2.3 Empirical Results
2.3.2 In-sample Results: Return Predictability in International Stock
First, we discuss the results of the in-sample analysis of return predictability in international stock markets. The only subjective element of the BACE approach is the choice of the a-priori expected model size k, i.e. the researcher’s belief of how¯ many variables are a-priori likely to be included in the predictive model. We choose a rather moderate specification of this hyperparameter, consistent with the principle of parsimony prevailing in econometrics. We therefore set the a-priori expected model size tok¯= 2variables.13 This implies a prior probability of inclusion ofπ = 2/κ= 0.¯2 for each variable. The choice of the expected model size is linked to the a-priori model probabilityp(Mj)which is given asp(Mj) =πkj(1−π)κ−kj.14 It is important to note that a prior probability of inclusion smaller than 0.5 amounts to an a-priori down weighting of larger model specifications. This implies an additional penalty for highly parameterized models beside the penalty implied by the degree of freedom adjustment of the BIC.
The tables for the different stock markets, which will be discussed in the following, are all organized in the same way. Panel A and C are based on monthly data while Panel B and D present results for quarterly data. Panel A and B report results for the composite model with bias-corrected slope coefficients. π|ydenotes the posterior probability of inclusion for each variable. The posterior probability of inclusion is defined as the total sum of the posterior probabilities of all models, in which the particular variable is included; it is computed asC0P, whereCis a2κ×κmatrix denoting inclusion (exclusion) of a particular variable in model j by 1 (0), and P is a 2κ×1 vector containing the posterior model probabilitiesp(Mj|y). Posterior means of the predictive coefficients in the weighted model based on Eq. (2.4) are reported in the second column of Panels A/B.
The third and fourth column report posterior Bayesian t-ratios. Following Avramov (2002), we report both t-ratios based on posterior standard deviations which ignore model uncertainty and t-ratios adjusted for model uncertainty (see discussion in Section 2).
We also assess the robustness of the different predictive variables according to two other criteria. In Panels A/B we report the proportion of cases when the coefficient on a particular variable (every time it is included in one of thej = 1,· · ·,2κmodels) has the same sign as the posterior mean in the composite model (denoted as sgn prob. in the tables). Furthermore, we also report the fraction of cases across the different models when a classical t-statistic for the particular variable is greater than two in absolute value. This statistic serves as another indicator of the robustness or fragility of a particular predictive variable (Sala-i-Martin, Doppelhofer, and Miller, 2004). Panels C and D, presents the five top-performing model specifications which receive the highest posterior probability of all models. The models are defined by inclusion (1) or exclusion (0) of the specific variable. Moreover, the corresponding posterior model probabilities and the adjustedR2 of the five top models are also reported.
France
Estimation results for the French stock market are provided in Table 2.2. As Panel A (monthly predictive regressions) shows, the only variable for which the posterior probability of inclusion π|y rises, compared to the prior probability of inclusion, is the relative bond rate RBR. In the case of the other variables, inspection of the data leads us to retract our prior opinion about their usefulness. Panel C reports monthly results for the five best-performing model specifications. After having seen the data, the model which includes RBR as a single predictive variable receives a posterior model probability of more than 50%, which is greatly higher than the one of the next best model specifications. A negative relation of the realative bond rate and expected excess returns is reasonable from an economic point of view, given that higher yields on long-term bonds are typically reflected in a higher level of corporate loan rates and thus may have a negative impact on subsequent real activity. The relative bond rate together with the output gap is also significant according to a posterior t-ratio.
