The following diagrams show which assumptions are required to get which implications in the finite and asymptotic scenarios.

[![Finite OLS Assumptions][1]][1]

[![asymptotic OLS assumptions][2]][2]

# Linear Regression Assumptions: Key Points #
Generally the assumptions can be broken down into what we need for our coefficient estimators

 1.  to be right on average--unbiased--or right with infinite data--consistent and
 2. to follow a certain distribution so we can know how precisely we are measuring them.

## Unbiasedness / Consistency ##

We want our coefficients to be right on average (unbiased) or at least right if we have a lot of data (consistent).

If you want unbiased coefficients, the key assumption is strict exogeneity. This means that the average value of the error term in the regression is 0 given the covariates used in the regression.

For consistent coefficients, the key assumption is “predetermined regressors” which is implied by: "there is no correlation between the error term and any of the covariates of the regression" if a constant is included in the regression.

Strictly speaking, there is no way to confirm these assumptions are right without randomly assigning the covariate whose coefficient you want to get right. Without random assignment, you have to make a qualitative argument that the assumptions are met. However, if you make a scatter plot of residuals on the y axis and the predicted outcome value on the x axis and there is a systematic trend away from 0, that’s a sign this assumption (or the linearity assumption) is not met.

Assumptions are also important to understand the precision of coefficient estimates.

## Understanding the Precision of the Coefficients ##
Homoskedasticity and normality are not needed for unbiased/consistent coefficients. You only need these additional assumptions if you want to get a sense of the precision with which you are measuring your coefficients with shortcut methods (e.g. F tests). However, you can always use heteroskedasticity robust standard errors, bootstrapping, or randomization inference to understand precision instead (descriptions and examples of these latter procedures can be found in my post [here][3]).


  [1]: https://i.sstatic.net/VUQAQ.png
  [2]: https://i.sstatic.net/7Fa14.png
  [3]: https://towardsdatascience.com/practical-experiment-fundamentals-all-data-scientists-should-know-f11c77fea1b2