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

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Introduction

Research question: Does Expected Returns exhibit high persistence, typical of long memory structures ? Motivation: Koijen and Van Binsbergen (2010):

µt+1 − δ0 = δ1(µt − δ0) + εµ

t+1 ; δ1 = 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.

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Fractional Integration and Long Memory

Granger (1980) aggregation: Long memory properties stems from aggregation of micro-processes. µj,t+1 = ajµj,t + bjwt....j = 1, ..N ...aj ∈ [0, 1] E(µt+1) = µt+1 = 1

N N

∑

j=1

• bj

1−ajL

• wt

µt+1 E(b)

1 1−aLdF(a)wt

where F(a) represents a Beta(p, q) distribution. Solving the above, and setting d = 1 − q, we have the interesting result that: µt+1 ∼ I(d) d is known as the fractional difference parameter or at times a measure of long memory.

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ARFIMA(p,d,q)

An ARFIMA(p, d, q) process, xt for t =1....T may be written as: ϕ(L)xt = θ(L)(1 − L)−dηt (1) ηt is white noise: (ηt ∼ N(0, σ2

η)

ϕ(L) and θ(L) are respectively equal to (1 − ϕL − ϕ2L2 − · · · − ϕpLp) and (1 − θL − θ2L2 − · · · − θpLq). Persistence is measured by d. The following definitions apply: 0 < d < 0.5 (The process is long memory and exhibits persistence)

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ARFIMA(p,d,q)

An ARFIMA(p, d, q) process, xt for t =1....T may be written as: ϕ(L)xt = θ(L)(1 − L)−dηt (1) ηt is white noise: (ηt ∼ N(0, σ2

η)

ϕ(L) and θ(L) are respectively equal to (1 − ϕL − ϕ2L2 − · · · − ϕpLp) and (1 − θL − θ2L2 − · · · − θpLq). 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)

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ARFIMA(p,d,q)

An ARFIMA(p, d, q) process, xt for t =1....T may be written as: ϕ(L)xt = θ(L)(1 − L)−dηt (1) ηt is white noise: (ηt ∼ N(0, σ2

η)

ϕ(L) and θ(L) are respectively equal to (1 − ϕL − ϕ2L2 − · · · − ϕpLp) and (1 − θL − θ2L2 − · · · − θpLq). 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)

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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: rt+1 κ + ρpdt+1 + ∆dt+1 − pdt (2) pdt = log(PDt), pd = E(pdt), κ = log(1 + exp(pd)) − ρpd and ρ =

exp(pd) 1+exp(pd).

Define : Et(rt+1) = µt and Et(∆dt+1) = gt Both are latent variables (not directly observed). We can think of adopting parametric specifications of µt and gt. (For instance Koijen and Van Binsbergen used AR(1) specification. I use the following specification: µt+1 = δ0 + ϕ(L)−1θ(L)(1 − L)−dεµ

t+1

(3) gt+1 = γ0 + γ1gt + εg

t+1

(4)

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Transformation of ARFIMA(1,d,0) into AR plus noise.

(3) may be written as an autoregressive process of infinite order: εµ

t+1 = θ(L)

ϕ(L)(1 − L)dµt =

∞

∑

j=0

πjµt−j = π(L)µt Consider truncating

∞

∑

j=0

πjµt−j to

p

∑

j=0

πjµt−j such that the tails are 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 − γ1L)(1 − L)dµt = εµ

t+1

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State Space Model

Measurement equation: pdt+1

• κ

1 − ρ + B(gt − µt) ∆dt+1 = gt + εd

t+1

State Equations: µt+1 = δ0 + ϕ(L)−1θ(L)(1 − L)−dεµ

t+1

(5) gt+1 = γ0 + γ1gt + εg

t+1

(6) Apply simple Kalman Filter to this multivariate model and obtain likelihood Objective is to optimize the vector of parameters: Θ = (γ0, δ0, γ1, δ1, σg, σµ, σd, ρg µ, ρgd, ρµd, dµ)

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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 γ0 0.021 0.014 0.003 0.081 0.019 0.012 0.018 0.001 δ0 0.055 0.019 0.029 0.074 0.046 0.020 0.05 0.001 γ1 0.11 0.119 0.165 0.210 0.395 0.203 0.326 0.108 δ1 0.921 0.050 0.158 0.031 0.929 0.049 0.118 0.033 d1 − − 0.475 0.050 − − 0.457 0.06 σg 0.052 0.016 0.109 0.001 0.05 0.014 0.048 0.193 σµ 0.02 0.010 0.044 0.001 0.015 0.009 0.044 0.030 σd 0.092 0.055 0.007 0.001 0.014 0.040 0.007 0.043 ρg µ 0.576 0.088 0.212 0.001 0.62 0.126 0.345 0.494 ρµd −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.

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1926-2008

Figure: Plot of Expected Returns for AR(1) and ARFIMA(1,d,0) and Realized Returns for 1926-2008 using Dividend Data.

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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)

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