# Model-Free Time-Aggregated Predictions for Econometric Datasets

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## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. The Existing NoVaS Method

#### 2.2. A New Method with Less Parameters

**Remark (The advantage of removing the ${\tilde{a}}_{0}$ term):**First, after removing the ${\tilde{a}}_{0}$ term, the prediction of the NoVaS method under the ${L}_{2}$ criterion is more stable. More details will be shown in Section 2.3. Second, the suggestion of removing ${\tilde{a}}_{0}$ can also lead to less time complexity of our new method. The reason for this phenomenon is simple. If we consider the limiting distribution of $\left\{{W}_{t}\right\}$ series, $1/\sqrt{{\tilde{a}}_{0}}$ is required to be larger than or equal to 3 to ensure that $\left\{{W}_{t}\right\}$ has a sufficiently large range, i.e., ${\tilde{a}}_{0}$ is required to be less than or equal to 0.111 (recall that the mass of standard normal data is within $[-3,3]$). However, the optimal combination of NoVaS coefficients may not render a suitable ${\tilde{a}}_{0}$. For this situation, we need to increase the NoVaS transformation order p and repeat the normalizing and variance-stabilizing process till ${\tilde{a}}_{0}$ in the optimal combination of coefficients is suitable. This repeating process definitely increases the computation workload.

Algorithm 1: The h-step ahead prediction for the GE-NoVaS-without-${\tilde{a}}_{0}$ method. |

Step 1 Define a grid of possible $\alpha $ values, $\{{\alpha}_{k};\phantom{\rule{3.33333pt}{0ex}}k=1,\cdots ,K\}$. Fix $\alpha ={\alpha}_{k}$, then calculate the optimal combination of ${\alpha}_{k},{a}_{1},\cdots ,{a}_{p}$ of the GE-NoVaS-without-${\tilde{a}}_{0}$ method, which minimizes $\left|Kurtosis\right({W}_{t})-3|$. |

Step 2 Derive the analytic form of Equation (11) using ${\alpha}_{k},{a}_{1},\cdots ,{a}_{p}$ from the first step. |

Step 3 Generate $\{{W}_{n+1}^{*},\cdots ,{W}_{n+h}^{*}\}$ M times from a standard normal distribution or the empirical distribution ${\widehat{F}}_{w}$. Plug $\{{W}_{n+1}^{*},\cdots ,{W}_{n+h}^{*}\}$ into the analytic form of Equation (11) to obtain M pseudo-values $\{{\widehat{Y}}_{n+h,1}^{*},\cdots ,{\widehat{Y}}_{n+h,M}^{*}\}$. |

Step 4 Calculate the optimal predictor of $g\left({Y}_{n+h}\right)$ by taking the sample mean (under ${L}_{2}$ risk criterion) or sample median (under ${L}_{1}$ risk criterion) of the set $\{g\left({\widehat{Y}}_{n+h,1}^{*}\right),\cdots ,g\left({\widehat{Y}}_{n+h,M}^{*}\right)\}$. |

Step 5 Repeat above steps with different $\alpha $ values from $\{{\alpha}_{k};\phantom{\rule{3.33333pt}{0ex}}k=1,\cdots ,K\}$ to get K prediction results. |

