[Paper Review] Seasonal fractional long-memory processes. A semiparametric estimation approach

Valdério Anselmo Reisen, Wilfredo Palma+2 more|arXiv (Cornell University)|Nov 25, 2010

Financial Risk and Volatility Modeling21 references3 citations

TL;DR

This paper proposes a semiparametric estimation method for seasonal fractional long-memory processes with two distinct seasonal periods using a multilinear regression approach on the log-spectral density. The method accurately estimates dual fractional differencing parameters $d_1$ and $d_2$, with Monte Carlo experiments and real data (PM₁₀) confirming robust performance and applicability to environmental time series with complex seasonal and long-memory dynamics.

ABSTRACT

This paper explores seasonal and long-memory time series properties by using the seasonal fractional ARIMA model when the seasonal data has one and two seasonal periods and short-memory counterparts. The stationarity and invertibility parameter conditions are established for the model studied. To estimate the memory parameters, the method given in Reisen, Rodrigues and Palma (2006 a,b) is generalized here to deal with a time series with two seasonal fractional long-memory parameters. The asymptotic properties are established and the accuracy of the method is investigated through Monte Carlo experiments. The good performance of the estimator indicates that it can be an alternative competitive procedure to estimate seasonal long-memory time series data. Artificial and PM10 series were considered as examples of applications of the proposed estimation method.

Motivation & Objective

• To develop a semiparametric estimation method for seasonal ARFIMA models with two seasonal fractional differencing parameters.
• To establish stationarity and invertibility conditions for the proposed model with multiple seasonal periods.
• To extend the Reisen et al. (2006) estimation framework to handle two seasonal long-memory components simultaneously.
• To evaluate the asymptotic properties and finite-sample performance of the estimator through Monte Carlo simulations.
• To demonstrate the method’s practical utility using real-world environmental data, such as daily PM₁₀ concentrations.

Proposed method

• The method employs a multilinear regression of the log-spectral density on frequency, using a semiparametric approach to estimate the fractional differencing parameters $d_1$ and $d_2$.
• The spectral density model is specified as $f(\lambda) = f^*(\lambda) |\lambda|^{-2d_1} \prod_{i=1}^{2} |\lambda - \lambda_{ij}|^{-2d_i}$, capturing two seasonal long-memory components.
• Estimates of $d_1$ and $d_2$ are derived from the slope of the log-spectral regression, with bandwidth selection critical for accuracy.
• The method is applied to both artificial data and real PM₁₀ concentration data to validate performance and model fit.
• Model order selection for the innovation process is performed using the Akaike Information Criterion (AIC), favoring an MA(1) component after filtering.
• The final model is SARFIMA(0, $d_1$, 1)×(0, $d_2$, 0)₇ with estimated $\hat{d}_1 = 0.1918$, $\hat{d}_2 = 0.1798$, and $\hat{\theta} = -0.2673$.

Experimental results

Research questions

• RQ1 Can a semiparametric estimator effectively estimate two distinct seasonal fractional differencing parameters in a long-memory time series?
• RQ2 How do the asymptotic and finite-sample properties of the estimator perform under varying sample sizes and parameter configurations?
• RQ3 Does the proposed method outperform or remain competitive with parametric alternatives in estimating seasonal long-memory dynamics?
• RQ4 Can the method reliably detect and quantify long-memory behavior in environmental time series with multiple seasonal cycles?
• RQ5 What is the impact of bandwidth selection on the stability and accuracy of the estimated memory parameters?

Key findings

• For the PM₁₀ data, the optimal bandwidth parameter $\alpha = 0.54$ yielded stable estimates $\hat{d}_1 = 0.1918$ and $\hat{d}_2 = 0.1798$, with low standard errors and tight confidence intervals.
• The F-test strongly rejected the null hypothesis $H_0: \mathbf{d} = \mathbf{0}$, indicating significant long-memory behavior in both seasonal components.
• The AIC criterion selected the SARFIMA(0, $d_1$, 1)×(0, $d_2$, 0)₇ model with $\hat{\theta} = -0.2673$ and standard error 0.0214, indicating a good fit for the filtered innovation process.
• The estimated coefficients $\hat{\pi}_j^*$ decayed rapidly, with $\hat{\pi}_{731}^* \approx 10^{-6}$, confirming negligible contribution from distant lags in the AR(∞) representation.
• Residual analysis showed no anomalies, with most correlations falling within confidence bounds, supporting the adequacy of the final model.
• Monte Carlo experiments demonstrated that the estimator maintains high accuracy even for sample sizes as large as 1080, with stable and consistent parameter estimates.