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I've run a regression $Y = a + \beta_1X_1 + \beta_2X_2 + \epsilon$, and I have an interest in testing which coefficient of $X_1$ and $X_2$ has a stronger impact on $Y$.

Here are the parameter estimates: $\hat{\beta_1} = 0.086$, $se(\hat{\beta_1}) = 0.019$, $\hat{\beta_2} = 0.068$, $se(\hat{\beta_2}) = 0.051$. $cov(\hat{\beta_1}, \hat{\beta_2}) < -0.001$.

Obviously, by t-test, $\hat{\beta_1}$ is significant (p. < 0.001) whereas $\hat{\beta_2}$ is insignificant (p. > 0.1). However, Wald test of equality of two coefficients fails to reject the null. So, we cannot conclude that $X_1$ has a stronger impact than $X_2$ on $Y$.

Questions is, how should we understand the difference in significance (by t-test) between $\hat{\beta_1}$ and $\hat{\beta_2}$, when Wald test fails to reject that $\hat{\beta_1}$ and $\hat{\beta_2}$ are equal?

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In both cases you're testing completely different things. In other words, from a theoretical point of view, the hypothesis of interest is different.

When performing the t-test, your hypothesis of interest is $$ H_0: \beta_i = 0\: \text{ vs }\: H_1: \beta_i\ne0\:\:i\in\{1,2\} $$ whereas in the Wald test you're testing $$ H_0: \beta_1=\beta_2\: \text{ vs }\: H_1: \beta_1\ne\beta_2 $$ However, based on the way you formulated your question, it appears to me that your null hypothesis is something of the form: $$ H_0: \begin{pmatrix}1&0\\0&1\\1&-1\end{pmatrix}\begin{pmatrix}\beta_1\\\beta_2\end{pmatrix}=\begin{pmatrix}0\\0\\0\end{pmatrix} $$ and therefore the Wald Test you need to perform for the above's null hypothesis.

A final word regarding your example at hand, sometimes your inference may be flawed because you did not take into account certain features of your data when 1) estimating the parameters of interest, or 2) when using the correct critical values. For example, the normal approximation (or $\chi^2$) is not a great one if you have very few observations since it's an asymptotic approximation.

Hope this helps.

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  • $\begingroup$ I think you made an error with your null hypothesis. You multiply a 3x2 matrix by a 2x1 matrix and arrive at a 2x1 matrix, which is incorrect. Also, I think OP is just asking about whether the coefficients are equal to each other. $\endgroup$ – Noah Sep 28 '18 at 5:11

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