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Uniform convergence of kernel averages under fixed design with heterogeneous dependent data

Danilo Hiroshi Matsuoka, Hudson da Silva Torrent

Abstract

We provide uniform convergence rates for kernel averages on $[0,1]$ under equally-spaced fixed design points of the form $x_{t,T}=t/T,\ t\in\{1,\dotsc, T\},\ T\in\mathbb{N}$. The rates of weak and strong uniform consistency are derived under strong mixing and moment conditions and do not require stationarity. The analysis exploits the grid structure and thus complements existing random-design results such as those of Hansen (2008) and Kristensen (2009), which rely on density-based conditioning arguments. The framework accommodates dependent triangular arrays and is particularly relevant for nonparametric methods applied to time series observed on deterministic grids. As an application, we derive uniform convergence rates for the local linear estimator in a nonparametric regression model with time-varying autoregressive errors. The theoretical results are illustrated through Monte Carlo experiments and an empirical application.

Uniform convergence of kernel averages under fixed design with heterogeneous dependent data

Abstract

We provide uniform convergence rates for kernel averages on under equally-spaced fixed design points of the form . The rates of weak and strong uniform consistency are derived under strong mixing and moment conditions and do not require stationarity. The analysis exploits the grid structure and thus complements existing random-design results such as those of Hansen (2008) and Kristensen (2009), which rely on density-based conditioning arguments. The framework accommodates dependent triangular arrays and is particularly relevant for nonparametric methods applied to time series observed on deterministic grids. As an application, we derive uniform convergence rates for the local linear estimator in a nonparametric regression model with time-varying autoregressive errors. The theoretical results are illustrated through Monte Carlo experiments and an empirical application.
Paper Structure (8 sections, 14 theorems, 137 equations, 5 figures, 3 tables)

This paper contains 8 sections, 14 theorems, 137 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. General results for kernel averages
  3. Uniform convergence in probability
  4. Almost sure uniform convergence
  5. Application to a nonparametric regression model
  6. Monte Carlo simulations
  7. Empirical application to sea level anomalies
  8. Conclusion

Key Result

Theorem 1

Assume that A.1$-$A.3 hold. Fix $c>0$ and suppose that $s>2$. Define Let $\Theta_T=\{\gamma\in\mathbb{R}^m: \|\gamma\|\leq d_T\}$ with $d_T=T^r$ and $r=\min\{c,(1-\theta)/(2\lambda)\}$. Suppose that and that the bandwidth satisfies If, in addition, the following bounds hold then

Figures (5)

  • Figure 1: Boxplots of the values $\mathrm{ASE}_r(\hat{g})$ and $\mathrm{ASE}_r(\hat{\phi})$.
  • Figure 2: Monthly sea level anomalies of the Black Sea.
  • Figure 3: First step residuals diagnostics.
  • Figure 4: Estimates of the autoregressive function $\hat{\phi}$.
  • Figure 5: Final residuals diagnostics.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 3.1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Corollary 1
  • Lemma 5
  • ...and 4 more