How Our Stagflation Scenario Probabilities are Derived Using Multinomial Logistic Regression

 

What is Logistic Regression (LR)?

What it does:

• It predicts the probability of an outcome (like yes/no, default/no default) from input features (like income, age, credit score).
• Instead of drawing a straight line through the data, it fits an S‑shaped curve that maps any input to a probability between 0 and 1.

Interpreting the image above

• Feature space: Think of each axis as one standardized input (e.g., core CPI z-score vs. unemployment z-score). Real models use more than 2 features; this is a 2D illustration.
• Multinomial logistic regression: For each scenario k, it learns a linear score Score_k = w_k · x + b_k. Applying softmax to all four scores yields a probability surface per class.
• Boundaries: The class decision boundaries are linear (straight lines in 2D), but probabilities change smoothly across them, which is the S-curve behavior generalizing to multiple classes.

Why 4 classes but only 2 features in the chart

• The model has four scenarios (classes): Inflation-first, Soft-landing, Recession-first, and Simultaneous-shock. Multinomial logistic regression learns one linear score per class and converts them to probabilities with softmax.
• The visualization uses only two features on purpose so you can see the decision regions in a 2D plot. It’s a didactic projection of a higher‑dimensional model into two axes, not a statement that the real model only has two inputs.

How it works (intuitively)

• It looks for a weighted combination of your inputs that best separates the two groups.
• Those weights are the model’s “importance” for each input.
• The weighted sum is passed through an S‑curve (the logistic/sigmoid function) to turn it into a probability.

What the numbers mean

• Coefficients (weights): Positive means “more of this feature increases the probability of the event,” negative means it decreases it (assuming the feature was standardized).
• Odds/log-odds: Under the hood, the model is linear in log-odds. You don’t need to use that day-to-day, but it’s why coefficients are easy to interpret in terms of “how much the odds change.”

Why it’s useful

• Calibrated probabilities: You get not just a class label, but how confident the model is.
• Interpretable: You can see which features push the probability up or down.
• Stable with regularization: Add L2/L1 penalties to prevent overfitting and keep weights sensible.

Why multinomial?

• Because we have 4 scenarios for stagflation: Recession first, Inflation first, Simultaneous recession with Inflation, soft landing
• Binary logistic regression: Two outcomes (e.g., recession vs. no recession).
• Multinomial logistic regression: More than two outcomes (e.g., our four scenario classes); it estimates a set of weights per class and outputs a probability for each.

A tiny example

• Inputs: inflation, unemployment, credit spread.
• The model learns weights like (+ for inflation, + for unemployment, + for credit spreads) for a “recession risk” class.
• Given this week’s data, it computes a weighted sum, runs it through the S‑curve, and says “there’s a 34% chance of recession-first.”
• In short: logistic regression finds a simple, interpretable rule that converts inputs into well-behaved probabilities for one or more classes. It’s popular because it’s fast, transparent, and often good enough—especially when you value explainability.

How logistic regression sets weights

• Setup: We standardize each input (z-scores vs 2010–present) so coefficients are directly comparable. We then fit a multinomial logistic regression to classify each week into one of four scenarios (Inflation-first, Soft-landing, Recession-first, Simultaneous-shock).
• Weights = coefficients: The model estimates coefficients that link each input to the log-odds of each scenario. Larger absolute coefficients mean that input moves the probability more (given standardization).
• Regularization and stability: We use L2 regularization to prevent overfitting and stabilize weights; outliers are clipped before fitting.
• From weights to probabilities: For a given week, scenario scores are the linear combination of inputs times their coefficients; applying softmax turns those scores into probabilities. We apply a small temporal smoothing on the probability outputs to reduce whipsaws.

Notes

These weights reflect the latest calibration: initial coefficients from multinomial logistic regression and information value screening, then a small expert adjustment to reflect robustness and interpretability. We review quarterly and when regime shifts are detected.

How to map this to our case

Think of the axes as three standardized features, e.g.:

X: Core CPI (z-score)

Y: Unemployment (z-score)

Z: 2s10s curve (z-score)

Our real model uses 7 features, so the true decision “surfaces” live in 7D. This 3D plot helps you visualize how multinomial logistic classification carves space with linear boundaries, even though we can’t draw all 7 dimensions.