NY Fed Probit Model: Converting T10Y3M into Recession Probability

The Federal Reserve Bank of New York’s probit model converts the T10Y3M spread value into a 12-month conditional recession probability, through a normal cumulative distribution function calibrated by maximum likelihood on the 1959-present history, and published monthly on the Recession Probability Page since 2006.

Understanding this quantitative formalization requires grasping its mathematical form, its empirical calibration, and its operational limits — beyond the monthly published number.

Mathematical formulation of the model

The NY Fed probit model takes the standard analytical form of a binary conditional probability regression. The dependent variable is a recession indicator equal to 1 if month t+12 falls within a recession period dated by the National Bureau of Economic Research, and 0 otherwise. The explanatory variable is the monthly average of the T10Y3M spread at month t — that is, the daily average of the spread across business days within the calendar month.

The link function between explanatory variable and estimated probability is the cumulative distribution function of the standard normal, denoted Φ. The model writes P(recession_{t+12} = 1 | T10Y3M_t) = Φ(α + β × T10Y3M_t), where α and β are the coefficients to estimate. The twelve-month lag between explanatory and dependent variables reflects the predictive horizon retained — the NY Fed estimates recession probability over the next twelve months conditional on the current spread value.

The use of the normal cumulative distribution rather than a logistic function (logit) or an identity function (linear) reflects two analytical properties. First, Φ constrains output between 0 and 1, ensuring probabilistic interpretation of the prediction. Second, standard probit converges with logit in practical terms across most macroeconomic samples, but the probit convention is the dominant academic convention in empirical monetary research since Estrella and Hardouvelis’s 1991 work published in the Journal of Finance.

Coefficient estimation by maximum likelihood

The two coefficients α (intercept) and β (spread sensitivity) are estimated by maximizing the joint likelihood function over the available historical sample. Concretely, one seeks the values of α and β that maximize the probability of observing the empirical sequence of NBER recessions from January 1959 — start of the T10Y3M series on FRED — through the month preceding current publication.

Across the 1959-2024 reference sample, the NY Fed-published coefficients are approximately α = -0.55 and β = -0.72. The negative β reflects that a lower T10Y3M (deeper inversion) raises recession probability — a relationship economically consistent with the transmission channel through bank credit contraction. The direct interpretation: a T10Y3M of -1 percent (100-basis-point inversion) yields a conditional probability Φ(-0.55 + 0.72) = Φ(0.17) ≈ 0.57, i.e., 57 percent probability of recession within the next twelve months.

The temporal stability of these coefficients constitutes a strong empirical argument in favor of model robustness. Between 1996 — date of the first formal publication by Estrella and Mishkin — and 2024, the published α and β values have varied by less than 10 percent of their central value, despite the addition of approximately twenty-eight years of data and three additional recessions (2001, 2008, 2020). This stability is rare for empirical macroeconomic models, whose coefficients often drift significantly with data additions.

NY Fed calibration published since 2006

The Federal Reserve Bank of New York has been publishing monthly model updates on its dedicated Recession Probability page since June 2006. The publication includes three elements: the monthly average T10Y3M value over the closed month, the 12-month conditional recession probability computed by the model, and a historical chart of probability since 1959 with NBER recession bands overlaid.

The publication calendar follows monthly Treasury-series close: generally between the 5th and 15th of the following month, with variable delays depending on business days. Model coefficients are re-estimated once per year, typically in January or February, to incorporate prior-year data. This annual re-estimation ensures the model benefits from all available observations without producing noisy monthly re-estimation. The detail of the T10Y3M signal and its institutional adoption by the NY Fed as canonical barometer is covered in the cluster’s central hub.

An independent replication of the model is documented by Eco3min economists in the detailed replication of the NY Fed probit model, which reproduces the estimated coefficients from raw FRED series and compares the obtained values with official publications. The observed numerical gaps are on the order of one hundredth of a percent, consistent with minor convention differences (DTB3 vs DGS3MO, daily vs monthly frequency, business-day treatment).

Step-by-step numerical replication

To replicate the model identically, four operational steps are necessary. The first downloads the DGS10 and DGS3MO series from FRED at daily frequency, over the full 1959-present history. The NY Fed uses DGS3MO rather than DTB3 — this is the constant-maturity convention underlying the official model, a technical distinction detailed in the precise meaning of T10Y3M and its two conventions. The second step builds the monthly average T10Y3M spread = DGS10 – DGS3MO over business days within each calendar month.

