Wavelet Based Time Series Models with Time-Varying Thresholds

Abstract: This paper develops a threshold model with a time-varying threshold, represented using a wavelet series expansion. The model adequately captures irregular and abrupt variations, as well as smooth changes in the threshold parameter, allowing greater flexibility than Fourier-based approaches. Simulation experiments and real-data applications are used to evaluate the model's performance.

• The paper introduces wavelet expansions for time-varying thresholds, using conditional least squares and numerical optimization to model both gradual changes and abrupt regime shifts.
• Simulation results with 2,048 observations show low estimation error for smooth and discontinuous thresholds, while bootstrap confidence intervals achieve approximately 92–95% coverage.
• The USD/ARS application shows that the wavelet model removes residual serial correlation, with Ljung–Box p-values of 0.9320 and 0.9965 at lags 20 and 30, outperforming constant and Fourier thresholds.

Motivation and contribution

Threshold autoregressive (TAR) models, introduced by Tong, partition the state space of a process into regimes governed by a threshold parameter. In the standard self-exciting TAR (SETAR) specification, this threshold is assumed constant over time—an assumption that is frequently untenable in economic and financial settings, where whether a variable is "high" or "low" is relative to other evolving quantities. Ignoring threshold variation can produce model misspecification and biased estimation. Yang et al.'s Fourier-approximated time-varying threshold addresses part of this problem, but trigonometric expansions are inherently global: they represent gradual, periodic variation well yet handle abrupt jumps poorly.

This paper proposes replacing the Fourier approximation with a wavelet series expansion for the threshold function $\gamma(t/T)$. Wavelets provide simultaneous time–frequency localization and compact support, so the resulting model accommodates both smooth drifts and sudden discontinuities in the threshold—precisely the situations arising from market interventions or rapid policy shifts that motivate the work.

Model specification and estimation

The proposed two-regime model is

$$y(t) = \begin{cases} \phi_0^{(1)} + \phi_1^{(1)}y(t-1) + \epsilon(t), & y(t-1) \leq \gamma_\theta(t/T) \ \phi_0^{(2)} + \phi_1^{(2)}y(t-1) + \epsilon(t), & y(t-1) > \gamma_\theta(t/T) \end{cases}$$

with $\epsilon(t) \sim \text{iid}(0,\sigma^2)$ and ergodicity conditions $\phi_1^{(1)} < 1$, $\phi_1^{(2)} < 1$, $\phi_1^{(1)}\phi_1^{(2)} < 1$. The threshold is expanded as

$$\gamma_\theta\left(\tfrac{t}{T}\right) = c_{00}\varphi_{00}\left(\tfrac{t}{T}\right) + \sum_{j=0}^{J-1}\sum_{k=0}^{2^j-1} d_{jk}\psi_{jk}\left(\tfrac{t}{T}\right),$$

