#### 3.1. The Data

In this paper, we extend the analysis of

Kyriakou et al. (

2019b), who considered the forecasting of long-term stock returns, to conditional variance predictions. Thus, we base our predictions on the same annual US data set which is provided by Robert Shiller and can be downloaded from

http://www.econ.yale.edu/~shiller/data.htm. It includes, among other variables, the Standard and Poor’s (S&P) Composite Stock Price Index, the consumer price index, and interest rate data from 1872 to 2019. We use here an updated and revised version of

Shiller (

1989, chp. 26), which provides a detailed description of the data. Note that the risk-free rate in this data set (based on the six-month commercial paper rate until 1997 and afterwards on the six-month certificate of deposit rate, secondary market) was discontinued in 2013. We follow the strategy of

Welch and Goyal (

2008) and replace it by an annual yield that is based on the six-month Treasury-bill rate, secondary market, from

https://fred.stlouisfed.org/series/TB6MS. This new series is only available from 1958 to 2019. In the absence of information prior to 1958, we had to estimate it. To this end, we regressed the Treasury-bill rate on the risk-free rate from Shiller’s data for the overlapping period 1958 to 2013, which yielded

with an

${R}^{2}$ of 98.6%. Therefore, we instrumented the risk-free rate from 1872 to 1957 with the predicted regression equation. The correlation between the actual Treasury-bill rate and the predictions for the estimation period is 99.3%.

Table 1 displays standard descriptive statistics for one-year and five-year returns as well as the available covariates.

#### 3.2. Single Benchmarking Approach

In this section, we consider a single benchmarking approach as in

Kyriakou et al. (

2019a,

2019b), i.e., only the dependent variable

${S}_{t}$ is benchmark adjusted, as shown in (

1), while the independent variable(s) is (are) measured on the original (nominal) scale. The models (

2) and (

5) are estimated in both steps with a local-linear kernel smoother using the quartic kernel. The optimal bandwidths are chosen by cross-validation, i.e., by maximizing the corresponding validation measure given by (

8). Given that we apply a local-linear smoother, it should be kept in mind that the nonparametric method can estimate linear functions without any bias. Thus, the linear model is automatically embedded in our approach. This is an important observation as the linear model is the usual benchmark in financial applications. In addition, in case that the true (but in advance) unknown function is really linear, our approach would exactly pick the line against all other functional alternatives. We study the

${R}_{V,\nu}^{2}$ values based on different validated scenarios shown for the one-year horizon in

Table 2 and the five-year horizon in

Table 3. Here, the same predictive variables

${X}_{t-1}$ are used in both steps of our approach. Note that we have only about 150 observations in our records. The small sample size clearly limits the complexity of our analysis in the sense of using higher dimensional vectors of explanatory variables. In what follows, we consider only one- and two-dimensional models. For a discussion on sparsely distributed annual observations in higher dimensions and ways to circumvent the curse-of-dimensionality, see, for example,

Kyriakou et al. (

2019a).

Overall, we find for the one-year horizon that only a few variables have small positive validated

${R}_{V,\nu}^{2}$’s and thus possibly some low explanatory power. For example, for the benchmarks

${B}^{\left(R\right)}$,

${B}^{\left(L\right)}$, and

${B}^{\left(E\right)}$, the excess stock return has the largest validated

${R}_{V,\nu}^{2}$ values for one-dimensional models (2.2%, 2.4%, and 1.5%). This finding would support an ARCH-type variance structure. For the inflation benchmark

${B}^{\left(C\right)}$, the model with the long-term interest rate produces the largest validated

${R}_{V,\nu}^{2}$ of 0.5%. When we apply the bootstrap tests introduced in

Section 2.5, the KNY-test does not reject the null of no predictability for all cases at the 5%-level. The SNS-test rejects the null only for the

${Y}_{t-1}^{\left(A\right)}$ covariate under the benchmarks

${B}^{\left(R\right)}$,

${B}^{\left(L\right)}$ and

${B}^{\left(E\right)}$ at the 5%-level.

14 Note that the two-dimensional models do not add predictive power as the validated

${R}_{V,\nu}^{2}$ values remain in the same low range.

