# Confidence intervals in multivariate linear regression

I am fitting my data to a multivariate linear regression $$Y = BX + \Xi$$, where the response is bivariate $$Y\in R^{n\times 2}$$, and the predictor is uni-variate but elevated to the projective plane to account for the intercept $$X\in R^{n\times 2}$$.

Now, finding the best fit reduces to $$\hat B = (X^T X)^{-1}X^T Y$$.

But I am interested in finding a $$0.7$$ confidence region around $$\hat B$$. How do I do that?

## 2 Answers

This t-value has a Student's t-distribution with $$n-2$$ degrees of freedom. Using it we can construct a confidence interval for $$\beta$$:

$$\beta \in \left[\widehat\beta - s_{\widehat\beta} t^*_{n - 2},\ \widehat\beta + s_{\widehat\beta} t^*_{n - 2}\right]$$

at confidence level $$1-\gamma$$, where $$t^*_{n - 2}$$ is the $$(1-\frac{\gamma}{2})$$-th quantile of the $$t_{n−2}$$ distribution.

• So this would work if the regression I was doing was 1D, but I want to extend it to the case where the response variable is 2D – Jsevillamol Jan 25 '20 at 20:51

Bayesian linear regression can provide an estimate for the confidence region for a linear regression estimate.

• Can you walk me through the math? What prior do I take? How do I update it efficiently? What is the resulting confidence region based on the posterior? – Jsevillamol Mar 24 '20 at 15:08