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Robust Econometric Inference for Stock Return
Predictability
Alex Kostakis (MBS), Tassos Magdalinos (Southampton) and Michalis
Stamatogiannis (Bath)
Alex Kostakis, MBS
2nd ISNPS, Cadiz
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
1 / 28
Presentation Outline
Motivation and Related Literature
This paper and our contributions
Econometric methodology, Wald test and small sample properties
Predictability tests for US stock returns
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
2 / 28
Motivation
Vast literature on stock return predictability via lagged …nancial
variables (Keim and Stambaugh, 1986, Fama and French, 1988,
Campbell and Shiller, 1988)
An unsettled debate (see 2008 RFS special issue). Lettau and
Ludvigson (2001) vs. Goyal and Welch (2008)
Massive implications ("new facts") for modern …nance (Cochrane,
1999): EMH and time-varying premia, dynamic (strategic) asset
allocation, conditional asset pricing models and conditional alphas
Various econometric issues related to inference in predictive
regressions (Stambaugh, 1999):
1
2
Commonly used regressors are highly persistent series uncertainty
regarding their order of integration (biased inference)
Innovations of predictor’s and return’s regressions are correlated
(endogeneity)
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
3 / 28
Related Studies
Campbell and Yogo (2006), Torous et al. (2004), Hjalmarsson
(2011). Setup:
yt
xt
= µ + Axt 1 + εt
= Rn xt 1 + ut
Local-to-unity assumption for xt , i.e. Rn = 1 + c/n, test H0 : A = 0
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
4 / 28
Related Studies
Campbell and Yogo (2006), Torous et al. (2004), Hjalmarsson
(2011). Setup:
yt
xt
= µ + Axt 1 + εt
= Rn xt 1 + ut
Local-to-unity assumption for xt , i.e. Rn = 1 + c/n, test H0 : A = 0
t-statistic has non-standard limit distribution + depends on c
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
4 / 28
Related Studies
Campbell and Yogo (2006), Torous et al. (2004), Hjalmarsson
(2011). Setup:
yt
xt
= µ + Axt 1 + εt
= Rn xt 1 + ut
Local-to-unity assumption for xt , i.e. Rn = 1 + c/n, test H0 : A = 0
t-statistic has non-standard limit distribution + depends on c
Problems of existing methods:
Inference is speci…c to local-to-unity assumption (not valid for less
persistent regressors, e.g. Rn = 1 + c/nα , α < 1)
cˆOLS inconsistent complicated inference (need to construct
Bonferroni bounds for t-statistics based on con…dence interval for c)
Statistics developed for scalar regressor. Cannot conduct joint inference
on A in multiple regression
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
4 / 28
This paper
We propose a new econometric methodology, IVX estimation, for
predictive regressions (Magdalinos and Phillips, 2012):
1
Robusti…es inference with respect to the unknown degree of
regressors’persistence
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
5 / 28
This paper
We propose a new econometric methodology, IVX estimation, for
predictive regressions (Magdalinos and Phillips, 2012):
1
Robusti…es inference with respect to the unknown degree of
regressors’persistence
2
Accommodates multivariate systems of predictive regressions
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
5 / 28
This paper
We propose a new econometric methodology, IVX estimation, for
predictive regressions (Magdalinos and Phillips, 2012):
1
Robusti…es inference with respect to the unknown degree of
regressors’persistence
2
Accommodates multivariate systems of predictive regressions
3
Easy to implement (Wald test, chi-square inference, no need to
construct Bonferroni bounds)
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
5 / 28
This paper
We propose a new econometric methodology, IVX estimation, for
predictive regressions (Magdalinos and Phillips, 2012):
1
Robusti…es inference with respect to the unknown degree of
regressors’persistence
2
Accommodates multivariate systems of predictive regressions
3
Easy to implement (Wald test, chi-square inference, no need to
construct Bonferroni bounds)
We apply this methodology to conduct a series of predictability tests
Predictability evidence for ntis, e/p and b/m ratios
Evidence almost entirely disappears in the post-1952 period with the
exception of cay
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
5 / 28
Allowing for an arbitrary degree of persistence
We consider the following multivariate system:
yt
xt
Rn
= µ + Axt 1 + εt ,
= Rn xt 1 + ut ,
C
= IK + α for some α
n
(Alex Kostakis, MBS)
0, C = diag(c1 , ..., cK ), ci
Stock Return Predictability
2nd ISNPS, Cadiz
0
6 / 28
Allowing for an arbitrary degree of persistence
