Fourier Nonlinear ARDL (Fourier NARDL)
Fourier Nonlinear Autoregressive Distributed Lag Model · Also known as: Fourier NARDL, Fourier nonlinear ARDL, F-NARDL, Fourier asymmetric ARDL
Fourier NARDL extends the Nonlinear ARDL (NARDL) bounds-testing framework by adding Fourier trigonometric terms to the error-correction equation, allowing the model to capture smooth, gradual structural breaks in the long-run relationship without requiring the researcher to know or specify the break date in advance.
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When to use it
Use Fourier NARDL when you have time-series data and theory or preliminary evidence suggests: (1) the long-run relationship is asymmetric — positive and negative changes in the driver affect the outcome differently; and (2) the relationship may have shifted gradually over the sample period due to policy changes, regime transitions, or slow structural evolution. It is appropriate when you cannot pinpoint a specific break date. Do not use it when you have a small sample (fewer than about 60 observations), as adding Fourier terms reduces degrees of freedom noticeably. If asymmetry is not theoretically justified, prefer the standard Fourier ARDL bounds test. If breaks are sharp and well-dated, a Zivot-Andrews or Bai-Perron approach is more parsimonious.
Strengths & limitations
- Simultaneously tests for asymmetric cointegration and accommodates smooth structural change without requiring a known break date.
- Fourier approximation is flexible: a single low-frequency term can capture a wide variety of smooth, nonlinear deterministic trends.
- Inherits the NARDL advantage of working with a mix of I(0) and I(1) variables, avoiding the need for pre-testing all variables to the same integration order.
- Dynamic multipliers provide intuitive visual evidence of the asymmetric adjustment process and its speed.
- Bounds-testing approach remains valid regardless of whether regressors are strictly I(1) or I(0), making it robust to integration uncertainty.
- Requires a reasonably long time series; adding Fourier components reduces degrees of freedom, making the model unreliable for samples below roughly 60 observations.
- Selection of the optimal Fourier frequency k via grid search is informal and may over-fit in small samples.
- The joint null tested by the F-bounds statistic conflates no cointegration with no asymmetry, making it difficult to isolate which restriction drives rejection.
- Critical values for the Fourier-augmented bounds test differ from standard ARDL tables and must be obtained by simulation; tabulated values for all specifications may not be readily available.
- Interpretation requires care: apparent asymmetry may partly reflect the Fourier correction rather than genuine behavioural asymmetry.
Frequently asked
How do I choose the Fourier frequency k?
Estimate the model for each integer value of k from 1 to some maximum (typically k ≤ 3 or 5) and choose the value that minimises the residual sum of squares. Some software reports the optimal k automatically.
Can Fourier NARDL be used with panel data?
Yes. Panel extensions of Fourier NARDL exist and have been applied in the literature, though they are less standardised than the single-equation version and require simulation-based critical values appropriate for the panel dimension.
What is the difference between Fourier NARDL and Fourier ARDL?
Fourier ARDL assumes symmetric adjustment — positive and negative changes in the regressor have equal effects. Fourier NARDL relaxes this by decomposing the regressor into positive and negative partial sums, testing for both asymmetry and smooth structural change simultaneously.
Do I still need to test for unit roots before running Fourier NARDL?
The NARDL bounds test is valid for regressors that are I(0) or I(1), so full pre-testing is not required. However, you must confirm that no variable is I(2); standard ADF or PP tests (or their Fourier variants) suffice for this check.
How do I interpret the dynamic multipliers?
Dynamic multipliers show the cumulative response of y to a unit positive shock in x+ versus a unit negative shock in x− over the subsequent h periods. Diverging multiplier paths provide visual evidence of asymmetry; the width of the confidence bands indicates estimation uncertainty.
Sources
- Shin, Y., Yu, B., & Greenwood-Nimmo, M. (2014). Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ARDL framework. In R. C. Sickles & W. C. Horrace (Eds.), Festschrift in Honor of Peter Schmidt (pp. 281–314). Springer. link ↗
- Becker, R., Enders, W., & Lee, J. (2006). A stationarity test in the presence of an unknown number of smooth breaks. Journal of Time Series Analysis, 27(3), 381–409. link ↗
How to cite this page
ScholarGate. (2026, June 3). Fourier Nonlinear Autoregressive Distributed Lag Model. ScholarGate. https://scholargate.app/en/econometrics/fourier-nardl
Which method?
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