Return Predictability

What drives stock market prices? A present-value decomposition and application of AR models

. Lochstoer

UCLA Anderson School of Management

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Winter 2022

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1 Stock Market Predictability

I Forecasting regressions

I The Dividend-Yield

I Cross-equation Restrictions (the Present-Value restriction)

2 References

3 Appendix: Background on Optimal Forecasting

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What drives stock price movements?

Seminal paper by Nobel prize winner

Do stock prices move too much to be justiÖed by subsequent dividends?

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Stock return predictability

Let Rt+1 denote the simple return on the aggregate market, e.g. the CRSP-VW index.

Let Dt denote aggregate dividends and dt = log(Dt ).

The ratio Dt /Pt is called the dividend yield while Pt /Dt is called the

price-to-dividend ratio

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Stock return predictability

A forecasting regression is a regression of an outcome at time t + j (with j > 0) using an predictor variable known at time t:

yt+j=α+βxt+εt+j, fort=1,…,T Table: Return Predictability

Regression slope

t-stat [2.309] [2.621] [1.989] 0.203

HAC t-stat [2.395] [2.726] [2.075] 0.185

R2 0.062 0.078 0.047 0.001

Rt+1 =a+b(D/P)t +εt+1

Rt+1 Rtf =a+b(D/P)t +εt+1 rt+1 =ar +br(dp)t +εrt+1

∆dt+1 = ad + bd (dp)t + εdt+1

3.498 3.933 0.105 0.008

Notes: Annual Data. Sample 1927-2009. Rt+1 is the real return on the CRSP-VW index. rt+1 denotes logs of the real return. Rtf+1 denotes the return on the real risk-free.

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Interpretation

an increase in the dividend yield of 1 percentage point in deviation from its mean increases the expected real return by 3.49 percentage points (per annum).

note: when returns are regressed on lagged persistent variables such as the dividend/yield, the disturbances are correlated with the regressorís innovation; this tends to create an upward bias in the case of dividend-yield regressions and is called Stambaugh bias; see Stambaugh (1999).

Stambaugh bias implies that OLS coe¢ cients are estimated to be too high.

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Relation between Regressions

note that the log dividend/yield in deviation from its mean is (to a Örst-order Taylor expansion) given by:

dpt =Dt/Pt /(D/P)

where D/P is the (unconditional) average dividend/price ratio

so we can state the return regression :

rt+1 = ar +br dpt +ut+1

as follows:

rt+1 = ar +br Dt/Pt / (D/P)+ut+1 the average dividend yield D/P is .035

so the implied coe¢ cient for the regression with the dividend yield is br = .105/.035 = 3.00

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Dividend Yield

1943 1957 1971 1984

Dividend Yield

dividend/price ratio

Dividend Yield on CRSP-VW (AMEX-NASDAQ-NYSE). Annual data. 1926-2009.

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Real Returns

1916 1930 1943

1957 1971 1984

Real Returns

real returns

Real Returns on CRSP-VW (AMEX-NASDAQ-NYSE). Annual data. 1926-2009.

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Dividend Growth

0.5 0.4 0.3 0.2 0.1

0 -0.1 -0.2 -0.3 -0.4

1916 1930 1943

1957 1971 1984

Div idend Growth

dividend growth

Dividend Growth on CRSP-VW (AMEX-NASDAQ-NYSE). Annual data. 1926-2009.

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Structural Break in 1991

Lettau and (2007) Önd a structural break in log dividend yield in 1991.

deÖned adjusted dividend yield:

dfpt = dptdp1

dpt = dptdp2

where dp1 denotes the mean in the Örst sample 1926-1991 and dp2 denotes

the mean in the second sample 1992-2009.

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Log Dividend Yield

0.8 0.6 0.4 0.2

0 -0.2 -0.4 -0.6 -0.8 -1 -1.2

Adjusted log Div idend Y ield

Demeaned log Dividend Yield dfpt on CRSP-VW (AMEX-NASDAQ-NYSE) with break in 1991. Annual data. 1926-2009.

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adjusted dp

log dividend yield

Return Predictability

Table: Return Predictability

Regression slope t-stat HC t-stat R2 rt+1 = ar + br (dfp)t + εt+1 0.267 [3.118] [3.667] 0.107 ∆dt+1 = ad + bd (dfp)t + εt+1 0.039 [0.624] [0.736] 0.004

Notes: Annual Data. Sample 1927-2009. Rt+1 is the real return on the CRSP-VW index. Rt+1 denotes logs of the real return. Rtf+1 denotes the return on the real risk-free.

