Linear VARs assume that a one-standard-deviation monetary shock produces the same output response in a deep recession as in a boom. That symmetry is a maintained hypothesis, not a finding. Threshold VARs drop it. The idea traces to Tong's (1990) self-exciting threshold autoregression (SETAR) for univariate series and was extended to the multivariate case by Tsay (1998). Hansen (1996) supplied the inferential machinery: because the threshold parameter vanishes under the null of linearity, conventional Wald and LM tests have non-standard distributions, and Hansen's bootstrapped sup-LM statistic became the workhorse test for threshold existence.

The mechanism is direct. Pick an observable variable - the output gap, credit spread, or transition indicator - and call it the threshold variable. The model searches over candidate cutoff values and splits the sample into two (or more) regimes. Each regime gets its own VAR coefficient matrix and its own innovation covariance. The switch is instantaneous: once the threshold variable crosses the cutoff, the entire parameter vector flips. This is what separates the Threshold VAR from its smooth-transition cousin, which allows gradual migration between regimes.

Estimation proceeds by concentrated least squares. For each candidate threshold, the sample splits and regime-specific OLS is fast. The concentrated sum of squared residuals is then minimized over the threshold grid. Hansen (1997, 2000) showed that the threshold estimator is super-consistent under regularity conditions, converging at rate T rather than the usual root-T. Confidence intervals for the threshold come from inverting a likelihood-ratio statistic, not from the asymptotic normal approximation.

Central banks and fiscal authorities use Threshold VARs to study asymmetric policy transmission. Balke (2000) documented that credit-channel effects strengthen when the credit spread exceeds a threshold, a finding that linear models miss entirely. Lo and Zivot (2001) applied the framework to interest-rate dynamics. More recent work nests Threshold VARs inside Bayesian frameworks with stochastic volatility to handle both regime switches and time-varying uncertainty.

The Threshold VAR stacks two (or M) linear VARs side by side, with a selection rule governed by a single observable variable and an estimated cutoff. Inputs are the same n-variable dataset that a linear VAR would require, plus a choice of threshold variable $q_t$, which can be one of the endogenous variables or an exogenous indicator. The delay parameter d controls how many periods back the threshold variable is measured: the regime at time t depends on $q_{t-d}$, not $q_t$, to avoid simultaneity.

Conditional on the threshold gamma and delay d, each regime m has its own intercept vector c^(m), lag-coefficient matrices A_1^(m) through A_p^(m), and innovation covariance Sigma^(m). The model switches between these parameter sets discontinuously at gamma. The concentrated least-squares objective scans a grid of candidate gamma values, evaluates the regime-specific residual sum of squares at each candidate, and picks the minimizer. The grid is trimmed so that each regime contains at least a fixed fraction (commonly 15 percent) of the observations.

Output from the Threshold VAR includes regime-specific impulse response functions, which are computed by feeding a shock into each regime's coefficient matrix separately. Generalized impulse responses (Koop, Pesaran, Potter 1996) account for the possibility that a shock pushes the system across the threshold, creating regime-dependent, history-dependent, and sign-dependent dynamics that linear IRFs cannot capture. The regime classification itself - which observations belong to which regime - is a direct, deterministic output of the estimated threshold.

The Federal Reserve Bank of St. Louis and the Bank of England use Threshold VARs to study asymmetric monetary transmission. Balke (2000) showed that tight credit conditions (high credit spread regime) amplify the real effects of interest rate shocks while loose conditions dampen them. This finding directly informed the financial-accelerator literature and motivated regime-dependent stress testing in central bank forecasting suites.

Fiscal multiplier research uses Threshold VARs with the output gap as the threshold variable to test whether government spending multipliers are larger in recessions than in expansions. Auerbach and Gorodnichenko (2012) popularized a smooth-transition variant, but the sharp-threshold version remains common in robustness exercises because it avoids the functional-form choice for the transition function.

The Threshold VAR has a hard limit: it requires the regime driver to be observable and the transition to be instantaneous. When regimes are latent (e.g., financial stress that is not captured by a single spread), Markov-switching models are more appropriate. When the transition is gradual (e.g., a slow shift in monetary policy stance), the smooth-transition regression nests the Threshold VAR and should be preferred. With small samples, the sup-LM test has limited power, and the trimming requirement can prevent detection of rare extreme regimes.

Extensions include multiple thresholds (three or more regimes), Bayesian estimation with conjugate priors on regime coefficients and a discrete prior on gamma, and combination with time-varying parameters within regimes. Panel threshold VARs (Seo and Shin 2016) extend the framework to cross-country data, estimating a common threshold while allowing country-specific coefficients.