Notation

The notation in PyBLP is a customized amalgamation of the notation employed by Berry, Levinsohn, and Pakes (1995), Nevo (2000), Morrow and Skerlos (2011), Grigolon and Verboven (2014), and others.

Indices

Index

Description

\(j\)

Products

\(t\)

Markets

\(i\)

Agents/individuals

\(f\)

Firms

\(h\)

Nests

\(c\)

Clusters

\(m\)

Micro moments

Dimensions/Sets

Dimension/Set

Description

\(T\)

Markets

\(N\)

Products across all markets

\(F\)

Firms across all markets

\(I\)

Agents across all markets

\(J_t\)

Products in market \(t\)

\(F_t\)

Firms in market \(t\)

\(J_{ft}\)

Products produced by firm \(f\) in market \(t\)

\(I_t\)

Agents in market \(t\)

\(K_1\)

Demand-side linear product characteristics

\(K_1^\text{ex}\)

Exogenous demand-side linear product characteristics

\(K_1^\text{en}\)

Endogenous demand-side linear product characteristics

\(K_2\)

Demand-side nonlinear product characteristics

\(K_3\)

Supply-side product characteristics

\(K_3^\text{ex}\)

Exogenous supply-side product characteristics

\(K_3^\text{en}\)

Endogenous supply-side product characteristics

\(D\)

Demographic variables

\(M_D\)

Demand-side instruments

\(M_S\)

Supply-side instruments

\(M_M\)

Micro moments

\(T_m\)

Markets over which micro moment \(m\) is averaged

\(T_{mn}\)

Markets over which micro moments \(m\) and \(n\) are both averaged

\(M\)

All moments

\(E_D\)

Absorbed dimensions of demand-side fixed effects

\(E_S\)

Absorbed dimensions of supply-side fixed effects

\(H\)

Nesting groups

\(J_{ht}\)

Products in nesting group \(h\) and market \(t\)

\(C\)

Clusters

\(J_{ct}\)

Products in cluster \(c\) and market \(t\)

Matrices, Vectors, and Scalars

Symbol

Dimensions

Description

\(X_1\)

\(N \times K_1\)

Demand-side linear product characteristics

\(X_1^\text{ex}\)

\(N \times K_1^\text{ex}\)

Exogenous demand-side linear product characteristics

\(X_1^\text{en}\)

\(N \times K_1^\text{en}\)

Endogenous demand-side linear product characteristics

\(X_2\)

\(N \times K_2\)

Demand-side Nonlinear product characteristics

\(X_3\)

\(N \times K_3\)

Supply-side product characteristics

\(X_3^\text{ex}\)

\(N \times K_3^\text{ex}\)

Exogenous supply-side product characteristics

\(X_3^\text{en}\)

\(N \times K_3^\text{en}\)

Endogenous supply-side product characteristics

\(\xi\)

\(N \times 1\)

Unobserved demand-side product characteristics

\(\omega\)

\(N \times 1\)

Unobserved supply-side product characteristics

\(p\)

\(N \times 1\)

Prices

\(s\) (\(s_{jt}\))

\(N \times 1\)

Market shares

\(s\) (\(s_{ht}\))

\(H \times 1\)

Group shares in a market \(t\)

\(s\) (\(s_{ijt}\))

\(N \times I_t\)

Choice probabilities in a market \(t\)

\(c\)

\(N \times 1\)

Marginal costs

\(\tilde{c}\)

\(N \times 1\)

Linear or log-linear marginal costs, \(c\) or \(\log c\)

\(\eta\)

\(N \times 1\)

Markup term from the BLP-markup equation

\(\zeta\)

\(N \times 1\)

Markup term from the \(\zeta\)-markup equation

\(\mathscr{H}\)

\(J_t \times J_t\)

Ownership or product holdings matrix in market \(t\)

\(\kappa\)

\(F_t \times F_t\)

Cooperation matrix in market \(t\)

\(\Delta\)

\(J_t \times J_t\)

Intra-firm matrix of (negative) demand derivatives in market \(t\)

\(\Lambda\)

\(J_t \times J_t\)

Diagonal matrix used to decompose \(\eta\) and \(\zeta\) in market \(t\)

\(\Gamma\)

\(J_t \times J_t\)

Another matrix used to decompose \(\eta\) and \(\zeta\) in market \(t\)

\(d\)

\(I_t \times D\)

