Nonlinear VAR Model
Nonlinear Vector Autoregression Model · Also known as: NLVAR, nonlinear vector autoregression, threshold VAR, TVAR
The Nonlinear VAR (NLVAR) model extends the standard vector autoregression by allowing the dynamic relationships among multiple time series to switch or change smoothly depending on an observed threshold variable, a latent regime state, or a smooth transition function. It is used when economic systems exhibit asymmetric responses, regime shifts, or state-dependent dynamics that a linear VAR cannot capture.
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When to use it
Use a Nonlinear VAR when you have multivariate time series that exhibit asymmetric dynamics, regime changes, or state-dependent behavior — for example, studying how monetary policy effects differ between recessions and expansions, or how commodity price shocks propagate differently in high- and low-volatility periods. A prerequisite is that a formal linearity test rejects the standard linear VAR. Do not use NLVAR when the linear VAR fits the data adequately, when sample sizes are small (the nonlinear model has many more parameters), or when no clear economic rationale exists for regime-switching behavior.
Strengths & limitations
- Captures asymmetric and state-dependent dynamics that a standard linear VAR misses.
- Threshold VAR, MS-VAR, and STVAR variants cover a wide range of economic nonlinearities.
- State-dependent impulse response functions provide richer policy-relevant insights.
- Formally testable against the linear null, so adoption is grounded in data evidence.
- Well-suited to macro-financial applications where crisis and normal-times behavior differ sharply.
- Substantially more parameters than a linear VAR, requiring larger samples to estimate reliably.
- Threshold and regime selection involves additional specification choices that affect results.
- Impulse response functions must be simulated rather than computed analytically, which is computationally intensive.
- Markov-switching VAR requires assumptions about the number of regimes, which is hard to determine from data alone.
- Interpretation of regimes and transitions can be ambiguous without strong economic theory.
Frequently asked
What is the difference between a threshold VAR and a Markov-switching VAR?
In a threshold VAR, regime switches are determined by an observed variable crossing a known threshold — regimes are data-observable. In a Markov-switching VAR, regime membership is latent and probabilistic, governed by an unobserved Markov chain. TVAR is interpretively simpler; MS-VAR is more flexible when no natural threshold variable exists.
How do I choose the number of regimes?
For threshold VAR, linearity tests suggest two regimes as a starting point; information criteria (AIC, BIC) help select among alternatives. For MS-VAR, likelihood ratio tests with Davies' bound correction or information criteria are used; two or three regimes are most common in practice.
Are standard impulse response functions valid for a nonlinear VAR?
No. In a nonlinear VAR, responses depend on the current regime, shock size, and sign — the principle of superposition does not hold. Generalized impulse response functions (GIRFs) proposed by Koop et al. (1996) are the standard approach; they average over the distribution of initial conditions.
How large a sample do I need?
A nonlinear VAR with k variables, p lags, and m regimes has roughly m × k²p free parameters. As a rule of thumb, each regime should contain at least 20–30 observations; total sample size of 100 or more is recommended for even a modest two-regime bivariate model.
Can I test whether the linear VAR is adequate before estimating the nonlinear model?
Yes — and you should. For threshold VAR, Tsay's (1998) multivariate portmanteau-type test and Hansen's (1996) sup-Wald test are standard. For smooth-transition VAR, LM tests against a linear null are available. Proceeding to the nonlinear model only when linearity is rejected keeps the analysis grounded.
Sources
- Tsay, R. S. (1998). Testing and modeling multivariate threshold models. Journal of the American Statistical Association, 93(443), 1188–1202. DOI: 10.1080/01621459.1998.10473779 ↗
- Krolzig, H.-M. (1997). Markov-Switching Vector Autoregressions: Modelling, Statistical Inference, and Application to Business Cycle Analysis. Springer. ISBN: 978-3540628644
How to cite this page
ScholarGate. (2026, June 3). Nonlinear Vector Autoregression Model. ScholarGate. https://scholargate.app/en/econometrics/nonlinear-var-model
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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