We consider the following multivariate system:
yt
xt
Rn
= µ + Axt 1 + εt ,
= Rn xt 1 + ut ,
C
= IK + α for some α
n
0, C = diag(c1 , ..., cK ), ci
0
xt unit root if C = 0 or α > 1
xt local-to-unit root if C < 0 and α = 1
xt mildly integrated if C < 0 and α 2 (0, 1)
xt stationary if C < 0 and α = 0
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
6 / 28
Allowing for an arbitrary degree of persistence
We consider the following multivariate system:
yt
xt
Rn
= µ + Axt 1 + εt ,
= Rn xt 1 + ut ,
C
= IK + α for some α
n
0, C = diag(c1 , ..., cK ), ci
0
xt unit root if C = 0 or α > 1
xt local-to-unit root if C < 0 and α = 1
xt mildly integrated if C < 0 and α 2 (0, 1)
xt stationary if C < 0 and α = 0
Denoting Yt = yt
y¯n , Xt = xt
x¯n
Yt = AXt
(Alex Kostakis, MBS)
1
and Et = εt
1
ε¯ n :
+ Et
Stock Return Predictability
2nd ISNPS, Cadiz
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Innovations covariance structure
Model innovations (εt ): i.i.d. sequence (0, Σεε )
Regressor innovations (ut ): stationary linear process
∞
∑ Cj et
ut =
j
et
iid (0, Σe )
j =0
Short run covariance matrices for εt and ut (sample estimates):
Σεε = E εt εt0 , Σεu = E εt ut0 , Σuu = E ut ut0 , δ = Corr (εt , ut )
Long run covariance matrices (HAC estimation):
Ωuu =
∞
∑
h= ∞
E ut ut0
h
0
, Ωεu = Σεu + Λuε
, Λuε =
∞
∑E
ut εt0
h
h =1
Extension: Asymptotic results hold under conditional
heteroscedasticity, except for pure stationary regressors
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
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IVX instrument construction
IVX approach uses ∆xt to construct mildly integrated instruments
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
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IVX instrument construction
IVX approach uses ∆xt to construct mildly integrated instruments
Choose β 2 (0, 1), Cz < 0 (implementation: β = 0.95, Cz =
IK )
Rnz = IK + Cz /n β
IVX instruments:
z˜t = Rnz z˜t
(Alex Kostakis, MBS)
1
+ ∆xt
Stock Return Predictability
2nd ISNPS, Cadiz
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IVX instrument construction
IVX approach uses ∆xt to construct mildly integrated instruments
Choose β 2 (0, 1), Cz < 0 (implementation: β = 0.95, Cz =
IK )
Rnz = IK + Cz /n β
IVX instruments:
z˜t = Rnz z˜t
1
+ ∆xt
z˜t behaves as an n β -mildly integrated process when β < α
z˜t
xt (nα -mildly integrated) when α < β
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
8 / 28
IVX instrument construction
IVX approach uses ∆xt to construct mildly integrated instruments
Choose β 2 (0, 1), Cz < 0 (implementation: β = 0.95, Cz =
IK )
Rnz = IK + Cz /n β
IVX instruments:
z˜t = Rnz z˜t
1
+ ∆xt
z˜t behaves as an n β -mildly integrated process when β < α
z˜t
xt (nα -mildly integrated) when α < β
0
For Y = (Y10 , ..., Yn0 ) the IVX estimator is
˜ n = Y 0 Z˜ X 0 Z˜
A
(Alex Kostakis, MBS)
Stock Return Predictability
1
2nd ISNPS, Cadiz
8 / 28
Mixed Normality Theorem
(i) If β < min (α, 1):
n
1+β
2
˜n
vec A
˜ uu1 0 Cz VCz Cz Ψ
˜ uu1
A ) MN 0, Ψ
Σεε ,
as n ! ∞, where
8
R1
>
< Ωuu + R0 B u dBu0 under unit root
1
˜ uu =
Ψ
Ωuu + 0 J C dJC0 if local-to-unit root
>
: Ω +V C
if mildly integrated
uu
C
R∞
R∞
VC = 0 e rC Ωuu e rC dr and VCz = 0 e rCz Ωuu e rCz dr .
1 +α
˜ n A ) N 0, V 1 Σεε
(ii) If α 2 (0, β) then n 2 vec A
C
(iii) If α = β > 0,
1 +α
˜ n A ) N 0, V 1 C 1 VC C 1 (V0 ) 1 Σεε , where
n 2 vec A
R∞
V = 0 e rC VC e rCz dr
p
˜ n A ) N 0, (Ex1 x 0 ) 1 Σεε
(iv) If α = 0 then nvec A
1
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
9 / 28
IVX Wald test
Testing for general linear restrictions:
H0 : Hvec (A) = h
(Alex Kostakis, MBS)
rank (H ) = r
Stock Return Predictability
2nd ISNPS, Cadiz
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IVX Wald test
Testing for general linear restrictions:
H0 : Hvec (A) = h
rank (H ) = r
Wald statistic:
˜n
Wn = HvecA
˜ n is the IVX estimator and
where A
h
1
QH = H Z˜ 0 X
0
˜n
h QH 1 HvecA
i h
Im M X 0 Z˜
M = Z˜ 0 Z˜ Σˆ εε nz¯n 1 z¯n0
ˆ FM = Σˆ εε Ω
ˆ εu Ω
ˆ uu1 Ω
ˆ 0εu
Ω
(Alex Kostakis, MBS)
1
Stock Return Predictability
ˆ FM
Ω
h
1
i
Im H 0
2nd ISNPS, Cadiz
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IVX Wald test
Testing for general linear restrictions:
H0 : Hvec (A) = h
rank (H ) = r
Wald statistic:
˜n
Wn = HvecA
˜ n is the IVX estimator and
where A
h
1
QH = H Z˜ 0 X
0
˜n
h QH 1 HvecA
i h
Im M X 0 Z˜
M = Z˜ 0 Z˜ Σˆ εε nz¯n 1 z¯n0
ˆ FM = Σˆ εε Ω
ˆ εu Ω
ˆ uu1 Ω
ˆ 0εu
Ω
1
ˆ FM
Ω
h
1
i
Im H 0
Theorem
For all α
0
Wn ) χ 2 ( r )
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
10 / 28
Advantages of the proposed approach
Inference valid for a large class of persistent regressors with roots
Rn = IK + C /nα α > 0
Equivalently, for C !
