I have the following question, any hint would be really welcome:
I am trying to conduct a two-country comparison by running two separate regressions, one for each country, and testing $H_0:\,b_1=b_2$ (using Wald tests), where $b_1$ is the coefficient of an explanatory variable in regression group 1 and $b_2$ is the coefficient of the same explanatory variable in regression group 2 (the two groups have unequal size). However, I have came up with the following outcome: $b_1$ is statistically significant, $b_2$ is statistically insignicant, and $H_0$ cannot be rejected, i.e. the difference between the two coefficients is statistically insignificant.
I know that this outcome is possible in terms of statistics, but it sounds a bit "problematic" in terms of logic. How could someone sensibly justify it?
Best Answer
Conceptually, estimates are fuzzy. Think of them like this on an invisible horizontal number line:
(I don't show the number line itself because the actual values, as we will see, do not matter. You need only understand the metaphor that numerical difference = horizontal distance.)
$b_1$ is shown in cyan; it has a relatively small horizontal spread. The value it is estimating should be somewhere beneath the darker cyan points, possibly under the lighter cyan ones.
$b_2$ is shown in red; it has a relatively large horizontal spread. The value it is estimating should be somewhere beneath the darker red points, possibly under the lighter red ones.
For "significance," both are being compared to a definite number, shown by the horizontal location of the black line. (Usually this number is zero for slopes, but it need not be.)
$b_1$ is "significant" because, despite being fuzzy, it is clearly separated from the line.
$b_2$ is not "significant" because, although the value it estimates may differ from the line, it is so fuzzy that the separation is not clear.
$b_1$ is not significantly different from $b_2$ because they overlap substantially.