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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 π‘π‘Ÿπ‘œπ‘—π‘’π‘π‘‘π‘’π‘‘π»π΄π‘‡=𝐡0+𝐡1π‘₯1+....+𝐡𝐾𝑋𝑖. 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.

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  • $\begingroup$ The previous question was closed because you specifically told us that the duplicate question answered your Q. So why did you told us that? If that was not the case I would not close it as a duplicate because in my opinion the two questions were not the same and your question is different, but then you yourself said that the linked answer answered your question $\endgroup$
    – 1muflon1 ♦
    Nov 17 '20 at 14:43
  • $\begingroup$ Wait so the answer that was submitted in the last question was the exact same to that post? $\endgroup$
    – Breakr14
    Nov 17 '20 at 14:46
  • $\begingroup$ The Q that was submitted in the comment by KennyLJ was the Q marked as a duplicate. i.e. this one economics.stackexchange.com/questions/3372/…. Does this linked question and answer answer your q? If not then I think your question is different from that one but if yes then this would be duplicate $\endgroup$
    – 1muflon1 ♦
    Nov 17 '20 at 14:48
  • $\begingroup$ Then no it doesn't, that post talks just about specific standard measures of economic significance. What I am looking for is the terms of statistical and economic significance of the coefficient, and the equations and terms that correlate too said significance. I've been having a hard time comprehending jumping back and forth between 4 projects in a weeks span. $\endgroup$
    – Breakr14
    Nov 17 '20 at 14:51
  • $\begingroup$ Unless that question is talking about Statistical significance of coefficient, but isn't directly referring to it $\endgroup$
    – Breakr14
    Nov 17 '20 at 14:52
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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.

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  • $\begingroup$ Thank you! My professor asked us an extra credit question of "Why one does not necessarily mean the other?" Isn't it because they look at two different things in the equation and for one of them it shows the importance the variable is to the estimated equation while the other one does not? $\endgroup$
    – Breakr14
    Nov 17 '20 at 17:53
  • $\begingroup$ @Breakr14 well that is explained in the second half of my answer for example model with R^=0.99 would have excellent fit, and model coefficient 0.01 would be statistically significant if standard error is (0.0001). But once you look at the economic meaning of the coefficient, for example in above as the effect of extra year of education on wages would you consider spending extra year of education for increase in daily wage of 0.01 - i.e. one cent per day for extra year in school economically significant? I would not - from economic perspective the effect is so small it might as well be zero $\endgroup$
    – 1muflon1 ♦
    Nov 17 '20 at 17:58

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