# Conditional Variance Forecasts for Long-Term Stock Returns

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## Abstract

**:**

## 1. Introduction

## 2. A Framework for Conditional Variance Prediction

#### 2.1. One-Year Predictions

#### 2.2. Longer-Horizon Predictions

^{,}7 For a discussion on asymptotic properties of our nonparametric estimators of model (5) and (6), see Section 2.3 in Kyriakou et al. (2019b).

#### 2.3. Alternative Ways in Estimating the Conditional Variance Function

#### 2.4. The Validation Criterion for the Choice of Smoothing Parameters and Model Selection

#### 2.5. A Bootstrap-Test: No Predictability vs. Predictability of the Conditional Variance

## 3. Empirical Application: Conditional Variance Prediction for Stock Returns in Excess of Different Benchmarks

#### 3.1. The Data

#### 3.2. Single Benchmarking Approach

#### 3.3. Full Benchmarking Approach

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

^{,}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.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1 | Our methodology of validating a fully nonparametric structure can be viewed as one of the simplest and therefore also most transparent version of machine learning; see Section 2 of Kyriakou et al. (2019a) for more details justifying the label machine learning for our approach. |

2 | Note that the use of a different ML method would come with the cost of losing interpretability, smoothness, or flexibility due to restrictions on the functional form. A comparison of different ML techniques in finding that one which gives the best predictions, wins an investment horse-race out-of-sample, or being the most robust method over different periods is out of the scope of our work. |

3 | The choice of the one-year horizon is related to the frequency of the data. In contrast, the five-year horizon is arbitrary but is intended to be a starting point for actuarial long-term models for real-income savings. Other horizons and related questions remain for future research. |

4 | Note that the set of explanatory variables in (2) could be different or overlapping for the mean and variance function. |

5 | For a description and statistical properties of the local-linear smoother, see, for example, Section 2.3 in Kyriakou et al. (2019b). Note further that the smoothing parameters h and g are separately chosen in each step. |

6 | Our flexible location-scale model in (5), could be easily extended to time-lags of higher order. However, in the empirical application in Section 3, we see that, for example, for real-earnings—the main driver of real-returns—an AR1-type model is ideally suited. This is in line with findings from Kothari et al. (2006). Note further that one might expect risk and return to be somehow related (see, for example, Merton 1973). The parametric GARCH-in-Mean process captures this idea (Linton and Yan 2011). However, the inclusion of an interaction of mean and variance in a fully nonparametric fashion is out of the scope of this paper. To our knowledge, only semiparametric versions where either the mean or variance function is modeled parametrically can be found in the literature, see, for example, Pagan and Hong (1991); Pagan and Ullah (1988); Linton and Perron (2003). |

7 | For possible solutions to the problem of autocorrelation, see, for example, Xiao et al. (2003); Su and Ullah (2006); Linton and Mammen (2008), or more recently Geller and Neumann (2018). The implementation and analysis of these techniques remain for future research. In our approach, we account for autocorrelation in the validation criterion with a leave-k-out strategy, where $k=2T-1$; see Section 2.4. |

8 | It does not estimate the volatility function as efficiently as if the true mean were known. |

9 | Examples of these variants are: (i) Applying a local-linear kernel smoother in both stages (Fan and Yao 1998). The result is again not necessarily nonnegative but asymptotically fully adaptive to the unknown mean function. (ii) Using the local exponential estimator to ensure nonnegativity (Ziegelmann 2002). (iii) Implementing a combined estimator (a multiplicative bias reduction technique), where a parametric guide captures some roughness features of the unknown variance function (Glad 1998; Mishra et al. 2010). (iv) Utilising a re-weighted local constant estimator maximising the empirical likelihood such that it becomes a bias-reducing moment restriction (Xu and Phillips 2011). |

10 | Those results are available upon request by the authors. |

11 | There is also a lack of studies using the difference-sequence method in a random design and in multivariate problems as in our case. |

12 | Model selection in the sense of composition of the set of explanatory variables. |

13 | The symbol ${1\phantom{\rule{-2.5pt}{0ex}}\mathrm{I}}_{A}$ denotes the indicator function of an appropriate condition A, i.e., it is one when A is true and zero otherwise. |

14 | The tests were conducted with 1000 repetitions at the 5% significance level for a selected number of cases. We do not present the p-values of the tests to save space. The results are available upon request by the authors. |

