## 4.15 Inference in Practice

I have covered the main approaches to inference in this section. I’d like to make a couple of concluding comments. First, all of the approaches discussed here (standard errors, t-statistics, p-values, and confidence intervals) are very closely related (in some sense, they are just alternative ways to report the same information). They all rely heavily on establishing asymptotic normality of the estimate of the parameter of interest — in fact, this is why we were interested in asymptotic normality in the first place. My sense is that the most common thing to report (at least in economics) is an estimate of the parameter of interest (e.g., $$\hat{\theta}$$ or $$\bar{Y}$$) along with its standard error. If you know this information, you (or your reader) can easily compute any of the other expressions that we’ve considered in this section.

Another important thing to mention is that there is often a distinction between statistical significance and economic significance.

In the next chapter, we’ll start to think about the effect of one variable on another (e.g., the effect of some economic policy on some outcome of interest). By far the most common null hypothesis in this case is that “the effect” is equal to 0. However, in economics/social sciences/business applications, there probably aren’t too many cases where (i) it would be interesting enough to consider the effect of one variable on another (ii) while simultaneously the effect is literally equal to 0. Since, all else equal, standard errors get smaller with more observations, as datasets in economics tend to get larger over time, we tend to find more statistically significant effects. This doesn’t mean that effects are getting bigger or more important — just that we are able to detect smaller and smaller effects if we have enough data. And most questions in economics involve more than just answering the binary question: does variable $$X$$ have any effect at all on variable $$Y$$? For example, if you are trying to evaluate the effect of some economic policy, it is usually more helpful to think in terms of a cost-benefit analysis — what are the benefits or the policy relative to the costs and these sorts of comparisons inherently involve thinking about magnitudes of effects.

A more succinct way to say all this is: the effect of one variable on another can be both “statistically significant” and “economically” small at the same time. Alternatively, if you do not have much data or the data is very “noisy”, it may be possible that there are relatively large effects, but that the estimates are not statistically significant (i.e., you are not able to detect them very well with the data that you have). Therefore, it is important to not become too fixated on statistical significance and to additionally think carefully about the magnitudes of estimates.