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Longer Horizons

we run the following regression of k period holding returns on the dividend yield:

∑ r t + i = a r + b rk ( d p ) t + ε t + k

as you increase the horizon k, the slope coe¢ cients brk increase and the R2

Note: in this case, you should account for autocorrelation of residuals up to

and including k 1 observations apart mechanically induced by the overlap I The next couple of slides shows how to do this using HAC standard errors

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HAC robust standard errors

If OLS residuals exihibit heteroskedasticity and/or autocorrelation (and, potentially, non-normality), OLS is still consistent

I But, not e¢ cient

I Maximum likelihood is the e¢ cient method in large samples

I OLS is maximum likelihood only when errors are i.i.d. normally distributed

If we still choose OLS (as a linear regression is pretty robust and parsimonious), we need to adjust the standard errors

I HAC (heteroskedasticity and autocorrelation adjusted) standard errors

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HAC robust standard errors: theory

Please refer back to the “Note on Asymptotic Standard Errors” I posted earlier (which have already read)

Recall, for the case of Asymptotic OLS

yt =xtβ+εt, fort=1,…,T

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asymptotically 1 01

! N 0,TExtxt SExtxt

and βˆT is the estimate of β in a sample of length T

∑∞ htt0jttji

HAC robust standard errors: theory

If the residuals are correlated across q leads and lags and zero thereafter

corr εt,εtj 6= 0 for jjj q =0 forjjj>q

∑q htt0jttji

These are called Hansen-Hodrick standard errors (see next slide)

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Hansen-Hodrick standard errors

1T0 RT(v;β)=T ∑xtxtvεtεtv

where the estimate of the spectral density matrix is

SˆT =RT 0;βˆT+ ∑q hRT v;βˆT+RT v;βˆT0i

The estimate of the covariance matrix is then

Est.Asy.Var βˆT = T XT0 XT 1 SˆT XT0 XT 1

where capital xt, Xt, is a T K matrix with títh row equal to xt

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Newey-West standard errors

Newey and West (1987) solve an issue for the Hansen-Hodrick standard errors

The estimated variance covariance matrix of βˆ can be non-positive deÖnite

I.e., not invertible, “negative variance”

To ensure a positive-deÖnite covariance matrix, downweight estimated autocorrelations more the farther from the 0íth lag:

SˆT =RT 0;βˆT+ ∑q q+1v hRT v;βˆT+RT v;βˆT0i v =1 q + 1

The Newey-West covariance matrix is then

Est.Asy.Var βˆ = T XT0 XT 1 SˆT XT0 XT 1

For Newey-West (NW) standard errors, should use (k 1) 1.5 or so due to the downweighting in the NW procedure

Note that NW with 0 lags overlap is the same as White standard errors

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Long-Horizon Return Predictability with Dividend Yield

Table: Return Predictability

Horizon 1 2 3 4 5

0.105 [1.989]

0.199 [2.692]

0.250 [2.976]

0.282 [3.046]

0.323 [3.232]

Notes: Annual Data. Sample 1927-2009. Forecasting regression of ∑ki=1 rt+i on the log dividend yield.

[2.036] 0.047

[2.399] 0.083

[2.578] 0.101

[2.573] 0.106

[2.600] 0.119

∑ki=1 rt+i denotes the sum of k years of logs of the real return.

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Longer Horizons

we run the following regression of k period holding returns on the dividend yield:

∑ t+i r rk f t t+k

r =a +b (dp) +ε i=1

as you increase the horizon k, the slope coe¢ cients brk increase and the R2 increase

Consider the 5 year horizon (next slide) where bˆr5 = 0.826. An increase in the dividend yield of 1 percentage point in deviation from its mean increases the expected real return by 23.71 percentage points (=.826/.035) or 4.74 percentage points (per annum).

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Long-Horizon Return Predictability with Adj. Div. Yield

Table: Return Predictability

Horizon 1 2 3 4 5

0.267 [3.137]

0.478 [4.075]

0.661 [5.179]

0.750 [5.578]

0.826 [5.892]

Notes: Annual Data. Sample 1927-2009. Forecasting regression of ∑ki=1 rt+i on the adjusted log dividend

[3.480] 0.107

[3.960] 0.170

[4.977] 0.251

[4.559] 0.283

[4.157] 0.308

yield. ∑ki=1 rt+i denotes the sum of k years of logs of the real return.

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5-year return Forecast

1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

1916 1930 1943

1957 1971 1984

5-year log return forecast using Adjusted log Dividend Yield on CRSP-VW (AMEX-NASDAQ-NYSE). Annual data. 1926-2009.

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Longer Horizons

the R2 in the regression of k period holding returns on the dividend yield is given by:

R2(k) = V[Et[rt+1]+…+Et[rt+k]] V[rt+1 +rt+2 +…+rt+k]

this grows at rate k initially because

I realized returns are negatively autocorrelated

I predicted returns are positively autocorrelated

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Linearizing the returns

consider the return on an asset:

P +D Dt+1 (1+PDt+1)

Rt+1 t+1 t+1 = Dt Pt

pdt denotes the log price-dividend ratio

pdt = pt dt = log Pt ,

where price is measured at the end of the period and the dividend áow is over the same period.

