BLPestimatoR
Performs a BLP Demand Estimation

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BLPestimatoR - Package for Demand Estimation

BLPestimatoR - Package for Demand Estimation
In BLPestimatoR: Performs a BLP Demand Estimation 

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

Intro

BLPestimatoR provides an efficient estimation algorithm to perform the demand estimation described in @BLP1995. The routine uses analytic gradients and offers a large number of optimization routines and implemented integration methods as discussed in @Brunner2017.

This extended documentation demonstrates the steps of a typical demand estimation with the package:

For this purpose the well-known training datasets for the cereal market [@Nevo2001] and the car market [@BLP1995] are included in the package. Loading the package is therefore the very first step of the demand estimation:

library(BLPestimatoR)

Data

Model

Since version 0.1.6 the model is provided in R’s formula syntax and consists of five parts. The variable to be explained is given by observed market shares. Explanatory variables are grouped into four (possibly overlapping) categories separated by |:

The first part of this documentation starts with the cereal data example from @Nevo2001. Nevo's model can be translated into the following formula syntax:

nevos_model <- as.formula("share ~  price + productdummy |
    0+ productdummy |
    price + sugar + mushy |
    0+ IV1 + IV2 + IV3 + IV4 + IV5 + IV6 + IV7 + IV8 + IV9 + IV10 + 
    IV11 + IV12 + IV13 + IV14 + IV15 + IV16 + IV17 + IV18 + IV19 + IV20")

The model is directly related to consumer $i$'s indirect utility from purchasing cereal $j$ in market $t$:

$$u_{ijt}=\sum_{m=1}^M x^{(m)}{jt} \beta{i,m}+\xi_{jt}+\epsilon_{ijt} \;\; \text{with}$$ $$\beta_{i,m}= \bar{\beta}m + \sum{r=1}^R \gamma_{m,r} d_{i,r} + \sigma_m \nu_{i,m}$$ and

$$\theta_2 = \begin{pmatrix} \sigma_1 & \gamma_{1,1} & \cdots & \gamma_{1,R} \ \sigma_2 & \gamma_{2,1} & \cdots & \gamma_{2,R} \ \vdots & \vdots & \ddots & \vdots \ \sigma_M & \gamma_{M,1} & \cdots & \gamma_{M,R} \end{pmatrix}$$

Dataframe

Product related variables are collected in the dataframe productData with the following requirements:

A variable that uniquely identifies a product in a market (product_identifier) is optional, but enhances clarity (interpreting elasticities, for example, is much easier). market_identifier and product_identifier together uniquely identify an observation, which is used by the function update_BLP_data to update any variable in the data (in this case product_identifier is mandatory).

In the cereal example, this gives the following dataframe:

head(productData_cereal)

Integration Draws

The arguments related to the numerical integration problem are of particular importance when providing own integration draws and weights, which is most relevant for observed heterogeneity (for unobserved heterogeneity, the straightforward approach is the use of automatic integration).

In the cereal data, both, observed and unobserved heterogeneity, is used for the random coefficients. Starting with observed heterogeneity, user provided draws are collected in a list. Each list entry must be named according to the name of a demographic. Each entry contains the following variables:

In the cereal example, observed heterogeneity is provided as follows (list names correspond to the demographics):

demographicData_cereal$income[1:4, 1:5]

demographicData_cereal$incomesq[1:4, 1:5]

demographicData_cereal$age[1:4, 1:5]

demographicData_cereal$child[1:4, 1:5]

If demographic input (demographicData) is missing, the estimation routine considers only coefficients for unobserved heterogeneity. This can be done by already implemented integration methods via integration_method as shown in the estimation section. In Nevo's cereal example however, a specific set of 20 draws is given. For this situation, draws are also provided as a list (list names correspond to the formula's random coefficients and each list entry has a variable market_identifier):

originalDraws_cereal$constant[1:4, 1:5]

# renaming constants:
names(originalDraws_cereal)[1] <- "(Intercept)"

originalDraws_cereal$price[1:4, 1:5]

originalDraws_cereal$sugar[1:4, 1:5]

originalDraws_cereal$mushy[1:4, 1:5]

As demonstrated above, list entries for draws of constants must be named (Intercept). Other names of list entries must match the random coefficients specified in the formula.

