Probit Regression

Fit a Probit Regression of whether dependent_ticker’s return is positive on independent_tickers.

As with get_logistic_regression, the dependent variable is turned into a binary outcome (1 when the return is positive, 0 otherwise). The probability of that outcome is modelled through the standard normal cumulative distribution function instead of the logistic function.

The formula is as follows:

\[\text{yₜ} = 1 \;\text{if the dependent ticker's}\; \text{column} (\text{Return by default}) \text{is above}\; 0,\;\; \text{else 0}\] \[P(\text{yₜ} = 1 | \text{xₜ}) = \Phi(\text{xₜ'beta}),\;\; \text{with xₜ the independent tickers' column plus a constant}\] \[\text{betâ} = \text{argmax}\; \text{Σ} \left[\text{yₜ} \cdot \ln(\text{pₜ}) + (1 - \text{yₜ}) \cdot \ln(1 - \text{pₜ})\right],\;\; \text{solved by Newton-Raphson (statsmodels Probit)}\] \[\text{Pseudo}\; R ^{2} = 1 - \text{ln}\; L (\text{betâ}) / \text{ln}\; \text{L₀} (\text{McFadden}),\;\; \text{with L₀ the likelihood of a constant-only model}\]

Where Phi is the standard normal CDF. The model is fitted by Maximum Likelihood. A positive coefficient means a higher probability of a positive return, but the size of a coefficient is not a log-odds ratio as in the Logit model. In practice Probit and Logit give very similar fitted probabilities; Probit is the convention in parts of the econometrics literature and underlies two-step estimators such as the Heckman selection correction.

For more information about the method, see the following paper:

Bliss, C.I. (1934). “The Method of Probits.” Science, 79(2037), 38-39.

Also known as: probit model.

Probit Regression in Python or via the MCP

Install the Finance Toolkit with:

pip install financetoolkit -U

Then call get_probit_regression as shown below.

from financetoolkit import Toolkit

toolkit = Toolkit(
    ["TSLA", "QQQ"],
    api_key="FINANCIAL_MODELING_PREP_KEY",
    start_date="2025-01-01",
    end_date="2025-12-31",
)

# TSLA (the first ticker) is dependent; QQQ is independent
toolkit.econometrics.get_probit_regression()

Or, after setting up the Finance Toolkit MCP, ask:

How well did the daily returns of the Invesco QQQ Trust predict whether Tesla rose on a day in 2025, according to a probit model?

Which returns:

	Coefficient	Std. Error	z-Statistic	P-Value
Intercept	-0.0848	0.0913	-0.9294	0.3527
QQQ	78.1324	11.1444	7.0109	0

Using Your Own Data

get_probit_regression runs on any data, not only prices: ratios, economic indicators, yield curves or your own dataset. Pass a DataFrame, a Series or a dictionary of them to Econometrics as data, with the dates in the rows or the columns, and set changes to how its changes are calculated: "percentage", "log", "difference" for rates and ratios, or None when the data are changes already. For example, for the monthly unemployment rates of the United States and Germany:

from financetoolkit import Economics, Econometrics

economics = Economics(start_date="2000-01-01")

unemployment = economics.get_unemployment_rate(
    countries=["United States", "Germany"], period="monthly"
)

econometrics = Econometrics(data=unemployment, changes="difference")

econometrics.get_probit_regression()

Parameters

get_probit_regression accepts the following parameters.

Parameter	Type	Description	Example
`dependent_ticker`	`str \| None`	The dependent asset (whose up/down direction is predicted). Defaults to None, meaning the Toolkit instance’s first ticker.	`"AAPL"`
`independent_tickers`	`str \| list[str] \| None`	The independent (predictor) asset(s). Defaults to None, meaning every other ticker in the Toolkit instance besides `dependent_ticker`.	`["MSFT", "Benchmark"]`
`include_benchmark`	`bool`	Whether to include “Benchmark” in the default independent ticker(s) (has no effect when independent_tickers is given explicitly). Defaults to False.	`True`
`period`	`str`	The data frequency (daily, weekly, monthly, quarterly, or yearly). Defaults to “daily”.	`"weekly"`
`column`	`str`	The historical data column to derive returns from. Defaults to “Return”.	`"Return"`
`add_constant`	`bool`	Whether to include an intercept. Defaults to True.	`False`
`rounding`	`int \| None`	The number of decimals to round the results to. Defaults to None.	`2`

Probit Regression in Python or via the MCP

Using Your Own Data

Parameters

Related Regression