Bayesian Analysis of Random Coefficient Logit Models Using Aggregate Data
rbayesBLP implements a hybrid MCMC algorithm for aggregate level sales data in a market with differentiated products. bayesm version 3.1-0 and prior verions contain an error when using instruments with this function; this will be fixed in a future version.
rbayesBLP(Data, Prior, Mcmc)
Data |
list(X, share, J, Z) |
Prior |
list(sigmasqR, theta_hat, A, deltabar, Ad, nu0, s0_sq, VOmega) |
Mcmc |
list(R, keep, nprint, H, initial_theta_bar, initial_r, initial_tau_sq, initial_Omega, initial_delta, s, cand_cov, tol) |
A list containing:
thetabardraw |
K x R/keep matrix of random coefficient mean draws |
Sigmadraw |
K*K x R/keep matrix of random coefficient variance draws |
rdraw |
K*K x R/keep matrix of r draws (same information as in |
tausqdraw |
R/keep x 1 vector of aggregate demand shock variance draws |
Omegadraw |
2*2 x R/keep matrix of correlated endogenous shock variance draws |
deltadraw |
I x R/keep matrix of endogenous structural equation coefficient draws |
acceptrate |
scalor of acceptance rate of Metropolis-Hasting |
s |
scale parameter used for Metropolis-Hasting |
cand_cov |
var-cov matrix used for Metropolis-Hasting |
Data = list(X, share, J, Z) [Z optional]
J: |
number of alternatives, excluding an outside option |
X: |
J*T x K matrix (no outside option, which is normalized to 0). |
If IV is used, the last column of X is the endogeneous variable. |
|
share: |
J*T vector (no outside option). |
Note that both the share vector and the X matrix are organized by the jt index. |
|
| j varies faster than t, i.e. (j=1,t=1), (j=2,t=1), ..., (j=J,T=1), ..., (j=J,t=T) | |
Z: |
J*T x I matrix of instrumental variables (optional) |
Prior = list(sigmasqR, theta_hat, A, deltabar, Ad, nu0, s0_sq, VOmega) [optional]
sigmasqR: |
K*(K+1)/2 vector for r prior variance (def: diffuse prior for Σ) |
theta_hat: |
K vector for θ_bar prior mean (def: 0 vector) |
A: |
K x K matrix for θ_bar prior precision (def: 0.01*diag(K)) |
deltabar: |
I vector for δ prior mean (def: 0 vector) |
Ad: |
I x I matrix for δ prior precision (def: 0.01*diag(I)) |
nu0: |
d.f. parameter for τ_sq and Ω prior (def: K+1) |
s0_sq: |
scale parameter for τ_sq prior (def: 1) |
VOmega: |
2 x 2 matrix parameter for Ω prior (def: matrix(c(1,0.5,0.5,1),2,2))
|
Mcmc = list(R, keep, nprint, H, initial_theta_bar, initial_r, initial_tau_sq, initial_Omega, initial_delta, s, cand_cov, tol) [only R and H required]
R: |
number of MCMC draws |
keep: |
MCMC thinning parameter -- keep every keepth draw (def: 1) |
nprint: |
print the estimated time remaining for every nprint'th draw (def: 100, set to 0 for no print) |
H: |
number of random draws used for Monte-Carlo integration |
initial_theta_bar: |
initial value of θ_bar (def: 0 vector) |
initial_r: |
initial value of r (def: 0 vector) |
initial_tau_sq: |
initial value of τ_sq (def: 0.1) |
initial_Omega: |
initial value of Ω (def: diag(2)) |
initial_delta: |
initial value of δ (def: 0 vector) |
s: |
scale parameter of Metropolis-Hasting increment (def: automatically tuned) |
cand_cov: |
var-cov matrix of Metropolis-Hasting increment (def: automatically tuned) |
tol: |
convergence tolerance for the contraction mapping (def: 1e-6) |
u_ijt = X_jt θ_i + η_jt + e_ijt
e_ijt ~ type I Extreme Value (logit)
θ_i ~ N(θ_bar, Σ)
η_jt ~ N(0, τ_sq)
This structure implies a logit model for each consumer (θ).
Aggregate shares (share) are produced by integrating this consumer level
logit model over the assumed normal distribution of θ.
r ~ N(0, diag(sigmasqR))
θ_bar ~ N(θ_hat, A^-1)
τ_sq ~ nu0*s0_sq / χ^2 (nu0)
Note: we observe the aggregate level market share, not individual level choices.
