Figure 2a, b and c summarize the different patterns between US, Germany and Japan for the observed historical long-term (10y) equity premiums with monthly observations. It looks that there is not a single pattern between different countries, however, in recent times (after 1990s) the probability of a positive premium increases over previous periods (after 1960s). This issue has no direct relationship with high or low sovereign interest rate environments.
In addition, Fig. 3a, b and c show similar behavior between the long-term (10y) and medium term (5y) historical equity premium. Thus, length of a project looks not a determinant driver for the observed equity premium.
Tables 2, 3, 4 and 5 display the estimated values of d from the model given by Eqs. (1) and (2), for both the risk premium series and their volatility (proxied by their squared first differences), jointly with the 95% confidence intervals of the non-rejection values of d using Robinson’s (1994) tests. We consider three different specifications for Eq. (1): i) α = β = 0 (i.e., no deterministic components); ii) β = 0 (i.e., an intercept only); iii) α and β freely estimated from the data (i.e., including both an intercept and a linear time trend). We also make two alternative assumptions about the residuals, namely that they follow in turn a white noise or an autocorrelated process, in the latter case the non-parametric model of Bloomfield (1973) for weakly autocorrelated errors being estimated. In each case the values of d in bold are those from our preferred specification, our model selection criterion being the statistical significance of the other estimated parameters according to their t-values.
Tables 2 and 3 show the results obtained for the persistence of the risk premium under the assumption of white noise and autocorrelated disturbances respectively. In both cases the selected specification includes an intercept only. It can be seen that with white noise errors (Table 2) the null hypothesis of I(1) or a unit root cannot be rejected in the majority of cases; it is only rejected (in favour of orders of integration which are above 1) in the case of the US for the 10, 5 and 2 year time horizons with monthly data, and also for Germany and Japan for the 5 year time span with monthly data. In all other cases d is not statistically different from 1, which supports the random walk hypothesis; there is no evidence of mean reversion (d < 1) in any single case. When assuming autocorrelation in the disturbances (Table 3) the estimated values of d are slightly lower but the unit root null hypothesis can still not be rejected in any case. This I(1) behaviour is consistent with market efficiency.
Next we analyse persistence in the volatility of the risk premium (measured by its squared first differences). Tables 4 and 5 report the estimated values of d with their confidence bands, again for the two cases of white noise and autocorrelated errors respectively. The two sets of results are very similar, most of the values of d lying in the interval (0, 0.5) and implying stationary long-memory behaviour. There are only two cases when the I(0) hypothesis of short memory cannot be rejected, namely for Japan and the US with a 5-year span and monthly data. We also find a significant time trend in the case of the US with a 10-year span and monthly data.
Finally, we consider the possibility of structural breaks. Given the similarity between the monthly and the weekly results for d for all series and the fact that the US is the largest economy with the longest time span, we decided to focus on the US case at the monthly frequency with 10-year, 5-year and 2-year spans. Specifically, we carry out the Bai and Perron (2003) and Gil-Alana’s (2008) tests for multiple breaks. Both suggest the presence of two, three and four breaks for the monthly data over a 10, 5 and 2-year span respectively. The specific break dates are displayed in Table 6 and are the following: 1974 m09 and 1997 m11 for the 10-year span; 1981 m11, 1997 m05 and 2007 m1 for the 5-year span, and 1982 m11, 1989 m04, 1997 m06 and 2008 m03 for the 2-year span, and broadly coincide with the 1973–74 oil crisis, the early US 1980s recession resulting from the Fed’s contractionary monetary policy, the 1997 Asian financial crisis, and the 2007 global financial crisis. One of the detected breaks in the 2-year sample corresponds to the 1998 Savings and Loan crisis. As for the volatility series, a single break is detected, in 1974 for the 10-year sample, and in 2003 for the other two.
Having detected some breaks in the series of interest, we re-estimate the differencing parameter d for each of the subsamples to see if it has changed over time. Its estimated values for both the risk premium and its volatility under the alternative assumptions of white noise and autocorrelated residuals are reported in Tables 7 and 8 respectively. In the former case (see Table 7) there is no evidence of mean reversion in the risk premium, and a slight increase in persistence in the second and third subsamples. As for volatility, there is a sizeable increase in the case of the 10-year sample, a slight one in the case of the 5-year sample, and a decrease in the case of the 2-year one.
When allowing for autocorrelation in the residuals (see Table 8), the values of d are generally smaller though the confidence intervals are much wider, such that the I(1) hypothesis cannot be rejected in any single case. Thus, once more, there is no evidence of mean reversion in the risk premium. As for volatility, its persistence increases in the case of the 10-year sample, and a decrease in the other two.