# 程序代写 Return Predictability – cscodehelp代写

Return Predictability
What drives stock market prices? A present-value decomposition and application of AR models
. Lochstoer
UCLA Anderson School of Management

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.
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and10ap/p6li0cat

Structural Break in 1991
Lettau and (2007) Önd a structural break in log dividend yield in 1991.
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.
. 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.
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and12ap/p6li0cat
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.
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and13ap/p6li0cat

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
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and14ap/p6li0cat

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
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and15ap/p6li0cat

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
∑∞ 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)
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and17ap/p6li0cat

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;βˆT0i
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
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and18ap/p6li0cat

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;βˆT0i 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
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and19ap/p6li0cat

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).
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and21ap/p6li0cat

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.
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and22ap/p6li0cat

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.
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and23ap/p6li0cat

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
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and24ap/p6li0cat

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
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and25ap/p6li0cat

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! . Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and26ap/p6li0cat 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 . Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and27ap/p6li0cat 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 . Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and28ap/p6li0cat 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 . Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and29ap/p6li0cat Variance Decomposition The variance decomposition of the log price/dividend ratio is the di§erence between two regression slope coe¢ cients: covpdt,∑∞ ρj1∆dt+j covpdt,∑∞ 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)? . 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. . Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and31ap/p6li0cat 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: Covdpt,∑∞ ρj1∆dt+j Covdpt,∑∞ ρj1rt+j j=1 j=1 1 = V(dpt) + V(dpt) = βd+βr where βd and βr are implicitly deÖned in the above. . Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and32ap/p6li0cat 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. . Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and33ap/p6li0cat 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 . Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and34ap/p6li0cat 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 . Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and35ap/p6li0cat 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 . Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and36ap/p6li0cat Cross-Equation Restrictions . Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 6 Return Predictability What drives stock market prices? A present-value Wdeicnotemrp2o0s2it2ion and37ap/p6li0cat Variance Decomposition slope coe¢ cients in predictability regressions represent fractions of variance due to disc 程序代写 CS代考 加微信: cscodehelp QQ: 2235208643 Email: kyit630461@163.com

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