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I'm trying to test whether the coefficient for one independent variable ($X_1$) is larger than coefficient for another variable ($X_2$) in predicting the dependent variable ($Y$). For example, my hypothesis states: The effect of $X_1$ on $Y$ is larger than the effect of $X_2$ on $Y$. In my regression analysis, it turns out that $X_1$ is significant while $X_2$ is not.

In this case, can I conclude that the effect of $X_1$ on $Y$ is larger that that of $X_2$? If I use the test command in Stata, the two effects are not statistically different. Is the test still meaningful when one variable is not significant? I look forward to your input.

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You can conclude from multiple regression that $Y$ relates more strongly or consistently to $X_1$ than to $X_2$ in your sample, but your slope coefficients are parameter estimates, so if you're looking to infer the same conclusion about the population, the test for differences in the coefficients is the appropriate one to interpret. Yes, it is still appropriate when the test of either individual coefficient is insignificant.

The default null hypothesis significance test for a regression coefficient compares it to zero, which presumably is not the value of your parameter estimate $\hat\beta_{X_2}$. Thus if, e.g., $\hat\beta_{X_1}=.3, p<.05$, but $\hat\beta_{X_2}=.1,p>.05$, you might expect $\hat\beta_{X_1}$ to differ more significantly from zero than from 0.1.

Furthermore, the uncertainty in your estimate should be taken into account, whereas there is no standard error of the null hypothesis. To extend the previous example, even a $\hat\beta_{X_2}=-.1$ would admit values of $\beta_{X_2}>0$ within its confidence interval. Thus your test of differences in the coefficients may have two reasons to be more conservative than your tests of individual coefficients.

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  • $\begingroup$ Hope I used the $\hat{\rm hats}$ appropriately; let me know (or just edit them out) if they're only appropriate for predictions and not parameter estimates... $\endgroup$
    – Nick Stauner
    Jul 21, 2014 at 23:21
  • $\begingroup$ Thank you for your answer, Nick! This is very helpful. I have a follow-up question. 1) Does the same apply for poisson and negative binomial regression, not just OLS? 2) If the units of two independent variables are different, is there a way to compare the standardized coefficients of these two? Thank you so much! $\endgroup$
    – esc123
    Jul 21, 2014 at 23:44
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    $\begingroup$ Glad to help :) 1. AFAIK, yes, this logic applies equally to all generalized linear models. 2. Yes, standardizing the variables before entering them into regression will remove differences in scale, permitting direct comparison of the resultant standardized coefficients. N.B. Different methods of comparing coefficients apply when they are estimated in one common model vs. two (or more) separate models. $\endgroup$
    – Nick Stauner
    Jul 21, 2014 at 23:53

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