I had a discussion with a colleague about a research project he is involved in.
We're living in a regression framework, where the colleague aims to infer about the causal effect of an independent variable X on a dependent variable Y. A model is assumed:
$Y_i = \beta_0 + \beta_1X_i + ... +\epsilon_i$
where the typical assumptions are made about the distributions and the error etc.
Now, the colleague's 'personal' research hypothesis is that the true coefficient $\beta_1$ should be very small or even 0. Meaning: X has little or no effect on Y.
To learn about his hypothesis, he estimates the model parameters (coefficients) by using least squares, obtaining an estimate $\hat\beta_1$.
After obtaining the estimate a hypothesis test is conducted:
$H_0: \beta_1 = 0$ and the p-value of the test statistic is observed.
This is the standard test which is displayed after conducting simple ols, by using ordinary programs like R or Stata.
Now, my colleague argues that observing a very low estimate for the coefficient e.g $\hat\beta_1$ = 0.1 in combination with a very low p-value e. g. p=0.001 (highly significant) would be strong evidence for his personal hypothesis of the true $\beta_1$ being very small or 0.
He also says, that the Case 1 example values would be stronger evidence in favor of his hypothesis compared with an example where the coefficient estimate would be some higher value e.g. $\hat\beta_1$ = 0.6 and the p-value would be e.g. p=0.95 (highly insignificant).
To break the argument down in one sentence: The claim is the case 1 values offer stronger evidence in favor of the 'personal' hypothesis that the true $\beta_1$ is very small or 0 compared with the values of case 2; meaning, X has little or no effect on Y.
Now I understand the argument behind the point but I don't really agree on this because in case 1 we reject the hypothesis that the true coefficient is 0, whereas in case 2 we cannot reject the hypothesis that it is actually 0.
Naturally, the question is:
Which case of values should a researcher desire who wants to obtain evidence in favor of their personal hypothesis of the true coefficient being very small or 0? (The personal hypothesis is that X has little or no effect on Y.)
Note: On purpose, I leave the expression 'very small or 0' as vague as it is. I also accept answers which demand that it cannot be answered without making this statement more explicit. Hoping for a discussion where the statement is as vague as it is though, because I often observe this in applied empirical discussions.