update_reg function

Update coefficient vector of multiple linear regression

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

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.

Returns

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

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

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

and mean vector

. Note the analogy of $\mu_1$ to the generalized least squares estimator

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)