also: note that

dpt = pdt

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Log-Linearizing returns

Campbell and Shiller (1989) log-linearization of the return equation around the (unconditional) mean log price/dividend ratio delivers the following expression for log returns:

rt+1 = ∆dt+1 +ρpdt+1 +k pdt,

with linearization coe¢ cients ρ and k that depend on the mean of the log

price/dividend ratio pd: epd

ρ = epd + 1 < 1 (the k coe¢ cient not important) this expression is an approximation of an identity. It must hold!
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The log of the price/dividend ratio
pdt = ∆dt+1 +ρpdt+1 +k rt+1 iterating forward on the linearized return equation
imposing a no-bubble condition:
lim ρjpdt+j =0
j!∞ expression for the log price/dividend ratio:
z }| { z }| {
discount rate
pdt = constant + ∑ ρj1∆dt+j ∑ ρj1rt+j
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Price/Dividend Ratios
Price/dividend ratios can only move if they predict returns or cash áows:
z }| {z }| {
a high price-to-dividend ratio pdt implies that dividends are expected to
increase or future returns (discount rates) are expected to decline
cash áow discount rate
t t " ∑∞ j 1 t + j # t " ∑∞ j 1 t + j # pd = constant + E ρ ∆d E ρ r
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The pd equation (without expectations) implies that the variance of the price/dividend ratio equals:
"∑∞ j1 # "∑∞ j1 # ∑∞ j1 ∑∞ j1 ! V[pdt] = V ρ ∆dt+j +V ρ rt+j 2cov ρ rt+j, ρ ∆dt+j
j=1 j=1 j=1
j=1 ∑∞ j1
∑∞ j1 = cov ρ
! ∑∞j1! ∑∞j1!
= cov pdt, ρ ∆dt+j cov pdt, ρ rt+j
Campbell and Shiller: the price/dividend ratio has to predict future (long-run) returns and/or dividends if it moves around!
I the evidence that it predicts returns seems stronger than the evidence that it predicts cash áows
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Variance Decomposition
The variance decomposition of the log price/dividend ratio is the di§erence between two regression slope coe¢ cients:
covpdt,∑∞ ρj1∆dt+j covpdt,∑∞ ρj1rt+j j=1 j=1
1 = V [pdt ] V [pdt ]
Is the variance of the pd-ratio driven by variation in expected cash áows or
expected returns (i.e., discount rates)?
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Price/Dividend Ratios
Price/dividend ratios predict future returns.
So do the term spread, the default spread and T-bill rates. The R2 increase with the forecasting horizon.
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Variance Decomposition
Recall that dpt = pdt . Thus., the dpt equation (without expectations) implies that the the slope coe¢ cients in a regression of discounted returns and dividend growth on dpt satisfy the following restriction:
Covdpt,∑∞ ρj1∆dt+j Covdpt,∑∞ ρj1rt+j j=1 j=1
1 = V(dpt) + V(dpt) = βd+βr
where βd and βr are implicitly deÖned in the above.
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Vector autoregressions (VAR)
consider 1st order restricted VAR:
rt+1 = ∆dt+1 = dt+1 pt+1 =
ar +brdpt +εrt+1
ad +bddpt +εdt+1
adp +φdpt +εdp t+1
remember we log-linearized an identity to get:
rt+1 = ∆dt+1 +ρpdt+1 +κ0 pdt.
ρ= epd 1+epd
this implies that there exists a deterministic relationship between these variables.
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Cross-equation restrictions
take expectations:
Et[rt+1] = Et[∆dt+1]+ρEt[pdt+1]+κ0 pdt.
go back to the 1st order VAR:
Et [rt+1] Et [∆dt+1] Et[dt+1 pt+1]
= ar + br dpt = ad + bd dpt = adp +φdpt
this implies that:
ar +brdpt = ad +bddpt ρ(adp +φdpt)+κ0 pdt
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Cross-equation restrictions
rt+1 = ∆dt+1 = dpt+1 =
ar +brdpt +εrt+1
ad +bddpt +εdt+1
a +φdpt +εdp dp t+1
) the coe¢ cients in these three equations must obey: br = bd + 1 φρ
or equivalently that the following is true:
br bd =1
I the Örst term is the slope coe¢ cient in the regression of the discount rate component on the dp-ratio
I the second term is the slope coe¢ cient in the regression of the cash áow component on the dp-ratio
I we show this on the next slide
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Slope coe¢ cient background math
Consider two hypothetical regressions:
1 the cash áow component on the dp-ratio:
ρE∆d =α+βdp+ε ∑∞ j 1 t t + j d d t t
Substitute in for future dividend growth using the VAR speciÖcation (note
error term equals zero always):
ad +bdEt dpt+j1
∞ j1 j1 bd =c+∑ρ bdφ dpt =c+1ρφdpt.
j=1 Thus, β
cov (dpt ,∑∞ ρj 1 ∆d j=1
bd (and c is a constant term).
2 the discount rate component on the dp-ratio:
ρEr =α+βdp+ε
∑∞ j 1 t t + j r r t t j=1
cov(dpt,∑∞ ρj1r )
Similar math as above yields β = j=1 t+j = br .
r V [dpt ] 1ρφ
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Cross-Equation Restrictions
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Variance Decomposition
slope coe¢ cients in predictability regressions represent fractions of variance due to disc
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