#### 2.3. The Potential Instability of the GE-NoVaS Method

## 3. Data Analysis and Results

#### 3.1. Simulation Study

**Model 1:**Time-varying GARCH(1,1) with Gaussian errors${X}_{t}={\sigma}_{t}{\u03f5}_{t},\phantom{\rule{3.33333pt}{0ex}}{\sigma}_{t}^{2}=0.00001+{\beta}_{1,t}{\sigma}_{t-1}^{2}+{\alpha}_{1,t}{X}_{t-1}^{2},\phantom{\rule{3.33333pt}{0ex}}\left\{{\u03f5}_{t}\right\}\sim i.i.d.\phantom{\rule{3.33333pt}{0ex}}N(0,1)$${\alpha}_{1,t}=0.1-0.05t/n$; ${\beta}_{1,t}=0.73+0.2t/n,\phantom{\rule{3.33333pt}{0ex}}n=250$**Model 2:**Standard GARCH(1,1) with Gaussian errors${X}_{t}={\sigma}_{t}{\u03f5}_{t},\phantom{\rule{3.33333pt}{0ex}}{\sigma}_{t}^{2}=0.00001+0.73{\sigma}_{t-1}^{2}+0.1{X}_{t-1}^{2},\phantom{\rule{3.33333pt}{0ex}}\left\{{\u03f5}_{t}\right\}\sim i.i.d.\phantom{\rule{3.33333pt}{0ex}}N(0,1)$**Model 3:**(Another) Standard GARCH(1,1) with Gaussian errors${X}_{t}={\sigma}_{t}{\u03f5}_{t},\phantom{\rule{3.33333pt}{0ex}}{\sigma}_{t}^{2}=0.00001+0.8895{\sigma}_{t-1}^{2}+0.1{X}_{t-1}^{2},\phantom{\rule{3.33333pt}{0ex}}\left\{{\u03f5}_{t}\right\}\sim i.i.d.\phantom{\rule{3.33333pt}{0ex}}N(0,1)$**Model 4:**Standard GARCH(1,1) with Student-t errors${X}_{t}={\sigma}_{t}{\u03f5}_{t},$$\phantom{\rule{3.33333pt}{0ex}}{\sigma}_{t}^{2}=0.00001+0.73{\sigma}_{t-1}^{2}+0.1{X}_{t-1}^{2},$$\phantom{\rule{3.33333pt}{0ex}}\left\{{\u03f5}_{t}\right\}\sim i.i.d.\phantom{\rule{3.33333pt}{0ex}}t$$\mathrm{distribution}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}\mathrm{five}\phantom{\rule{4.pt}{0ex}}\mathrm{degrees}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{freedom}$

**Result analysis:**From the first block of Table 1, we can read that both NoVaS methods are superior to the GARCH(1,1) model. Although these simulated datasets are generated from GARCH(1,1)-type models, the GE-NoVaS-without-${\tilde{a}}_{0}$ method can bring around 66% and 48% improvements compared to the GARCH(1,1) model for 5-step-ahead time-aggregated predictions of Model-4 and Model-1 data, respectively. Notably, GARCH(1,1) brings poor results for the 30-step-ahead time-aggregated predictions of Model-4 simulated data, which implies that such a classical method is impaired by error accumulation problems when long-term predictions are required. On the other hand, the model-free NoVaS method can avoid this issue. Taking a closer look at these results, we can observe that almost all optimal results come from applying the GE-NoVaS-without-${\tilde{a}}_{0}$ method. Moreover, the GE-NoVaS method is surpassed by GARCH(1,1) when forecasting 30-step-ahead time-aggregated Model-2 data. On the other hand, the GE-NoVaS-without-${\tilde{a}}_{0}$ method provides consistently stable results. These results imply that the GE-NoVaS-without-${\tilde{a}}_{0}$ method dominates the GE-NoVaS method when predicting long-term or short-term time-aggregated predictions. Besides, using the same generated models from the previous study of the NoVaS method [4] ensures fairness. Additionally, with simulation implementations, the ability against model misspecification of NoVaS methods is verified in Appendix A.

#### 3.2. A Few Real Datasets

- 2-year period data: 2018∼2019 stock price data.
- 1-year period data: 2019 stock price and index data.
- 1-year period volatile data due to pandemic: 11.2019∼10.2020 stock price, currency and index data.

