Notation¶
The notation in PyBLP is a customized amalgamation of the notation employed by Berry, Levinsohn, and Pakes (1995), Nevo (2000a), 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 |
\(M_C\) |
Covariance 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 |
\(N_m\) |
Observations underlying observed micro moment value \(m\). |
\(M\) |
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, transposed) 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 \(t\) |
\(a\) |
\(I_t \times J_t\) |
Agent-specific product availability 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 |
\(Z_C\) |
\(N \times M_C\) |
Covariance 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 |
\(g_C\) |
\(N \times M_C\) |
Covariance moments |
\(g_M\) |
\(I \times M_M\) |
Micro moments |
\(g\) (\(g_{jt}\)) |
\(N \times (M_D + M_S + M_C)\) |
Demand-side, supply-side, and covariance moments |
\(g\) (\(g_c\)) |
\(C \times (M_D + M_S + M_C)\) |
Clustered demand-side, supply-side, and covariance moments |
\(\bar{g}_D\) |
\(M_D \times 1\) |
Averaged demand-side moments |
\(\bar{g}_S\) |
\(M_S \times 1\) |
Averaged supply-side moments |
\(\bar{g}_C\) |
\(M_C \times 1\) |
Averaged covariance 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 |