15 | Note that until now we have used the same set of covariates in both steps of our analysis to reduce the overwhelming number of models. It is also clear that not all combinations of variables are practically relevant. Now, we relax this restriction for the model with the highest predictive power for the returns. |

16 | Table 6 and Table 7 also present the results for the short- and long-term interest benchmarks ${B}^{\left(R\right)}$ and ${B}^{\left(L\right)}$. However, it is again hard to find predictability at all in these cases. Note that the benchmark using the earnings-by-price variable ${B}^{\left(E\right)}$ is not applicable since it matches the covariate and the benchmark in the first step. |

17 | Here, we use the Sharpe-ratio for the comparison. From Table 1, we get ${\overline{Y}}^{\left(C\right)}=6.41\%$ and divide it either by 18.05% or by 16.91%. We obtain 0.355 and 0.379, which corresponds to a difference of 2.4% points. |

18 | Here, we use again the Sharpe-ratio for the comparison. From Table 1, we get ${\overline{Y}}^{\left(C\right)}=32.34\%$ and divide it either by 36.42% or by 34.08%. We obtain 0.888 and 0.949, which corresponds to a difference of 6.1% points. |

19 | The estimated coefficient is significant at the 0.1%-level (with a corresponding standard error of 0.08), the residual standard error of the regression is 0.0572, and its ${R}^{2}$ has a value of 0.357. |

20 | The following values are used for the calculation of the current real earnings-by-price: $P=2976.74$, $E=135.53$, ${B}^{\left(C\right)}=1.0173$. |

**Figure 1.**Double inflation benchmark. Relation between real stock returns and real earnings-by-price. Estimated nonparametric function $\widehat{m}$ (red solid line) and historical average (dashed green line).

**Left**: one-year horizon.

**Right**: five-year horizon. Period: 1872–2019. Data: annual S&P 500.

**Figure 2.**Standardized predicted stock returns in excess of the inflation benchmark (based on the model using earnings-by-price as covariate for mean-prediction; double benchmarking). Histogram, kernel density estimate (red), and fitted normal distribution (green).

**Left**: one-year horizon.

**Right**: five-year horizon. Period: 1872–2019. Data: annual S&P 500.

Max | Min | Mean | Sd | Skew | Exc. kurt | |
---|---|---|---|---|---|---|

S&P stock price index P | 2789.80 | 3.25 | 277.58 | 558.13 | 2.43 | 5.50 |

Dividend accruing to index D | 53.75 | 0.18 | 6.04 | 10.56 | 2.45 | 6.00 |

Earnings accruing to index E | 132.39 | 0.16 | 13.96 | 26.31 | 2.43 | 5.35 |

Dividend-by-price d | 9.88 | 1.17 | 4.31 | 1.71 | 0.46 | 0.25 |

Earnings-by-price e | 17.75 | 1.72 | 7.28 | 2.75 | 1.05 | 1.39 |

Short-term interest rate r | 14.93 | 0.07 | 3.97 | 2.50 | 0.96 | 2.34 |

Long-term interest rate l | 14.59 | 1.88 | 4.53 | 2.27 | 1.81 | 3.63 |

Inflation $\pi $ | 20.69 | −15.65 | 2.23 | 5.96 | 0.26 | 1.60 |

Spread s | 3.64 | −3.71 | 0.56 | 1.32 | −0.05 | 0.02 |

One-year excess stock returns ${Y}^{\left(R\right)}$ | 42.39 | −58.26 | 4.58 | 17.28 | −0.57 | 0.68 |

One-year excess stock returns ${Y}^{\left(C\right)}$ | 54.04 | −48.81 | 6.41 | 18.05 | −0.40 | 0.64 |

Five-year excess stock returns ${Z}^{\left(R\right)}$ | 107.27 | −78.54 | 23.49 | 36.69 | −0.14 | −0.37 |

Five-year excess stock returns ${Z}^{\left(C\right)}$ | 122.96 | −57.34 | 32.34 | 36.42 | −0.05 | −0.40 |

**Table 2.**Predictive power for the variance of one-year excess stock returns ${Y}_{t}^{\left(A\right)}$: the single benchmarking approach. The prediction problem is defined in (2). The same predictive variables ${X}_{t-1}$ are used in the predictions for the conditional mean and variance function. The predictive power (%) is measured by ${R}_{V,\nu}^{2}$ as defined in (8). The benchmarks ${B}^{\left(A\right)}$ considered are based on the short-term interest rate ($A\equiv R$), long-term interest rate ($A\equiv L$), earnings-by-price ratio ($A\equiv E$), and consumer price index ($A\equiv C$). The predictive variables used are ${X}_{t-1}$, given by the dividend-by-price ratio ${d}_{t-1}$, earnings-by-price ratio ${e}_{t-1}$, short-term interest rate ${r}_{t-1}$, long-term interest rate ${l}_{t-1}$, inflation ${\pi}_{t-1}$, term spread ${s}_{t-1}$, excess stock return ${Y}_{t-1}^{\left(A\right)}$, or the possible different pairwise combinations as indicated.

