pyblp.ProblemResults¶

class
pyblp.
ProblemResults
¶ Results of a solved BLP problem.
Many results are class attributes. Other postestimation outputs be computed by calling class methods.
Note
Methods in this class that compute one or more postestimation output per market support
parallel()
processing. If multiprocessing is used, marketbymarket computation of each postestimation output will be distributed among the processes.
last_results
¶ ProblemResults
from the last GMM step. Type
ProblemResults

step
¶ GMM step that created these results.
 Type
int

optimization_time
¶ Number of seconds it took the optimization routine to finish.
 Type
float

cumulative_optimization_time
¶ Sum of
ProblemResults.optimization_time
for this step and all prior steps. Type
float

total_time
¶ Sum of
ProblemResults.optimization_time
and the number of seconds it took to set up the GMM step and compute results after optimization had finished. Type
float

cumulative_total_time
¶ Sum of
ProblemResults.total_time
for this step and all prior steps. Type
float

converged
¶ Whether the optimization routine converged.
 Type
bool

cumulative_converged
¶ Whether the optimization routine converged for this step and all prior steps.
 Type
bool

optimization_iterations
¶ Number of major iterations completed by the optimization routine.
 Type
int

cumulative_optimization_iterations
¶ Sum of
ProblemResults.optimization_iterations
for this step and all prior steps. Type
int

objective_evaluations
¶ Number of GMM objective evaluations.
 Type
int

cumulative_objective_evaluations
¶ Sum of
ProblemResults.objective_evaluations
for this step and all prior steps. Type
int

fp_converged
¶ Flags for convergence of the iteration routine used to compute \(\delta(\theta)\) in each market during each objective evaluation. Rows are in the same order as
Problem.unique_market_ids
and column indices correspond to objective evaluations. Type
ndarray

cumulative_fp_converged
¶ Concatenation of
ProblemResults.fp_converged
for this step and all prior steps. Type
ndarray

fp_iterations
¶ Number of major iterations completed by the iteration routine used to compute \(\delta(\theta)\) in each market during each objective evaluation. Rows are in the same order as
Problem.unique_market_ids
and column indices correspond to objective evaluations. Type
ndarray

cumulative_fp_iterations
¶ Concatenation of
ProblemResults.fp_iterations
for this step and all prior steps. Type
ndarray

contraction_evaluations
¶ Number of times the contraction used to compute \(\delta(\theta)\) was evaluated in each market during each objective evaluation. Rows are in the same order as
Problem.unique_market_ids
and column indices correspond to objective evaluations. Type
ndarray

cumulative_contraction_evaluations
¶ Concatenation of
ProblemResults.contraction_evaluations
for this step and all prior steps. Type
ndarray

parameters
¶ Stacked parameters in the following order: \(\hat{\theta}\), concentrated out elements of \(\hat{\beta}\), and concentrated out elements of \(\hat{\gamma}\).
 Type
ndarray

parameter_covariances
¶ Estimated covariance matrix for \(\sqrt{N}(\hat{\theta}  \theta)\), in which \(\theta\) are the stacked parameters. Standard errors are extracted from the diagonal of this matrix. Note that the estimated covariance matrix of \(\hat{\theta}\) is the same as this, but divided by \(N\). Parameter covariances are not estimated during the first step of twostep GMM.
 Type
ndarray

theta
¶ Estimated unfixed parameters, \(\hat{\theta}\), in the following order: \(\hat{\Sigma}\), \(\hat{\Pi}\), \(\hat{\rho}\), nonconcentrated out elements from \(\hat{\beta}\), and nonconcentrated out elements from \(\hat{\gamma}\).
 Type
ndarray

sigma
¶ Estimated Cholesky root of the covariance matrix for unobserved taste heterogeneity, \(\hat{\Sigma}\).
 Type
ndarray