Note: r is the vector of nonzero elements of cholesky root of Σ. Instead of Σ we draw r, which is one-to-one correspondence with the positive-definite Σ.
u_ijt = X_jt θ_i + η_jt + e_ijt
e_ijt ~ type I Extreme Value (logit)
θ_i ~ N(θ_bar, Σ)
X_jt = [X_exo_jt, X_endo_jt]
X_endo_jt = Z_jt δ_jt + ζ_jt
vec(ζ_jt, η_jt) ~ N(0, Ω)
r ~ N(0, diag(sigmasqR))
θ_bar ~ N(θ_hat, A^-1)
δ ~ N(deltabar, Ad^-1)
Ω ~ IW(nu0, VOmega)
Step 1 (Σ):
Given θ_bar and τ_sq, draw r via Metropolis-Hasting.
Covert the drawn r to Σ.
Note: if user does not specify the Metropolis-Hasting increment parameters
(s and cand_cov), rbayesBLP automatically tunes the parameters.
Step 2 without IV (θ_bar, τ_sq):
Given Σ, draw θ_bar and τ_sq via Gibbs sampler.
Step 2 with IV (θ_bar, δ, Ω):
Given Σ, draw θ_bar, δ, and Ω via IV Gibbs sampler.
r_cand = r_old + s*N(0,cand_cov)
Fix the candidate covariance matrix as cand_cov0 = diag(rep(0.1, K), rep(1, K*(K-1)/2)).
Start from s0 = 2.38/sqrt(dim(r))
Repeat{
Run 500 MCMC chain.
If acceptance rate < 30% => update s1 = s0/5.
If acceptance rate > 50% => update s1 = s0*3.
(Store r draws if acceptance rate is 20~80%.)
s0 = s1
} until acceptance rate is 30~50%
Scale matrix C = s1*sqrt(cand_cov0)
Correlation matrix R = Corr(r draws)
Use C*R*C as s^2*cand_cov.
Keunwoo Kim, Anderson School, UCLA, keunwoo.kim@gmail.com.
For further discussion, see Bayesian Analysis of Random Coefficient Logit Models Using Aggregate Data by Jiang, Manchanda, and Rossi, Journal of Econometrics, 2009.
if(nchar(Sys.getenv("LONG_TEST")) != 0) {
## Simulate aggregate level data
simulData <- function(para, others, Hbatch) {
# Hbatch does the integration for computing market shares
# in batches of size Hbatch
## parameters
theta_bar <- para$theta_bar
Sigma <- para$Sigma
tau_sq <- para$tau_sq
T <- others$T
J <- others$J
p <- others$p
H <- others$H
K <- J + p
## build X
X <- matrix(runif(T*J*p), T*J, p)
inter <- NULL
for (t in 1:T) { inter <- rbind(inter, diag(J)) }
X <- cbind(inter, X)
## draw eta ~ N(0, tau_sq)
eta <- rnorm(T*J)*sqrt(tau_sq)
X <- cbind(X, eta)
share <- rep(0, J*T)
for (HH in 1:(H/Hbatch)){
## draw theta ~ N(theta_bar, Sigma)
cho <- chol(Sigma)
theta <- matrix(rnorm(K*Hbatch), nrow=K, ncol=Hbatch)
theta <- t(cho)%*%theta + theta_bar
## utility
V <- X%*%rbind(theta, 1)
expV <- exp(V)
expSum <- matrix(colSums(matrix(expV, J, T*Hbatch)), T, Hbatch)
expSum <- expSum %x% matrix(1, J, 1)
choiceProb <- expV / (1 + expSum)
share <- share + rowSums(choiceProb) / H
}
## the last K+1'th column is eta, which is unobservable.
X <- X[,c(1:K)]
return (list(X=X, share=share))
}
## true parameter
theta_bar_true <- c(-2, -3, -4, -5)
Sigma_true <- rbind(c(3,2,1.5,1), c(2,4,-1,1.5), c(1.5,-1,4,-0.5), c(1,1.5,-0.5,3))
cho <- chol(Sigma_true)
r_true <- c(log(diag(cho)), cho[1,2:4], cho[2,3:4], cho[3,4])
tau_sq_true <- 1
## simulate data
set.seed(66)
T <- 300
J <- 3
p <- 1
K <- 4
H <- 1000000
Hbatch <- 5000
dat <- simulData(para=list(theta_bar=theta_bar_true, Sigma=Sigma_true, tau_sq=tau_sq_true),
others=list(T=T, J=J, p=p, H=H), Hbatch)
X <- dat$X
share <- dat$share
## Mcmc run
R <- 2000
H <- 50
Data1 <- list(X=X, share=share, J=J)
Mcmc1 <- list(R=R, H=H, nprint=0)
set.seed(66)
out <- rbayesBLP(Data=Data1, Mcmc=Mcmc1)
## acceptance rate
out$acceptrate
## summary of draws
summary(out$thetabardraw)
summary(out$Sigmadraw)
summary(out$tausqdraw)
### plotting draws
plot(out$thetabardraw)
plot(out$Sigmadraw)
plot(out$tausqdraw)
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