Benchmark ${\mathit{B}}^{\left(\mathit{A}\right)}$ | Explanatory Variable(s) ${\mathit{X}}_{\mathit{t}-1}$ | ||||||
---|---|---|---|---|---|---|---|

${Y}^{\left(A\right)}$ | d | e | r | l | $\pi $ | s | |

Short-term rate | 2.2 | −1.1 | −0.6 | −0.3 | 0.3 | −1.2 | −0.1 |

Long-term rate | 2.4 | −1.2 | −0.6 | 0.3 | 0.6 | −1.4 | −0.1 |

Earnings-by-price | 1.5 | −1.3 | −0.7 | −0.1 | 0.5 | −1.4 | 0.1 |

Inflation | 0.2 | 0.1 | −1.3 | −0.4 | 0.5 | −1.2 | −0.6 |

$({Y}^{\left(A\right)},d)$ | $({Y}^{\left(A\right)},e)$ | $({Y}^{\left(A\right)},r)$ | $({Y}^{\left(A\right)},l)$ | $({Y}^{\left(A\right)},\pi )$ | $({Y}^{\left(A\right)},s)$ | ||

Short-term rate | 2.4 | 1.9 | 1.1 | 2.2 | 0.1 | 0.3 | |

Long-term rate | 1.5 | 1.4 | 1.1 | 2.1 | −0.2 | 0.1 | |

Earnings-by-price | 1.6 | 1.4 | 0.9 | 2.0 | −0.2 | 0.1 | |

Inflation | −1.0 | −1.1 | −0.6 | 0.6 | −2.1 | −1.0 | |

$(d,e)$ | $(d,r)$ | $(d,l)$ | $(d,\pi )$ | $(d,s)$ | |||

Short-term rate | −2.1 | −1.5 | −0.8 | −2.4 | −1.5 | ||

Long-term rate | −2.0 | −1.1 | −0.6 | −2.2 | −1.5 | ||

Earnings-by-price | −1.9 | −1.4 | −0.7 | −2.3 | −1.5 | ||

Inflation | −0.4 | −1.0 | −0.2 | −2.3 | −1.3 | ||

$(e,r)$ | $(e,l)$ | $(e,\pi )$ | $(e,s)$ | ||||

Short-term rate | −1.0 | −0.4 | −2.3 | −0.8 | |||

Long-term rate | −0.6 | −0.2 | −2.2 | −0.8 | |||

Earnings-by-price | −1.0 | −0.2 | −2.2 | −0.8 | |||

Inflation | −1.7 | −0.9 | −2.2 | −1.6 | |||

$(r,l)$ | $(r,\pi )$ | $(r,s)$ | |||||

Short-term rate | 1.3 | −1.5 | 1.4 | ||||

Long-term rate | 1.3 | −1.0 | 1.4 | ||||

Earnings-by-price | 1.4 | −1.5 | 1.6 | ||||

Inflation | 1.3 | −1.5 | 1.2 | ||||

$(l,\pi )$ | $(l,s)$ | ||||||

Short-term rate | −1.2 | 1.4 | |||||

Long-term rate | −0.9 | 1.4 | |||||

Earnings-by-price | −1.0 | 1.6 | |||||

Inflation | −0.9 | 1.3 | |||||

$(\pi ,s)$ | |||||||

Short-term rate | 0.2 | ||||||

Long-term rate | 0.2 | ||||||

Earnings-by-price | −0.6 | ||||||

Inflation | −0.1 |

**Table 3.**Predictive power for the variance of five-year excess stock returns ${Z}_{t}^{\left(A\right)}$: the single benchmarking approach. The prediction problem is defined in (5). The same predictive variables ${X}_{t-1}$ are used in the predictions for the conditional mean and variance function. Additional notes: see Table 2.