**Remark (One ARCH-type model for non-stationary data):**Since our stationarity tests suggest that some series may not be stationary, we can consider applying ARCH-without-intercept, which is a variant of the ARCH model. This variant is non-stationary but stable in the sense that the observed process has non-degenerated distribution. Moreover, it appears to be an alternative to common stationary but highly persistent GARCH models [18]. Inspired by this ARCH-type model, the NoVaS method may be further improved by removing the corresponding intercept term $\alpha {s}_{t-1}^{2}$ in Equations (1) and (6). More empirical experiments could be conducted along this direction.**Result analysis:**From the last three blocks of Table 1, there is no optimal result that comes from the GARCH(1,1) method. When the target data are short and volatile, GARCH(1,1) gives poor results for 30-step-ahead time-aggregated predictions, such as the volatile Djones, CADJPY and IBM cases. Among the two NoVaS methods, the GE-NoVaS-without-${\tilde{a}}_{0}$ method outperforms the GE-NoVaS method for the three types of real-world data. More specifically, around 70% and 30% improvements are created by our new method compared to the existing GE-NoVaS method when forecasting 30-step-ahead time-aggregated volatile Djones and CADJPY data, respectively. We should also notice that the GE-NoVaS method is again surpassed by the GARCH(1,1) model on 30-step-ahead aggregated predictions of 2018∼2019 BAC data. On the other hand, the GE-NoVaS-without-${\tilde{a}}_{0}$ method performs stably. These comprehensive prediction comparisons cover the shortage of empirical analyses of NoVaS methods, and imply that NoVaS-type methods are indeed valid and efficient for real-world short- or long-term predictions of three main types of econometric data. See Appendix A for more results.

#### 3.3. Statistical Significance

## 4. Summary

- Existing GE-NoVaS and new GE-NoVaS-without-${\tilde{a}}_{0}$ methods provide substantial improvements for time-aggregated prediction, which hints towards the stability of NoVaS-type methods for providing long-horizon inferences.
- Our new method has superior performance to the GE-NoVaS method, especially for shorter sample sizes or more volatile data. This is significant given that GARCH-type models are difficult to estimate in shorter samples.
- We provide a statistical hypothesis test that shows that our model provides a more parsimonious fit, especially for long-term time-aggregated predictions.

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Additional Simulation Study and Data Analysis Results

#### Appendix A.1. Additional Simulation Study: Model Misspecification

**Model 5:**Another time-varying GARCH(1,1) with Gaussian errors${X}_{t}={\sigma}_{t}{\u03f5}_{t},\phantom{\rule{3.33333pt}{0ex}}{\sigma}_{t}^{2}={\omega}_{0,t}+{\beta}_{1,t}{\sigma}_{t-1}^{2}+{\alpha}_{1,t}{X}_{t-1}^{2},\phantom{\rule{3.33333pt}{0ex}}\left\{{\u03f5}_{t}\right\}\sim i.i.d.\phantom{\rule{3.33333pt}{0ex}}N(0,1)$${g}_{t}=t/n;{\omega}_{0,t}=-4sin\left(0.5\pi {g}_{t}\right)+5;{\alpha}_{1,t}=-1{({g}_{t}-0.3)}^{2}+0.5;{\beta}_{1,t}=0.2sin\left(0.5\pi {g}_{t}\right)+0.2,\phantom{\rule{3.33333pt}{0ex}}n=250$**Model 6:**Exponential GARCH(1,1) with Gaussian errors${X}_{t}={\sigma}_{t}{\u03f5}_{t},\phantom{\rule{3.33333pt}{0ex}}log\left({\sigma}_{t}^{2}\right)=0.00001+0.8895log\left({\sigma}_{t-1}^{2}\right)+0.1{\u03f5}_{t-1}+0.3\left(\right|{\u03f5}_{t-1}|-E|{\u03f5}_{t-1}\left|\right),$$\phantom{\rule{3.33333pt}{0ex}}\left\{{\u03f5}_{t}\right\}\sim i.i.d.\phantom{\rule{3.33333pt}{0ex}}N(0,1)$**Model 7:**GJR-GARCH(1,1) with Gaussian errors${X}_{t}={\sigma}_{t}{\u03f5}_{t},\phantom{\rule{3.33333pt}{0ex}}{\sigma}_{t}^{2}=0.00001+0.5{\sigma}_{t-1}^{2}+0.5{X}_{t-1}^{2}-0.5{I}_{t-1}{X}_{t-1}^{2},\phantom{\rule{3.33333pt}{0ex}}\left\{{\u03f5}_{t}\right\}\sim i.i.d.\phantom{\rule{3.33333pt}{0ex}}N(0,1){I}_{t}=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{if}\phantom{\rule{3.33333pt}{0ex}}{X}_{t}\le 0;{I}_{t}=0\phantom{\rule{3.33333pt}{0ex}}\mathrm{otherwise}$**Model 8:**Another GJR-GARCH(1,1) with Gaussian errors${X}_{t}={\sigma}_{t}{\u03f5}_{t},\phantom{\rule{3.33333pt}{0ex}}{\sigma}_{t}^{2}=0.00001+0.73{\sigma}_{t-1}^{2}+0.1{X}_{t-1}^{2}+0.3{I}_{t-1}{X}_{t-1}^{2},\phantom{\rule{3.33333pt}{0ex}}\left\{{\u03f5}_{t}\right\}\sim i.i.d.\phantom{\rule{3.33333pt}{0ex}}N(0,1){I}_{t}=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{if}\phantom{\rule{3.33333pt}{0ex}}{X}_{t}\le 0;{I}_{t}=0\phantom{\rule{3.33333pt}{0ex}}\mathrm{otherwise}$