Calling BLP_data

Calling BLP_data structures and prepares the data for estimation and creates the data object:

productData_cereal$startingGuessesDelta <- c(log(w_guesses_cereal)) # include orig. draws in the product data

cereal_data <- BLP_data(
  model = nevos_model,
  market_identifier = "cdid",
  par_delta = "startingGuessesDelta",
  product_identifier = "product_id",
  productData = productData_cereal,
  demographic_draws = demographicData_cereal,
  blp_inner_tol = 1e-6, blp_inner_maxit = 5000,
  integration_draws = originalDraws_cereal,
  integration_weights = rep(1 / 20, 20)
)

The arguments in greater detail:

If you decide to update your data later, you can use the function update_BLP_data.

Estimation

Starting guesses

The provided set of starting guesses par_theta2 is matched with formula input and demographic data:

These requirements are demonstrated with a set of exemplary starting guesses:

# before:
theta_guesses_cereal
theta_guesses_cereal[theta_guesses_cereal == 0] <- NA
colnames(theta_guesses_cereal) <- c("unobs_sd", "income", "incomesq", "age", "child")
rownames(theta_guesses_cereal) <- c("(Intercept)", "price", "sugar", "mushy")

# correctly named:
theta_guesses_cereal

Calling estimateBLP

The following code performs the demand estimation:

cereal_est <- estimateBLP(
  blp_data = cereal_data,
  par_theta2 = theta_guesses_cereal,
  solver_method = "BFGS", solver_maxit = 1000, solver_reltol = 1e-6,
  standardError = "heteroskedastic",
  extremumCheck = FALSE,
  printLevel = 1
)

summary(cereal_est)

The arguments in greater detail:

Many of these arguments have default values. In the following setting you see a minimum of necessary arguments with an automatic generation of integration draws and just unobserved heterogeneity. The summary output informs you about the most important default values.

cereal_data2 <- BLP_data(
  model = nevos_model,
  market_identifier = "cdid",

  product_identifier = "product_id",
  productData = productData_cereal,
  integration_method = "MLHS",
  integration_accuracy = 20, integration_seed = 213
)

cereal_est2 <- estimateBLP(blp_data = cereal_data2, printLevel = 1)

summary(cereal_est2)

Postestimation

Standard Errors

Standard errors can be computed with three options that control for the unobserved characteristic $\xi$, which consists of $N$ elements. $\Omega$ denotes the variance covariance matrix of $\xi$.

$$\Omega = \begin{pmatrix} \Sigma_1 & 0 & \dots & 0\ 0 & \Sigma_2 & & 0\ \vdots & & \ddots & 0\ 0 & 0 & 0 & \Sigma_M \ \end{pmatrix}$$

Elasticities

The following code demonstrates the calculation of elasticities for the estimation object cereal_est. 

# extract parameters from output
theta1_price <- cereal_est$theta_lin["price", ]
theta2 <- matrix(NA, nrow = 4, ncol = 5)
colnames(theta2) <- c("unobs_sd", "income", "incomesq", "age", "child")
rownames(theta2) <- c("(Intercept)", "price", "sugar", "mushy")
for (i in 1:13) {
  theta2[cereal_est$indices[i, 1], cereal_est$indices[i, 2]] <- cereal_est$theta_rc[i]
}

delta_data <- data.frame(
  "product_id" = cereal_data$parameters$product_id,
  "cdid" = cereal_data$parameters$market_id_char_in,
  "startingGuessesDelta" = cereal_est$delta
)
# always use update_BLP_data() to update data object to maintain consistent data
cereal_data <- update_BLP_data(
  data_update = delta_data,
  blp_data = cereal_data
)

shareObj <- getShareInfo(
  blp_data = cereal_data,
  par_theta2 = theta2,
  printLevel = 1
)

get_elasticities(
  blp_data = cereal_data,
  share_info = shareObj,
  theta_lin = theta1_price,
  variable = "price",
  products = c("cereal_1", "cereal_4"),
  market = "market_2"
)

The value of the elasticity matrix in row $j$ and column $i$ for a variable $x$, gives the effect of a change in product $i$'s characteristic $x$ on the share of product $j$.

Modular Examples

Further analysis like incorporating a supply side or performing a merger simulation often requires access to building blocks of the BLP algorithm. The following wrappers insure correct data inputs and access the internal functions of the algorithm.

In the following, you find an example of the contraction mapping and an evaluation of the GMM function at the starting guess:

delta_eval <- getDelta_wrap(
  blp_data = cereal_data,
  par_theta2 = theta_guesses_cereal,
  printLevel = 4
)

productData_cereal$startingGuessesDelta[1:6]
delta_eval$delta[1:6]
delta_eval$counter

gmm <- gmm_obj_wrap(
  blp_data = cereal_data,
  par_theta2 = theta_guesses_cereal,
  printLevel = 2
)

gmm$local_min

Printed distances in the contraction mapping are maximum absolute distances between the current vector of mean utilities and the previous one.

