rbayesBLP {bayesm}R Documentation

Bayesian Analysis of Random Coefficient Logit Models Using Aggregate Data

Description

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.

Usage

rbayesBLP(Data, Prior, Mcmc)

Arguments

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)

Value

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 Sigmadraw)

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

Argument Details

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)

Model Details

Model and Priors (without IV):

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.

Model and Priors (with IV):

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)

MCMC and Tuning Details:

MCMC Algorithm:

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.

Tuning Metropolis-Hastings algorithm:

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.

Author(s)

Keunwoo Kim, Anderson School, UCLA, keunwoo.kim@gmail.com.

References

For further discussion, see Bayesian Analysis of Random Coefficient Logit Models Using Aggregate Data by Jiang, Manchanda, and Rossi, Journal of Econometrics, 2009.

Examples

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)
}

[Package bayesm version 3.1-4 Index]