How to rely on CLT without standardizing coefficients? How can a person rely on Central Limit Theorem if he/she does not want to standardize coefficients in order to stick to the original interpretation?
If I cannot rely on CLT then my inference results are unreliable. So do I have to choose between keeping the original interpretation and the reliability of the inference, or am I missing something?
 A: The usual Central Limit Theorem is as follows. Let $Y_1,\ldots,Y_n$ be an IID sample with $E(Y_i)=\mu,\text{var}(Y_i)=\sigma^2$ for all $i$, where both $\mu,\sigma^2$ are finite. Furthermore, let $\bar Y$ be the sample average. Then
$$
\frac{\sqrt{n}(\bar Y-\mu)}{\sigma}\overset{d}{\to}N(0,1),
$$
where $\overset{d}{\to}$ means convergence in distribution.
Now, in the limit, or when $n$ is sufficiently large, we can replace $\overset{d}{\to}$ by $\sim$.
Thus, in the limit or for large $n$, using properties of the normal distribution we have
$$
\bar Y \sim N(\mu, \sigma^2/n).
$$
To be more precise, only in the limit we have exact equality in distribution (at least, at all the continuity points of the distribution of the sequence) whereas for large $n$ we only have an approximate equality.
In your case, let $\hat \beta$ the the OLS estimator and $\hat{\text{var}}(\hat\beta)$ the estimated variance. Then by the same token,
$$
\frac{\hat\beta-\beta}{\sqrt{\hat{\text{var}}(\hat\beta)}}\overset{d}{\to} N(0,1),
$$
thus for large sample sizes
$$
\hat\beta\sim N(\beta, \hat{\text{var}}(\hat\beta)).
$$
So the normal limiting distribution holds also in the nonstandardized case, again, exactly in the limit and only approximately for large $n$.
A: You require standardizing (in a particular fashion, not necessarily to standard normal) for the CLT itself, otherwise you won't get convergence in distribution.
However, to "use" the CLT to argue that some sum or average should be approximately normal* -- you don't need to standardize, since if the standardized version is approximately normal, so is the unstandardized version.

* how this is to work** without knowing some pretty particular things about the distribution is unclear. The sense in which the distribution of some mean (say) at finite $n$ and a normal distribution are close*** is not necessarily particularly helpful for every possible purpose (e.g. knowing that two cdfs are 'within' epsilon of each other at each point may not be particularly helpful if the thing you need isn't directly related to that -- distributions with very similar cdfs may have some quite distinct properties even though their cdfs are alike).
** 'work' in the sense of being confident that the approximation will lead to sufficient accuracy for your purposes in some particular calculation
*** the value of this closeness at finite $n$ for a particular parent distribution isn't actually given by the CLT either. [The Berry-Esseen inequality does given a bound on the difference in cdf in terms of a scaled absolute third moment, $E(|X-\mu|^3)/\sigma^3$ and the square root of the sample size. No doubt other kinds of bounds should be possible, though offhand I can't name any.]