Robustness of a particular variable can also be assessed by the sign certainty probability which measures the fraction of cases where the coefficient on the particular variable (when included in one of the 2κ Models) has the same sign as its coefficient in the weighted model. According to this criterion, the relative bond rate is again rather successful. The relative bond rate (RBR), the term spread (TRM), industrial production growth (IPG) and the output gap (GAP) all have sign certainty probabilities exceeding 90%, whereas several other popular predictors such as the dividend yield perform clearly worse. However, Table 2.2 also makes clear that none of the variables remains significant when the additional variability of estimates across models is accounted for.15 Panels B and D show that the evidence for predictability in the French stock market is somewhat weaker in the quarterly case. Again, only the relative bond rate receives a posterior probability of inclusion larger than 0.¯2. It is also worth noting that the
15This is a general result which holds for almost all predictive variables and almost all stock markets
earnings yield performs relatively well in-terms of sign certainty in the quarterly case.
Germany
Table 2.3 provides estimation results for the German stock market. As can be seen in Panel A and C of Table 2.3, predictability of monthly stock returns is fairly weak on statistical grounds. The case for predictability is clearly less pronounced than in the French stock market discussed in the previous subsection. The model receiving the highest posterior probability is the one without any lagged state variables (i.i.d.
case). None of the variables in the monthly model receives a higher posterior inclusion probability compared to the prior inclusion probability ofπ= 0.¯2. Among the variables considered only the relative bond rate (RBR) and the output gap (GAP) may be consid-ered as significant according to a Bayesian t-ratio, but this does not hold true when the dispersion of coefficients across models is considered.
Similar to the French case, the relative bond rate is rather important in the quarterly regressions (Panel B of Table 2.3) where the probability of inclusion rises after having seen the data. Evidence for predictability with quarterly data is somewhat stronger than for monthly data. This can be seen from the result in Panel D that the most likely quarterly model is now the one which includes the relative bond rate. This model achieves an adjustedR2 of about 5% in the quarterly regressions, which is quite high for the stock return predictability literature. Several variables appear quite robust with regard to sign certainty: The term spread (TRM), the relative bond rate (RBR), industrial production growth (IPG), and the two valuation ratios (LDY, LEY) have the same sign as the posterior mean in the composite model in more than 90% of all models in which they are included.
Table2.2:EstimationResults,In-Sample:France anelA:CompositeModel,MonthlyPanelB:CompositeModel,Quarterly π|ypost.t-ratiot-ratiosgnfractionπ|ypost.t-ratiot-ratiosgnfraction mean(adj)prob.|t|>2mean(adj)prob.|t|>2 0.0330.0101.2050.5570.9690.027TRM0.0360.0281.0080.5110.9490.004 0.042-0.016-1.535-0.5720.4730.016RTB0.064-0.070-1.462-0.6010.6250.023 0.743-0.958-3.494-1.8371.0000.340RBR0.641-2.400-3.146-1.4611.0000.414 0.017-0.001-0.404-0.2720.1800.117INF0.0210.0020.3290.1790.8910.453 0.0170.0010.8800.4240.9450.246IPG0.030-0.002-0.204-0.1270.0390.238 0.0420.0221.3110.5810.4410.000LRV0.0310.0240.5980.3760.2500.000 Y0.0030.0010.2450.1970.4340.000LDY0.0150.0340.7720.4380.5000.000 0.0030.0010.1900.1590.7420.000LEY0.0180.0510.8900.4750.9100.000 0.116-0.031-2.347-0.7640.9490.172GAP0.129-0.104-2.031-0.7320.8160.027 anelC:Top5Models,MonthlyPanelD:Top5Models,Quarterly 00000TRM00000 00001RTB00010 10010RBR10001 00000INF00000 00000IPG00000 00010LRV00000 Y00000LDY00000 00000LEY00000 00100GAP00101 0.6350.1160.0790.0310.028Prob0.5270.1620.0850.0400.025 0.0270.0000.0170.0320.012¯R2 0.0650.0000.0380.0270.068 A(monthly)andB(quarterly)reportestimationresultsforthecompositemodel.Thecoefficientsintheweightedmodelarethecoefficientsin modelsweightedbytheposteriormodelprobabilities.Posteriorprobabilitiesofinclusionarecalculatedasthesumoftheposteriorprobabilities whichincludetherespectivevariable.Bayesiant-ratiosarereported,withoutandwithadjustmentformodeluncertainty(adj).PanelC andD(quarterly)displaythefivebest-performingmodelspecifications(highestposteriormodelprobability),where0indicatesexclusionand1 oftherespectivepredictivevariable.AlsotheadjustedR2 andtheposteriormodelprobabilitiesofthemodelswhichreceivethehighestposterior arereported.Thesetofpredictorscomprisesthetermspread(TRM),theshort-terminterestraterelativetoits12-monthmoving (RTB),along-termgovernmentbondyieldrelativetoits12-monthmovingaverage(RBR),annualinflationrate(INF),annualgrowthofindustrial (IPG),(log)realizedvolatility(LRV),(log)dividendyield(LDY),(log)earningsyield(LEY),outputgap(GAP).
