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...

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

(see this link Adding coefficients to obtain interaction effects - what to do with SEs?)

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

Thank you very much.


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...

  • $\begingroup$ 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). $\endgroup$
    – whuber
    Jan 16, 2013 at 16:03

2 Answers 2


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.


Maybe I misunderstood your question but if you want to test for the global (net?) impact of your IVs on your DV you can use the usual global F test of your regression.

Given your repeated measure design, another way to analyze your data could be the application of a latent growth modeling. With this kind of models you can take into account the correlation between measurement times, test if there is a significant "global" mean change and test interesting hypotheses on how people change as a function of time and, more in line with your design, predict another variable (DV) with the change variable as IV. So I think using latent growth modeling can be very useful.

  • $\begingroup$ Thank you. I search just the formula for computing the Standard deviations of the sum of several coefficients (yes global impact). My director ask me to do this with my results. I'm sure that is very easy but i found nothing that could help me. $\endgroup$
    – user19727
    Jan 16, 2013 at 11:05

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