Economic and Statistical Significance Of Coefficient

Question

I want to know If I am comprehending these terms correctly

Summarizing the difference between economic and statistical significance of coefficients (Describing the terms, process of assessing each with formulas) Statistical Significance is driven from a large estimate or small standard error where we look at t-tests or p-values to determine whether or not to reject a null hypothesis. While Economic significance looks more at the magnitude and sign of the estimated coefficient, if the numbers turn out to be so small then the x variables do not really affect the y variable.

The process of assessing with formulas For Statistical significance wouldn't assessing the process with formulas just include the formulas to calculate t-test and p-values to determine if we can or cant reject the null. Such as the t*, critical values, p-value

For Economic significance isn't it 𝑝𝑟𝑜𝑗𝑒𝑐𝑡𝑒𝑑𝐻𝐴𝑇=𝐵0+𝐵1𝑥1+....+𝐵𝐾𝑋𝑖. and dealing with 𝑅2?

Question 2 Why one does not necessarily mean the other?

(For Whoever marked my last one as a duplicate I don't see why, My question asks about Statistical Significance and Economic while the other one just talked about economic, and I looked at his question and feel maybe it could answer 10% of my question) If whoever first commented could respond again, the answer was so helpful I wanted to show my friends but got burdened this morning waking up to seeing the post was gone.

The previous question got "duplicated" by this "Is there any standard measure of economic significance?" post.

Answer 1

Statistical significance refers to situation where we can say the coefficient is statistically significant at some level based on test statistics (in regression $t$-statistics given by $\hat{\beta}/(s.e.(\hat{\beta}))$) and corresponding $p$-value.

Economic significance indeed depends on the magnitude of the coefficient. As pointed out in this question by KennyLJ sometimes rule of thumb when measuring economic significance is to look at if one standard deviation change in $x$ would result in 'sufficiently large' (which could be 1/2 of standard deviation or also 1 standard deviation change in $y$ - this is very context and field dependent - what is economically significant simply requires good knowledge of wider economic literature).

For example, if we would estimate regression of daily wages ($w$), measured in ${\\\$}$, on education $(E)$, measured in years of education:

$$w_i = \beta_0 + \beta_1 E_i + e_i $$

and if we would estimate that $\hat{\beta_1} = 0.01$, and this estimate would be statistically significant at $1\%$ due to $t\text{-stat}=10$, we would say the coefficient is statistically significant but economically insignificant because if one extra year of education on average increases daily wages just by ${\\\$}0.01$ and that is nothing. Even if the coefficient is statistically significant from economic perspective it might as well be just zero. However, if $\hat{\beta_1} = 20$ meaning that on average one year of education increases daily wage by ${\\\$}20$, it would be both statistically and economically significant because having extra $20$ dollars per day thanks to one year of education is non-trivial.

Next the estimated wage $\hat{w_i}= \hat{\beta_0} + \hat{\beta_1} E_i$ would be just that - an estimated/predicted value by the model given what $E_i$ is. Since the predicted value is derived from estimates of $\beta_0$ and $\beta_2$ it can tell you about statistical significance of the result only as much as just looking at the betas individually.

Lastly $R^2$ is measure that tells you what amount of variation in the sample can the model explain. For example, $R^2=0.25$ would tell you that your model can explain. This is important to take into consideration but again does not tell you about economic significance of the result. For example, it is possible to have results where betas are for all practical perspective very close to zero, but there is so little variation in the data that they are significant and you get excellent fit (high $R^2$), but that would not necessarily made the result economically significant if the magnitude of the effect is relatively miniscule.