Table2.3:EstimationResults,In-Sample:Germany PanelA:CompositeModel,MonthlyPanelB:CompositeModel,Quarterly π|ypost.t-ratiot-ratiosgnfractionπ|ypost.t-ratiot-ratiosgnfraction mean(adj)prob.|t|>2mean(adj)prob.|t|>2 TRM0.0290.0061.1980.5540.9340.469TRM0.0380.0210.9800.5040.9220.418 RTB0.025-0.006-1.064-0.5200.7660.039RTB0.043-0.032-0.905-0.4730.7770.066 RBR0.188-0.196-2.418-0.8161.0000.285RBR0.467-1.896-2.683-1.1031.0000.527 INF0.015-0.001-0.342-0.2540.4880.000INF0.026-0.004-0.272-0.2120.5040.000 IPG0.017-0.000-0.241-0.1520.0660.273IPG0.0310.0010.1720.1050.9490.297 LRV0.0210.0040.7610.4350.1600.430LRV0.0390.0330.8360.4620.2500.000 LDY0.005-0.003-0.855-0.4621.0000.004LDY0.007-0.006-0.292-0.2260.9380.000 LEY0.0130.0000.0320.0310.6090.000LEY0.0270.0580.5760.3690.9650.000 GAP0.089-0.016-1.981-0.7070.8160.133GAP0.144-0.093-1.995-0.7370.6760.078 PanelC:Top5Models,MonthlyPanelD:Top5Models,Quarterly TRM00010TRM00000 RTB00001RTB00001 RBR01000RBR10010 INF00000INF00000 IPG00000IPG00000 LRV00000LRV00000 LDY00000LDY00000 LEY00000LEY00000 GAP00100GAP00110 Prob0.6350.1630.0750.0220.020Prob0.3640.3220.0930.0260.024 ¯R2 0.0000.0130.0080.0020.001¯R2 0.0520.0000.0310.0600.009 Note:PanelA(monthly)andB(quarterly)reportestimationresultsforthecompositemodel.Thecoefficientsintheweightedmodelarethecoefficientsin individualmodelsweightedbytheposteriormodelprobabilities.Posteriorprobabilitiesofinclusionarecalculatedasthesumoftheposteriorprobabilities ofthemodelswhichincludetherespectivevariable.Bayesiant-ratiosarereported,withoutandwithadjustmentformodeluncertainty(adj).PanelC (monthly)andD(quarterly)displaythefivebest-performingmodelspecifications(highestposteriormodelprobability),where0indicatesexclusionand1 inclusionoftherespectivepredictivevariable.AlsotheadjustedR2 andtheposteriormodelprobabilitiesofthemodelswhichreceivethehighestposterior modelprobabilityarereported.Thesetofpredictorscomprisesthetermspread(TRM),theshort-terminterestraterelativetoits12-monthmoving average(RTB),along-termgovernmentbondyieldrelativetoits12-monthmovingaverage(RBR),annualinflationrate(INF),annualgrowthofindustrial production(IPG),(log)realizedvolatility(LRV),(log)dividendyield(LDY),(log)earningsyield(LEY),outputgap(GAP).
