"Ordinary Least Squares Fitting" and "Maximum Likelihood Fitting" bullets are identical

[codeisnotcode]

In https://github.com/GeostatsGuy/DataScience_Interactive_Python/blob/main/Interactive_Model_Fitting.ipynb
the bullets (near the top) for "Ordinary Least Squares Fitting" and "Maximum Likelihood Fitting" are identical and equation (1) is the same as (2) - both calculate variable Beta-hat^OLS.

So I asked ChatGPT; feel free to use its response.

User
What is the difference between "Ordinary Least Squares Fitting" and "Maximum Likelihood Fitting"

ChatGPT
"Ordinary Least Squares (OLS) fitting" and "Maximum Likelihood (ML) fitting" are both methods used in statistics and econometrics to estimate the parameters of a model based on observed data. While they share some similarities, they differ in their underlying principles and assumptions.

Principle:
OLS: OLS fitting aims to minimize the sum of the squared differences between the observed values and the values predicted by the model.
ML: ML fitting aims to maximize the likelihood function, which is a measure of how likely the observed data are given the parameter values of the model.

Assumptions:
OLS: OLS assumes that the errors (the differences between the observed values and the predicted values) are normally distributed with a mean of zero and constant variance.
ML: ML makes fewer assumptions about the distribution of errors and instead focuses on maximizing the likelihood of observing the actual data under the assumed distribution.

Application:
OLS: OLS is commonly used in linear regression models, where the relationship between the independent variables and the dependent variable is assumed to be linear.
ML: ML is more general and can be applied to a wider range of models, including models with non-linear relationships and models with different distributions of errors.

Efficiency:
ML: ML estimation tends to be more efficient than OLS estimation when the assumptions of the ML method are met. This means that ML estimators often have smaller variances and are closer to the true parameter values.
OLS: OLS estimation can be less efficient if its assumptions are violated, leading to biased and inconsistent parameter estimates.

In summary, while both OLS and ML fitting are methods used for parameter estimation in statistical modeling, they differ in their underlying principles, assumptions, and applications. OLS is more restrictive in its assumptions and application but is simpler and easier to implement, while ML is more flexible and efficient but requires fewer assumptions and can handle a wider range of models.

[GeostatsGuy]

Thank you for the assistance!