CARACTERÍSTICAS DE LA COMPETENCIA DIRECTA E INDIRECTA

The results of the previous section provide convincing evidence that the dispersion in options traders’ beliefs can significantly predict future excess stock market returns in-sample (IS). In this section, we evaluate the out-of-sample (OS) performance of our dispersion in beliefs measures following Lettau and Ludvigson (2001), Goyal and Welch (2003, 2008), Guo (2006), and Campbell and Thompson (2008) among others. The purpose of this exercise is to assess the usefulness of the dispersion in

options trading volume across strike prices for an investor who has access only to real time data when making her forecasts and also to gauge regression parameter instability over time. Following the literature we mainly rely on OS regressions of 1-month horizon but for robustness purposes we also report results for 3- and 6- month horizons, keeping in mind the relatively low statistical power of OS regression analysis compared to IS analysis (Inoue and Kilian, 2004).

As in Goyal and Welch (2008), Campbell and Thompson (2008), Rapach, Strauss and Zhou (2010), and Ferreira and Santa-Clara (2011) we estimate the model in equation (5.3) recursively using the first s = s0...T −h observations and based on

the estimated parameters we form our OS forecasts for the expected excess market return using the concurrent values of the predictor variables:

b

res+h,h =αbs,h+βb 0

s,hzs. (5.4)

The initial estimation period is from 1996:01-1999:12 and the first prediction is made for 2000:01. This way we create a series of TOS OS forecasts that is compared to a

series of recursively estimated historical averages, which correspond to OS forecasts of a restricted model with only a constant as a regressor. We employ four measures to assess the OS predictability performance of our dispersion in expectations measures.

The first measure is the OSR2 _{denoted by R}2

OS which takes the form:

R2_OS = 1− M SEU M SER , (5.5) where M SEU = _T1 OS PT−h t=s (ret+h,h−rebt+h,h) 2

is the mean square error of the unre- stricted model andM SER= _T1

OS

PT−h

t=s (ret+h,h−reet+h,h)

2

is the mean square error of the restricted model with re_et+h,h being the recursively estimated historical av-

erage. R2

OS takes positive values whenever the unrestricted model outperforms the

The second measure of OS performance is the F-test from McCracken (2007):

M SE−F = (TOS−h+ 1)

M SER−M SEU M SEU

, (5.6)

which tests whether M SEU is statistically significantly lower than M SER.

The third OS performance test is the encompassing test of Clark and McCracken (2001): EN C−N EW = (TOS −h+ 1) TOS PT−h t=s (ret+h,h−rebt+h,h) 2₋ (ret+h,h−rebt+h,h) (ret+h,h−reet+h,h) M SEU , (5.7)

which examines whether the restricted model encompasses the unrestricted model, meaning that the unrestricted model does not improve the forecasting ability of the restricted model. Statistical inference for theM SE−F and theEN C−N EW tests relies on the critical values derived by McCracken (2007) and Clark and McCracken (2001) using Monte Carlo simulations.

The final measure of OS forecasting performance is the constrained OS R2 _de-

noted by R2

C−OS suggested by Campbell and Thompson (2008). This measure is the

same with R2

OS apart from the fact that it sets the OS forecasts of the unrestricted

model equal to zero whenever they take negative values. Therefore, an investor’s real time equity premium prediction becomes in accordance with standard asset pricing theory.

Table 5.5 presents the results for 1-, 3- and 6-month horizon OS predictability. In the case of 1-month horizon, DISP and DISP* exhibit positiveR2

OSs of 1.70% and

1.55% respectively. For both measures, theM SE−F test rejects at 5% level the null hypothesis that the mean square error of the unrestricted model is equal to the mean square error of the restricted model while theEN C−N EW test rejects at 5% level the null hypothesis that the restricted model encompasses the unrestricted model. When we impose the restriction of positive expected equity premium the results are improved for both dispersion in beliefs measures, with R2

DISP and 2.44% for DISP*. Turning to the rest of the predictors, only VRP provides a positiveR2

OS of 7.96%. Moreover, theM SE−F andEN C−N EW tests strongly

reject the respective null hypotheses at 5% level. Since univariate analysis suggests that only the dispersion in options trading volume across strikes and VRP have significant OS forecasting performance, we proceed by combining the two dispersion in options traders’ beliefs measures with VRP. The results show that the bivariate models increase the R2

OS, which becomes 9.06% in the regression including DISP

and VRP and 8.56% in the case of DISP* and VRP confirming that the information content of the dispersion in options traders’ expectations is different from that of VRP. Moreover, the M SE −F and EN C−N EW tests reject the respective null hypotheses even more decisively.

The results for the 3-month horizon are similar to those for the 1-month horizon but stronger for both dispersion in beliefs measures and the VRP. This is in line with the IS regression results presented in the previous section. In particular, DISP (DISP*) has an R2

OS of 3.37% (3.04%) while VRP has an R2OS of 12.47%. The M SE−F and the EN C−N EW tests prodive even stronger evidence against the respective null hypotheses. As in the 1-month horizon analysis, apart from DISP, DISP* and VRP, none of the other predictors exhibit positive R2

OSs. Furthermore,

the bivariate model of the dispersion in options traders’ expectations with VRP is even more successful in OS return predictability. The results for the 6-month horizon are in the same vein with the evidence from the other horizons. In particular, the

R2

OSs of DISP and DISP* remain positive while theM SE−F and theEN C−N EW

tests still reject the respective null hypotheses at 5% level. Moreover, except for VRP none of the other alternative predictors provide a positive R2

OS, while the

combination of DISP (or DISP*) with VRP offers even stronger OS predictability. Overall, the empirical evidence regarding OS return predictability suggests that only the dispersion in options traders’ expectations and VRP are successful predic- tors at short horizons and that their predictive power is enhanced when they are combined in one bivariate model.