T. Niero

Applied quantitative research in financial econometrics and machine learning

Keywords: fractional differencing · parameter stability · structural breaks · regime-conditional inference · stationary bootstrap · financial time series · statistical methodology · machine learning in finance

Research statement

Financial models depend on the transformations applied before estimation. This project studies one such transformation: the Fixed-width Fractional Differencing operator used to construct stationary features from persistent financial time series.

The question is operational. A preprocessing rule should reduce non-stationarity without erasing the dependence structure that may carry information. When the volatility environment changes, a single operator calibrated on a long sample may fail on both margins: under-transforming one window and over-transforming another. The work reframes this trade-off as an operator-selection problem and asks whether the selected order remains stable across volatility states defined inside VIX-based regimes.

The contribution is methodological. The object of inference is the stability of the preprocessing operator, not the estimation of a latent memory parameter.

Current project research in progress

Testing the d-Invariance Hypothesis for the Fixed-Width Fractional Differencing Operator Across Volatility States Using VIX

Method: structural-break detection, nonparametric segmentation, fixed-width fractional differencing, stationary bootstrap

The fractional differencing operator \((1-L)^d\) generalises integer differencing to non-integer orders (Granger and Joyeux, 1980; Hosking, 1981). Its weights decay rather than terminate, so the operator can reduce non-stationarity while retaining more autocorrelation structure than first differencing. In finite samples, the infinite sequence of weights requires truncation. López de Prado (2018, Chapter 5) gives a practical implementation through the Fixed-width Fractional Differencing method, which retains weights up to the largest lag \(L^*\) satisfying \(|\omega_{L^*}(d)| \geq \tau\):

\[ \tilde{X}_t = \sum_{k=0}^{L^*} \omega_k(d)\, X_{t-k}, \qquad L^* = \max\{k \in \mathbb{N}: |\omega_k(d)| \geq \tau\}. \]

In this design, \(d\) is not estimated inside a predictive model. A deterministic stationarity rule selects the minimum admissible order \(d^*\) before model estimation. The selected order depends on the observed series, the truncation threshold \(\tau\), the candidate grid, and the stationarity criterion. Throughout, \(d\) denotes the operator parameter and \(d^*\) denotes the data-dependent selected order.

The pipeline uses a two-level boundary hierarchy. Structural-break detection on VIX defines parent regimes. Nonparametric segmentation inside each regime defines child volatility states (McGonigle and Cho, 2025). The d-invariance hypothesis is tested across the units of an admissible partition. For a fixed partition \(\mathcal{P} = \{S_1, \ldots, S_K\}\):

\[ H_0:\ d^*_1 = d^*_2 = \cdots = d^*_K, \qquad H_1:\ \exists\ k < l \ \text{such that}\ d^*_k \neq d^*_l. \]

Inference uses a stationary bootstrap drawn independently within each fixed segment, recentred around the observed selected orders, with the partition treated as fixed (Politis and Romano, 1994). Validation and test observations never enter operator selection.

The test is conditional on the supplied partition. Joint inference over boundary estimation and operator selection is outside the public repository scope and is reserved for the publication in progress.

The repository linked below contains a public-safe implementation of the procedure: fixed-width fractional differencing utilities, boundary-consuming segmentation interfaces, the d-invariance statistic, leakage checks, synthetic examples, and unit tests. Empirical results are reserved for the publication in progress.

Repository: d-invariance-ffd · DOI: 10.5281/zenodo.20384552

Selected references

The references below frame the methodological context of the current project. Each entry corresponds to a primary source consulted directly.

Hosking, J. R. M. (1981). Fractional differencing. Biometrika, 68(1), 165–176. Establishes the fractional differencing operator and its autocorrelation properties.
Granger, C. W. J., & Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis, 1(1), 15–29. Introduces fractional differencing within autoregressive fractionally integrated models.
Diebold, F. X., & Inoue, A. (2001). Long memory and regime switching. Journal of Econometrics, 105(1), 131–159. Shows why long-memory behaviour and regime switching can be observationally close.
Beran, J., & Terrin, N. (1996). Testing for a change of the long-memory parameter. Biometrika, 83(3), 627–638. http://www.jstor.org/stable/2337514 Provides a precedent for testing stability in memory-related parameters.
McGonigle, E. T., & Cho, H. (2025). Nonparametric data segmentation in multivariate time series via joint characteristic functions. Biometrika, 112(2), asaf024. Provides the segmentation method used as an external boundary-generating component.
Politis, D. N., & Romano, J. P. (1994). The stationary bootstrap. Journal of the American Statistical Association, 89(428), 1303–1313. Provides the bootstrap procedure used for weakly dependent time series.
Tsay, W.-J., & Härdle, W. K. (2009). A generalized ARFIMA process with Markov-switching fractional differencing parameter. Journal of Statistical Computation and Simulation, 79(5), 731–745. https://doi.org/10.1080/00949650801910239 Provides a state-dependent fractional-differencing specification and motivates testing whether the selected operator order varies across volatility states.
Lipton, A., & Reghai, A. (2023). SPX, VIX and scale-invariant LSV. SSRN Working Paper No. 4356976. Motivates state-aware financial model design and the use of scale-consistent volatility representations.
López de Prado, M. (2018). Advances in Financial Machine Learning. Wiley. Chapter 5 introduces the Fixed-width Fractional Differencing method.