Benchmark ${\mathit{B}}^{\left(\mathit{A}\right)}$ | Explanatory Variable(s) ${\mathit{X}}_{\mathit{t}-1}$ | ||||||
---|---|---|---|---|---|---|---|

${Y}^{\left(A\right)}$ | d | e | r | l | $\pi $ | s | |

Short-term rate | 0.6 | −1.7 | −1.7 | −1.2 | −1.0 | −2.0 | −3.0 |

Long-term rate | 0.0 | −1.5 | −1.3 | −1.2 | −1.1 | −1.2 | −2.7 |

Earnings-by-price | 0.8 | −1.8 | −1.1 | −1.8 | −2.7 | −0.3 | −3.8 |

Inflation | −1.0 | −3.8 | −4.7 | −0.7 | −1.5 | 1.4 | 0.5 |

$({Y}^{\left(A\right)},d)$ | $({Y}^{\left(A\right)},e)$ | $({Y}^{\left(A\right)},r)$ | $({Y}^{\left(A\right)},l)$ | $({Y}^{\left(A\right)},\pi )$ | $({Y}^{\left(A\right)},s)$ | ||

Short-term rate | −2.8 | −2.5 | −1.7 | −1.7 | −1.7 | −3.9 | |

Long-term rate | −2.5 | −2.1 | −1.6 | −1.8 | −1.2 | −3.4 | |

Earnings-by-price | −2.3 | −2.1 | −1.2 | −4.1 | 0.4 | −3.4 | |

Inflation | −5.1 | −4.7 | −1.5 | −2.6 | 0.4 | −0.9 | |

$(d,e)$ | $(d,r)$ | $(d,l)$ | $(d,\pi )$ | $(d,s)$ | |||

Short-term rate | −3.6 | −3.1 | −2.2 | −2.8 | −4.1 | ||

Long-term rate | −3.1 | −3.2 | −2.7 | −2.3 | −4.3 | ||

Earnings-by-price | −4.1 | −4.0 | −5.3 | −2.3 | −4.9 | ||

Inflation | −5.2 | −5.0 | −8.9 | −2.5 | −3.2 | ||

$(e,r)$ | $(e,l)$ | $(e,\pi )$ | $(e,s)$ | ||||

Short-term rate | −3.3 | −3.3 | −3.5 | −4.9 | |||

Long-term rate | −2.8 | −3.3 | −2.9 | −4.9 | |||

Earnings-by-price | −4.5 | −5.5 | −2.7 | −6.5 | |||

Inflation | −8.5 | −7.8 | −4.9 | −6.4 | |||

$(r,l)$ | $(r,\pi )$ | $(r,s)$ | |||||

Short-term rate | −3.8 | −1.7 | −3.9 | ||||

Long-term rate | −4.1 | −1.3 | −4.2 | ||||

Earnings-by-price | −5.3 | −1.9 | −5.4 | ||||

Inflation | −3.9 | 0.3 | −1.9 | ||||

$(l,\pi )$ | $(l,s)$ | ||||||

Short-term rate | −1.7 | −3.9 | |||||

Long-term rate | −1.3 | −4.2 | |||||

Earnings-by-price | −2.6 | −5.4 | |||||

Inflation | −1.2 | −1.8 | |||||

$(\pi ,s)$ | |||||||

Short-term rate | −4.4 | ||||||

Long-term rate | −3.5 | ||||||

Earnings-by-price | −4.8 | ||||||

Inflation | −0.1 |

**Table 4.**Predictive power for the variance of one-year excess stock returns ${Y}_{t}^{\left(A\right)}$: the double benchmarking approach. The prediction problem is defined in (14). The same predictive variables ${X}_{t-1}^{\left(A\right)}$ are used in the predictions for the conditional mean and variance. The predictive power (%) is measured by ${R}_{V,\nu}^{2}$ as defined in (8). The benchmarks ${B}^{\left(A\right)}$ considered are based on the short-term interest rate ($A\equiv R$), long-term interest rate ($A\equiv L$), earnings-by-price ratio ($A\equiv E$), and consumer price index ($A\equiv C$). The predictive variables used are ${X}_{t-1}^{\left(A\right)}$ using the indicated benchmark ${B}_{t-1}^{\left(A\right)}$ as shown in (16). ${X}_{t-1}$ are given by the dividend-by-price ratio ${d}_{t-1}$, earnings-by-price ratio ${e}_{t-1}$, short-term interest rate ${r}_{t-1}$, long-term interest rate ${l}_{t-1}$, inflation ${\pi}_{t-1}$, term spread ${s}_{t-1}$, excess stock return ${Y}_{t-1}^{\left(A\right)}$, or the possible different pairwise combinations as indicated. “–” are not applicable cases of matched covariate with benchmark. Note: ${s}^{\left(R\right)}$ and ${l}^{\left(R\right)}$ (and their combinations with $Y,d,e,\pi $) have the same ${R}_{V}^{2}$ by construction of the transformed spread according to (16). For example, ${s}_{t-1}^{\left(R\right)}=({l}_{t-1}-{r}_{t-1})/{B}_{t-1}^{\left(R\right)}=(1+{l}_{t-1})/(1+{r}_{t-1})-1$ and ${l}_{t-1}^{\left(R\right)}=(1+{l}_{t-1})/(1+{r}_{t-1})$. The case of ${s}^{\left(L\right)}$ and ${r}^{\left(L\right)}$ is similar.