Japan
Results for the Japanese stock market are given in Table 2.4. As for Germany, there is no compelling evidence that monthly stock returns in Japan are predictable: The model with clearly the highest posterior probability in Panel C is the model with no explanatory variables (i.i.d.-model). The output gap (GAP) and the relative bond rate (RBR) are somewhat marginally important, but their explanatory power is fairly low.
Note also that industrial production growth (IPG) and inflation (INF) are quite robust in terms of sign certainty probability.
With quarterly data, the evidence for predictability is even more modest. Again the model which does not include any predictors receives the highest probability a-posteriori.
Only the output gap receives a higher posterior probability of inclusion than expected a-priori (Panel D of Table 2.4). However, model uncertainty again plays a substantial role as evinced by the adjusted Bayesian t-ratios. It is also worth noting that according to the sign certainty measure, the output gap must be considered as a rather fragile predictor.
United Kingdom
Table 2.5 reveals, that both for monthly and quarterly predictive regressions, the case for return predictability in the United Kingdom is quite weak. Panel C shows, that the largest posterior probability in the monthly regressions is assigned to the i.i.d.-model (as in the case of monthly regressions for Germany and Japan). Contrary to the countries discussed so far, interest rate variables do not show up among the most prominent predictors, which confirms the recent findings by Giot and Petitjean (2006) based on univariate return prediction models. By contrast, the dividend yield (LDY) has some predictive content for future stock returns in the UK. Yet, as before, accounting for model uncertainty greatly reduces the evidence for predictability and explanatory power of return prediction models in the UK is rather low.
Table2.4:EstimationResults,In-Sample:Japan PanelA:CompositeModel,MonthlyPanelB:CompositeModel,Quarterly π|ypost.t-ratiot-ratiosgnfractionπ|ypost.t-ratiot-ratiosgnfraction mean(adj)prob.|t|>2mean(adj)prob.|t|>2 TRM0.0150.0020.4500.3030.8090.309TRM0.023-0.003-0.166-0.1370.4340.066 RTB0.055-0.022-1.694-0.6480.5740.105RTB0.035-0.021-0.696-0.3770.5820.102 RBR0.192-0.194-2.479-0.8260.8240.051RBR0.048-0.083-1.255-0.5710.6640.000 INF0.030-0.002-1.391-0.5921.0000.371INF0.026-0.003-0.521-0.3450.8830.031 IPG0.0230.0010.9440.4240.9100.336IPG0.0410.0070.9410.4050.8950.348 LRV0.022-0.006-1.017-0.5100.3750.000LRV0.025-0.013-0.527-0.3480.7030.027 LDY0.003-0.001-0.584-0.3650.7930.230LDY0.0170.0030.0800.0710.4180.027 LEY0.005-0.001-0.296-0.1850.2380.480LEY0.0200.0120.3070.2350.8400.500 GAP0.256-0.043-2.603-0.8870.4920.219GAP0.309-0.182-2.448-0.9090.6450.195 PanelC:Top5Models,MonthlyPanelD:Top5Models,Quarterly TRM00000TRM00000 RTB00010RTB00010 RBR00100RBR00100 INF00001INF00000 IPG00000IPG00001 LRV00000LRV00000 LDY00000LDY00000 LEY00000LEY00000 GAP01000GAP01001 Prob0.4670.2130.1630.0450.017Prob0.5310.2530.0340.0240.019 ¯R2 0.0000.0150.0140.0070.002¯R2 0.0000.0370.0070.0010.047 Note:PanelA(monthly)andB(quarterly)reportestimationresultsforthecompositemodel.Thecoefficientsintheweightedmodelarethecoefficientsin individualmodelsweightedbytheposteriormodelprobabilities.Posteriorprobabilitiesofinclusionarecalculatedasthesumoftheposteriorprobabilities ofthemodelswhichincludetherespectivevariable.Bayesiant-ratiosarereported,withoutandwithadjustmentformodeluncertainty(adj).PanelC (monthly)andD(quarterly)displaythefivebest-performingmodelspecifications(highestposteriormodelprobability),where0indicatesexclusionand1 inclusionoftherespectivepredictivevariable.AlsotheadjustedR2 andtheposteriormodelprobabilitiesofthemodelswhichreceivethehighestposterior modelprobabilityarereported.Thesetofpredictorscomprisesthetermspread(TRM),theshort-terminterestraterelativetoits12-monthmoving average(RTB),along-termgovernmentbondyieldrelativetoits12-monthmovingaverage(RBR),annualinflationrate(INF),annualgrowthofindustrial production(IPG),(log)realizedvolatility(LRV),(log)dividendyield(LDY),(log)earningsyield(LEY),outputgap(GAP).
