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Consider the following model $y_i = \beta_0+ \beta_1x_{i1}+ \beta_2x_{i2}+ \epsilon_i$ where $\epsilon_i \overset{iid}{\sim} N(0, \sigma^2)$.

How do I test the null hypothesis that $\beta_1 + 10\beta_2 = 0$?

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Quoting from Wikipedia:

The variance of a sum of two random variables is given by $$\operatorname{Var}(aX+bY)=a^2\operatorname{Var}(X)+b^2\operatorname{Var}(Y)+2ab\operatorname{Cov}(X,Y)$$

where $\operatorname{Cov(⋅, ⋅)}$ is the covariance.

In your case, substitute $a = 1$, $b=10$, $X = \beta_1$, and $Y=\beta_2$. Statistical software will typically provide a variance-covariance matrix for the coefficients of a linear regression model (although you might have to look into the documentation to figure out how to find it). The individual coefficient variances are along the diagonal, the $(\beta_1,\beta_1)$ and $(\beta_2,\beta_2)$ entries respectively. The $(\beta_1,\beta_2)$ entry of that matrix provides the desired covariance value.

The square root of that variance of the weighted sum of the coefficients is the estimate of its standard deviation. With a sufficiently large number of cases, if $\beta_1+10\beta_2$ is more than 1.96 standard deviations away from 0, then you have a significant difference at p < 0.05 in a 2-sided z-test (equivalent to a Wald test). With a smaller number of cases you could use a t-test .

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