Nonlinear ARCH Model (NARCH)
Nonlinear Autoregressive Conditional Heteroscedasticity Model · Also known as: NARCH, Nonlinear ARCH, nonlinear conditional heteroscedasticity model, NARCH model
The Nonlinear ARCH (NARCH) model, introduced by Higgins and Bera (1992), extends Engle's original ARCH framework by allowing the power transformation of volatility to be estimated from the data rather than fixed at two. This flexibility captures a broader class of volatility dynamics observed in financial and macroeconomic time series.
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When to use it
Use the Nonlinear ARCH model when preliminary analysis suggests that standard ARCH (delta = 2) fits poorly — for instance, when the volatility response to shocks appears asymmetric in magnitude or when standardised residual diagnostics indicate remaining structure. It is appropriate for financial returns, inflation, or exchange-rate series with moderately long samples (at least several hundred observations) needed to identify the power parameter reliably. Avoid it when the sample is small (fewer than ~200 observations), since delta is hard to estimate precisely. Also avoid it when leverage effects (asymmetry between positive and negative shocks) are the main concern — GJR-GARCH or EGARCH are better suited for that purpose.
Strengths & limitations
- Nests standard ARCH as a special case, so it can be compared via a likelihood ratio test.
- The estimated power parameter delta provides direct evidence on the shape of the volatility response function without imposing an arbitrary functional form.
- Flexible enough to capture both sublinear and superlinear shock responses within a single parsimonious model.
- Straightforward maximum likelihood estimation using standard numerical optimisers available in most econometric software.
- Empirically motivated: Higgins and Bera found delta significantly different from two in several macro and financial series.
- The power parameter delta is often estimated imprecisely in small or moderately sized samples, producing wide confidence intervals.
- Does not separately model asymmetric responses to positive versus negative shocks (leverage effects) — a structural limitation relative to EGARCH or GJR-GARCH.
- Adding delta as a free parameter increases the risk of numerical convergence problems during likelihood maximisation.
- Less widely implemented than GARCH variants in standard econometric packages, making replication and benchmarking harder.
- Interpretation of delta is not always straightforward for practitioners unfamiliar with power-transformation models.
Frequently asked
How does NARCH differ from standard ARCH?
Standard ARCH uses the square of past innovations (delta = 2) to form the variance equation. NARCH estimates delta freely from the data, so the volatility response to shocks can be sublinear (delta < 2) or superlinear (delta > 2). ARCH is a special case of NARCH nested at delta = 2.
Is NARCH the same as APARCH?
They are closely related. NARCH (Higgins & Bera, 1992) introduced the power parameter in the pure ARCH setting. APARCH (Ding, Granger & Engle, 1993) extended the idea to a GARCH-type framework and also incorporated asymmetric (leverage) effects. NARCH can be viewed as the symmetric, pure-ARCH precursor to APARCH.
What value of delta is typically found in practice?
Empirical studies often find delta estimates in the range of 1 to 1.5 for financial return series — below two — suggesting that large shocks have somewhat less than a quadratic impact on future volatility. However, estimates vary substantially across assets and sample periods.
How do I test whether the nonlinear specification is necessary?
Estimate both the NARCH model and the nested ARCH model (delta constrained to two), then compute a likelihood ratio test statistic as twice the difference in log-likelihoods. Under the null that delta = 2, this statistic follows a chi-squared distribution with one degree of freedom.
Can I use NARCH for GARCH-type persistence?
The original NARCH specification does not include lagged conditional variances, so it does not directly capture GARCH-type long-run persistence. For persistent volatility, consider APARCH or a power-GARCH extension that adds lagged variance terms.
Sources
- Higgins, M. L., & Bera, A. K. (1992). A class of nonlinear ARCH models. International Economic Review, 33(1), 137-158. DOI: 10.2307/2526988 ↗
- Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4), 987-1007. DOI: 10.2307/1912773 ↗
How to cite this page
ScholarGate. (2026, June 3). Nonlinear Autoregressive Conditional Heteroscedasticity Model. ScholarGate. https://scholargate.app/en/econometrics/nonlinear-arch-model
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