United States
As shown by Table 2.6, evidence for in-sample return predictability is clearly stronger in the US compared to other international stock markets such as Germany, Japan or the UK. Variables which appear important after having seen the data include the relative bond rate (RBR) and, most notably, the output gap (GAP). The output gap is the only variable which can be considered as a significant predictor once model uncertainty is accounted for. It receives a posterior probability of inclusion of more than 80%, which is a substantial upward revision of the prior probability of inclusion.16 The output gap also appears to be a less fragile predictor in the US compared to the other countries. It is also worth noting that the earnings yield (LEY) provides more explanatory power than the dividend yield (LDY). Several other variables – such as the relative bond rate (RBR), inflation (INF), and industrial production growth (IPG) – are important when model uncertainty is ignored, but lose their significance once model uncertainty is considered.
When we consider predictive models at a quarterly horizon, the output gap (GAP) again appears as an important variable a-posteriori and also survives the model uncertainty adjustment. Also note that the relative bond rate is less important in the quarterly regressions. Panels A and B further show that the earnings yield appears to be very robust with regard to sign certainty, which holds both in the monthly and the quarterly models.
Table2.5:EstimationResults,In-Sample:UnitedKingdom PanelA:CompositeModel,MonthlyPanelB:CompositeModel,Quarterly π|ypost.t-ratiot-ratiosgnfractionπ|ypost.t-ratiot-ratiosgnfraction mean(adj)prob.|t|>2mean(adj)prob.|t|>2 TRM0.0170.0010.6600.3990.9020.156TRM0.0290.0040.5580.3020.7620.105 RTB0.014-0.000-0.138-0.0890.1130.152RTB0.018-0.001-0.095-0.0490.0980.281 RBR0.026-0.010-1.345-0.5800.7810.148RBR0.026-0.011-0.708-0.4050.7850.063 INF0.0160.0000.5090.3170.5940.008INF0.0230.0020.8170.4260.8050.066 IPG0.0210.0021.2190.5560.9260.398IPG0.0230.0020.7710.3870.8750.328 LRV0.0210.0060.8830.4510.2500.012LRV0.033-0.019-1.056-0.5200.9220.414 LDY0.2030.3631.7250.7240.6330.020LDY0.5593.9742.1691.1010.9920.289 LEY0.0430.0300.9930.5091.0000.480LEY0.0830.2691.4290.6191.0000.441 GAP0.073-0.016-2.038-0.7010.3160.313GAP0.041-0.023-1.747-0.6210.3200.273 PanelC:Top5Models,MonthlyPanelD:Top5Models,Quarterly TRM00000TRM00000 RTB00000RTB00000 RBR00001RBR00000 INF00000INF00000 IPG00000IPG00000 LRV00000LRV00001 LDY01000LDY10001 LEY00010LEY00100 GAP00100GAP00010 Prob0.6040.1850.0600.0390.019Prob0.4810.2680.0700.0320.019 ¯R2 0.0000.0150.0070.0080.002¯R2 0.0620.0000.0360.0160.063 Note:PanelA(monthly)andB(quarterly)reportestimationresultsforthecompositemodel.Thecoefficientsintheweightedmodelarethecoefficientsin individualmodelsweightedbytheposteriormodelprobabilities.Posteriorprobabilitiesofinclusionarecalculatedasthesumoftheposteriorprobabilities ofthemodelswhichincludetherespectivevariable.Bayesiant-ratiosarereported,withoutandwithadjustmentformodeluncertainty(adj).PanelC (monthly)andD(quarterly)displaythefivebest-performingmodelspecifications(highestposteriormodelprobability),where0indicatesexclusionand1 inclusionoftherespectivepredictivevariable.AlsotheadjustedR2 andtheposteriormodelprobabilitiesofthemodelswhichreceivethehighestposterior modelprobabilityarereported.Thesetofpredictorscomprisesthetermspread(TRM),theshort-terminterestraterelativetoits12-monthmoving average(RTB),along-termgovernmentbondyieldrelativetoits12-monthmovingaverage(RBR),annualinflationrate(INF),annualgrowthofindustrial production(IPG),(log)realizedvolatility(LRV),(log)dividendyield(LDY),(log)earningsyield(LEY),outputgap(GAP).
