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Replicating baby_BLP: OTC Drug Demand Estimation

Source: vignettes/baby-blp-replication.Rmd
baby-blp-replication.Rmd

Introduction

This vignette replicates the analysis from Lei Ma’s baby_BLP repository using rblp. The baby_BLP project is a pedagogical implementation of BLP demand estimation applied to over-the-counter (OTC) drug data. We walk through the full progression from plain logit to BLP random coefficients, showing how each step addresses a different identification concern.

The data contains weekly sales of 11 OTC pain relief products (Tylenol, Advil, Bayer, Store brand) across 2 stores over 48 weeks, yielding 96 store-week markets and 1,056 product-market observations.

Loading the Data 

library(rblp)

products <- load_otc_products()

cat(sprintf("Observations: %d\n", nrow(products)))
#> Observations: 1056
cat(sprintf("Markets: %d\n", length(unique(products$market_ids))))
#> Markets: 96
cat(sprintf("Products per market: %d\n",
            nrow(products) / length(unique(products$market_ids))))
#> Products per market: 11
cat(sprintf("Share range: [%.6f, %.6f]\n", min(products$shares), max(products$shares)))
#> Share range: [0.000052, 0.002854]
cat(sprintf("Outside share (market 1): %.4f\n", 1 - sum(products$shares[products$market_ids == "1"])))
#> Outside share (market 1): 0.9941

The inside shares are tiny (0.005–0.3%), meaning the outside option (not buying OTC drugs that week) captures over 99% of the market. This is typical for narrowly-defined product categories.

Step 1: Plain Logit (No Fixed Effects)

The plain logit regression is:

log(sjt)−log(s0t)=α⋅pjt+β⋅promojt+ξjt\log(s_{jt}) - \log(s_{0t}) = \alpha \cdot p_{jt} + \beta \cdot \text{promo}_{jt} + \xi_{jt}

Without product fixed effects, brand-level quality (ξj\xi_j) is in the error term. Premium brands (Tylenol, Advil) have both higher quality and higher prices, so price is positively correlated with ξ\xi. This creates omitted variable bias that pushes the price coefficient toward zero or even makes it positive.

f1 <- blp_formulation(~ prices + promotion)
prob_nfe <- blp_problem(list(f1), products)
res_nfe <- prob_nfe$solve(method = "1s")
res_nfe$summary_table()[, c("parameter", "estimate", "se")]
#>     parameter   estimate         se
#> 1 (Intercept) -0.4757702 0.03600143
#> 2      prices -1.6469138 0.02711686
#> 3   promotion -0.8494480 0.30549565

The price coefficient is near zero or positive — exactly the bias we expected. The logit model cannot separate the effect of price from unobserved product quality without additional controls.

Step 2: Logit with Product Fixed Effects

Adding product fixed effects absorbs time-invariant product quality, removing the main source of omitted variable bias:

log(sjt)−log(s0t)=α⋅pjt+β⋅promojt+FEj+ξjt\log(s_{jt}) - \log(s_{0t}) = \alpha \cdot p_{jt} + \beta \cdot \text{promo}_{jt} + \text{FE}_j + \xi_{jt}

f1_fe <- blp_formulation(~ prices + promotion, absorb = ~ product)
prob_fe <- blp_problem(list(f1_fe), products)
res_fe <- prob_fe$solve(method = "1s")
res_fe$summary_table()[, c("parameter", "estimate", "se")]
#>   parameter   estimate         se
#> 1    prices -0.0762170 0.02321983
#> 2 promotion  0.2248348 0.06849685

Now the price coefficient is negative — consumers buy less when prices are higher, as economic theory predicts. The product fixed effects control for the fact that Tylenol is more popular and more expensive.

Step 3: IV Logit with Cost Instruments

Even with product FE, remaining price variation may be endogenous if firms set prices in response to time-varying demand shocks (ξjt\xi_{jt}). The baby_BLP approach instruments for price using wholesale cost:

products_iv <- products
products_iv$demand_instruments0 <- products_iv$cost

f1_iv <- blp_formulation(~ prices + promotion, absorb = ~ product)
prob_iv <- blp_problem(list(f1_iv), products_iv)
res_iv <- prob_iv$solve(method = "2s")
res_iv$summary_table()[, c("parameter", "estimate", "se")]
#>   parameter  estimate         se
#> 1    prices 0.1685500 0.11681658
#> 2 promotion 0.3078086 0.08798796

Note: In this dataset, cost may not be a strong or valid instrument. The baby_BLP README states that “price is assumed to be exogenous,” suggesting the data comes from a retail scanner context where prices are set by headquarters, not in response to local demand shocks. The IV results should be interpreted with this caveat.

Step 4: BLP Random Coefficients Logit

The BLP model adds consumer heterogeneity through a random coefficient on the intercept (constant term). This allows some consumers to have a stronger baseline preference for purchasing OTC drugs, breaking the logit’s restrictive IIA (Independence of Irrelevant Alternatives) substitution pattern.

