central limit theorem
What is wrong with the following sentence:
The Central Limit Theorem implies that, as the sample size grows, the error distribution approaches normality.
Am I correct by saying that it should in stead state the MEAN of the sample error approaches zero as sample size grows?
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.
A single random variable has a distribution; a sample mean from a random sample is a single random variable. Of course you can only observe its distribution by looking at multiple random samples (such as multiple sample means); then as the number of such samples increases the sample (empirical) cdf will approach the population distribution function. The standard error of the sample cdf about the population cdf decreases as the square root of sample size (quadruple the sample size and you halve the standard error).
In short, the number of samples you take (each of size $n$) has no impact on how close the distribution of sample means is to being normal ... only on how accurately you can see it when you look at a collection of sample means all from samples of the same size.
To see how close you are to normality at some sample size, you may need a substantial number of sample means. In simulation experiments it is common to look at thousands of such samples so as to get a good sense of the distributional shape.
The picture shows histograms of 20, 300 and 100000 sample means for samples of size n=30 from a skewed distribution. We have some sense of the broad shape in the first one, a somewhat clearer sense of it in the second one, but we get a pretty clear idea of the shape of this distribution of sample means in the third one, where we have a large number of realizations of the sample mean.
In this case sample means don't have close to a normal distribution; n=30 would not be sufficient to treat these means as approximately normally distributed (at least not for typical purposes).
If you want a good sense of how the tails of the distribution behave you may need considerably larger numbers of sample means.
However, when you're dealing with real data, you generally only get a single sample. You have to base your inference (whether you rely on the CLT or not) on that one sample.
You may have been misled about what the central limit theorem says.
The actual central limit theorem says nothing whatever about n=30 nor about any other finite sample size.
It is instead a theorem about the behaviour of standardized means (or sums) in the limit as n goes to infinity.
While it's true that (under certain conditions) sample means will be approximately normally distributed (in a particular sense of approximate) if the sample size is large enough, what constitutes 'large enough' for some purpose depends on several factors. As we see in the plot above, skewness can (for example) have a substantial impact on the approach to normality (if the population is skewed, the distribution of sample means is also skewed but less so with increasing sample size).
Best Answer
In its standard simplest form, the Central Limit Theorem (CLT) is a statement about the cumulative distribution function of the random variable $$Z_n = \frac{X_1 + X_2 + \cdots + X_n -n\mu}{\sigma \sqrt{n}}$$ where the $X_i$ are independent identically distributed random variables with mean $\mu$ and standard deviation $\sigma$. The CLT asserts that for each $a$, $-\infty < a < \infty$, $$F_{Z_n}(a) = P\left\{\frac{X_1 + X_2 + \cdots + X_n -n\mu}{\sigma \sqrt{n}} \leq a \right\} \to \Phi(a) = \int_{-\infty}^a \frac{e^{-x^2/2}}{\sqrt{2\pi}}\mathrm dx$$ as $n \to \infty$.
If by "error distribution" you mean the distribution function of $$Y_n = \left(\frac{1}{n}\sum_{i=1}^n X_i\right) -\mu = \frac{\sigma}{\sqrt{n}}Z_n,$$ that is, the difference of the sample mean $\bar{X} = n^{-1}\sum_iX_i$ and the population mean $\mu$, then the CLT certainly does not imply that $F_{Y_n}(\cdot)$ "approaches normality" as the sample size $n$ grows large in the usual sense of normality, though nitpickers may want to claim that the distribution is approaching a normal distribution with mean $0$ and standard deviation $0$ (often called a constant by statistically illiterate people).
On the other hand, the mean of the sample error is not a random variable but a constant (in fact, $0$ since the sample mean is an unbiased estimator of the population mean) and does not need to approach $0$; it is already there! I think what you meant to say is that the distribution $F_{Y_n}(a)$ of the sample error approaches the unit step function: $$F_{Y_n}(a) \to u(a) = \begin{cases}1, & \text{if}~a > 0,\\ 0 &\text{if}~a < 0,\end{cases}$$ which is certainly correct, and follows from the CLT, but also follows from results such as the weak law of large numbers which makes no assertions about the distribution of $Z_n$, only about $Y_n$.