Observed agent characteristics called demographics in market \(t\)

\(\nu\)

\(I_t \times K_2\)

Unobserved agent characteristics called integration nodes in market \(t\)

\(w\)

\(I_t \times 1\)

Integration weights in market \(t\)

\(\delta\)

\(N \times 1\)

Mean utility

\(\mu\)

\(J_t \times I_t\)

Agent-specific portion of utility in market \(t\)

\(\epsilon\)

\(N \times 1\)

Type I Extreme Value idiosyncratic preferences

\(\bar{\epsilon}\) (\(\bar{\epsilon}_{ijt}\))

\(N \times 1\)

Type I Extreme Value term used to decompose \(\epsilon\)

\(\bar{\epsilon}\) (\(\bar{\epsilon}_{iht}\))

\(N \times 1\)

Group-specific term used to decompose \(\epsilon\)

\(U\)

\(J_t \times I_t\)

Indirect utilities

\(V\) (\(V_{ijt}\))

\(J_t \times I_t\)

Indirect utilities minus \(\epsilon\)

\(V\) (\(V_{iht}\))

\(J_t \times I_t\)

Inclusive value of a nesting group

\(\pi\) (\(\pi_{jt}\))

\(N \times 1\)

Population-normalized gross expected profits

\(\pi\) (\(\pi_{ft}\))

\(F_t \times 1\)

Population-normalized gross expected profits of a firm in market \(t\)

\(\beta\)

\(K_1 \times 1\)

Demand-side linear parameters

\(\beta^\text{ex}\)

\(K_1^\text{ex} \times 1\)

Parameters in \(\beta\) on exogenous product characteristics

\(\alpha\)

\(K_1^\text{en} \times 1\)

Parameters in \(\beta\) on endogenous product characteristics

\(\Sigma\)

\(K_2 \times K_2\)

Cholesky root of the covariance matrix for unobserved taste heterogeneity

\(\Pi\)

\(K_2 \times D\)

Parameters that measures how agent tastes vary with demographics

\(\rho\)

\(H \times 1\)

Parameters that measures within nesting group correlation

\(\gamma\)

\(K_3 \times 1\)

Supply-side linear parameters

\(\gamma^\text{ex}\)

\(K_3^\text{ex} \times 1\)

Parameters in \(\gamma\) on exogenous product characteristics

\(\gamma^\text{en}\)

\(K_3^\text{en} \times 1\)

Parameters in \(\gamma\) on endogenous product characteristics

\(\theta\)

\(P \times 1\)

Parameters

\(Z_D\)

\(N \times M_D\)

Demand-side instruments

\(Z_S\)

\(N \times M_S\)

Supply-side instruments

\(W\)

\(M \times M\)

Weighting matrix

\(S\)

\(M \times M\)

Moment covariances

\(q\)

\(1 \times 1\)

Objective value

\(g_D\)

\(N \times M_D\)

Demand-side moments

\(g_S\)

\(N \times M_S\)

Supply-side moments

\(g_M\)

\(I \times M_M\)

Micro moments

\(g\) (\(g_{jt}\))

\(N \times (M_D + M_S)\)

Demand- and supply-side moments

\(g\) (\(g_c\))

\(C \times (M_D + M_S)\)

Clustered demand- and supply-side moments

\(\bar{g}_D\)

\(M_D \times 1\)

Averaged demand-side moments

\(\bar{g}_S\)

\(M_S \times 1\)

Averaged supply-side moments

\(\bar{g}_M\)

\(M_M \times 1\)

Averaged micro moments

\(\bar{g}\)

\(M \times 1\)

Averaged moments

\(\bar{G}\)

\(M \times P\)

Jacobian of the averaged moments with respect to \(\theta\)

\(\varepsilon\)

\(J_t \times J_t\)

Elasticities of demand in market \(t\)

\(\mathscr{D}\)

\(J_t \times J_t\)

Diversion ratios in market \(t\)

\(\bar{\mathscr{D}}\)

\(J_t \times J_t\)

Long-run diversion ratios in market \(t\)

\(\mathscr{M}\)

\(N \times 1\)

Markups

\(\mathscr{E}\)

\(1 \times 1\)

Aggregate elasticity of demand of a market

\(\text{CS}\)

\(1 \times 1\)

Population-normalized consumer surplus of a market

\(\text{HHI}\)

\(1 \times 1\)

Herfindahl-Hirschman Index of a market