∞ in a local-to-unity context
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
11 / 28
Advantages of the proposed approach
Inference valid for a large class of persistent regressors with roots
Rn = IK + C /nα α > 0
Equivalently, for C !
∞ in a local-to-unity context
Inference also valid for purely stationary regressors (α = 0) when εt is
an uncorrelated sequence
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
11 / 28
Advantages of the proposed approach
Inference valid for a large class of persistent regressors with roots
Rn = IK + C /nα α > 0
Equivalently, for C !
∞ in a local-to-unity context
Inference also valid for purely stationary regressors (α = 0) when εt is
an uncorrelated sequence
Standard (χ2 ) and direct (no Bonferroni bounds) inference
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
11 / 28
Advantages of the proposed approach
Inference valid for a large class of persistent regressors with roots
Rn = IK + C /nα α > 0
Equivalently, for C !
∞ in a local-to-unity context
Inference also valid for purely stationary regressors (α = 0) when εt is
an uncorrelated sequence
Standard (χ2 ) and direct (no Bonferroni bounds) inference
Joint inference: Various combinations of regressors/ regressands can
be tested
(Alex Kostakis, MBS)
Stock Return Predictability
2nd ISNPS, Cadiz
11 / 28
Nominal size 5%, no autocorrelation in u_t
IVX Wald vs Campbell-Yogo Q-test and Jansson-Moreira test
(Alex Kostakis, MBS)
Stock Return Predictability
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Nominal size 5%, autocorrelation=0.5 in u_t
(Alex Kostakis, MBS)
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Power plots, n=250, innovations corr=-0.95
(Alex Kostakis, MBS)
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Power plots, n=250, innovations corr=-0.5
(Alex Kostakis, MBS)
Stock Return Predictability
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Power plots, n=250, innovations corr=0
(Alex Kostakis, MBS)
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Multiple regressors, size for joint and individual signi…cance
Three regressors, di¤erent degrees of persistence
(Alex Kostakis, MBS)
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Multiple regressors, power for joint test
(Alex Kostakis, MBS)
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Size, conditionally heteroscedastic univariate DGP
(Alex Kostakis, MBS)
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Power, heteroscedastic DGP, n=1000, corr=-0.95
(Alex Kostakis, MBS)
Stock Return Predictability
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Power, heteroscedastic DGP, n=1000, corr=-0.5
(Alex Kostakis, MBS)
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Data and unit root tests
Commonly used persistent regressors and log excess market returns from
Goyal-Welch (2008). Period: 1927-2012. Quarterly data shown.
(Alex Kostakis, MBS)
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Univariate regressions, Full sample period
(Alex Kostakis, MBS)
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Univariate regressions, Post-1952
(Alex Kostakis, MBS)
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Multivariate regressions, Full sample period
(Alex Kostakis, MBS)
Stock Return Predictability
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Multivariate regressions, Post-1952
(Alex Kostakis, MBS)
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Latest extension: Long-horizon predictability tests
We are also interested in testing for long-horizon predictability via:
yt (K ) = µf + Axt
1
+ η f ,t
1
where yt (K ) = ∑K
i =0 yt +i is the K period cumulative stock return
Prevailing view (Campbell, 2000): Predictability becomes more
signi…cant as the horizon K increases
Standard LS tests may be severely biased (oversized) in the presence
of highly persistent regressors (Boudoukh et al., 2008)
We extend our framework to this case, introducing a K horizon IVX
estimator and the corresponding Wald test
Proposed test has excellent small sample properties
Main empirical result: If any, predictability typically weakens as the
predictive horizon increases
(Alex Kostakis, MBS)
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Conclusions
New econometric methodology for stock return predictability
Accommodates a very rich class of predictive regressors, allows for
joint tests and is easy to implement via a Wald test
Empirical application reveals predictive ability of net equity expansion,
e/p and b/m ratios over excess market returns
Nevertheless, predictability evidence almost entirely disappears in the
post-1952 period with the exception of consumption-wealth ratio
If any, predictability typically weakens as the predictive horizon
increases
Methodology applicable for predictability tests in other asset classes
(bond yields, FOREX returns) that use persistent regressors
(Alex Kostakis, MBS)
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