# Why is i.i.d. an OLS assumption?

Assume the following linear relationship: $$Y_i = \beta_0 + \beta_1 X_i + u_i$$, where $$Y_i$$ is the dependent variable, $$X_i$$ a single independent variable and $$u_i$$ the error term.

According to Stock & Watson (Introduction to Econometrics; Chapter 4), the second least squares assumption is that data should be i.i.d.

• I understand what this means, and know that data should be representative from the population. But I do not understand HOW exactly the violation of this assumption makes the OLS estimators biased/inconsistent. Is it just via the non-serial correlation of errors assumption?
• I disagree with the premise, at least with the title of your question. I can run an experiment fixing the values of $X$ and still use OLS to analyse the results. I don't need to sample randomly from any population for valid inference. This is a different question to the residuals being i.i.d. which is a key assumption. Mar 11 at 22:28
• @GeorgeSavva did I misunderstand SW's explanation, or you are saying you disagree with them as well? Mar 11 at 22:46
• Your link is broken so I don't know what they mean by random sampling. But it should be fairly obvious that we can choose the $X$ values in our regression however we like. And you've only cited them to support the i.i.d. assumption, not the random sampling aspect and I agree that the residuals of your model (rather than the data itself) should be i.i.d. Mar 11 at 22:59
• What is being randomly sampled are the conditional responses. As @George points out, by selecting the values of $X_i$ to use you are not randomly sampling from any joint distribution of $(X_i,Y_i).$ The heart of your question is the last part: what goes wrong when the conditional responses ("errors") are not i.i.d.? You can find many examples by searching our site, but think about some extreme cases. What if, for instance, all the errors are random but must be equal to one another? What does any such dataset look like and how will it mislead you?
– whuber
Mar 11 at 23:51
• @Dave It's useful to distinguish between observational data and errors in variables. The situations differ in their applications and conceptions of the data. In the former, we wish to estimate the conditional response distributions. In the latter, we wish to estimate the responses conditional on the true values of the explanatory variables, having observed values corrupted by substantial measurement error. This book errs in ruling out any application of regression methods to experimental data, where the explanatory variables are selected by the experimenter and are not random.
– whuber
Mar 12 at 15:18