Normally distributed errors and the central limit theorem

Question

In Wooldridge's Introductory Econometrics there's a quote:

The argument justifying the normal distribution for the errors usually runs something like this: because $u$ is the sum of many different unobserved factors affecting $y$, we can invoke the central limit theorem to conclude that $u$ has an approximate normal distribution.

This quote relates to one of the linear model assumptions, namely:

$u \sim N(μ, σ^2)$

where $u$ is the error term in the population model.

Now, as far as I know, Central Limit Theorem states that the distribution of

$Z_i=(\overline{Y_i}-μ)/(σ/√n)$

(where $\overline{Y_i}$ are averages of random samples drawn from any population with mean $μ$ and variance $σ^2$)

approaches that of a standard normal variable as $n \rightarrow \infty$.

Question:

Help me understand how asymptotic normality of $Z_i$ implies $u \sim N(μ, σ^2)$

Answer 1

This may be better appreciated by expressing the result of CLT in terms of sums of iid random variables. We have

$$\sqrt{n} \frac{ \bar{X} -\mu}{\sigma} \sim N(0, 1) \quad \text{asymptotically}$$

Multiply the quotient by $\frac{\sigma}{\sqrt{n}}$ and use the fact that $Var(cX) = c^2 Var(X)$ to get

$$\bar{X}-\mu \sim N\left(0, \frac{\sigma^2}{n} \right)$$

Now add $\mu$ to the LHS and use the fact that $\mathbb{E} \left[a X+\mu\right] = a \mathbb{E}[X] + \mu$ to obtain

$$\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i \sim N\left(\mu, \frac{\sigma^2}{n} \right)$$

Lastly, multiply by $n$ and use the above two results to see that

$$\sum_{i=1}^n X_i \sim N \left(n \mu, n\sigma^2 \right) $$

And what does this have to do with Wooldridge's statement? Well, if the error is the sum of many iid random variables then it will be approximately normally distributed, as just seen. But there is an issue here, namely that the unobserved factors will not necessarily be identically distributed and they might not even be independent!

Nevertheless, the CLT has been successfully extended to independent non-identically distributed random variables and even cases of mild dependence, under some additional regularity conditions. These are essentially conditions that guarantee that no term in the sum exerts disproportional influence on the asymptotic distribution, see also the wikipedia page on the CLT. You do not need to know these results of course; Wooldridge's aim is merely to provide intuition.

Hope this helps.