For any $\theta_2$, you can compute predicted shares: 

shareObj <- getShareInfo(
  blp_data = cereal_data,
  par_theta2 = theta_guesses_cereal,
  printLevel = 4
)

shareObj$shares[1:6]

The object contains a list of outputs that are useful for further economic analysis. For example, the list element sij contains share probabilities for every individual and needs to be given to calculate elasticities.

The gradient contains two important building blocks as explained in the appendix of @Nevo2001:

Both are used to compute the jacobian and are easy to obtain with the package as the following example demonstrates:

# market 2:
derivatives1 <- dstdtheta_wrap(
  blp_data = cereal_data,
  par_theta2 = theta_guesses_cereal,
  market = "market_2"
)
derivatives2 <- dstddelta_wrap(
  blp_data = cereal_data,
  par_theta2 = theta_guesses_cereal,
  market = "market_2"
)

jac_mkt2 <- -solve(derivatives2) %*% derivatives1

jac_mkt2[1:5, 1:4]

# all markets
jacobian_nevo <- getJacobian_wrap(
  blp_data = cereal_data,
  par_theta2 = theta_guesses_cereal,
  printLevel = 2
)

jacobian_nevo[25:29, 1:4] # compare to jac_mkt2

Another Example: Merger Analysis with BLP's car data

Analyzing a hypothetical merger is demonstrated by the car data of @BLP1995. In this case, the preparation of product data comprises the computation of instruments as a function of product characteristics of competitors' products (for details, check @BLP1995). This example is based on data and documentation of @KM2014. 

# add owner matix to productData
own_pre <- dummies_cars
colnames(own_pre) <- paste0("company", 1:26)
productData_cars <- cbind(productData_cars, own_pre)

# construct instruments
nobs <- nrow(productData_cars)
X <- data.frame(
  productData_cars$const, productData_cars$hpwt,
  productData_cars$air, productData_cars$mpg, productData_cars$space
)

sum_other <- matrix(NA, nobs, ncol(X))
sum_rival <- matrix(NA, nobs, ncol(X))
sum_total <- matrix(NA, nobs, ncol(X))

for (i in 1:nobs) {
  other_ind <- productData_cars$firmid == productData_cars$firmid[i] &
    productData_cars$cdid == productData_cars$cdid[i] &
    productData_cars$id != productData_cars$id[i]
  rival_ind <- productData_cars$firmid != productData_cars$firmid[i] &
    productData_cars$cdid == productData_cars$cdid[i]
  total_ind <- productData_cars$cdid == productData_cars$cdid[i]

  sum_other[i, ] <- colSums(X[other_ind == 1, ])
  sum_rival[i, ] <- colSums(X[rival_ind == 1, ])
  sum_total[i, ] <- colSums(X[total_ind == 1, ])
}

colnames(sum_other) <- paste0("IV", 1:5)
colnames(sum_rival) <- paste0("IV", 6:10)
productData_cars <- cbind(productData_cars, sum_other, sum_rival)
head(productData_cars)

# To show similarities between implementations of other authors,
# the variable "const" is used, although constants are considered by default.
blps_model <- as.formula("share ~  0 + const + price + hpwt + air + mpg + space |
                        0 + const + hpwt + air + mpg + space |
                        0 + price + const + hpwt + air + mpg |
                        0 + IV1 + IV2 + IV3 + IV4 + IV5 + IV6 + IV7 + IV8 + IV9 + IV10")

car_data <- BLP_data(
  model = blps_model,
  market_identifier = "cdid",
  product_identifier = "id",
  additional_variables = paste0("company", 1:26), # check reordering works
  productData = productData_cars,
  blp_inner_tol = 1e-9,
  blp_inner_maxit = 5000,
  integration_method = "MLHS",
  integration_accuracy = 50, integration_seed = 48
)

In the next step, starting guesses for random coefficients are generated from a standard normal distribution. The estimation of the model works like before.

set.seed(121)
theta_guesses <- matrix(rnorm(5))
rownames(theta_guesses) <- c("price", "const", "hpwt", "air", "mpg")
colnames(theta_guesses) <- "unobs_sd"


car_est <- estimateBLP(
  blp_data = car_data,
  par_theta2 = theta_guesses,
  solver_method = "BFGS", solver_maxit = 1000, solver_reltol = 1e-6,
  extremumCheck = FALSE, printLevel = 0
)

summary(car_est)

Next, all parameters that are required by the subsequent merger analysis are extracted. Note that all extracted data is based on the estimation object car_est or the data object car_data to maintain data consistency (for example, the order of data in product_data_cars might differ from car_data). Moreover, mean utilities are updated in car_data by the values in the estimation object car_est.