sigma_squared
¶ Estimated covariance matrix for unobserved taste heterogeneity, \(\hat{\Sigma}\hat{\Sigma}'\).
 Type
ndarray

pi
¶ Estimated parameters that measures how agent tastes vary with demographics, \(\hat{\Pi}\).
 Type
ndarray

rho
¶ Estimated parameters that measure within nesting group correlations, \(\hat{\rho}\).
 Type
ndarray

beta
¶ Estimated demandside linear parameters, \(\hat{\beta}\).
 Type
ndarray

gamma
¶ Estimated supplyside linear parameters, \(\hat{\gamma}\).
 Type
ndarray

sigma_se
¶ Estimated standard errors for \(\hat{\Sigma}\), which are not estimated in the first step of twostep GMM.
 Type
ndarray

sigma_squared_se
¶ Estimated standard errors for \(\hat{\Sigma}\hat{\Sigma}'\), which are computed with the delta method, and are not estimated in the first step of twostep GMM.
 Type
ndarray

pi_se
¶ Estimated standard errors for \(\hat{\Pi}\), which are not estimated in the first step of twostep GMM.
 Type
ndarray

rho_se
¶ Estimated standard errors for \(\hat{\rho}\), which are not estimated in the first step of twostep GMM.
 Type
ndarray

beta_se
¶ Estimated standard errors for \(\hat{\beta}\), which are not estimated in the first step of twostep GMM.
 Type
ndarray

gamma_se
¶ Estimated standard errors for \(\hat{\gamma}\), which are not estimated in the first step of twostep GMM.
 Type
ndarray

sigma_bounds
¶ Bounds for \(\Sigma\) that were used during optimization, which are of the form
(lb, ub)
. Type
tuple

pi_bounds
¶ Bounds for \(\Pi\) that were used during optimization, which are of the form
(lb, ub)
. Type
tuple

rho_bounds
¶ Bounds for \(\rho\) that were used during optimization, which are of the form
(lb, ub)
. Type
tuple

beta_bounds
¶ Bounds for \(\beta\) that were used during optimization, which are of the form
(lb, ub)
. Type
tuple

gamma_bounds
¶ Bounds for \(\gamma\) that were used during optimization, which are of the form
(lb, ub)
. Type
tuple

sigma_labels
¶ Variable labels for rows and columns of \(\Sigma\), which are derived from the formulation for \(X_2\).
 Type
list of str

pi_labels
¶ Variable labels for columns of \(\Pi\), which are derived from the formulation for demographics.
 Type
list of str

rho_labels
¶ Variable labels for \(\rho\). If \(\rho\) is not a scalar, this is
Problem.unique_nesting_ids
. Type
list of str

beta_labels
¶ Variable labels for \(\beta\), which are derived from the formulation for \(X_1\).
 Type
list of str

gamma_labels
¶ Variable labels for \(\gamma\), which are derived from the formulation for \(X_3\).
 Type
list of str

delta
¶ Estimated mean utility, \(\delta(\hat{\theta})\).
 Type
ndarray
Vector of booleans indicator whether the associated simulated shares were clipped during the last fixed point iteration to compute \(\delta(\hat{\theta})\). All elements will be
False
ifshares_bounds
inProblem.solve()
is disabled (by default shares are bounded from below by a small number to alleviate issues with underflow and negative shares). Type
ndarray

tilde_costs
¶ Estimated transformed marginal costs, \(\tilde{c}(\hat{\theta})\) from (9). If
costs_bounds
were specified inProblem.solve()
, \(c\) may have been clipped. Type
ndarray

clipped_costs
¶ Vector of booleans indicating whether the associated marginal costs were clipped. All elements will be
False
ifcosts_bounds
inProblem.solve()
was not specified. Type
ndarray

xi
¶ Estimated unobserved demandside product characteristics, \(\xi(\hat{\theta})\), or equivalently, the demandside structural error term. When there are demandside fixed effects, this is \(\Delta\xi(\hat{\theta})\) in (32). That is, fixed effects are not included.
 Type
ndarray

