and θ ( L ) are respectively equal to ( 1 − ϕ L − ϕ 2 L 2 − · · · − ϕ p L p ) and ( 1 − θ L − θ 2 L 2 − · · · − θ p L q ) . Persistence is measured by d . The following definitions apply: 0 < d < 0 . 5 (The process is long memory and exhibits persistence) − 0 . 5 < d < 0 (The process is antipersistent) d > 0 . 5 (The process is nonstationary) (University of Exeter) 21st January 2012 4 / 11

Present Value Campbell and Shiller (1989) show from the present value of price that the price dividend ratio encompasses expected returns and expected dividend growth: r t + 1 � κ + ρ pd t + 1 + ∆ d t + 1 − pd t (2) pd t = log ( PD t ) , pd = E ( pd t ) , κ = log ( 1 + exp ( pd )) − ρ pd and exp ( pd ) ρ = 1 + exp ( pd ) . Define : E t ( r t + 1 ) = µ t and E t ( ∆ d t + 1 ) = g t Both are latent variables (not directly observed). We can think of adopting parametric specifications of µ t and g t . (For instance Koijen and Van Binsbergen used AR(1) specification. I use the following specification: δ 0 + ϕ ( L ) − 1 θ ( L )( 1 − L ) − d ε µ = µ t + 1 (3) t + 1 γ 0 + γ 1 g t + ε g g t + 1 = (4) t + 1 (University of Exeter) 21st January 2012 5 / 11

Transformation of ARFIMA(1,d,0) into AR plus noise. ( 3 ) may be written as an autoregressive process of infinite order: ∞ t + 1 = θ ( L ) ε µ ϕ ( L )( 1 − L ) d µ t = π j µ t − j = π ( L ) µ t ∑ j = 0 ∞ p Consider truncating ∑ π j µ t − j to ∑ π j µ t − j such that the tails are j = 0 j = 0 equal to zero. the choice of p depends on many factors, but primarily the autocorrelation structure must be the same. Usually large p needs to be chosen. Personal Monte Carlo experiment shows that setting p = 20 for T = 200 captures autocorrelation structure for d = 0 . 4. The ARFIMA(1 , d , 0 ) can specifically be written as: ( 1 − γ 1 L )( 1 − L ) d µ t = ε µ t + 1 (University of Exeter) 21st January 2012 6 / 11

State Space Model Measurement equation: κ � 1 − ρ + B ( g t − µ t ) pd t + 1 g t + ε d ∆ d t + 1 = t + 1 State Equations: δ 0 + ϕ ( L ) − 1 θ ( L )( 1 − L ) − d ε µ µ t + 1 = (5) t + 1 γ 0 + γ 1 g t + ε g g t + 1 = (6) t + 1 Apply simple Kalman Filter to this multivariate model and obtain likelihood Objective is to optimize the vector of parameters: Θ = ( γ 0 , δ 0 , γ 1 , δ 1 , σ g , σ µ , σ d , ρ g µ , ρ gd , ρ µ d , d µ ) (University of Exeter) 21st January 2012 7 / 11

Results Price Dividend Ratio/Dividend Growth 1926-2008 1946-2008 AR(1) ARFIMA(1,d,0) AR(1) ARFIMA(1,d,0) Parameters PARAM SE PARAM SE PARAM SE PARAM SE γ 0 0 . 021 0 . 014 0 . 003 0 . 081 0 . 019 0 . 012 0 . 018 0 . 001 δ 0 0 . 055 0 . 019 0 . 029 0 . 074 0 . 046 0 . 020 0 . 05 0 . 001 γ 1 0 . 11 0 . 119 0 . 165 0 . 210 0 . 395 0 . 203 0 . 326 0 . 108 δ 1 0 . 921 0 . 050 0 . 158 0 . 031 0 . 929 0 . 049 0 . 118 0 . 033 − − − − d 1 0 . 475 0 . 050 0 . 457 0 . 06 σ g 0 . 052 0 . 016 0 . 109 0 . 001 0 . 05 0 . 014 0 . 048 0 . 193 σ µ 0 . 02 0 . 010 0 . 044 0 . 001 0 . 015 0 . 009 0 . 044 0 . 030 σ d 0 . 092 0 . 055 0 . 007 0 . 001 0 . 014 0 . 040 0 . 007 0 . 043 ρ g µ 0 . 576 0 . 088 0 . 212 0 . 001 0 . 62 0 . 126 0 . 345 0 . 494 ρ µ d − 0 . 046 0 . 001 − 0 . 166 0 . 001 − 0 . 055 0 . 685 − 0 . 002 0 . 250 Log-Likelihood − 103 . 79 − 110 . 45 − 130 . 07 − 126 . 83 Table: Estimation of AR(1) and ARFIMA(1,d,0) Model for Dividend data. The parameters optimized are from the previously defined parameter set (AR(1)) and (ARFIMA(1,d,0)) over two periods. (University of Exeter) 21st January 2012 8 / 11

1946-2008 Figure: Plot of Expected Returns for AR(1) and ARFIMA(1,d,0): Sample 1946-2008 using Dividend Data. (University of Exeter) 21st January 2012 9 / 11

1926-2008 Figure: Plot of Expected Returns for AR(1) and ARFIMA(1,d,0) and Realized Returns for 1926-2008 using Dividend Data. (University of Exeter) 21st January 2012 10 / 11

Other results/Conclusion: ARFIMA( p , d , q ) model is not stable for such a small sample size. Monte Carlo shows that if true specification follows a Markov Switching model (with equal probability of being in bullish or bearish regime), estimated parameters suffer more in the case of ARFIMA( p , d , q ). Problem is that given the latent structure of expected returns, performance evaluation is difficult. Test of long memory using Skip sampling method (Davidson and Rambaccussing 2012) show that process is a pure long memory process. Weak predictability of Returns from both AR and ARFIMA model. (Does not beat the Price Dividend Ratio) (University of Exeter) 21st January 2012 11 / 11