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Asymptotic Uncertainty in the Estimation of Frequency Domain Causal Effects for Linear Processes

Nicolas-Domenic Reiter, Jonas Wahl, Gabriele C. Hegerl, Jakob Runge

Abstract

Structural vector autoregressive (SVAR) processes are commonly used time series models to identify and quantify causal interactions between dynamically interacting processes from observational data. The causal relationships between these processes can be effectively represented by a finite directed process graph - a graph that connects two processes whenever there is a direct delayed or simultaneous effect between them. Recent research has introduced a framework for quantifying frequency domain causal effects along paths on the process graph. This framework allows to identify how the spectral density of one process is contributing to the spectral density of another. In the current work, we characterise the asymptotic distribution of causal effect and spectral contribution estimators in terms of algebraic relations dictated by the process graph. Based on the asymptotic distribution we construct approximate confidence intervals and Wald type hypothesis tests for the estimated effects and spectral contributions. Under the assumption of causal sufficiency, we consider the class of differentiable estimators for frequency domain causal quantities, and within this class we identify the asymptotically optimal estimator. We illustrate the frequency domain Wald tests and uncertainty approximation on synthetic data, and apply them to analyse the impact of the 10 to 11 year solar cycle on the North Atlantic Oscillation (NAO). Our results confirm a significant effect of the solar cycle on the NAO at the 10 to 11 year time scale.

Asymptotic Uncertainty in the Estimation of Frequency Domain Causal Effects for Linear Processes

Abstract

Structural vector autoregressive (SVAR) processes are commonly used time series models to identify and quantify causal interactions between dynamically interacting processes from observational data. The causal relationships between these processes can be effectively represented by a finite directed process graph - a graph that connects two processes whenever there is a direct delayed or simultaneous effect between them. Recent research has introduced a framework for quantifying frequency domain causal effects along paths on the process graph. This framework allows to identify how the spectral density of one process is contributing to the spectral density of another. In the current work, we characterise the asymptotic distribution of causal effect and spectral contribution estimators in terms of algebraic relations dictated by the process graph. Based on the asymptotic distribution we construct approximate confidence intervals and Wald type hypothesis tests for the estimated effects and spectral contributions. Under the assumption of causal sufficiency, we consider the class of differentiable estimators for frequency domain causal quantities, and within this class we identify the asymptotically optimal estimator. We illustrate the frequency domain Wald tests and uncertainty approximation on synthetic data, and apply them to analyse the impact of the 10 to 11 year solar cycle on the North Atlantic Oscillation (NAO). Our results confirm a significant effect of the solar cycle on the NAO at the 10 to 11 year time scale.

Paper Structure

This paper contains 27 sections, 15 theorems, 131 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphical modelling
  4. Structural vector autoregressive process processes
  5. Structural causal models of stochastic processes
  6. Asymptotic distribution of causal effect estimators in the frequency domain
  7. Asymptotic uncertainty in the estimation of link functions
  8. The asymptotic covariance of the causal effect estimator
  9. The asymptotic covariance of the spectral contribution estimator
  10. Asymptotic significance tests and confidence regions
  11. Asymptotic efficiency
  12. Impact of the solar cycle on the NAO
  13. Background
  14. Data
  15. Result
  16. ...and 12 more sections

Key Result

Proposition 1

Suppose $\mathbf{L}_v$ satisfies the AL2 assumption assumption: AL2. Then for generic choices of SVAR coefficients $\Phi$ and for all but finitely many $z\in S^1$, the asymptotically normal OLS estimator $\hat{H}_{\bullet, w}(z)$, i.e. has positive definite asymptotic covariance matrix. The asymptotic covariance matrix is block structured into $2\times 2$-matrices indexed by tuples of $\mathrm{pa

Figures (7)

  • Figure 1: A process graph (left) together with an excerpt of its underlying time series graph (right).
  • Figure 2: A process graph representing causal interactions among the processes $V= \{u_1, u_2, v, m, w\}$. Each vertex is annotated by the function generating the internal dynamics, and each edge is annotated by the associated link function.
  • Figure 3: Statistical analysis of the causal structure of the process shown in Figure \ref{['fig:process graph']} based on a synthetically generated time series of length $500$. Analysis of the estimates for direct mediated and total effects of $v$ on $w$ are shown on the left. The analysis of the spectral contribution of $\mathrm{anc}(v)$ on $w$ via $\Pi = \mathrm{P}(v,w)$ is displayed on the right.
  • Figure 4: The process graph based on which the solar effect on NAO is investigated, where S stands for solar activity represented by sun spot numbers.
  • Figure 5: The statistical analysis of the solar influence on the NAO for oscillations with periods between 8 and 13 years.
  • ...and 2 more figures

Theorems & Definitions (32)

  • Example 1
  • Example 2
  • Proposition 1
  • Proposition 2
  • Example 3
  • Proposition 3
  • Proposition 4
  • Example 4
  • Remark 1
  • Example 5
  • ...and 22 more