# Standard deviation of the sum of regression coefficients

I'm doing OLS estimation with an independent variable lagged as t-1, t-2, t-3, and t-4 (four beta coefficients). I would like to have the sum of these coefficients for interpreting the net impact of these variables on the endogenous one (for example the net influence of the US's GDP growth lagged in t-1, t-2, t-3, and t-4 on the Thai GDP growth).

How can I compute the standard deviation of this new coefficient for computing a t-test?

I have found this formula but i'm not sure...

$$SE_{b_{new}} = \sqrt{SE_1^2 + SE_2^2+2Cov(b_1,b_2)}$$

But how do I calculate the covariance of two (four in my case) coefficients? Does that make any sense?

Thank you very much.

Ok,

I need the covariance matrix of coefficient to retrieve the cov(b1,b2) cov(b1,b3)...

After this, i should be able to compute the standard errors of the sum of coefficients with the formula above...

• It is difficult to know what "net influence" might mean when the other variables will be strongly correlated with their sum. Perhaps what you want to do is a simple regression of Thai GDP growth against the sum of the four lagged US GDP growth variables (without any other variables included).
– whuber
Jan 16, 2013 at 16:03

I think what you might be looking for is this formula: For a vector $A$ and a constant $c$, to test the hypothesis $H_0: A\beta=c$, use the following statistic: $$\frac{(A\hat{\beta}-c)^T (A(X^T X)^{-1}A^T)(A\hat{\beta}-c)}{\sum \varepsilon_i/(n-p)}$$ which has an $F_{(1,n-p)}$ distribution under the null hypothesis. You can use the same statistic to calculate a confidence interval for $c$.
In your case the vector $A$ would be just a $(p\times 1)$ vector of 1's.