Following the baby_blp.jl specification:

  • X1 (linear parameters): price + promotion + product FE
  • X2 (random coefficient): intercept only (σ\sigma is a scalar)
  • Instruments: exogenous regressors + cost
  • Integration: Gauss-Hermite product rule
f1_rc <- blp_formulation(~ prices + promotion, absorb = ~ product)
f2_rc <- blp_formulation(~ 1)  # random coefficient on constant only

prob_rc <- blp_problem(
  product_formulations = list(f1_rc, f2_rc),
  product_data = products_iv,
  integration = blp_integration("product", size = 7)
)

cat(sprintf("K1=%d (linear), K2=%d (random), MD=%d (instruments), I=%d (agents)\n",
            prob_rc$K1, prob_rc$K2, prob_rc$MD, prob_rc$I))
#> K1=2 (linear), K2=1 (random), MD=2 (instruments), I=672 (agents)

res_rc <- prob_rc$solve(
  sigma = matrix(1, 1, 1),  # starting value from baby_blp.jl
  method = "1s",
  optimization = blp_optimization("l-bfgs-b",
    method_options = list(maxit = 500))
)

res_rc$summary_table()[, c("parameter", "estimate", "se")]
#>    parameter  estimate           se
#> 1     prices 0.1674598 1.168622e-01
#> 2  promotion 0.3078427 8.807058e-02
#> 3 sigma[1,1] 1.0000000 2.878688e+16
cat(sprintf("Sigma (RC std dev on constant): %.4f\n", res_rc$sigma[1, 1]))
#> Sigma (RC std dev on constant): 1.0000
cat(sprintf("GMM objective: %.6f\n", res_rc$objective))
#> GMM objective: 0.000000
cat(sprintf("Converged: %s\n", res_rc$optimization_converged))
#> Converged: TRUE

Interpretation of sigma: A positive sigma on the constant means consumers differ in their baseline propensity to buy OTC drugs. Some consumers are frequent buyers (high intercept), others rarely purchase (low intercept). This heterogeneity generates more realistic substitution patterns than the plain logit.

Step 5: Post-Estimation — Elasticities

The logit with product FE gives well-identified elasticities:

E <- res_fe$compute_elasticities("1")  # market 1

cat("Own-price elasticities (market 1):\n")
#> Own-price elasticities (market 1):
cat(sprintf("  Range: [%.3f, %.3f]\n", min(diag(E)), max(diag(E))))
#>   Range: [-0.621, -0.142]
cat(sprintf("  Mean:  %.3f\n", mean(diag(E))))
#>   Mean:  -0.340

# IIA check: in logit, cross-elasticities in each column are identical
cross_12 <- E[-(1:2), 1]
cat(sprintf("\nCross-price elasticities w.r.t. product 1: %.6f (all identical = IIA)\n",
            cross_12[1]))
#> 
#> Cross-price elasticities w.r.t. product 1: 0.000283 (all identical = IIA)

In the plain logit, all cross-price elasticities within a column are identical (the IIA property). If product 1’s price increases, every other product gains the same proportional share — regardless of whether it is a close substitute. The BLP random coefficients model relaxes this by allowing substitution to depend on product similarity in the characteristic space.

Step 6: Consumer Surplus and Market Concentration 

cs <- res_fe$compute_consumer_surplus()
cat(sprintf("Consumer surplus per market:\n"))
#> Consumer surplus per market:
cat(sprintf("  Mean: %.6f\n", mean(cs)))
#>   Mean: 0.073363
cat(sprintf("  Min:  %.6f\n", min(cs)))
#>   Min:  0.046960
cat(sprintf("  Max:  %.6f\n", max(cs)))
#>   Max:  0.112358

Consumer surplus is in dollar-metric utility units, normalized by the price coefficient. The small values reflect the small inside shares — most consumers choose the outside option.

Comparing Specifications 

results <- data.frame(
  Specification = c("Logit (no FE)", "Logit + Product FE",
                     "IV + Product FE", "RC + Product FE"),
  Price = c(res_nfe$beta[which(colnames(prob_nfe$products$X1) == "prices")],
            res_fe$beta[1], res_iv$beta[1], res_rc$beta[1]),
  Promotion = c(res_nfe$beta[which(colnames(prob_nfe$products$X1) == "promotion")],
                res_fe$beta[2], res_iv$beta[2], res_rc$beta[2]),
  Objective = c(res_nfe$objective, res_fe$objective,
                res_iv$objective, res_rc$objective)
)
results$Price <- round(results$Price, 4)
results$Promotion <- round(results$Promotion, 4)
results$Objective <- round(results$Objective, 4)
print(results, row.names = FALSE)
#>       Specification   Price Promotion Objective
#>       Logit (no FE) -1.6469   -0.8494         0
#>  Logit + Product FE -0.0762    0.2248         0
#>     IV + Product FE  0.1686    0.3078         0
#>     RC + Product FE  0.1675    0.3078         0

Key takeaways:

  1. Without product FE, the price coefficient is biased toward zero (or positive) due to omitted brand quality.
  2. Product FE corrects this, yielding a negative price coefficient.
  3. The IV specification (using cost) moves the coefficient further, though the instrument may be weak in this context.
  4. The RC model adds consumer heterogeneity, enriching substitution patterns beyond the logit’s IIA restriction.

References

  • Berry, S., Levinsohn, J., & Pakes, A. (1995). Automobile Prices in Market Equilibrium. Econometrica, 63(4), 841–890.
  • Conlon, C., & Gortmaker, J. (2020). Best Practices for Differentiated Products Demand Estimation with pyblp. RAND Journal of Economics.
  • Lei Ma, baby_BLP: https://github.com/leima0521/baby_BLP
  • Nevo, A. (2000). A Practitioner’s Guide to Estimation of Random-Coefficients Logit Models of Demand. Journal of Economics & Management Strategy, 9(4), 513–548.