# Confidence interval for linear combination of regression coefficients that are calculated from different variables

I would like to estimate a confidence interval for a linear combination of regression coefficients that come from several linear regression models that are calculated from different but correlated variables. For instance, suppose $$y_1$$ and $$x_1$$ represent cloud cover and temperature anomalies at one level of the atmosphere as measured by a satellite, and $$y_2$$ and $$x_2$$ represent cloud cover and temperature anomalies in a lower level level of the atmosphere. I expect $$x_1$$ to be correlated with $$x_2$$. I also expect $$y_1$$ to be negatively correlated with $$y_2$$ because the satellite only views the highest clouds. I use OLS regression to determine the relationship

$$y_i = \beta_i x_i + \epsilon$$ for $$i \in {1,2}$$

The standard error for the regression slopes is $$\sigma_1$$ and $$\sigma_2$$. Now I want to calculate a linear combination of the regression slopes: $$C=a_1 \beta_1 +a_2 \beta_2$$ where $$a_1$$ and $$a_2$$ are real-valued constants. I think that the standard error for $$C$$ can be calculated from the relationship

$$\sigma_C^2 = a_1^2 \sigma_1^2 + a_2^2 \sigma_2^2 + 2a_1 a_2 \sigma_{1,2}$$

where $$\sigma_{1,2}$$ represents the covariance between $$\beta_1$$ and $$\beta_2$$. My main question is how does one estimate $$\sigma_{1,2}$$ given that $$\beta_1$$ and $$\beta_2$$ are calculated from different variables?

• Of possible interest: seemingly unrelated regressions.
– chl
Nov 10 '20 at 18:54
• Answer: you can't estimate $\sigma_{1,2}$ with the information given. It doesn't even exist unless your measurements are all paired with one another: that is, each time you have $(x_1,y_1)$ you also have $(x_2,y_2).$ Now if that's the case, there's an interesting question here if you would change your inquiry to exploring how to estimate $a_1\beta_1+a_2\beta_2$ rather than trying to combine two separate estimates.
– whuber
Nov 10 '20 at 20:00
• Thanks for your reply. The measurements are indeed all paired with one another. Every time I have a measurement of $(x_1,y_1)$ I also have a measurement of $(x_2,y_2)$. In terms of changing the inquiry to estimate $a_1 \beta_1 + a_2 \beta_2$ differently, it would be ideal to combine the two separate estimates of $\beta_1$ and $\beta_2$ because decomposing the linear combination this way would help to understand the physical system. But if doing the analysis that way makes it impossible to quantify uncertainty, then I need to figure out another way, of course. Many thanks. Nov 10 '20 at 20:29

You may be able to avoid the problem of estimating $$\sigma_{12}$$ altogether.

Just to be clear, here is my understanding of your setup. You have a sample of quadruples $$(X_1,X_2,Y_1,Y_2).$$ Your model of it is that

\begin{aligned} Y_1 &= \beta_1 X_1 + \varepsilon_1 \\ Y_2 &= \beta_2 X_2 + \varepsilon_2 \end{aligned}

with $$(\varepsilon_1,\varepsilon_2)$$ independent of $$(X_1,X_2).$$ The covariance matrix of the errors exists, but is unknown, and can be written in terms of three parameters as

$$\operatorname{Cov}\pmatrix{\varepsilon_1\\\varepsilon_2} = \pmatrix{\sigma_1^2&\sigma_{12}\\\sigma_{12}&\sigma_2^2}.$$

You have stipulated numbers $$a_i$$ (they are not estimated from the data) and you wish to estimate

$$\gamma = a_1\beta_1 + a_2\beta_2.$$

One direct way to estimate this quantity is to note that your model implies the relation

$$a_1Y_1 + a_2Y_2 = a_1\beta_1 X_1 + a_2\beta_2 X_2 + (a_1\varepsilon_1 + a_2\varepsilon_2).$$

Under the foregoing assumptions this is a standard regression model for data $$(X_1,X_2, a_1Y_1+a_2Y_2)$$ which you might fit with, say, ordinary least squares (or any other regression technique appropriate for the assumed distribution of $$a_1\varepsilon_1 + a_2\varepsilon_2)$$). That procedure yields estimates $$\widehat{a_i\beta_i}$$ of the two coefficients, whose sum estimates $$\gamma,$$ along with an estimated covariance matrix $$\widehat\Sigma$$ for those estimates, from which you may extract the standard error as

$$\operatorname{se}(\widehat{\gamma}) = \pmatrix{1&1}\widehat\Sigma\pmatrix{1\\1}.$$

It sounds like you might be supposing the $$\varepsilon_i$$ are negatively correlated: that is, $$\sigma_{12}\lt 0.$$ If so, and if $$a_1a_2 \gt 0,$$ the error variance in this model is

$$\operatorname{Var}(a_1\varepsilon_1 + a_2\varepsilon_2) = a_1\sigma_1^2 + a_2\sigma_2^2 + a_1a_2\sigma_{12} \lt a_1\sigma_1^2 + a_2\sigma_2^2,$$

which is a nice thing to have: it means that this linear combination of the $$Y_i$$ tends to cancel out the errors, giving better estimates of $$\gamma$$ than if you separately regressed the $$Y_i$$ against the $$X_i.$$