# 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 |