# 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 |