GE-NoVaS | GE-NoVaS-without-${\tilde{\mathit{a}}}_{0}$ | GARCH(1,1) | |
---|---|---|---|

M5-1step | 0.91538 | 0.83168 | 1.00000 |

M5-5steps | 0.49169 | 0.43772 | 1.00000 |

M5-30steps | 0.25009 | 0.22659 | 1.00000 |

M6-1step | 0.95939 | 0.94661 | 1.00000 |

M6-5steps | 0.93594 | 0.84719 | 1.00000 |

M6-30steps | 0.84401 | 0.70301 | 1.00000 |

M7-1step | 0.84813 | 0.73553 | 1.00000 |

M7-5steps | 0.50849 | 0.46618 | 1.00000 |

M7-30steps | 0.06832 | 0.06479 | 1.00000 |

M8-1step | 0.79561 | 0.76586 | 1.00000 |

M8-5steps | 0.48028 | 0.38107 | 1.00000 |

M8-30steps | 0.00977 | 0.00918 | 1.00000 |

#### Appendix A.2. Additional Data Analysis: 1-Year Datasets

GE-NoVaS | GE-NoVaS-without-${\tilde{\mathit{a}}}_{0}$ | GARCH(1,1) | |
---|---|---|---|

2019-MCD-1step | 0.95959 | 0.93141 | 1.00000 |

2019-MCD-5steps | 1.00723 | 0.90061 | 1.00000 |

2019-MCD-30steps | 1.05239 | 0.80805 | 1.00000 |

2019-BAC-1step | 1.04272 | 0.97757 | 1.00000 |

2019-BAC-5steps | 1.22761 | 0.89571 | 1.00000 |

2019-BAC-30steps | 1.45020 | 1.01175 | 1.00000 |

2019-MSFT-1step | 1.03308 | 0.98469 | 1.00000 |

2019-MSFT-5steps | 1.22340 | 1.02387 | 1.00000 |

2019-MSFT-30steps | 1.23020 | 0.97585 | 1.00000 |

2019-TSLA-1step | 1.00428 | 0.98646 | 1.00000 |

2019-TSLA-5steps | 1.06610 | 0.97523 | 1.00000 |

2019-TSLA-30steps | 2.00623 | 0.87158 | 1.00000 |

2019-Bitcoin-1step | 0.89929 | 0.86795 | 1.00000 |

2019-Bitcoin-5steps | 0.62312 | 0.55620 | 1.00000 |

2019-Bitcoin-30steps | 0.00733 | 0.00624 | 1.00000 |

2019-Nasdaq-1step | 0.99960 | 0.93558 | 1.00000 |

2019-Nasdaq-5steps | 1.15282 | 0.84459 | 1.00000 |

2019-Nasdaq-30steps | 0.68994 | 0.58924 | 1.00000 |

2019-NYSE-1step | 0.92486 | 0.90407 | 1.00000 |

2019-NYSE-5steps | 0.86249 | 0.69822 | 1.00000 |

2019-NYSE-30steps | 0.22122 | 0.18173 | 1.00000 |

2019-Smallcap-1step | 1.02041 | 0.98731 | 1.00000 |

