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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.

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  • $\begingroup$ Just be consistent in the symbols when you post a thread $\endgroup$
    – Louise
    Jul 2 at 22:43
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Yes it is important indeed to use precise language to clarify methods and issues, especially in econometrics, a discipline in which fundamental but confusing nuances between different types of variables: some are endogenous, other exogenous, observed, unobserved, random, conditionally random, etc. I did not often encounter such subtleties in mathematics (or even statistics).

Regarding the parameter vector $\beta$: it is fixed but unknown, and so it makes no sense to ask ourself whether it is significantly different from zero, because $\beta$ is not random, just unknown by the econometrician (sometimes the $\beta$ are known by the individuals or the firms, which raises interesting methodological issues, but they are not known by the econometrician). Instead, our estimate $\widehat{\beta}$ of $\beta$ is random, because it is inferred from the data which are random (or realisations of random variables). We use $\widehat{\beta}$ together with its variance, to build plausible confidence intervals (or ellipsoids) for the true (but unobserved) values of $\beta$. Hypotheses are stated in terms of what we really would like to know: the true value of $\beta$ (once again, $\beta$ is unobserved but deterministic). For instance we state: "$H_0: \beta_2=0$" or equivalently "the tariff on imported steel has no impact on domestic steel production". As we cannot observe $\beta_2$, we compute an estimate of it $\widehat{\beta}_2$ which we use to ask ourselves whether it is plausible that $\beta_2=0$.

Regarding $x_j$, in a model linear in both $\beta_j$ and $x_j$, it is equivalent to say that $x_j$ has a significant impact on $y$ (at a given threshold) and that the associated parameter $\beta_j$ is significantly different from zero. My impression is that there is no ambiguity in Wooldridge's textbooks where a lot of care is given to the choice of the appropriate wording. Unfortunately we are often sloppy and indeed say ambiguous things like: "$x$ is statistically significant, or statistically different from zero, at the 5% level" while we actually mean "the effect of $x$ is statistically significant" and not $x$ itself. Wooldridge correctly writes "We usually say..." which does not mean that this is the best practice.

EDIT: The two claims:
(i) "$X_j$ is statistically different from zero"
(ii) "$X_j$ has a impact which is statistically different from zero"
are logically independent, in the sense that in a data generating process, all 4 possible cases can occurr: it is possible that

  • (i) and (ii) are satisfied,
  • either (i) or (ii) are satisfied,
  • neither (i) nor (ii) are satisfied

So yes, in a linear regression, it is wrong saying that "$x_j$ is statistically significant" instead of "$\widehat{\beta}_j$ is statistically significant", but this abuse of language is so common, that even most respected scholars may use it from time to time. In our lectures, however, we should do our best and fight against this habit.

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    $\begingroup$ Thinking of $\beta$ as fixed and $\hat{\beta}$ as random is helpful. However, I am still confused as to what the best practice should be. "$\beta$ is statistically significant" and "$\beta$ is statistically different from zero" seem correct to me: Although we do not know whether $\beta$ is truly different than zero, we can say that it is statistically different from zero. "$x$ is statistically significant" or "$x$ is statistically different from zero" seem definitely wrong, while "the effect of $x$ is ss" seems equivalent to the statement that "$\beta$ is ss." $\endgroup$
    – Austin D.
    Jul 4 at 1:22
  • $\begingroup$ @Austin D.: saying that "$\beta_j$ is statistically significant" is an abuse of language, because $\beta_j$ is fixed (but unknown). It has a variance of zero, and is different from zero with probability 1 ($\beta_j=0$ has a measure zero in \mathbb{R}). Instead, I prefer "$\widehat{\beta}_j$ is statistically significant". $\endgroup$
    – Bertrand
    Jul 4 at 10:51
  • $\begingroup$ @Austin D.: I have edited my answer and added a paragraph at the end to address your point about statistical significance of "$x_j$ versus $\widehat{\beta}_j$". $\endgroup$
    – Bertrand
    Jul 4 at 11:05
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    $\begingroup$ I see, I think I understand now. We want to say that the estimator $\hat{\beta}$ is statistically significant (or is statistically different from zero) because we should imagine that if we were to take random samples over and over again that the estimator would give a result different than 0 in 95% of the re-draws (using $\alpha=0.05$). $\endgroup$
    – Austin D.
    Jul 5 at 17:26

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