The third step constructs the NBER recession indicator variable. The recession dates published by the NBER Business Cycle Dating Committee define closed intervals [peak, trough] during which the U.S. economy is in recession. The indicator variable takes value 1 if the examined month falls within such an interval, 0 otherwise. Constructing the twelve-month-lagged indicator means associating each month t with the indicator value at month t+12 — this lagged indicator constitutes the dependent variable of the probit model.

The fourth step estimates the α and β coefficients by maximum likelihood. In practice, this estimation is done via a numerical solver (Python statsmodels, R glm with binomial probit family, Stata probit) that maximizes joint log-likelihood. The values obtained should reproduce within 1 percent the coefficients published by the NY Fed on the same sample.

Once coefficients are estimated, the conditional probability at a given month is computed directly as Φ(α + β × T10Y3M_t). The value of Φ for a given argument is tabulated in all statistics libraries, or can be numerically approximated by integration of the standard normal density. For verification, the NY Fed also publishes monthly probability values since 1959, allowing comparison of the replication with the official reference.

Model limits and the 2022-2026 academic debate

The NY Fed univariate probit model is not exempt from critique, and the 2022-2026 academic debate has amplified certain operational limits that the post-inversion non-recession of 2022-2024 has made salient.

The first limit is parsimony. A univariate model ignores by construction all other variables that could inform on the economic cycle: industrial production, employment, financial conditions indicators, sentiment indicators. The defense by Estrella and Mishkin (1996) remains valid — adding explanatory variables does not improve out-of-sample predictive power — but it applies to the historical calibration sample, not necessarily to structurally different economic regimes.

The second limit concerns potential signal distortion by post-2008 quantitative easing policies. Affine term-structure models such as the Adrian-Crump-Moench estimation maintained by the NY Fed suggest that the 10-year term premium remained markedly negative across 2022-2024 (between -50 and -100 basis points), which would make T10Y3M more negative than it would be under a normalized term-premium regime. The debate is not settled, but it weakens the direct quantitative reading of the probit number.

The third limit is the absence of fiscal policy consideration. The probit model embeds no federal-deficit variable, even though the coexistence of monetary tightening (which transmits the recession signal via the credit channel) with strong fiscal expansion (which supports aggregate demand) can partly neutralize the contractionary effect. This configuration was not dominant across the eight calibration recessions (1968-2020), but it was central across 2022-2024 — federal deficit between 5 and 7 percent of GDP, unprecedented historically outside recessions or conflicts. The detail of this economic anomaly is documented in the 2022-2024 T10Y3M inversion, the longest since 1980, where the model peaked at 71 percent conditional probability without an actual recession. How the copper-gold ratio tracks the 10-year yield places this observation in its macro frame.

A fourth, more subtle limit deserves note: the probit specification produces a smooth probability function rather than a discrete recession signal. The conventional reading thresholds (30 percent as alert, 50 percent as strong signal) are post-hoc interpretive conventions, not properties endogenous to the model. The model itself outputs a continuous probability that requires editorial framing to translate into operational guidance. In the 2022-2024 episode, the probability oscillated between 50 and 71 percent for sixteen consecutive months — a sustained alert that historically would have been followed by recession, but that produced no NBER-dated turning point. The interpretation challenge is then less about the model itself than about how to read sustained high probability without confirmed recession.

These limits do not disqualify the probit model as a barometer — it remains the only single-variable indicator to have predicted seven of the eight U.S. recessions since 1968 with a true-positive rate above 87 percent. But they impose a conditional reading: the probit number correctly estimates recession probability in economic regimes comparable to historical calibration. The mechanism is taken apart in the Fed model for equity valuation. In an atypical regime — like 2022-2024, marked by massive fiscal expansion and post-pandemic excess savings — the probit number should be read with an additional margin of uncertainty not encoded in the model. The standard reading grid of central-bank policy and rate-cycle transmission thus benefits from combining the probit signal with effective-transmission indicators (SLOOS, financial conditions, labor market).

Last updated — 14 June 2026