2019-Smallcap-5steps | 1.15868 | 0.87700 | 1.00000 |

2019-Samllcap-30steps | 1.30467 | 0.88825 | 1.00000 |

2019-BSE-1step | 0.70667 | 0.67694 | 1.00000 |

2019-BSE-5steps | 0.25675 | 0.23665 | 1.00000 |

2019-BSE-30steps | 0.03764 | 0.02890 | 1.00000 |

2019-BIST-1step | 0.96807 | 0.95467 | 1.00000 |

2019-BIST-5steps | 0.98944 | 0.82898 | 1.00000 |

2019-BIST-30steps | 2.21996 | 0.88511 | 1.00000 |

#### Appendix A.3. Additional Data Analysis: Volatile 1-Year Datasets

GE-NoVaS | GE-NoVaS-without-${\tilde{\mathit{a}}}_{0}$ | GARCH(1,1) | |
---|---|---|---|

11.2019∼10.2020-MCD-1step | 0.51755 | 0.58018 | 1.00000 |

11.2019∼10.2020-MCD-5steps | 0.10725 | 0.17887 | 1.00000 |

11.2019∼10.2020-MCD-30steps | 3.32 × ${10}^{-5}$ | 7.48 × ${10}^{-6}$ | 1.00000 |

11.2019∼10.2020-AMZN-1step | 0.97099 | 0.90200 | 1.00000 |

11.2019∼10.2020-AMZN-5steps | 0.88705 | 0.71789 | 1.00000 |

11.2019∼10.2020-AMZN-30steps | 0.58124 | 0.53460 | 1.00000 |

11.2019∼10.2020-SBUX-1step | 0.68206 | 0.69943 | 1.00000 |

11.2019∼10.2020-SBUX-5steps | 0.24255 | 0.30528 | 1.00000 |

11.2019∼10.2020-SBUX-30steps | 0.00499 | 0.00289 | 1.00000 |

11.2019∼10.2020-MSFT-1step | 0.80133 | 0.84502 | 1.00000 |

11.2019∼10.2020-MSFT-5steps | 0.35567 | 0.37528 | 1.00000 |

11.2019∼10.2020-MSFT-30steps | 0.01342 | 0.00732 | 1.00000 |

11.2019∼10.2020-EURJPY-1step | 0.95093 | 0.94206 | 1.00000 |

11.2019∼10.2020-EURJPY-5steps | 0.76182 | 0.76727 | 1.00000 |

11.2019∼10.2020-EURJPY-30steps | 0.16202 | 0.15350 | 1.00000 |

11.2019∼10.2020-CNYJPY-1step | 0.77812 | 0.79877 | 1.00000 |

11.2019∼10.2020-CNYJPY-5steps | 0.38875 | 0.40569 | 1.00000 |

11.2019∼10.2020-CNYJPY-30steps | 0.08398 | 0.06270 | 1.00000 |

11.2019∼10.2020-Smallcap-1step | 0.58170 | 0.60931 | 1.00000 |

11.2019∼10.2020-Smallcap-5steps | 0.10270 | 0.10337 | 1.00000 |

11.2019∼10.2020-Smallcap-30steps | 7.00 × ${10}^{-5}$ | 5.96 × ${10}^{-5}$ | 1.00000 |