Benchmark ${\mathit{B}}^{\left(\mathit{A}\right)}$ | Explanatory Variable(s) ${\mathit{X}}_{\mathit{t}-1}$ | ||||||
---|---|---|---|---|---|---|---|

${Y}^{\left(A\right)}$ | ${d}^{\left(A\right)}$ | ${e}^{\left(A\right)}$ | ${r}^{\left(A\right)}$ | ${l}^{\left(A\right)}$ | ${\pi}^{\left(A\right)}$ | ${s}^{\left(A\right)}$ | |

Short-term rate | 2.2 | −0.3 | 0.7 | – | −0.2 | 0.1 | −0.2 |

Long-term rate | 2.4 | 0.2 | −0.5 | −0.1 | – | −0.2 | −0.1 |

Earnings-by-price | 1.5 | −0.2 | – | 0.6 | −0.2 | −0.7 | 0.0 |

Inflation | 0.2 | −0.9 | −1.2 | −0.3 | −0.2 | – | −0.7 |

$({Y}^{\left(A\right)},{d}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{e}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{r}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{l}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{s}^{\left(A\right)})$ | ||

Short-term rate | 0.8 | 0.7 | – | 0.2 | 0.1 | 0.2 | |

Long-term rate | 1.3 | 3.0 | 0.1 | – | −0.3 | 0.1 | |

Earnings-by-price | 0.2 | – | 0.7 | 2.5 | 0.0 | 0.1 | |

Inflation | −3.1 | −1.4 | −1.5 | −1.9 | – | −1.0 | |

$({d}^{\left(A\right)},{e}^{\left(A\right)})$ | $({d}^{\left(A\right)},{r}^{\left(A\right)})$ | $({d}^{\left(A\right)},{l}^{\left(A\right)})$ | $({d}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({d}^{\left(A\right)},{s}^{\left(A\right)})$ | |||

Short-term rate | −1.3 | – | 0.9 | 0.0 | 0.9 | ||

Long-term rate | −1.0 | 0.9 | – | −0.7 | 0.9 | ||

Earnings-by-price | – | −0.3 | −0.8 | −1.8 | 0.4 | ||

Inflation | −1.9 | 0.7 | 1.6 | – | −0.7 | ||

$({e}^{\left(A\right)},{r}^{\left(A\right)})$ | $({e}^{\left(A\right)},{l}^{\left(A\right)})$ | $({e}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({e}^{\left(A\right)},{s}^{\left(A\right)})$ | ||||

Short-term rate | – | −0.4 | −2.6 | −0.4 | |||

Long-term rate | −0.6 | – | −2.5 | −0.6 | |||

Earnings-by-price | – | – | – | – | |||

Inflation | −1.6 | −1.5 | – | −1.6 | |||

$({r}^{\left(A\right)},{l}^{\left(A\right)})$ | $({r}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({r}^{\left(A\right)},{s}^{\left(A\right)})$ | |||||

Short-term rate | – | – | – | ||||

Long-term rate | – | −1.2 | – | ||||

Earnings-by-price | −0.5 | −2.1 | −0.3 | ||||

Inflation | −1.9 | – | −1.6 | ||||

$({l}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({l}^{\left(A\right)},{s}^{\left(A\right)})$ | ||||||

Short-term rate | −1.4 | – | |||||

Long-term rate | – | – | |||||

Earnings-by-price | −2.5 | −0.5 | |||||

Inflation | – | −1.7 | |||||

$({\pi}^{\left(A\right)},{s}^{\left(A\right)})$ | |||||||

Short-term rate | −1.4 | ||||||

Long-term rate | −1.2 | ||||||

Earnings-by-price | −1.6 | ||||||

Inflation | – |

**Table 5.**Predictive power for the variance of five-year excess stock returns ${Z}_{t}^{\left(A\right)}$: the double benchmarking approach. The prediction problem is defined in (15). The same predictive variables ${X}_{t-1}^{\left(A\right)}$ are used in the predictions for the conditional mean and variance. Additional notes: see Table 4.