Table2.6:EstimationResults,In-Sample:UnitedStates anelA:CompositeModel,MonthlyPanelB:CompositeModel,Quarterly π|ypost.t-ratiot-ratiosgnfractionπ|ypost.t-ratiot-ratiosgnfraction mean(adj)prob.|t|>2mean(adj)prob.|t|>2 0.0140.0010.3870.2540.7890.059TRM0.022-0.000-0.035-0.0260.2460.172 0.025-0.007-1.450-0.5780.8980.168RTB0.0250.0000.0250.0130.9100.258 0.302-0.278-3.127-0.9821.0000.809RBR0.060-0.089-1.622-0.6420.3360.035 0.086-0.031-3.787-0.8460.9180.457INF0.011-0.002-0.912-0.4700.8160.059 0.059-0.006-2.443-0.7010.8520.254IPG0.028-0.001-0.142-0.0670.2890.148 0.0100.0000.0170.0140.1170.000LRV0.0670.0581.2350.5741.0000.078 Y0.0300.0110.6950.2860.5270.172LDY0.0010.0010.5120.3300.5080.000 0.1170.2283.2110.8380.9960.594LEY0.0060.0141.4450.5931.0000.090 0.805-0.181-3.904-2.1791.0000.621GAP0.938-0.659-3.829-2.8240.9960.457 anelC:Top5Models,MonthlyPanelD:Top5Models,Quarterly 00000TRM00001 00000RTB00000 01100RBR00100 00011INF00000 00010IPG00000 00000LRV01000 Y00000LDY00000 00011LEY00000 11001GAP11101 0.5440.1360.1030.0340.025Prob0.7770.0550.0370.0240.016 0.0290.0370.0230.0460.046¯R2 0.0690.0810.0720.0000.065 A(monthly)andB(quarterly)reportestimationresultsforthecompositemodel.Thecoefficientsintheweightedmodelarethecoefficientsin modelsweightedbytheposteriormodelprobabilities.Posteriorprobabilitiesofinclusionarecalculatedasthesumoftheposteriorprobabilities whichincludetherespectivevariable.Bayesiant-ratiosarereported,withoutandwithadjustmentformodeluncertainty(adj).PanelC andD(quarterly)displaythefivebest-performingmodelspecifications(highestposteriormodelprobability),where0indicatesexclusionand1 oftherespectivepredictivevariable.AlsotheadjustedR2 andtheposteriormodelprobabilitiesofthemodelswhichreceivethehighestposterior arereported.Thesetofpredictorscomprisesthetermspread(TRM),theshort-terminterestraterelativetoits12-monthmoving (RTB),along-termgovernmentbondyieldrelativetoits12-monthmovingaverage(RBR),annualinflationrate(INF),annualgrowthofindustrial (IPG),(log)realizedvolatility(LRV),(log)dividendyield(LDY),(log)earningsyield(LEY),outputgap(GAP).