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Suppose the regression specification is $$y_i=\beta_0+\beta_1x_i+\epsilon_i,$$ No matter $x_i$ is stochastic or not, we will need the assumption that $\epsilon_i$ is distributed the same for all $i$. However, if $x_i$ is a stochastic random variable rather than a fixed-value, another assumption is needed, namely the disturbance term has zero conditional expectation; in other words, $\epsilon_i$ is distributed independent of $x_i$.

My question is how does this assumption even make a difference in practice? I feel like in practice, there is no way to assess whether $\epsilon_i$ is distributed independent or dependent of $x_i$ since we only have one observation of $(x_i,y_i)$ for each $i$.

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In practice the difference is huge. The exogenous assumption that you refer to requires that the errors are not correlated with regressors. If they're correlated then you can't rely on the regressions with stochastic regressors.

For instance, in observational studies, such as pretty much all economics, you do not control the regressors. You can not set US GDP to a desired level, you can only observe it. Hence, in the model where GDP is a regressor, you want errors to be independent of GDP, because in this model you can only assume stochastic regressors.

When your errors are correlated with regressors you get endogeneity issue. There are ways to handle it, such as using lagged regressors or instrumental variables.

In econometrics a textbook example is the impact of the exogenous price on the demand. We're talking about typical demand-supply equations. Here, the problem is that the prices also depend on the supply. Hence, there is an endogeneity issue, which any econometrician will promptly point out. This is to answer your question on feasibility of testing the assumption.

Once you figured that endogeneity is here, you may look for a so called instrumental variable. These are regressors which are correlated with the price but not with demand, i.e. something that may impact the supply, for instance. If the demand is for oranges, then maybe a temperature in Florida in Spring would be a suitable instrument, because it's going to impact supply of oranges - and price - but not the demand. So, you plug this instrument into the regression and tease out the impact of the price on demand

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Note that we do not require $\epsilon_i$ to be have the same distribution for all $i$. Unequal variance can be handled through weighted least squares or standard errors made robust to heteroskedasticity, while correlations among the error term can be handled using Huber-White standard errors.

I agree that we can never assess whether $\epsilon_i$ is correlated with $x_i$. In my current work, the covariate of interest is typically randomly assigned, so we can assert that it is independent of the error term. Other included regressors might not be, but they too are uncorrelated with the regressor of interest and so do not affect the estimate of its coefficient.

My formal training is in economics, where observational studies are more common. There we appeal to outside knowledge to assess this assumption. For example, the regression of wages on years of schooling does not estimate the parameters of a conditional expectation because the error term contains things like motivation, which are correlated with years of schooling. A great deal of effort in economics is put into identifying credible variation, though ultimately the credibility of such observational analyses is debatable.

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