## Pre-Merger data
own_pre <- as.matrix(car_data$data$additional_data[, paste0("company", 1:26)])
delta_pre <- car_est$delta
theta1_price <- car_est$theta_lin["price", ]
theta2_price <- car_est$theta_rc["unobs_sd*price"]
theta2_all <- matrix(car_est$theta_rc)
rownames(theta2_all) <- c("price", "const", "hpwt", "air", "mpg")
colnames(theta2_all) <- "unobs_sd"

## update mean utility in data ( always use update_BLP_data() to update data object to maintain consistent data )
delta_data <- data.frame(
  "id" = car_data$parameters$product_id,
  "cdid" = car_data$parameters$market_id,
  "delta" = delta_pre
)
car_data_updated <- update_BLP_data(
  data_update = delta_data,
  blp_data = car_data
)

In the next step, an estimate for marginal costs $mc$ before the merger is computed. The following is based on the FOC of a Bertrand equilibrium with prices $p$ before the merger:

$$ p^{pre} - \widehat{mc} = \Omega^{pre}(p^{pre})^{-1} \hat{s}(p^{pre}) $$ $\Omega^{pre}(p^{pre})^{-1}$ is defined marketwise as the inverse of

$$ \Omega^{pre}(p^{pre}) = \pmatrix{ -\frac{\partial s_{1}}{\partial p_{1}} (p^{pre}) \cdot D_{1,1} & -\frac{\partial s_{2}}{\partial p_{1}} (p^{pre}) \cdot D_{1,2} & \cdots & -\frac{\partial s_{j}}{\partial p_{1}} (p^{pre}) \cdot D_{1,j} & \cdots & -\frac{\partial s_{J}}{\partial p_{1}} (p^{pre}) \cdot D_{1,J}\ -\frac{\partial s_{1}}{\partial p_{2}} (p^{pre}) \cdot D_{2,1} & -\frac{\partial s_{2}}{\partial p_{2}} (p^{pre}) \cdot D_{2,2} & \cdots & -\frac{\partial s_{j}}{\partial p_{2}} (p^{pre}) \cdot D_{2,j} & \cdots & -\frac{\partial s_{J}}{\partial p_{2}} (p^{pre}) \cdot D_{2,J}\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots\ -\frac{\partial s_{1}}{\partial p_{k}} (p^{pre}) \cdot D_{k,1} & -\frac{\partial s_{2}}{\partial p_{k}} (p^{pre}) \cdot D_{k,2} & \cdots & -\frac{\partial s_{j}}{\partial p_{k}} (p^{pre}) \cdot D_{k,j} & \cdots & -\frac{\partial s_{J}}{\partial p_{k}} (p^{pre}) \cdot D_{k,J}\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots\ -\frac{\partial s_{1}}{\partial p_{J}} (p^{pre}) \cdot D_{J,1} & -\frac{\partial s_{2}}{\partial p_{J}} (p^{pre}) \cdot D_{J,2} & \cdots & -\frac{\partial s_{j}}{\partial p_{J}} (p^{pre}) \cdot D_{J,j}& \cdots & -\frac{\partial s_{J}}{\partial p_{J}} (p^{pre}) \cdot D_{J,J}\ } $$

with $$D_{k,j} = \begin{cases} 1 & \text{if products k and j are produced by the same firm} \ 0 & \text{otherwise} \ \end{cases}$$

Partial derivatives $\frac{\partial s_j}{\partial p_k}$ can be calculated based on the elasticity $\eta_{jk} = \frac{\partial s_j }{\partial p_k }\frac{ p_k}{ s_j}$, so $$ \frac{\partial s_j}{\partial p_k} = \eta_{jk} \cdot \frac{ s_j}{ p_k} $$

In the following code chunk, these objects in a market i are labeled as follows:

