Model card — Recession-probability models¶

recession_probit

A binary-choice model for a 0/1 recession indicator y_t: it maps a linear index of leading variables — the term spread above all — into a recession probability through a probit or logit link. The default is the textbook static model (Phi(x_t' beta)); flipping dynamic=True fits the Kauppi-Saikkonen (2008) dynamic probit, whose index carries its own autoregressive lag so that last period's recession pressure raises this period's.

`recession_probit`¶

What it estimates. The coefficients beta of a binary-choice model for the probability that period t is a recession,

P(y_t = 1 | x_t) = F(x_t' beta),

with F = Phi (standard normal CDF, link="probit") or F = Lambda (logistic CDF, link="logit"), fit by exact-likelihood maximum likelihood. With dynamic=True the index is instead autoregressive,

index_t = w + x_t' b + rho * index_{t-1},   P(y_t = 1) = Phi(index_t),

the Kauppi-Saikkonen (2008) dynamic probit: rho measures how much recession pressure persists from one period to the next, and the estimator returns it alongside the intercept w and the covariate slopes b.

Assumptions. The indicator is genuinely binary (y_t in {0, 1}), the predictors are observed leading variables (the model does not lag them for you), and the link is correctly specified. The static MLE requires that the classes be not perfectly separable — a predictor that splits recessions from expansions cleanly drives the coefficient to infinity and there is no finite maximum (reported as a separation error). The dynamic index is assumed stationary (|rho| < 1); it is initialized at its stationary mean w / (1 - rho).

When to use (and when not). Use it to turn leading indicators into a calibrated recession probability — the canonical Estrella-Mishkin term-spread model is exactly recession_probit(y, [const, spread]). Reach for dynamic=True when recessions are persistent (a downturn this quarter makes one next quarter more likely) beyond what the covariates already capture; on such data it fits strictly better because it nests the static probit at rho = 0. Do not use it for a continuous target (this is classification, not regression), and do not expect the dynamic model to help on data with no index persistence — rho will simply estimate near zero.

Key arguments and defaults (and why). link="probit" is the default and the convention in the recession-forecasting literature; "logit" fits the same data with a logistic link (coefficients are on a different scale — roughly 1.6x the probit's — but fitted probabilities are close). dynamic=False is the static model; dynamic=True switches to the Kauppi-Saikkonen dynamic probit (probit only — link is ignored). The one design rule that bites: x must include a constant column for the static model, and must NOT include one for the dynamic model — the dynamic estimator supplies its own intercept w, and a redundant constant column makes the information matrix singular.

How to read the output. params are the coefficients (static: in the column order of x; dynamic: laid out [w, b_0, .., b_{m-1}, rho]), with matching bse (standard errors from the inverse observed information) and zstats (params / bse, an asymptotic z-test). probabilities is the fitted P(y_t = 1) path — the object you actually plot or threshold. loglik is the maximized log-likelihood and pseudo_r2 is McFadden's 1 - loglik/loglik_null (0 at the intercept-only fit, 1 at a perfect one; values around 0.2-0.4 are "good" for binary choice, not the 0.9 you might expect from an R²). The dynamic fit additionally exposes w, beta, and rho as named scalars/vectors, plus converged (the optimizer flag). A rho near 1 flattens the likelihood surface and can leave converged=False at a perfectly good optimum — read the estimates and loglik, not the flag alone.

Failure modes. Complete or quasi-complete separation (a predictor that perfectly classifies) has no finite MLE and raises a separation error — coarsen or drop the offending predictor. Forgetting the constant column in the static model biases every coefficient; including one in the dynamic model makes the information matrix singular. A degenerate response (all-0 or all-1 — no recessions in the sample) cannot be fit. Because the dynamic model is optimized by a derivative-free simplex over a recursive index, treat a huge |rho| or a converged=False flag as a prompt to inspect loglik, not an automatic error.

Validated against. statsmodels' Probit and Logit maximum-likelihood estimators (sm.Probit(y, X).fit(), sm.Logit(y, X).fit()) on a fixed simulated dataset — point estimates, analytic-Hessian standard errors, log-likelihood, and McFadden pseudo-R² match to ~1e-5 (fixtures/tsecon- recession.json), a genuine cross-implementation check by a separate code path. The dynamic probit has no statsmodels counterpart and is validated property-only in the crate's Rust suite: on data from a known dynamic-probit DGP it recovers rho and b within Monte-Carlo bands, and its log-likelihood exceeds the static probit's on persistent data.

References. Estrella & Mishkin (1998, term-spread probit); Kauppi & Saikkonen (2008, dynamic probit); McFadden (1974, pseudo-R²).

Static model — the term-spread recession probit (x includes a constant):

import numpy as np, tsecon

rng = np.random.default_rng(0)
n = 600
spread = np.zeros(n)                           # term spread: a persistent (AR1) leading indicator
for t in range(1, n):
    spread[t] = 0.6 * spread[t - 1] + rng.standard_normal()

index = -0.7 - 0.9 * spread                    # inverted curve (low spread) -> higher recession risk
y = (index + rng.standard_normal(n) > 0).astype(float)   # latent-probit draw: P(y=1) = Phi(index)

X = np.column_stack([np.ones(n), spread])      # STATIC model: X MUST include the constant column
fit = tsecon.recession_probit(y, X, link="probit")
print("params [const, spread]:", np.round(fit["params"], 3))   # ~[-0.76, -0.82] (true -0.7, -0.9)
print("z-stats               :", np.round(fit["zstats"], 3))
print(f"pseudo-R2 = {fit['pseudo_r2']:.3f}   converged: {fit['converged']}")

logit = tsecon.recession_probit(y, X, link="logit")            # same data, logistic link
print("logit params          :", np.round(logit["params"], 3))  # rescaled ~1.6x the probit's

Dynamic model — Kauppi-Saikkonen probit (x omits the constant; the model estimates its own intercept w):

import numpy as np, tsecon

rng = np.random.default_rng(1)
n = 1200
w, b, rho, px = -0.3, 1.0, 0.6, 0.5
x = np.zeros(n)
for t in range(1, n):
    x[t] = px * x[t - 1] + rng.standard_normal()   # a persistent predictor
y = np.zeros(n)
prev = w / (1 - rho)                               # stationary-mean initialization
for t in range(n):
    idx = w + b * x[t] + rho * prev                # the index carries its own lag
    y[t] = 1.0 if idx + rng.standard_normal() > 0 else 0.0
    prev = idx

Xd = x.reshape(-1, 1)                              # DYNAMIC model: NO constant column
dyn = tsecon.recession_probit(y, Xd, dynamic=True)
print("params [w, b, rho]:", np.round(dyn["params"], 3))   # ~[-0.25, 1.00, 0.61] (true -0.3, 1.0, 0.6)
print(f"rho = {dyn['rho']:.3f}   pseudo-R2 = {dyn['pseudo_r2']:.3f}")

stat = tsecon.recession_probit(y, np.column_stack([np.ones(n), x]))   # static comparison
print(f"loglik  dynamic {dyn['loglik']:.1f}  >=  static {stat['loglik']:.1f}")  # dynamic fits better

Model card — Recession-probability models¶

recession_probit¶

`recession_probit`¶