I've been reading Jeffrey Wooldridge's textbook Introductory Econometrics: A Modern Approach (7th edition) in preparation for a class I will be teaching. I've appreciated the precise language he uses, for example, the distinction between an "estimator" and an "estimate", why "estimate an OLS model" is wrong, and that a hypothesis test is used for statistical inference about the true parameter. However, I've run into a rare case of imprecise language, and now I can't figure out which is correct. The issue involves "statistical significance." On page 127 of his textbook, Wooldridge writes:
We usually say that x is statistically significant, or statistically different from zero, at the 5% level. [bold emphasis mine]
My intuition tells me we should say: $\beta$ is statistically significant or statistically different from zero. It doesn't make sense to say that x is statistically different than zero when we care about the effect of x. Perhaps it is OK to say: x is statistically significant OR $\beta$ is statistically different from zero.
To make matters worse, on page 129, he writes:
We say that $\hat \beta$ is statistically different than zero at the appropriate significance level.
This to me seems very wrong, since he made a point to say that the hypothesis test allows for inference about $\beta$ not $\hat \beta$. Earlier, on pages 122-123, he writes:
We are not testing hypotheses about the estimates from a particular sample. Thus, it never makes sense to state a null hypothesis as $\hat \beta$ = 0 or, even worse, as 0.237 = 0 when the estimate of a parameter is 0.237 in the sample.
Could someone help clear up this confusion? I think it is important as a profession to use precise language, as Wooldridge himself argues on page 97 (and on Twitter) regarding the incorrect use of the phrase "estimating an OLS model":
The problem with using imprecise language is that it leads to vagueness on the most important considerations: what assumptions are being made on the underlying linear model? The issue of the assumptions we are using is conceptually different from the estimator we wind up applying.