Benchmark ${\mathit{B}}^{\left(\mathit{A}\right)}$ | Explanatory Variable(s) ${\mathit{X}}_{\mathit{t}-1}$ | ||||||
---|---|---|---|---|---|---|---|

${Y}^{\left(A\right)}$ | ${d}^{\left(A\right)}$ | ${e}^{\left(A\right)}$ | ${r}^{\left(A\right)}$ | ${l}^{\left(A\right)}$ | ${\pi}^{\left(A\right)}$ | ${s}^{\left(A\right)}$ | |

Short-term rate | 0.6 | −2.2 | −3.2 | – | −3.1 | −3.2 | −3.1 |

Long-term rate | 0.0 | −3.4 | −2.8 | −2.8 | – | −1.3 | −2.8 |

Earnings-by-price | 0.8 | 1.8 | – | −2.3 | −3.2 | 0.6 | −3.8 |

Inflation | −1.0 | 1.6 | 0.3 | 0.6 | 1.6 | – | 0.3 |

$({Y}^{\left(A\right)},{d}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{e}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{r}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{l}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{s}^{\left(A\right)})$ | ||

Short-term rate | −2.1 | −4.3 | – | −4.0 | −1.2 | −4.0 | |

Long-term rate | −3.8 | −3.2 | −3.6 | – | −1.1 | −3.6 | |

Earnings-by-price | 1.1 | – | −2.8 | −3.8 | −0.5 | −3.4 | |

Inflation | 0.3 | −0.8 | −0.3 | 0.4 | – | −1.0 | |

$({d}^{\left(A\right)},{e}^{\left(A\right)})$ | $({d}^{\left(A\right)},{r}^{\left(A\right)})$ | $({d}^{\left(A\right)},{l}^{\left(A\right)})$ | $({d}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({d}^{\left(A\right)},{s}^{\left(A\right)})$ | |||

Short-term rate | −3.7 | – | −5.4 | −2.1 | −5.4 | ||

Long-term rate | −4.2 | −5.8 | – | −3.3 | −5.8 | ||

Earnings-by-price | – | −0.4 | −2.6 | 0.3 | −3.3 | ||

Inflation | −4.3 | −0.2 | −0.8 | – | −0.8 | ||

$({e}^{\left(A\right)},{r}^{\left(A\right)})$ | $({e}^{\left(A\right)},{l}^{\left(A\right)})$ | $({e}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({e}^{\left(A\right)},{s}^{\left(A\right)})$ | ||||

Short-term rate | – | −5.9 | −4.9 | −5.9 | |||

Long-term rate | −6.1 | – | −4.1 | −6.1 | |||

Earnings-by-price | – | – | – | – | |||

Inflation | −4.8 | −4.1 | – | −2.1 | |||

$({r}^{\left(A\right)},{l}^{\left(A\right)})$ | $({r}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({r}^{\left(A\right)},{s}^{\left(A\right)})$ | |||||

Short-term rate | – | – | – | ||||

Long-term rate | – | −2.3 | – | ||||

Earnings-by-price | −6.3 | −3.2 | −6.1 | ||||

Inflation | −1.0 | – | 0.5 | ||||

$({l}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({l}^{\left(A\right)},{s}^{\left(A\right)})$ | ||||||

