Predictive Regressions: A Present-Value Approach (van Binsbergen and Koijen, 2009) October 5th, 2009
Overview Key ingredients: Results: Draw inference from the Campbell and Shiller (1988) present value model Instead of using standard OLS estimation, treat the conditional expected returns and expected dividend rates as latent variables estimation by Kalman filter Both expected returns and expected dividend growth are time-varying and persistent Expected returns are more persistent Filtered series for expected returns and expected dividend growth are good predictors of realized returns and realized dividend growth
Returns and Dividend Growth Define the total log return on the aggregate stock market: ( ) Pt+1 + D t+1 r t+1 log = µ t + ε r t+1 (2) Assume the expected log return follows an AR(1) process: µ t+1 = δ 0 + δ 1 (µ t δ 0 ) + ε µ t+1 (3) Define log dividend growth: ( ) Dt+1 d t+1 log (4) = g t + ε D t+1 (5) Assume expected log dividend growth follows an AR(1) processes: g t+1 = γ 0 + γ 1 (g t γ 0 ) + ε g t+1 (6) P t D t (1)
Dividend Reinvestment: 2 Methods Authors consider annual times series to avoid dividend seasonality, so have to take stance on how dividends are reinvested through the year. Market-invested dividends vs cash-invested (30-day T-bill) Market-invested twice as volatile as cash-invested (12.3% vs 6.2%). Returns calculated using either method very highly correlated. Assumed expected cash-invested dividend growth is AR(1): g t+1 = γ 0 + γ 1 (g t γ 0 ) + ε g t+1 (7) Consider reduced-form representation of market-invested dividends: D M t+1 = D t+1 exp(ε M t+1) (8) Implies expected market-invested dividend growth is ARMA(1,1): g M t+1 E t+1 [ d M t+2] (9) = γ 0 + γ 1 (g M t γ 0 ) + ε g t+1 + γ 1ε M t ε M t+1 (10)
pd t approximately affine in µ t and g t Following Campbell-Shiller, log-linearize return definition: r t+1 κ + ρpd t+1 + d t+1 pd t (11) Iterating this difference equation, and taking conditional expectations: pd t κ 1 ρ + ρ j 1 E t [ d t+j r t+j ] (12) j=1 Applying the AR(1) assumptions on µ t and g t : pd t A B 1 (µ t δ 0 ) + B 2 (g t γ 0 ) (13) where B 1 = 1 1 ρδ 1 and B 2 = 1 1 ργ 1
Estimation Method Substitute pd t into expected return dynamics: ĝ t+1 = γ 1 ĝ t + ε g t+1 d t+1 = γ 0 + ĝ t + ε D t+1 + ε M t+1 εm t pd t+1 = (1 δ 1 )A + B 2 (γ 1 δ 1 )ĝ t + δ 1 pd t B 1 ε µ t+1 + B 2ε g t+1 ε M t+1 + δ 1ε M t where hats denote demeaned variables, and bold font denotes extra terms for market-invested dividends. All equations are linear, so compute likelihood using Kalman filter. Estimation follows by maximum likelihood.
Data Monthly returns on value-weighted portfolio of NYSE, Amex, and Nasdaq stocks Construct annual time series for prices and aggregate dividends (2 methods) 1946-2007
Estimation Results: cash-invested dividends
Estimation Results: market-invested dividends
Summary New: model conditional expected returns and conditional expected dividend growth rates as latent processes, and use filtering techniques to estimate them. Both expected returns and expected dividend growth are persistent, but expected returns are more persistent, are are good predictors of realized returns and realized dividend growth. Choice of dividend reinvestment not innocuous.
Discussion I: Latent-variables vs OLS OLS: d t+1 = a d + β d pd t + ε d,ols t+1 (14) r t+1 = a r + β r pd t + ε r,ols t+1 (15)
Discussion I: Latent-variables vs OLS Using Kalman filter in stationary state, can obtain Wold decomposition: d t+1 = a0 d + a1,ipd d t i + a2,id d t i + ε d t+1 (16) i=0 i=0 i=0 r t+1 = a0 r + a1,ipd r t i + a2,id r t i + ε r t+1 (17) Kalman filter aggregates information in a parsimonious way, expanding the information set without increasing the number of parameters. i=0
Discussion I: Latent-variables vs OLS Expected cash-invested dividend growth
Discussion I: Latent-variables vs OLS Expected market-invested dividend growth
Discussion II: Assumption on dynamics not innocuous Formal specification test rejects g t = AR(1) and g M t = AR(1), over g t = AR(1) and g M t = ARMA(1,1). But if g M t assumed AR(1): MLE: persistence of expected market-invested div. growth negative Maximizing likelihood maximizing R 2 What if expected cash-invested dividend growth, g t, is not AR(1)?
Discussion III: Estimation bias Bootstrap: 1,000 samples with same number of observations as the data, drawn from unconditional distribution of state variables.