RAKOTOMAROLAHY

Encompassing tests compare two non-nested competing models to determine whether one model contains all the relevant information captured by the other. They were developed based on the assumption of independence, particularly in cases involving parametric and non-parametric regression methods. However, this assumption is too restrictive since econometric dynamic models and time series rarely exhibit perfect independence. One challenge would be extending results for i.i.d. variables to dependent processes. We are interested in the φ-mixing dependence measure because of its properties, such as the fast decay rate. Consequently, the Central Limit Theorem (CLT) was previously obtained under significantly weaker mild conditions than those required for other mixing notions. This leads to better asymptotic behavior propreties for test statistics, despite practical constraints of the φ-mixing condition compared to weaker dependence measures. We examine the encompassing test for linear parametric and nonparametric nearest neighbor regression methods for φ-mixing processes. We establish the asymptotic normality of the encompassing statistics associated with the encompassing hypotheses. Our results are comparable to those obtained using popular methods in the literature, such as kernel regression. We achieve convergence rates of order k−1/2 for the tests related to the nonparametric as rival model, and n−1/2 for the parametric as rival model. These rates are the same as in the i.i.d. case. The asymptotic variances of the tests are well-defined due to the fast decay of the φ-mixing coefficients. Unlike many tests with nonparametric regression methods, ours do not depend on any density.

We examine the statistical properties of nearest neighbor regression function estimation beyond the classical i.i.d assumption. Second-order properties of this estimator for uniformly mixing processes have been derived in previous studies. Nevertheless, uniform mixing is a very strong form of dependence that is difficult to achieve. In contrast, strong mixing conditions are satisfied by a broad class of stochastic processes commonly encountered in theoretical and applied time series modeling. This paper focuses on the analysis of the second-order properties of the nearest neighbor regression estimator under strong mixing dependence measure. Under appropriate regularity conditions, including strong mixing assumptions and smoothness of the underlying density functions, we derive expressions for both the bias and the quadratic mean squared error (QMSE) of the nearest neighbor regression estimator with a uniform weighting scheme for estimating the unknown conditional mean function. Our results demonstrate that the QMSE of the nearest neighbor estimator attains the minimax-optimal rate for estimating a p-smooth regression function in a d-dimensional embedding space. The theoretical analysis integrates the dependence structure specific to strongly mixing processes with the geometric characteristics of k-nearest neighborhoods. This combination enables the identification of an optimal choice for the number of nearest neighbors that effectively balances the trade-off between bias and variance. Overall, these findings provide a rigorous theoretical foundation for the application of nearest neighbor regression methods to short-range dependent data. Furthermore, the explicit bias and variance characterizations lay the groundwork for establishing asymptotic normality, thereby enabling the construction of valid confidence intervals and supporting reliable statistical inference in dependent data settings.

The purpose of this study is to conduct an empirical comparative study of volatility models for three of the most popular cryptocurrencies. We study the volatility of the following cryptocurrencies: Bitcoin, Ethereum, and Litecoin. We consider the GARCH-type, boosting-family-tree-based ensemble learning, and ANFIS volatility models for these financial crypto-assets, which some have claimed capture stylized facts about cryptocurrency volatility well. We conduct comparative studies on in-sample and out-of-sample empirical analyses. The results show that tree-based ensemble learning delivers better forecast accuracy. Nevertheless, the performance of some GARCH-type volatility models is relatively close to that of the best model on both training and evaluation samples.

Ensemble learning techniques continue to show greater interest in forecasting the volatility of cryptocurrency assets. In particular, XGBoost, an ensemble learning technique, has been shown in recent studies to provide the most accurate forecast of Bitcoin volatility. However, the performance of XGBoost largely depends on the tuning of its hyperparameters. In this study, we examine the effectiveness of the Bayesian optimization method for tuning the XGBoost hyperparameters for Bitcoin volatility forecasting. We chose to explore this method rather than the most commonly used manual, grid, and random hyperparameter choices due to its ability to predict the most promising areas of hyperparameter spaces through exploitation and exploration using acquisition functions, as well as its ability to minimize error with a reduced amount of time and resources required to find an optimal configuration. The obtained XGBoost configuration improves the forecast accuracy of Bitcoin volatility. Our empirical results, based on letting the data speak for itself, could be used for a comparative study on Bitcoin volatility forecasting. This would also be important for volatility trading, option pricing, and managing portfolios related to Bitcoin.

This paper proposes prediction of the bitcoin return direction with logistic, discriminant analysis and machine learning classification techniques. It extends the prediction of the bitcoin return direction using exogenous macroeconomic and financial variables which have been investigated as drivers of bitcoin return. We also use google trends as proxy for investors interest on bitcoin. We consider those variables as predictors for bitcoin return direction. We conduct an in-sample and out-of-sample empirical analysis and achieve a misclassification error around 4% for in-sample evaluation and around 41% in out-of-sample empirical analysis. Ensemble learning trees based outperforms the other methods in both in-sample and out-of-sample analyses.

Encompassing test has been well developed for fully parametric modeling. In this study, we are interested on encompassing test for parametric and nonparametric regression methods. We consider linear regression for parametric modeling and nearest neighbor regression for nonparametric methods. We establish asymptotic normality of encompassing statistic associated to the encompassing hypotheses for the linear parametric method and the nonparametric nearest neighbor regression estimate. We also obtain convergence rate depending only on the number of neighbors 𝑘 while it depends on the number of observation 𝑛 and the bandwidth ℎ𝑛 for kernel method. We achieve the same convergence rate when ℎ𝑛=𝑘/𝑛. Moreover, asymptotic variance of the encompassing statistic associated to kernel regression depends on the density, this is not the case for nearest neighbor regression estimate.

Quantitative analysis of the impact of financial variables on real economic activity would be the purpose of this paper. Focusing on interest rate spread, stock market and exchange rate as financial variables, the strength of our study compared to the existing literature will be the consideration of the dynamic of these financial variables and the use of forecast combination. This approach would lead better understanding on the relationship between financial and economic spheres. Interest on using financial variables for real GDP predictor will be tested through forecast encompassing principles.

Inuence of some variables on economic activity need particular attention according to their impact on economic sphere in history. Especially, inuence of oil and housing variables on forecasting real Gross Domestic Product (GDP). Linear features on relationship have been exhibited in the literature. The aim of this paper is to address possible nonlinear relationship between these two economic variables and the real GDP.