Confidence intervals in multivariate linear regression

Question

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

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?

lcrmorin · Answer

Looking at https://en.wikipedia.org/wiki/Simple_linear_regression :

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

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

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

Brian Spiering · Answer

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

Confidence intervals in multivariate linear regression

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