Contrary to the mean prediction, where

Kyriakou et al. (

2019b) find that five-year predictability improves over the one-year case, we observe that the majority of predictor based volatility models do not surpass the constant volatility alternative for the five-year horizon. Even though some models produce small positive

${R}_{V,\nu}^{2}$ values, this time both the SNS- and the KNY-test do not reject the null of no predictability. Note that our results are in line with

Christoffersen and Diebold (

2000) who conclude that volatility forecastability may be much less important at longer horizons.

#### 3.3. Full Benchmarking Approach

In the next step, we consider the double benchmarking approach of

Kyriakou et al. (

2019a,

2019b) to analyze now whether transforming the explanatory variables can improve the predictions for the volatility function. Recall that fully nonparametric models suffer in general by the curse of dimensionality. Problems with sparsely distributed annual observations in higher dimensions, as in our framework, could be reduced or circumvented by importing more structure in the estimation process.

Here, we extend the study presented in

Section 3.2 transforming both the dependent and independent variables according to the same benchmark. To this end, in our full (double) benchmarking approach, the prediction problems are reformulated as

where we use transformed predictive variables

This approach can be interpreted as a simple way of reducing the dimensionality of the estimation procedure. The adjusted variable

${X}_{t-1}^{\left(A\right)}$ includes now an additional predictive variable, the benchmark itself. Results of this empirical study are presented for the one-year horizon in

Table 4 and for the five-year horizon in

Table 5.

We find that, in comparison to the single-benchmarking approach in the one-year case, the double benchmarking improves in 15 out of 82 models (in the sense of producing a positive and higher ${R}_{V,\nu}^{2}$ as before). However, predictability is still questionable. The best model under the long-term interest rate benchmark ${B}^{\left(L\right)}$ uses the pair $({Y}_{t-1}^{\left(L\right)},{e}_{t-1}^{\left(L\right)})$ and yields ${R}_{V,\nu}^{2}=3.0$, while the best model under ${B}^{\left(E\right)}$ uses the pair $({Y}_{t-1}^{\left(E\right)},{l}_{t-1}^{\left(E\right)})$ and yields ${R}_{V,\nu}^{2}=2.5$. The SNS-test rejects for both the null of no predictability, while the KNY-test does not. For the rest of the new combinations of predictive variables in all benchmarks, both tests again do not reject.

For the five-year case, we find that in comparison to the single-benchmarking the double benchmarking improves in 11 out of 82 models. The best model under ${B}^{\left(E\right)}$ uses ${d}_{t-1}^{\left(E\right)}$ and yields ${R}_{V,\nu}^{2}=1.8$, while under ${B}^{\left(C\right)}$ the covariates ${d}_{t-1}^{\left(C\right)}$ and ${l}_{t-1}^{\left(C\right)}$ both yield ${R}_{V,\nu}^{2}=1.6$. Nevertheless, we do not find any combination of covariates with statistically significant predictive power.

#### 3.4. Real-Income Long-Term Pension Prediction

In long-term pension planning or other asset allocation problems optimized with regard to real-income protection (

Gerrard et al. (

2019a,

2019b); (

Merton 2014)), the econometric models should reflect those needs and use covariates net-of-inflation. Therefore, we take the inflation benchmark

${B}^{\left(C\right)}$ and analyse in more detail the best model found by

Kyriakou et al. (

2019b), which uses the earnings-by-price variable for the mean prediction and produced a

${R}_{V,m}^{2}=12.2$ for the one-year horizon and

${R}_{V,m}^{2}=12.4$ for the five-year horizon (see

Kyriakou et al. 2019b, Tables 4 and 5) in the double benchmarking case. For this specific model, we are now interested in finding the set of covariates that best predicts the conditional variance.

15^{,}16 The empirical findings in terms of

${R}_{V,\nu}^{2}$ are shown for the one-year horizon in

Table 6 and the five-year horizon in

Table 7. For the one-year horizon, we find in the double benchmarking approach when inflation is the benchmark,

${B}^{\left(C\right)}$ that the dividend-by-price

${d}^{\left(C\right)}$ together with the short-term interest-rate

${r}^{\left(C\right)}$ or the long-term interest-rate

${l}^{\left(C\right)}$ are chosen as best predictive variables in terms of

${R}_{V,\nu}^{2}$ (2.9% and 2.0%). Note that these values are rather low and that the SNS-test does reject the null of no predictability for both models, while the KNY-test does not reject. For all other combinations and also the five-year case, we do not find evidence for statistical significant predictability of the conditional variance. Therefore, we conclude that the constant volatility model is appropriate for practical purposes.