Short-term rate | −3.4 | – | |||||

Long-term rate | – | – | |||||

Earnings-by-price | −3.6 | −6.2 | |||||

Inflation | – | 0.5 | |||||

$({\pi}^{\left(A\right)},{s}^{\left(A\right)})$ | |||||||

Short-term rate | −3.4 | ||||||

Long-term rate | −2.3 | ||||||

Earnings-by-price | −4.6 | ||||||

Inflation | – |

**Table 6.**Predictive power for the variance of one-year excess stock returns ${Y}_{t}^{\left(A\right)}$: the double benchmarking approach for the conditional mean model with earnings-by price as single covariate. The prediction problem is defined in (14). The predictive power (%) is measured by ${R}_{V,\nu}^{2}$ as defined in (8). The benchmarks ${B}^{\left(A\right)}$ considered are based on the short-term interest rate ($A\equiv R$), long-term interest rate ($A\equiv L$), and consumer price index ($A\equiv C$). The predictive variables used are ${X}_{t-1}^{\left(A\right)}$ using the indicated benchmark ${B}_{t-1}^{\left(A\right)}$ as shown in (16). ${X}_{t-1}$ are given by the dividend-by-price ratio ${d}_{t-1}$, earnings-by-price ratio ${e}_{t-1}$, short-term interest rate ${r}_{t-1}$, long-term interest rate ${l}_{t-1}$, inflation ${\pi}_{t-1}$, term spread ${s}_{t-1}$, excess stock return ${Y}_{t-1}^{\left(A\right)}$, or the possible different pairwise combinations as indicated. “–” are not applicable cases of matched covariate with benchmark. Note: ${s}^{\left(R\right)}$ and ${l}^{\left(R\right)}$ (and their combinations with $Y,d,e,\pi $) have the same ${R}_{V,\nu}^{2}$ by construction of the transformed spread according to (16). For example, ${s}_{t-1}^{\left(R\right)}=({l}_{t-1}-{r}_{t-1})/{B}_{t-1}^{\left(R\right)}=(1+{l}_{t-1})/(1+{r}_{t-1})-1$ and ${l}_{t-1}^{\left(R\right)}=(1+{l}_{t-1})/(1+{r}_{t-1})$. Similar is the case of ${s}^{\left(L\right)}$ and ${r}^{\left(L\right)}$.

Benchmark ${\mathit{B}}^{\left(\mathit{A}\right)}$ | Explanatory Variable(s) ${\mathit{X}}_{\mathit{t}-1}$ | ||||||
---|---|---|---|---|---|---|---|

${Y}^{\left(A\right)}$ | ${d}^{\left(A\right)}$ | ${e}^{\left(A\right)}$ | ${r}^{\left(A\right)}$ | ${l}^{\left(A\right)}$ | ${\pi}^{\left(A\right)}$ | ${s}^{\left(A\right)}$ | |

Short-term rate | 1.0 | 0.3 | 0.7 | – | 0.1 | −0.4 | 0.1 |

Long-term rate | 1.4 | 0.1 | −0.5 | 0.9 | – | −0.1 | 0.9 |

Inflation | 0.4 | −0.6 | −1.2 | −0.4 | −0.1 | – | 0.8 |

$({Y}^{\left(A\right)},{d}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{e}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{r}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{l}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{s}^{\left(A\right)})$ | ||

Short-term rate | 0.6 | 0.7 | – | 0.2 | −0.5 | 0.2 | |

Long-term rate | 0.7 | 2.0 | 0.7 | – | −0.6 | 0.7 | |

Inflation | −1.7 | −1.6 | −1.5 | −1.7 | – | −0.4 | |

$({d}^{\left(A\right)},{e}^{\left(A\right)})$ | $({d}^{\left(A\right)},{r}^{\left(A\right)})$ | $({d}^{\left(A\right)},{l}^{\left(A\right)})$ | $({d}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({d}^{\left(A\right)},{s}^{\left(A\right)})$ | |||

Short-term rate | 0.0 | – | −0.5 | −0.4 | −0.5 | ||

Long-term rate | −1.0 | 0.3 | – | −1.4 | 0.3 | ||

Inflation | −1.9 | 2.9 | 2.0 | – | 1.5 | ||

$({e}^{\left(A\right)},{r}^{\left(A\right)})$ | $({e}^{\left(A\right)},{l}^{\left(A\right)})$ | $({e}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({e}^{\left(A\right)},{s}^{\left(A\right)})$ | ||||

Short-term rate | – | 0.5 | −2.2 | 0.5 | |||

Long-term rate | −0.7 | – | −2.5 | −0.7 | |||

Inflation | −0.9 | −1.7 | – | −0.3 | |||

$({r}^{\left(A\right)},{l}^{\left(A\right)})$ | $({r}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({r}^{\left(A\right)},{s}^{\left(A\right)})$ | |||||

Short-term rate | – | – | – | ||||

Long-term rate | – | −0.4 | – | ||||

Inflation | −0.5 | – | 0.7 | ||||

$({l}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({l}^{\left(A\right)},{s}^{\left(A\right)})$ | ||||||

Short-term rate | 0.1 | – | |||||

Long-term rate | – | – | |||||

Inflation | – | −0.2 | |||||

$({\pi}^{\left(A\right)},{s}^{\left(A\right)})$ | |||||||

Short-term rate | 0.1 | ||||||

Long-term rate | −0.4 | ||||||

Inflation | – |

**Table 7.**Predictive power for the variance of five-year excess stock returns ${Z}_{t}^{\left(A\right)}$: the double benchmarking approach for the conditional mean model with earnings-by price as single covariate. The prediction problem is defined in (15). Additional notes: see Table 6.

