Robust Econometric Inference for Stock Return Predictability

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 Commonly used regressors are highly persistent series uncertainty regarding their order of integration (biased inference) 2 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: y t = µ + Ax t 1 + ε t x t = R n x t 1 + u t Local-to-unity assumption for x t, i.e. R n = 1 + c/n, test H 0 : 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. R n = 1 + c/n α, α < 1) ĉ OLS 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 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: y t = µ + Ax t 1 + ε t, x t = R n x t 1 + u t, R n = I K + C n α for some α 0, C = diag(c 1,..., c K ), c i 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: y t = µ + Ax t 1 + ε t, x t = R n x t 1 + u t, R n = I K + C n α for some α 0, C = diag(c 1,..., c K ), c i 0 x t unit root if C = 0 or α > 1 x t local-to-unit root if C < 0 and α = 1 x t mildly integrated if C < 0 and α 2 (0, 1) x t 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: y t = µ + Ax t 1 + ε t, x t = R n x t 1 + u t, R n = I K + C n α for some α 0, C = diag(c 1,..., c K ), c i 0 x t unit root if C = 0 or α > 1 x t local-to-unit root if C < 0 and α = 1 x t mildly integrated if C < 0 and α 2 (0, 1) x t stationary if C < 0 and α = 0 Denoting Y t = y t ȳ n, X t = x t x n 1 and E t = ε t ε n : Y t = AX t 1 + E t (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 6 / 28

Innovations covariance structure Model innovations (ε t ): i.i.d. sequence (0, Σ εε ) Regressor innovations (u t ): stationary linear process u t = C j e t j e t iid (0, Σ e ) j=0 Short run covariance matrices for ε t and u t (sample estimates): Σ εε = E ε t ε 0 t, Σεu = E ε t u 0 t, Σuu = E u t u 0 t, δ = Corr (εt, u t ) Long run covariance matrices (HAC estimation): Ω uu = E u t ut 0 h, Ωεu = Σ εu + Λ 0 uε, Λ uε = E u t ε 0 t h h= h=1 Extension: Asymptotic results hold under conditional heteroscedasticity, except for pure stationary regressors (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 7 / 28

IVX instrument construction IVX approach uses x t to construct mildly integrated instruments (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 8 / 28

IVX instrument construction IVX approach uses x t to construct mildly integrated instruments Choose β 2 (0, 1), C z < 0 (implementation: β = 0.95, C z = I K ) R nz = I K + C z /n β IVX instruments: z t = R nz z t 1 + x t z t behaves as an n β -mildly integrated process when β < α z t x t (n α -mildly integrated) when α < β (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 8 / 28

IVX instrument construction IVX approach uses x t to construct mildly integrated instruments Choose β 2 (0, 1), C z < 0 (implementation: β = 0.95, C z = I K ) R nz = I K + C z /n β IVX instruments: z t = R nz z t 1 + x t z t behaves as an n β -mildly integrated process when β < α z t x t (n α -mildly integrated) when α < β For Y = (Y1 0,..., Y n) 0 0 the IVX estimator is Ã n = Y 0 Z X 0 Z 1 (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 8 / 28

Mixed Normality Theorem (i) If β < min (α, 1): n 1+β 2 vec Ã n A ) MN 0, Ψ 1 uu 0 Cz V Cz C z Ψ 1 uu Σ εε, as n!, where 8 >< Ω uu + R 1 0 B udbu 0 under unit root Ψ uu = Ω uu + R 1 0 >: J C djc 0 if local-to-unit root Ω uu + V C C if mildly integrated V C = R 0 erc Ω uu e rc dr and V Cz = R 0 erc z Ω uu e rc z dr. (ii) If α 2 (0, β) then n 1+α 2 vec Ã n A ) N 0, VC 1 Σ εε (iii) If α = β > 0, n 1+α 2 vec Ã n A ) N 0, V 1 C 1 V C C 1 (V 0 ) 1 Σ εε, where V = R 0 erc V C e rc z dr (iv) If α = 0 then p nvec Ã n A ) N 0, (Ex 1 x1 0 ) 1 Σ εε (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 9 / 28

IVX Wald test Testing for general linear restrictions: H 0 : Hvec (A) = h rank (H) = r (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 10 / 28

IVX Wald test Testing for general linear restrictions: H 0 : Hvec (A) = h rank (H) = r Wald statistic: W n = HvecÃ n h 0 Q 1 H HvecÃ n h where Ã n is the IVX estimator and h Q H = H Z 0 X i h 1 Im M M = Z 0 Z ˆΣ εε n z n 1 z 0 n 1 ˆΩ FM ˆΩ FM = ˆΣ εε ˆΩ εu ˆΩ 1 uu ˆΩ 0 εu X 0 Z 1 Im i H 0 (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 10 / 28

IVX Wald test Testing for general linear restrictions: H 0 : Hvec (A) = h rank (H) = r Wald statistic: W n = HvecÃ n h 0 Q 1 H HvecÃ n h where Ã n is the IVX estimator and h Q H = H Z 0 X i h 1 Im M M = Z 0 Z ˆΣ εε n z n 1 z 0 n 1 ˆΩ FM ˆΩ FM = ˆΣ εε ˆΩ εu ˆΩ 1 uu ˆΩ 0 εu X 0 Z 1 Im i H 0 Theorem For all α 0 W n ) 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 R n = I K + C /n α 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 R n = I K + C /n α α > 0 Equivalently, for C! in a local-to-unity context Inference also valid for purely stationary regressors (α = 0) when ε t 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 R n = I K + C /n α α > 0 Equivalently, for C! in a local-to-unity context Inference also valid for purely stationary regressors (α = 0) when ε t is an uncorrelated sequence Standard (χ 2 ) 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 R n = I K + C /n α α > 0 Equivalently, for C! in a local-to-unity context Inference also valid for purely stationary regressors (α = 0) when ε t is an uncorrelated sequence Standard (χ 2 ) 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 2nd ISNPS, Cadiz 12 / 28

Nominal size 5%, autocorrelation=0.5 in u_t (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 13 / 28

Power plots, n=250, innovations corr=-0.95 (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 14 / 28

Power plots, n=250, innovations corr=-0.5 (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 15 / 28

Power plots, n=250, innovations corr=0 (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 16 / 28

Multiple regressors, size for joint and individual signi cance Three regressors, di erent degrees of persistence (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 17 / 28

Multiple regressors, power for joint test (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 18 / 28

Size, conditionally heteroscedastic univariate DGP (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 19 / 28

Power, heteroscedastic DGP, n=1000, corr=-0.95 (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 20 / 28

Power, heteroscedastic DGP, n=1000, corr=-0.5 (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 21 / 28

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) Stock Return Predictability 2nd ISNPS, Cadiz 22 / 28

Univariate regressions, Full sample period (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 23 / 28

Univariate regressions, Post-1952 (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 24 / 28

Multivariate regressions, Full sample period (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 25 / 28

Multivariate regressions, Post-1952 (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 26 / 28

Latest extension: Long-horizon predictability tests We are also interested in testing for long-horizon predictability via: y t (K ) = µ f + Ax t 1 + η f,t where y t (K ) = K 1 i=0 y t+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 estimator and the corresponding Wald test Proposed test has excellent small sample properties horizon IVX Main empirical result: If any, predictability typically weakens as the predictive horizon increases (Alex Kostakis, MBS) Stock Return Predictability 2nd ISNPS, Cadiz 27 / 28

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) Stock Return Predictability 2nd ISNPS, Cadiz 28 / 28