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Pivotal inference for linear predictions in stationary processes

Holger Dette, Sebastian Kühnert

Abstract

In this paper we develop pivotal inference for the final (FPE) and relative final prediction error (RFPE) of linear forecasts in stationary processes. Our approach is based on a self-normalizing technique and avoids the estimation of the asymptotic variances of the empirical autocovariances. We provide pivotal confidence intervals for the (R)FPE, develop estimates for the minimal order of a linear prediction that is required to obtain a prespecified forecasting accuracy and also propose (pivotal) statistical tests for the hypotheses that the (R)FPE exceeds a given threshold. Additionally, we provide pivotal uncertainty quantification for the commonly used coefficient of determination $R^2$ obtained from a linear prediction based on the past $p \geq 1$ observations and develop new (pivotal) inference tools for the partial autocorrelation, which do not require the assumption of an autoregressive process.

Pivotal inference for linear predictions in stationary processes

Abstract

In this paper we develop pivotal inference for the final (FPE) and relative final prediction error (RFPE) of linear forecasts in stationary processes. Our approach is based on a self-normalizing technique and avoids the estimation of the asymptotic variances of the empirical autocovariances. We provide pivotal confidence intervals for the (R)FPE, develop estimates for the minimal order of a linear prediction that is required to obtain a prespecified forecasting accuracy and also propose (pivotal) statistical tests for the hypotheses that the (R)FPE exceeds a given threshold. Additionally, we provide pivotal uncertainty quantification for the commonly used coefficient of determination obtained from a linear prediction based on the past observations and develop new (pivotal) inference tools for the partial autocorrelation, which do not require the assumption of an autoregressive process.

Paper Structure

This paper contains 16 sections, 10 theorems, 149 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Sequential estimation of the final prediction error
  3. The measure $S_p$ and the coefficient of determination $R^2_p$
  4. Confidence intervals and testing relevant hypotheses
  5. Estimating the order for linear predictions
  6. Order selection by hypotheses testing
  7. Finite sample properties
  8. Testing relevant hypotheses.
  9. Estimating the order for linear predictions.
  10. Relative improvement and partial autocorrelations
  11. Statistical consequences
  12. Partial autocorrelations
  13. Finite sample properties
  14. Multivariate setting
  15. Acknowledgements
  16. ...and 1 more sections

Key Result

Theorem 2.1

For the process in det94 it holds where $\mathcal{M}_{p,\boldsymbol{\gamma}_p}\in\mathop{\mathrm{\mathbb{R}}}\nolimits^{(p+1)\times(p+1)}$ is the lower triangular matrix from Eq. kue7 in the Appendix, $\Sigma\in\mathop{\mathrm{\mathbb{R}}}\nolimits^{(p+1)\times(p+1)}$ is the matrix in Eq. AN for vector of autocov, and $\pmb{\mathbb{

Figures (3)

  • Figure 1: Simulated rejection probabilities (y-axis) of the test \ref{['det102']} for the hypotheses \ref{['det103']} for various values of the threshold $\Delta$ (x-axis). Vertical lines indicate the boundary of the hypotheses, where $S_p=\Delta$, and horizontal lines mark the nominal level $\alpha = 5\%$. The data generating process is given by \ref{['eq:(X_k) lin process']} with two choices of coefficients given by \ref{['kue10']}.
  • Figure 2: Histograms of the estimator $\hat{p}$ for the lag $p^*$ defined in \ref{['2.18']}, where the nominal level is $\alpha = 10\%$. Upper panel: the process is given by the AR($5$) model defined in \ref{['kue13']}. The true value is given by $p^\ast = 3$ for $\nu=0.4$. Bottom panel: the process is given by the MA$(\infty)$ model defined in \ref{['kue10']} with polynomially decaying coefficients. The true value is given by $p^\ast = 5$ for $\nu=0.35.$
  • Figure 3: Empirical rejection probabilities (y-axis) of the test in \ref{['det86']} for the hypotheses in \ref{['det85']} with $p=1$. The data generating process is given by \ref{['det300']} with two choices of coefficients. Vertical lines indicate the true values of $\mathscr{S}_1$, while the horizontal line marks the nominal level $\alpha = 10\%$.

Theorems & Definitions (22)

  • Theorem 2.1
  • Corollary 2.1
  • Remark 2.1
  • Theorem 3.1
  • Corollary 3.1
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.2
  • Theorem 3.3
  • Theorem 3.4
  • ...and 12 more