Predictive Regressions: A Present Value...

Predictive Regressions: A Present Value Approach

Jules van Binsbergen and Ralph Koijen

Sept 25, 2007

Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach

Introduction

Most studies have relied on log approximations to derive thepresent value relationship between the Price-Dividend ratio,and future expected returns and dividend growth

Empirically, the Price-Dividend ratio seems to predict returnsbut not dividend growth

van Binsbergen and Koijen want to avoid the log linearapproximation framework, instead develop their own toanalyze these predictive relationships


Results

Develop a framework that does not rely on the logapproximation, Price-Dividend ratio is an exact linear functionof expected returns and expected dividend growth rates

Find that Price-Dividend ratio predicts both future returnsand dividend growth, R2 of 16% and 8% respectively.

Find evidence of a transient and persistent component of theexpected dividend growth rate, R2 rise to 18% and 16%


The Model

Price-Dividend Ratio: PDt = PtDt

Returns: Rt+1 = Pt+1

Pt−Dt

Expected returns: µ∗t = Et [Rt+1]− 1


Unobserved States

Do not observe the expected return sequence {µ∗t } orsequence of expected dividend growth rate {gt}

How one models these sequences is extremely important!


Unobserved States

Model Dividend Growth as:

Dt+1

Dt= (1 + gt)(1 + εDt+1)

Define µt such that:

1 + gt

1 + µ∗t= 1 + gt − µt

Introduce two new variables, gt and µt such that:

gt = γ0 + gt

µt = δ0 + µt


Unobserved States: Dynamics

Model these as non linear equations:

gt+1 = γ11 + γ0 − δ0

1 + γ0 − δ0 + gt − µtgt + εgt+1

µt+1 = δ11 + γ0 − δ0

1 + γ0 − δ0 + gt − µtµt + εµt+1


Unobserved States: Dynamics

Re-label as in the paper:

gt+1 = γ1t gt + εgt+1

µt+1 = δ1t µt + εµt+1

Where:γ1t = γ1λt

δ1t = δ1λt

λt =α

α + gt − µt

α = 1 + γ0 − δ0


Correlation

The covariance between innovations in expected dividendgrowth and expected return is:

Cov(εgt+1, εµt+1) = σgµ

The covariance between innovations in unexpected dividendgrowth and expected return is:

Cov(εDt+1, εµt+1) = σDµ

For identification, the covariance between innovations inunexpected and expected dividend growth is zero:

Cov(εDt+1, εgt+1) = σDg = 0


Price-Dividend Ratio

The Price-Dividend Ratio can be written as:

PDt = A + B1µt + B2gt

WhereA = (1− α +

σDµα

1− δ1α + σDµ)−1

B1 =−A

1− δ1α + σDµ

B2 =A + B1σDµ

1− γ1α


Why do variables move?

Changes in the dividend growth, returns, or the price dividendratio can be attributed to changes in expected returns,expected growth rates, or unexpected growth rates

Expected change in PDt is:

Et [PDt ]− PDt = B1(δ1t − 1)µt + B2(γ1t − 1)gt

Unexpected change in price dividend ratio is:

PDt+1 − Et [PDt+1] = B1εµt+1 + B2ε

gt+1


Unexpected Returns

Unexpected return is:

Rt+1 − (1 + µ∗t ) = ht

[B1ε

µt+1 + B2ε

gt+1 + Et [PDt+1]εDt+1

+B1(εdt+1εµt+1 − σDµ) + B2ε

Dt+1ε

gt+1

]Where:

ht =1 + gt

PDt − 1


Implications for Variance of Returns

If we assume that (εgt+1, εµt+1, ε

Dt+1) are jointly normal, then

the conditional variance of returns is:

σ2R,t = h2

t

[B2

1σ2µ + B2

2σ2g + (Et [PDt+1])2σ2

D + B21 (σ2

Dσ2µ + 2σ2

Dµ)

+B22σ

2gσ

2D + B1B2σ

2gµ + B1Et [PDt+1]σD,µ + B1B2σ

2Dσgµ

]

The conditional covariance between expected returns andunexpected returns is:

Covt(Rt+1 − Et [Rt+1], εµt+1) = ht(B1σ2µ + B2σµg + Et [PDt+1]σµd)


Data

Annual data, 1946-2005

Use total payout data as in some previous studies


Empirical Exercise

First, want to characterize small sample performance ofestimator given constant dividend growth and compare theirmaximum likelihood estimator to that of conventionalpredictive regressions

PV model: PDt = A + B1µt ⇒ L(Θ; ∆DT ,PDT )

Standard Predictive Regression: Rt+1 = α + βPDt + ηt+1

Can link the two as follows:

δ0 = α + βA1 = βB1


Empirical Exercise

Generate 10000 series of 60 observations, given some set ofparameters

Estimate model

Look at distribution of estimators


Results


Estimation of Full Model

Transition Equations:


µt+1 = δ1t µt + εgt+1

Measurement equations:

PDt = A + B1µt + B2gtDt+1

Dt= (1 + δ0 + gt) + (1 + δ0 + gt)εDt+1)


Estimating the Model

Want to evaluate the likelihood:

L(y t ; Θ) =T∏

t=1

l(yt |y t−1; Θ)

L(y t ; Θ) =T∏

t=1

∫ ∫l(yt |y t−1, εtg , g0,Θ)l(εtg , g0|y t−1,Θ)dεtgdg0

This is hard in a non linear world!


Estimating the Model

To evaluate the likelihood function use 1 of:

Unscented Kalman Filter

Particle Filter

Then maximize the likelihood and bootstrap for standard errors.


Estimation of Model

Can reduce the dimension by subbing out one of the transitionequations to get:


Measurement equations:

PDt+1 = A + δ1t [PDt − A− B2gt ] + B2γ1t gt + B1εµt+1 + B2ε

gt+1

Dt+1

Dt= (1 + δ0 + gt) + (1 + δ0 + gt)εDt+1)


Results


Expected Returns


Expected Dividend Growth Rates


Decomposing Unexpected Returns: First Order Effects


Decomposing Unexpected Returns: Second Order Effects


Variance Decomposition


Decomposing PDt


Multiple Growth Rates

Want to allow for two growth rates

gt = γ0 + g1t + g2t

git = γit git + εgi ,t+1

Now the price dividend ratio

PDt = A + B1µt + B2g1t + B3g2t


Results


Variance Decomposition


Price Dividend Decomposition


Unexpected Returns


Test for Persistent Growth Rate

Perform a likelihood ratio test, LR = 2(L1 − L0)

H0 : γ1 = σg1 = ρµg1 = 0 gives p value of 0.087

H0 : δ1 = σµ = ρµg1 = ρDµ = 0 is rejected


Conclusion

Developed a closed form present value model with timevarying expected dividend growth rates and expected returns

Find that the Price-Dividend ratio can predict both returnsand dividend growth rates

Decompose dividend growth rates and find a transient andpersistent component


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