11.2019∼10.2020-BSE-1step | 0.39493 | 0.39745 | 1.00000 |

11.2019∼10.2020-BSE-5steps | 0.03320 | 0.04109 | 1.00000 |

11.2019∼10.2020-BSE-30steps | 2.45 × ${10}^{-5}$ | 1.82 × ${10}^{-5}$ | 1.00000 |

11.2019∼10.2020-NYSE-1step | 0.55741 | 0.57174 | 1.00000 |

11.2019∼10.2020-NYSE-5steps | 0.08994 | 0.10182 | 1.00000 |

11.2019∼10.2020-NYSE-30steps | 1.36 × ${10}^{-5}$ | 6.64 × ${10}^{-6}$ | 1.00000 |

11.2019∼10.2020-USDXfuture-1step | 1.14621 | 0.99640 | 1.00000 |

11.2019∼10.2020-USDXfuture-5steps | 0.61075 | 0.54834 | 1.00000 |

11.2019∼10.2020-USDXfuture-30steps | 0.10723 | 0.10278 | 1.00000 |

11.2019∼10.2020-Nasdaq-1step | 0.71380 | 0.75350 | 1.00000 |

11.2019∼10.2020-Nasdaq-5steps | 0.29332 | 0.33519 | 1.00000 |

11.2019∼10.2020-Nasdaq-30steps | 0.01223 | 0.00599 | 1.00000 |

11.2019∼10.2020-Bovespa-1step | 0.60031 | 0.57558 | 1.00000 |

11.2019∼10.2020-Bovespa-5steps | 0.08603 | 0.07447 | 1.00000 |

11.2019∼10.2020-Bovespa-30steps | 6.87 × ${10}^{-6}$ | 2.04 × ${10}^{-6}$ | 1.00000 |

## Appendix B. Stationarity Test Results of Some Real-World Datasets

ADF | KPSS | PP | |
---|---|---|---|

2018∼2019 MCD | 0.01 | 0.10 | 0.01 |

2018∼2019 BAC | 0.01 | 0.10 | 0.01 |

2019 AAPL | 0.01 | 0.10 | 0.01 |

2019 Djones | 0.10 | 0.10 | 0.01 |

2019 SP500 | 0.18 | 0.10 | 0.01 |

11.2019∼10.2020 IBM | 0.31 | 0.05 | 0.01 |

11.2019∼10.2020 CADJPY | 0.01 | 0.10 | 0.01 |

11.2019∼10.2020 SP500 | 0.23 | 0.08 | 0.01 |

11.2019∼10.2020 Djones | 0.22 | 0.08 | 0.01 |

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**Figure 1.**Curves of the true and predicted time-aggregated squared log-returns from GE-NoVaS and GE-NoVaS-without-${\tilde{a}}_{0}$ methods.

GE-NoVaS | GE-NoVaS-without-${\tilde{\mathit{a}}}_{0}$ | GARCH(1,1) | p-Value(CW Test) | ||
---|---|---|---|---|---|

