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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
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value dWecionmtepro2s0it2io2n and1ap/p6li0cat
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
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value dWecionmtepro2s0it2io2n and2ap/p6li0cat
What drives stock price movements?
Seminal paper by Nobel prize winner
Do stock prices move too much to be justiÖed by subsequent dividends?
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value dWecionmtepro2s0it2io2n and3ap/p6li0cat
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
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value dWecionmtepro2s0it2io2n and4ap/p6li0cat
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.
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value dWecionmtepro2s0it2io2n and5ap/p6li0cat
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.
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value dWecionmtepro2s0it2io2n and6ap/p6li0cat
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
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value dWecionmtepro2s0it2io2n and7ap/p6li0cat
Dividend Yield
1943 1957 1971 1984
Dividend Yield
dividend/price ratio
Dividend Yield on CRSP-VW (AMEX-NASDAQ-NYSE). Annual data. 1926-2009.
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value dWecionmtepro2s0it2io2n and8ap/p6li0cat
Real Returns
1916 1930 1943
1957 1971 1984
Real Returns
real returns
Real Returns on CRSP-VW (AMEX-NASDAQ-NYSE). Annual data. 1926-2009.
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value dWecionmtepro2s0it2io2n and9ap/p6li0cat
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 = dpt dp1
dpt = dpt dp2
where dp1 denotes the mean in the Örst sample 1926-1991 and dp2 denotes
the mean in the second sample 1992-2009.
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and11ap/p6li0cat
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
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and16ap/p6li0cat
asymptotically 1 0 1
! N 0,TExtxt SExtxt
and βˆT is the estimate of β in a sample of length T
∑∞ htt0 jtt ji
HAC robust standard errors: theory
If the residuals are correlated across q leads and lags and zero thereafter
corr εt,εt j 6= 0 for jjj q =0 forjjj>q
∑q htt0 jtt ji
These are called Hansen-Hodrick standard errors (see next slide)
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Hansen-Hodrick standard errors
1T0 RT(v;β)=T ∑xtxt vεtεt v
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+1 v 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.
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and20ap/p6li0cat
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 + ∑ ρj 1∆dt+j ∑ ρj 1rt+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:
"∑∞ j 1 # "∑∞ j 1 # ∑∞ j 1 ∑∞ j 1 ! 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 ∑∞ j 1
∑∞ j 1 = cov ρ
! ∑∞j 1! ∑∞j 1!
= 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,∑∞ ρj 1∆dt+j covpdt,∑∞ ρj 1rt+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)?
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and30ap/p6li0cat
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,∑∞ ρj 1∆dt+j Covdpt,∑∞ ρj 1rt+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+j 1
∞ j 1 j 1 bd =c+∑ρ bdφ dpt =c+1 ρφdpt.
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,∑∞ ρj 1r )
Similar math as above yields β = j=1 t+j = br .
r V [dpt ] 1 ρφ
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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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