## calculate sij
shareObj <- getShareInfo(
  blp_data = car_data_updated,
  par_theta2 = theta2_all,
  printLevel = 0
)

## computation of marginal costs
market_id <- car_data$parameters$market_id
nmkt <- length(unique(market_id))
markups <- numeric(length(market_id))

sh <- shareObj$shares
prices_pre <- car_data$data$X_rand[, "price"]


for (i in 1:nmkt) {
  mkt_ind <- market_id == i
  share_i <- sh[ mkt_ind ]
  price_pre_i <- prices_pre[ mkt_ind ]
  scalar_i <- matrix(1 / share_i) %*% matrix(price_pre_i, nrow = 1)
  elasticities_i <- get_elasticities(
    blp_data = car_data_updated,
    share_info = shareObj,
    theta_lin = theta1_price,
    variable = "price",
    market = i,
    printLevel = 0
  )

  derivatives_i <- elasticities_i / scalar_i # partial derivatives of shares wrt price
  own_pre_i <- own_pre[ mkt_ind, ]
  own_prod_pre_i <- own_pre_i %*% t(own_pre_i) # if element (i,j) equals 1, that means that prod i and j are produced by same firm
  markups[mkt_ind] <- c(-solve(t(derivatives_i) * own_prod_pre_i) %*% share_i)
}
marg_cost <- prices_pre - markups

The ownership matrix is adjusted to implement a hypothetical merger between Chrysler and GM: 

# Merger between company 16 and 19 (i.e. GM and Chrysler)
prices_post <- numeric(2217)
own_post <- cbind(
  own_pre[, 1:15],
  own_pre[, 16] + own_pre[, 19],
  own_pre[, 17:18],
  own_pre[, 20:26]
)

To analyze the effect on prices the FOC of the new equilibrium must be solved: $$ p^{post} - \widehat{mc} = \Omega^{post}(p^{post})^{-1} \hat{s}(p^{post}) $$

The solution of this set of non-linear equations is obtained by the function foc_bertrand_mkt and the package nleqslv:

foc_bertrand_mkt <- function(par, own_prod, blp_data, mkt, marg_cost, theta_lin, theta_rc) {
  # argument par: candidate for post merger prices
  # arguments own_prod, blp_data, mkt, marg_cost, theta_lin, theta_rc: see previous code blocks

  # post merger updates: update the BLP_data object for market i
  tmp <- data.frame(
    "id" = blp_data$parameters$product_id,
    "cdid" = blp_data$parameters$market_id,
    "delta" = blp_data$data$delta,
    "price" = blp_data$data$X_rand[, "price"]
  )

  market_ind <- blp_data$parameters$market_id == mkt
  delta_old <- blp_data$data$delta
  prices_pre <- blp_data$data$X_rand[, "price"]
  tmp$price[ market_ind ] <- par
  tmp$delta[ market_ind ] <- delta_old[market_ind] - prices_pre[market_ind] * theta_lin + par * theta_lin


  new_blp_data <- update_BLP_data(
    blp_data = blp_data,
    data_update = tmp
  )

  ShareObj <- getShareInfo(
    blp_data = new_blp_data,
    par_theta2 = theta_rc,
    printLevel = 0
  )

  implied_shares <- as.matrix(ShareObj$shares[market_ind])

  elasticities_post_mkt <- get_elasticities(
    blp_data = new_blp_data,
    share_info = ShareObj,
    theta_lin = theta_lin,
    variable = "price",
    market = mkt,
    printLevel = 0
  )

  scalar_mkt <- matrix(1 / implied_shares) %*% matrix(par, nrow = 1)
  derivatives_mkt <- elasticities_post_mkt / scalar_mkt

  markups_post <- c(-solve(t(derivatives_mkt) * own_prod) %*% implied_shares)
  differences <- par - marg_cost[market_ind] - markups_post

  return(differences)
}

Finally, the function is used to compute the new equilibrium:

library(nleqslv) # to solve non linear first order conditions
for (i in 1:nmkt) {
  mkt_ind <- market_id == i
  own_post_i <- own_post[ mkt_ind, ]
  own_prod_post_i <- own_post_i %*% t(own_post_i)
  price_pre_i <- prices_pre[ mkt_ind ]

  solution <- nleqslv(
    x = price_pre_i, foc_bertrand_mkt, # startingguesses: price_pre_i
    own_prod = own_prod_post_i,
    blp_data = car_data_updated,
    mkt = i,
    marg_cost = marg_cost,
    theta_lin = theta1_price,
    theta_rc = theta2_all
  )

  prices_post[ market_id == i ] <- solution$x
}

References



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BLPestimatoR documentation built on Dec. 3, 2022, 5:07 p.m.

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