so the parameter vector $\boldsymbol{\theta}$ has dimension $2^J$. The Haar basis is natural on $[0,1)$; other Daubechies families (extremal phase D(N), least asymmetric LA(N)) require boundary handling via Hardle's mirroring method. Estimation follows Yang et al.: reparameterizing with indicator functions yields a linear regression in $$y(t) = \begin{cases} \phi_0^{(1)} + \phi_1^{(1)}y(t-1) + \epsilon(t), & y(t-1) \leq \gamma_\theta(t/T) \ \phi_0^{(2)} + \phi_1^{(2)}y(t-1) + \epsilon(t), & y(t-1) > \gamma_\theta(t/T) \end{cases}$$0 conditional on $$y(t) = \begin{cases} \phi_0^{(1)} + \phi_1^{(1)}y(t-1) + \epsilon(t), & y(t-1) \leq \gamma_\theta(t/T) \ \phi_0^{(2)} + \phi_1^{(2)}y(t-1) + \epsilon(t), & y(t-1) > \gamma_\theta(t/T) \end{cases}$$1, so $$y(t) = \begin{cases} \phi_0^{(1)} + \phi_1^{(1)}y(t-1) + \epsilon(t), & y(t-1) \leq \gamma_\theta(t/T) \ \phi_0^{(2)} + \phi_1^{(2)}y(t-1) + \epsilon(t), & y(t-1) > \gamma_\theta(t/T) \end{cases}$$2 is obtained by conditional least squares while $$y(t) = \begin{cases} \phi_0^{(1)} + \phi_1^{(1)}y(t-1) + \epsilon(t), & y(t-1) \leq \gamma_\theta(t/T) \ \phi_0^{(2)} + \phi_1^{(2)}y(t-1) + \epsilon(t), & y(t-1) > \gamma_\theta(t/T) \end{cases}$$3 minimizes the SSR over a compact parameter set, using DEoptim for initialization and R's optim for refinement. The resolution level $$y(t) = \begin{cases} \phi_0^{(1)} + \phi_1^{(1)}y(t-1) + \epsilon(t), & y(t-1) \leq \gamma_\theta(t/T) \ \phi_0^{(2)} + \phi_1^{(2)}y(t-1) + \epsilon(t), & y(t-1) > \gamma_\theta(t/T) \end{cases}$$4 is selected by minimizing RMSE over small candidate values ($$y(t) = \begin{cases} \phi_0^{(1)} + \phi_1^{(1)}y(t-1) + \epsilon(t), & y(t-1) \leq \gamma_\theta(t/T) \ \phi_0^{(2)} + \phi_1^{(2)}y(t-1) + \epsilon(t), & y(t-1) > \gamma_\theta(t/T) \end{cases}$$5).

Simulation evidence

Two designs with $$y(t) = \begin{cases} \phi_0^{(1)} + \phi_1^{(1)}y(t-1) + \epsilon(t), & y(t-1) \leq \gamma_\theta(t/T) \ \phi_0^{(2)} + \phi_1^{(2)}y(t-1) + \epsilon(t), & y(t-1) > \gamma_\theta(t/T) \end{cases}$$6 and 500 replications probe the two regimes of threshold behavior:

Parameter	Study 1: true	estimate	RMSE	Study 2: true	estimate	RMSE
$$y(t) = \begin{cases} \phi_0^{(1)} + \phi_1^{(1)}y(t-1) + \epsilon(t), & y(t-1) \leq \gamma_\theta(t/T) \ \phi_0^{(2)} + \phi_1^{(2)}y(t-1) + \epsilon(t), & y(t-1) > \gamma_\theta(t/T) \end{cases}$$7	0.5	0.4926	0.0429	0.5	0.4618	0.0842
$$y(t) = \begin{cases} \phi_0^{(1)} + \phi_1^{(1)}y(t-1) + \epsilon(t), & y(t-1) \leq \gamma_\theta(t/T) \ \phi_0^{(2)} + \phi_1^{(2)}y(t-1) + \epsilon(t), & y(t-1) > \gamma_\theta(t/T) \end{cases}$$8	−0.3	−0.3089	0.0448	0.3	0.2776	0.0560
$$y(t) = \begin{cases} \phi_0^{(1)} + \phi_1^{(1)}y(t-1) + \epsilon(t), & y(t-1) \leq \gamma_\theta(t/T) \ \phi_0^{(2)} + \phi_1^{(2)}y(t-1) + \epsilon(t), & y(t-1) > \gamma_\theta(t/T) \end{cases}$$9	1.0	1.0298	0.1285	−1.0	−0.9923	0.0421
$\epsilon(t) \sim \text{iid}(0,\sigma^2)$0	0.3	0.2904	0.0483	0.5	0.4898	0.0501
$\epsilon(t) \sim \text{iid}(0,\sigma^2)$1	2.0	1.9938	0.0642	1.0	1.0116	0.0334

Study 1 uses a step-function threshold (jumping between 1 and 1.5 at $\epsilon(t) \sim \text{iid}(0,\sigma^2)$2 and $\epsilon(t) \sim \text{iid}(0,\sigma^2)$3) estimated with the Haar wavelet at $\epsilon(t) \sim \text{iid}(0,\sigma^2)$4; Study 2 uses a smooth quadratic threshold estimated with LA(4) at the optimal $\epsilon(t) \sim \text{iid}(0,\sigma^2)$5. In both cases the autoregressive parameters and noise variance are recovered with small bias and low RMSE, and the estimated threshold curves track the true ones closely. The key claim—that wavelet thresholds match Fourier performance under smooth variation while also handling abrupt changes—is supported by these results, though the paper does not report a head-to-head comparison against the Fourier estimator within the simulations, so the claimed advantage over Fourier methods rests primarily on the real-data application.