Note further that the ratio in our validation criterion for the mean prediction,

${R}_{V,m}^{2}$, in (

8) compares the sample variance of the estimated residuals from our model based on earnings-by-price (the numerator) with the sample variance of the benchmarked stock returns (the denominator). For the one-year case, we find from

Table 1 the latter to be equal to

${0.1805}^{2}=0.03258$. A simple calculation using the corresponding

${R}_{V,m}=12.2\%$ leads then to

$0.03258(1-0.122)=0.02861$ or a standard deviation of 16.91% for returns based on the earnings-model. This means that the linear expression of real stock returns in terms of real earnings-by-price presented in

Kyriakou et al. (

2019b) as

gives on average 2.4% higher returns at the same risk as the historical mean

${\overline{Y}}^{\left(C\right)}$.

17 Similarly, for the five-year case, we get from

Table 1 that

${0.3642}^{2}=0.1326$. From the

${R}_{V,m}=12.4\%$, we obtain then

$0.1326(1-0.122)=0.1162$ or a standard deviation of 34.08% for returns based on the earnings-model. Thus, the linear expression of real stock returns in terms of real earnings-by-price presented in

Kyriakou et al. (

2019b) as

gives on average 6.1% higher returns at the same risk as the historical mean

${\overline{Y}}^{\left(C\right)}$.

18 Figure 1 shows the estimated nonparametric function

$\widehat{m}$ (red solid line) for the one-year horizon (left) and the five-year horizon (right) under the double inflation benchmark for the earnings-by-price covariate together with the corresponding historical mean (dashed green line).

Figure 2 depicts histograms and a kernel density estimate (red solid line) of the standardized predicted returns for the one-year horizon (left) and the five-year horizon (right). The similarity for both horizons is striking and driven by the fact that the ratio of the slope of the regression lines in (

17) and (

18) with the corresponding standard deviation given above yields almost the same value of 6.63.

Finally, we consider a simple mean-reverting autoregressive model of order one for the real earnings-by-price—the main drivers of real returns in Equations (

17) and (

18)—and estimate it with ordinary least squares (OLS)

19:

Note that, for the whole sample period (1872–2019), the mean and standard deviation of real earnings-by-price are 0.0524 and 0.0595, resp. Moreover, using the current (30/09/2019) value of real earnings-by-price of 0.0278, model (

19) predicts a change in real earnings-by-price of 0.0176, i.e., an expected value of real earnings-by-price of 0.0454 for 2020Q3, which is still below the long-term average.

20We subsequently calculate the correlation between the estimated residuals of models (

17) and (

19) to be −0.014. A standard stationary block-bootstrap (

Politis and Romano 1994) based on 10,000 repetitions and a block-length of 12 suggests that this correlation is not statistically significantly different from zero. The correlation structure between returns and their drivers is important while searching for optimal investment strategies in a dynamic market, see

Kim and Omberg (

1996).

Gerrard et al. (

2019c) follow the approach of

Kim and Omberg (

1996) in a long-term return setting and show that the above correlation is very hard to estimate with precision. Sometimes, it is negative and, with a slight change of data, it is positive, and a test would almost always provide that zero correlation cannot be rejected. When this added insight is provided that zero correlation significantly simplifies that technical calculation of the optimal dynamic strategy while significantly reducing parameter uncertainty, the conclusion seems clear: we should work with zero correlation unless there is a strong argument not to do that. In our case—which is a discrete analogue to the continuous models considered in

Gerrard et al. (

2019c) and

Kim and Omberg (

1996)—it is, therefore, comforting that we can provide a simple zero-correlation econometric model to guide the market dynamics. In further work, we expect the simple econometric model of this paper to be used while generalizing the non-dynamic new approach to pension products of

Gerrard et al. (

2019a,

2019b).