Simulated-1-year-data | Model-1-1step | 0.91369 | 0.88781 | 1.00000 | |

Model-1-5steps | 0.61001 | 0.52872 | 1.00000 | ||

Model-1-30steps | 0.77250 | 0.73604 | 1.00000 | ||

Model-2-1step | 0.97796 | 0.94635 | 1.00000 | ||

Model-2-5steps | 0.98127 | 0.96361 | 1.00000 | ||

Model-2-30steps | 1.38353 | 0.98872 | 1.00000 | ||

Model-3-1step | 0.99183 | 0.92829 | 1.00000 | ||

Model-3-5steps | 0.77088 | 0.67482 | 1.00000 | ||

Model-3-30steps | 0.79672 | 0.71003 | 1.00000 | ||

Model-4-1step | 0.83631 | 0.78087 | 1.00000 | ||

Model-4-5steps | 0.38296 | 0.34396 | 1.00000 | ||

Model-4-30steps | 0.00199 | 0.00201 | 1.00000 | ||

2-years-data | 2018∼2019-MCD-1step | 0.99631 | 0.99614 | 1.00000 | 0.00053 |

2018∼2019-MCD-5steps | 0.95403 | 0.92120 | 1.00000 | 0.03386 | |

2018∼2019-MCD-30steps | 0.75730 | 0.62618 | 1.00000 | 0.19691 | |

2018∼2019-BAC-1step | 0.98393 | 0.97966 | 1.00000 | 0.09568 | |

2018∼2019-BAC-5steps | 0.98885 | 0.95124 | 1.00000 | 0.07437 | |

2018∼2019-BAC-30steps | 1.14111 | 0.87414 | 1.00000 | 0.03643 | |

1-year-data | 2019-AAPL-1step | 0.84533 | 0.80948 | 1.00000 | 0.25096 |

2019-AAPL-5steps | 0.85401 | 0.68191 | 1.00000 | 0.06387 | |

2019-AAPL-30steps | 0.99043 | 0.73823 | 1.00000 | 0.17726 | |

2019-Djones-1step | 0.96752 | 0.96365 | 1.00000 | 0.34514 | |

2019-Djones-5steps | 0.98725 | 0.89542 | 1.00000 | 0.24529 | |

2019-Djones-30steps | 0.86333 | 0.80304 | 1.00000 | 0.23766 | |

2019-SP500-1step | 0.96978 | 0.92183 | 1.00000 | 0.45693 | |

2019-SP500-5steps | 0.96704 | 0.75579 | 1.00000 | 0.24402 | |

2019-SP500-30steps | 0.34389 | 0.29796 | 1.00000 | 0.08148 | |

Volatile-1-year-data | 11.2019∼10.2020-IBM-1step | 0.80222 | 0.80744 | 1.00000 | 0.16568 |

11.2019∼10.2020-IBM-5steps | 0.38933 | 0.40743 | 1.00000 | 0.03664 | |

11.2019∼10.2020-IBM-30steps | 0.01143 | 0.00918 | 1.00000 | 0.15364 | |

11.2019∼10.2020-CADJPY-1step | 0.46940 | 0.48712 | 1.00000 | 0.16230 | |

11.2019∼10.2020-CADJPY-5steps | 0.11678 | 0.13549 | 1.00000 | 0.06828 | |

11.2019∼10.2020-CADJPY-30steps | 0.00584 | 0.00394 | 1.00000 | 0.15174 | |

11.2019∼10.2020-SP500-1step | 0.97294 | 0.92349 | 1.00000 | 0.05536 | |

11.2019∼10.2020-SP500-5steps | 0.96590 | 0.75183 | 1.00000 | 0.17380 | |

11.2019∼10.2020-SP500-30steps | 0.34357 | 0.29793 | 1.00000 | 0.16022 | |

11.2019∼10.2020-Djones-1step | 0.56357 | 0.57550 | 1.00000 | 0.11099 | |

11.2019∼10.2020-Djones-5steps | 0.09810 | 0.11554 | 1.00000 | 0.45057 | |

11.2019∼10.2020-Djones-30steps | 4.32 × ${10}^{-5}$ | 1.24 × ${\mathbf{10}}^{-\mathbf{5}}$ | 1.00000 | 0.68487 |

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**MDPI and ACS Style**

Wu, K.; Karmakar, S.
Model-Free Time-Aggregated Predictions for Econometric Datasets. *Forecasting* **2021**, *3*, 920-933.
https://doi.org/10.3390/forecast3040055

**AMA Style**

Wu K, Karmakar S.
Model-Free Time-Aggregated Predictions for Econometric Datasets. *Forecasting*. 2021; 3(4):920-933.
https://doi.org/10.3390/forecast3040055

**Chicago/Turabian Style**

Wu, Kejin, and Sayar Karmakar.
2021. "Model-Free Time-Aggregated Predictions for Econometric Datasets" *Forecasting* 3, no. 4: 920-933.
https://doi.org/10.3390/forecast3040055