Bootstrap inference

The paper develops residual-based bootstrap confidence intervals: residuals from the fitted model are centered, resampled with replacement, and used to regenerate series via the fitted regime structure; percentile intervals are formed for the AR parameters, and sup-t simultaneous bands (following Montiel Olea–Plagborg-Møller style constructions) cover the entire threshold curve. Applied to the Study 1 design with $\epsilon(t) \sim \text{iid}(0,\sigma^2)$6 and 1000 Monte Carlo repetitions, coverage probabilities were 0.94, 0.95, 0.93, 0.94, and 0.92 for $\epsilon(t) \sim \text{iid}(0,\sigma^2)$7, respectively, and 0.94 for the threshold curve's simultaneous band—all close to nominal 95% levels. This indicates the inference procedure is reliable even when the threshold contains sharp jumps.

Empirical application: USD/ARS exchange rate

The method is applied to 2048 daily minima of the USD/ARS exchange rate (January 2016 to November 2023), first-differenced. A Hansen linearity test rejects linearity (p-value 0.01). Three models are compared via Ljung–Box tests on residuals at lags 20 and 30:

Model	p-value (lag 20)	p-value (lag 30)
Constant threshold	$\epsilon(t) \sim \text{iid}(0,\sigma^2)$8	$\epsilon(t) \sim \text{iid}(0,\sigma^2)$9
Fourier time-varying threshold	$\phi_1^{(1)} < 1$0	$\phi_1^{(1)} < 1$1
Wavelet time-varying threshold (D(5), $\phi_1^{(1)} < 1$2)	0.9320	0.9965

The constant-threshold SETAR leaves strong residual serial correlation, and the Fourier-threshold model performs worse still—its globally smooth sinusoidal threshold cannot track the peso's regime shifts, which included abrupt devaluation episodes. The wavelet-threshold model, by contrast, yields residuals with no detectable serial dependence, providing direct empirical support for the central claim that local basis functions outperform both fixed and Fourier thresholds on data with irregular threshold dynamics. Bootstrap intervals for the AR coefficients exclude zero in three of four cases, and the estimated threshold curve with sup-t bands shows pronounced movement consistent with Argentina's currency interventions. All Daubechies wavelets tried produced comparable RMSE and MAE, suggesting robustness to the choice within the family.

Limitations and open questions

Several caveats qualify the results. First, asymptotic theory for the estimator—consistency of $\phi_1^{(1)} < 1$3 and limiting distributions—is not established; inference relies entirely on the bootstrap, whose validity for this nonlinear, nonstandard estimation problem is assumed rather than proven. Second, the simulations use only $\phi_1^{(1)} < 1$4 and Gaussian innovations; performance under heavy tails, heteroskedasticity, or shorter samples relevant to typical macroeconomic applications is unexamined. Third, no formal model-selection criterion or test for choosing $\phi_1^{(1)} < 1$5 is provided beyond minimum RMSE, and the comparison with the Fourier alternative lacks simulation-level benchmarking. Finally, the extension to higher-order AR terms, more than two regimes, or other threshold model classes (e.g., threshold GARCH) is noted as promising but not carried out.

Conclusion

The paper contributes a wavelet-series parameterization of the time-varying threshold in TAR models, together with conditional least squares estimation and bootstrap inference including sup-t simultaneous bands for the threshold curve. Simulations show accurate recovery of both abruptly jumping and smoothly varying thresholds, and the USD/ARS application demonstrates decisively that the wavelet threshold eliminates residual serial correlation where constant and Fourier thresholds fail. The main open issues are the absence of asymptotic theory and broader robustness checks, both of which would strengthen the case for routine use of the method.