Benchmark ${\mathit{B}}^{\left(\mathit{A}\right)}$ | Explanatory Variable(s) ${\mathit{X}}_{\mathit{t}-1}$ | ||||||
---|---|---|---|---|---|---|---|

${Y}^{\left(A\right)}$ | ${d}^{\left(A\right)}$ | ${e}^{\left(A\right)}$ | ${r}^{\left(A\right)}$ | ${l}^{\left(A\right)}$ | ${\pi}^{\left(A\right)}$ | ${s}^{\left(A\right)}$ | |

Short-term rate | 0.1 | −1.8 | −3.2 | – | −4.5 | −2.5 | −4.5 |

Long-term rate | 0.6 | −3.9 | −2.8 | −4.2 | – | −1.1 | −4.2 |

Inflation | 0.0 | −0.1 | 0.3 | −0.4 | −0.1 | – | −2.6 |

$({Y}^{\left(A\right)},{d}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{e}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{r}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{l}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({Y}^{\left(A\right)},{s}^{\left(A\right)})$ | ||

Short-term rate | −1.7 | −4.6 | – | −5.7 | −3.7 | −5.7 | |

Long-term rate | −4.5 | −4.5 | −4.2 | – | −2.5 | −4.2 | |

Inflation | −1.9 | −1.8 | −1.9 | −1.7 | – | −3.9 | |

$({d}^{\left(A\right)},{e}^{\left(A\right)})$ | $({d}^{\left(A\right)},{r}^{\left(A\right)})$ | $({d}^{\left(A\right)},{l}^{\left(A\right)})$ | $({d}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({d}^{\left(A\right)},{s}^{\left(A\right)})$ | |||

Short-term rate | −6.2 | – | −7.1 | −4.3 | −7.1 | ||

Long-term rate | −4.5 | −7.9 | – | −5.2 | −7.9 | ||

Inflation | −3.9 | −2.1 | −3.2 | – | −2.8 | ||

$({e}^{\left(A\right)},{r}^{\left(A\right)})$ | $({e}^{\left(A\right)},{l}^{\left(A\right)})$ | $({e}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({e}^{\left(A\right)},{s}^{\left(A\right)})$ | ||||

Short-term rate | – | −8.1 | −5.8 | −8.1 | |||

Long-term rate | −6.6 | – | −4.9 | −6.6 | |||

Inflation | −2.8 | −3.4 | – | −2.6 | |||

$({r}^{\left(A\right)},{l}^{\left(A\right)})$ | $({r}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({r}^{\left(A\right)},{s}^{\left(A\right)})$ | |||||

Short-term rate | – | – | – | ||||

Long-term rate | – | −5.7 | – | ||||

Inflation | −3.0 | – | −3.1 | ||||

$({l}^{\left(A\right)},{\pi}^{\left(A\right)})$ | $({l}^{\left(A\right)},{s}^{\left(A\right)})$ | ||||||

Short-term rate | −6.5 | – | |||||

Long-term rate | – | – | |||||

Inflation | – | −3.0 | |||||

$({\pi}^{\left(A\right)},{s}^{\left(A\right)})$ | |||||||

Short-term rate | −6.5 | ||||||

Long-term rate | −5.7 | ||||||

Inflation | – |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mammen, E.; Nielsen, J.P.; Scholz, M.; Sperlich, S. Conditional Variance Forecasts for Long-Term Stock Returns. *Risks* **2019**, *7*, 113.
https://doi.org/10.3390/risks7040113

**AMA Style**

Mammen E, Nielsen JP, Scholz M, Sperlich S. Conditional Variance Forecasts for Long-Term Stock Returns. *Risks*. 2019; 7(4):113.
https://doi.org/10.3390/risks7040113

**Chicago/Turabian Style**

Mammen, Enno, Jens Perch Nielsen, Michael Scholz, and Stefan Sperlich. 2019. "Conditional Variance Forecasts for Long-Term Stock Returns" *Risks* 7, no. 4: 113.
https://doi.org/10.3390/risks7040113