# pyblp.DemographicCovarianceMoment¶

class pyblp.DemographicCovarianceMoment(X2_index, demographics_index, values, market_ids=None)

Configuration for micro moments that match covariances between product characteristics and demographics.

For example, micro data can sometimes be used to compute the sample covariance between a product characteristic $$x_{jt}$$ of an agent’s choice $$j$$, and a demographic such as income, $$y_{it}$$, amongst those agents who purchase an inside good. 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)$v_{mt} = \text{Cov}(y_{it}, z_{it})$

where conditional on choosing an inside good, the expected value of $$x_{jt}$$ for agent $$i$$ is

(2)$z_{it} = \sum_{j \in J_t} x_{jt}s_{ij(-0)t}$

where $$s_{ij(-0)t} = s_{ijt} / (1 - s_{i0t})$$ is the probability of $$i$$ choosing $$j$$ when the outside option is 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_index (int) – Column index of $$x_{jt}$$ in the matrix of demand-side nonlinear product characteristics, $$X_2$$. This should be between zero and $$K_2 - 1$$, inclusive.

• demographics_index (int) – Column index of the demographic $$y_{it}$$ (which can be any demographic, not just income) in the matrix of agent demographics, $$d$$. This should be between zero and $$D - 1$$, 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