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.

Dimensions and Sets

Symbol

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 \(t\)

\(I_t\)

Agents in market \(t\)

\(K_1\)

Linear product characteristics

\(K_1^x\)

Exogenous linear product characteristics

\(K_1^p\)

Endogenous linear product characteristics

\(K_2\)

Nonlinear product characteristics

\(K_3\)

Cost product characteristics

\(D\)

Demographic variables

\(M_D\)

Demand-side instruments, which is the number of exluded demand-side instruments plus \(K_1^x\)

\(M_S\)

Supply-side instruments, which is the number of exluded supply-side instruments plus \(K_3\)

\(E_D\)

Absorbed dimensions of demand-side fixed effects

\(E_S\)

Absorbed dimensions of supply-side fixed effects

\(H\)

Nesting groups

\(C\)

Clusters

\(P\)

Parameters

\(\mathscr{J}_{ft}\)

Set of products produced by firm \(f\) in market \(t\)

\(\mathscr{J}_{ht}\)

Set of products in nesting group \(h\) and market \(t\)

\(\mathscr{J}_c\)

Set of products in cluster \(c\)

Matrices, Vectors, and Scalars

Dimensions that can differ across markets are reported for a single market \(t\). Some notation differs depending on how it is indexed.

Symbol

Dimensions

Description

\(X_1\)

\(N \times K_1\)

Linear product characteristics

\(X_1^x\)

\(N \times K_1^x\)

Exogenous linear product characteristics

\(X_1^p\)

\(N \times K_1^p\)

Endogenous linear product characteristics

\(X_2\)

\(N \times K_2\)

Nonlinear product characteristics

\(X_3\)

\(N \times K_3\)

Cost 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

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

\(N \times 1\)

Marketshares

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

\(H \times 1\)

Group shares in a market \(t\)

\(s\) (\(s_{jti}\))

\(N \times I_t\)

Choice probabiltiies 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

\(O\)

\(J_t \times J_t\)

Ownership 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

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

\(N \times 1\)

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

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

\(N \times 1\)

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

\(U\)

\(J_t \times I_t\)

Indirect utilities

\(V\) (\(V_{jti}\))

\(J_t \times I_t\)

Indirect utilities minus \(\epsilon\)

\(V\) (\(V_{hti}\))

\(J_t \times I_t\)

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

\(\beta^x\)

\(K_1^x \times 1\)

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

\(\alpha\)

\(K_1^p \times 1\)

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

\(\theta\)

\(P \times 1\)

Parameters

\(Z_D\)

\(N \times M_D\)

Excluded demand-side instruments and \(X_1\), except for \(X_1^p\)

\(Z_S\)

\(N \times M_S\)

Excluded supply-side instruments and \(X_3\)

\(W\)

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

Weighting matrix

\(S\)

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

Sample moment covariances or inverse of the weighting matrix

\(q\)

\(1 \times 1\)

Objective value

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

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

Sample moments

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

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

Clustered sample moments

\(\bar{g}\)

\((M_D + M_S) \times 1\)

Sample moment conditions

\(\bar{G}\)

\((M_D + M_S) \times P\)

Jacobian of the sample moment conditions 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