Ordinary Least Squares (OLS)

Fit an Ordinary Least Squares (OLS) regression of dependent_ticker on independent_tickers, using the selected column (returns by default) of each ticker.

OLS estimates the coefficients that minimise the sum of squared residuals, i.e. the line (or plane) that best explains the dependent ticker from the independent tickers.

The formula is as follows:

Each coefficient is the expected change in dependent_ticker for a one-unit change in that independent ticker, holding the others fixed, and the intercept is what remains when all of them are zero. With returns this is the familiar way to estimate how strongly one asset moves with others (e.g. a beta against an index).

Under the classical assumptions (linearity, no perfect multicollinearity, homoskedastic and uncorrelated errors) OLS is the Best Linear Unbiased Estimator (Gauss-Markov). The coefficients stay unbiased when the errors are heteroskedastic, but the classical standard errors do not: check with get_breusch_pagan_test or get_white_test and, if needed, refit with a robust cov_type such as “HC1” (or “HC3” for small samples), or “HAC” when the errors are also autocorrelated.

For more information about the HAC covariance, see the following paper:

• Newey, W.K. & West, K.D. (1987). “A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.” Econometrica, 55(3), 703-708.

Also known as: linear regression, least squares regression.

Ordinary Least Squares (OLS) in Python or via the MCP

Install the Finance Toolkit with:

pip install financetoolkit -U

Then call get_ols as shown below.

from financetoolkit import Toolkit

toolkit = Toolkit(
    ["XOM", "USO"],
    api_key="FINANCIAL_MODELING_PREP_KEY",
    start_date="2021-01-01",
    end_date="2025-12-31",
)

# XOM (the first ticker) is dependent; USO and the Benchmark are independent.
# dependent_ticker and independent_tickers choose others, cov_type='HC1' gives
# heteroskedasticity-robust and cov_type='HAC' (with maxlags) Newey-West standard errors.
toolkit.econometrics.get_ols(include_benchmark=True)

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

Which returns:

Using Your Own Data

get_ols 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_ols()

Parameters

get_ols accepts the following parameters.

Related Regression