pyblp.DiversionCovarianceMoment

class pyblp.DiversionCovarianceMoment(X2_index1, X2_index2, values, market_ids=None)

Configuration for micro moments that match covariances between product characteristics of first and second choices.

For example, survey data can sometimes be used to compute the sample covariance between a product characteristic \(x_{jt}^{(1)}\) of an agent’s first choice \(j\) and either the same or a different product characteristic \(x_{kt}^{(2)}\) of the agent’s second choice \(k\) if \(j\) were removed from the choice set, amongst those agents whose first and second choices are both inside goods. With the value \(\mathscr{V}_{mt}\) of this sample covariance, a micro moment \(m\) in market \(t\) can be defined by \(g_{M,mt} = \mathscr{V}_{mt} - v_{mt}\) where

(1)\[v_{mt} = \text{Cov}(z_{it}^{(1)}, z_{it}^{(2)})\]

where conditional on purchasing inside goods, the expected values of \(x_{jt}^{(1)}\) and \(x_{kt}^{(2)}\) for agent \(i\) are

(2)\[z_{it}^{(1)} = \sum_{j \in J_t} x_{jt}^{(1)} s_{ij(-0)t}, \quad z_{it}^{(2)} = \sum_{j, k \in J_t} x_{kt}^{(2)} s_{ik(-0,j)t} s_{ij(-0)t}\]

where \(s_{ij(-0)t}\) is the probability of choosing \(j\) when the outside option is removed from the choice set and \(s_{ik(-0,j)t}\) is the probability of choosing \(k\) when both the outside option and \(j\) are removed from the choice set.

Integrals of these micro moments are averaged across a set \(T_m\) of markets, which gives \(\bar{g}_{M,m}\) in (34).

Parameters
  • X2_index1 (int) – Column index of \(x_{jt}^{(1)}\) in the matrix of demand-side nonlinear product characteristics, \(X_2\). This should be between zero and \(K_2 - 1\), inclusive.

  • X2_index2 (int) – Column index of \(x_{kt}^{(2)}\) in the matrix of demand-side nonlinear product characteristics, \(X_2\). This should be between zero and \(K_2 - 1\), inclusive.

  • values (float) – Values \(\mathscr{V}_{mt}\) of the statistic estimated from micro data. If a scalar is specified, then \(\mathscr{V}_{mt} = \mathscr{V}_m\) is assumed to be constant across all markets in which the moment is relevant. Otherwise, this should have as many elements as market_ids, or as the total number of markets if market_ids is None.

  • market_ids (array-like, optional) – Distinct market IDs over which the micro moments will be averaged to get \(\bar{g}_{M,m}\). These are also the only markets in which the moments will be computed. By default, the moments are computed for and averaged across all markets.

Examples

Methods