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I have this regression model,
$$\hat{Y}=\hat{a}X_1+\hat{b}X_2+\hat{c}$$
Both $X_1$ and $X_2$ are significant at 0.01 level. $X_1$ and $X_2$ have a same unit. Now I want to find a test that tells me whether $\hat{a}$ is significantly higher than $\hat{b}$ or not? Can I use a simple t-test for this comparison?
Yes, you can use a t-test!
Since regression coefficients are asymptotically Normal, their linear combination is also asymptotically Normal. First, notice that the test $H_0: \hat{a} = \hat{b}$ is equivalent to $H_0: \hat{a} - \hat{b} = 0$. Then you can get a null distribution on $\hat{a}-\hat{b}$,
$$\hat{a} - \hat{b} \sim N(0, Var(\hat{a}) + Var(\hat{b}) - 2Cov(\hat{a},\hat{b}))$$
Which means you can use a t-test with $n-3$ degrees of freedom on the test statistic,
$$\frac{\hat{b} - \hat{a}}{\sqrt{Var(\hat{a}) + Var(\hat{b}) - 2Cov(\hat{a},\hat{b}))}}$$.
R: cov.unscaled equals $(X^\prime X)^{-1}$. That matrix needs to be multiplied by an estimate of the residual variance in order to be an estimated covariance matrix. Instead, use vcov(object) to extract the variance-covariance matrix of the parameter estimate contained in object. If you would like to confirm this, compare sqrt(diag(vcov(fit))) to the "Std. Error" column produced by summary.
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