This function updates the coefficient vector of a multiple linear regression.

## Usage

update_reg(mu0, Tau0, XSigX, XSigU)

## Arguments

mu0

The mean vector of the normal prior distribution for the coefficient vector.

Tau0

The precision matrix (i.e. inverted covariance matrix) of the normal prior distribution for the coefficient vector.

XSigX

The matrix $$\sum_{n=1}^N X_n'\Sigma^{-1}X_n$$. See below for details.

XSigU

The vector $$\sum_{n=1}^N X_n'\Sigma^{-1}U_n$$. See below for details.

## Value

A vector, a draw from the normal posterior distribution of the coefficient vector in a multiple linear regression.

## Details

This function draws from the posterior distribution of $$\beta$$ in the linear utility equation $$U_n = X_n\beta + \epsilon_n,$$ where $$U_n$$ is the (latent, but here assumed to be known) utility vector of decider $$n = 1,\dots,N$$, $$X_n$$ is the design matrix build from the choice characteristics faced by $$n$$, $$\beta$$ is the unknown coefficient vector (this can be either the fixed coefficient vector $$\alpha$$ or the decider-specific coefficient vector $$\beta_n$$), and $$\epsilon_n$$ is the error term assumed to be normally distributed with mean $$0$$ and (known) covariance matrix $$\Sigma$$. A priori we assume the (conjugate) normal prior distribution $$\beta \sim N(\mu_0,T_0)$$ with mean vector $$\mu_0$$ and precision matrix (i.e. inverted covariance matrix) $$T_0$$. The posterior distribution for $$\beta$$ is normal with covariance matrix $$\Sigma_1 = (T_0 + \sum_{n=1}^N X_n'\Sigma^{-1}X_n)^{-1}$$ and mean vector $$\mu_1 = \Sigma_1(T_0\mu_0 + \sum_{n=1}^N X_n'\Sigma^{-1}U_n)$$. Note the analogy of $$\mu_1$$ to the generalized least squares estimator $$\hat{\beta}_{GLS} = (\sum_{n=1}^N X_n'\Sigma^{-1}X_n)^{-1} \sum_{n=1}^N X_n'\Sigma^{-1}U_n$$ which becomes weighted by the prior parameters $$\mu_0$$ and $$T_0$$.

## Examples

### true coefficient vector
beta_true <- matrix(c(-1,1), ncol=1)
### error term covariance matrix
Sigma <- matrix(c(1,0.5,0.2,0.5,1,0.2,0.2,0.2,2), ncol=3)
### draw data
N <- 100
X <- replicate(N, matrix(rnorm(6), ncol=2), simplify = FALSE)
eps <- replicate(N, rmvnorm(mu = c(0,0,0), Sigma = Sigma), simplify = FALSE)
U <- mapply(function(X, eps) X %*% beta_true + eps, X, eps, SIMPLIFY = FALSE)
### prior parameters for coefficient vector
mu0 <- c(0,0)
Tau0 <- diag(2)
### draw from posterior of coefficient vector
XSigX <- Reduce(+, lapply(X, function(X) t(X) %*% solve(Sigma) %*% X))
XSigU <- Reduce(+, mapply(function(X, U) t(X) %*% solve(Sigma) %*% U, X, U, SIMPLIFY = FALSE))
beta_draws <- replicate(100, update_reg(mu0, Tau0, XSigX, XSigU), simplify = TRUE)
rowMeans(beta_draws)
#>  -1.071996  0.986084