omega
¶ Estimated unobserved supplyside product characteristics, \(\omega(\hat{\theta})\), or equivalently, the supplyside structural error term. When there are supplyside fixed effects, this is \(\Delta\omega(\hat{\theta})\) in (32). That is, fixed effects are not included.
 Type
ndarray

micro_values
¶ Simulated micro moment values, \(v_{mt}\). Rows are in the same order as
Problem.unique_market_ids
. Columns are in the same order asProblemResults.micro
. If a micro moment is not computed in one or more markets, the associated values will benumpy.nan
. Type
ndarray

objective
¶ GMM objective value, \(q(\hat{\theta})\), defined in (10). If
scale_objective
wasTrue
inProblem.solve()
(which is the default), this value was scaled by \(N\) so that objective values are more comparable across different problem sizes. Note that in some of the BLP literature (and earlier versions of this package), this expression was previously scaled by \(N^2\). Type
float

xi_by_theta_jacobian
¶ Estimated \(\frac{\partial\xi}{\partial\theta} = \frac{\partial\delta}{\partial\theta}\), which is used to compute the gradient and standard errors.
 Type
ndarray

omega_by_theta_jacobian
¶ Estimated \(\frac{\partial\omega}{\partial\theta} = \frac{\partial\tilde{c}}{\partial\theta}\), which is used to compute the gradient and standard errors.
 Type
ndarray

micro_by_theta_jacobian
¶ Estimated \(\frac{\partial\bar{g}_M}{\partial\theta}\), which is used to compute the gradient and standard errors.
 Type
ndarray

gradient
¶ Gradient of the GMM objective, \(\nabla q(\hat{\theta})\), defined in (18). This is computed after the optimization routine finishes even if the routine was configured to not use analytic gradients.
 Type
ndarray

projected_gradient
¶ Projected gradient of the GMM objective. When there are no parameter bounds, this will always be equal to
ProblemResults.gradient
. Otherwise, if an element in \(\hat{\theta}\) is equal to its lower (upper) bound, the corresponding projected gradient value will be truncated at a maximum (minimum) of zero. Type
ndarray

projected_gradient_norm
¶ Infinity norm of
ProblemResults.projected_gradient
. Type
ndarray

hessian
¶ Estimated Hessian of the GMM objective. By default, this is computed with finite central differences after the optimization routine finishes.
 Type
ndarray

reduced_hessian
¶ Reduced Hessian of the GMM objective. When there are no parameter bounds, this will always be equal to
ProblemResults.hessian
. Otherwise, if an element in \(\hat{\theta}\) is equal to either its lower or upper bound, the corresponding row and column in the reduced Hessian will be all zeros. Type
ndarray

reduced_hessian_eigenvalues
¶ Eigenvalues of
ProblemResults.reduced_hessian
. Type
ndarray

W
¶ Weighting matrix, \(W\), used to compute these results.
 Type
ndarray
Examples
Methods
bootstrap
([draws, seed, iteration])Use a parametric bootstrap to create an empirical distribution of results.
compute_aggregate_elasticities
([factor, …])Estimate aggregate elasticities of demand, \(\mathscr{E}\), with respect to a variable, \(x\).
compute_approximate_prices
([firm_ids, …])Approximate equilibrium prices after firm or cost changes, \(p^*\), under the assumption that shares and their price derivatives are unaffected by such changes.
compute_consumer_surpluses
([prices, …])Estimate populationnormalized consumer surpluses, \(\text{CS}\).
compute_costs
([firm_ids, ownership, market_id])Estimate marginal costs, \(c\).
compute_delta
([agent_data, integration, …])Estimate mean utilities, \(\delta\).
compute_diversion_ratios
([name, market_id])Estimate matrices of diversion ratios, \(\mathscr{D}\), with respect to a variable, \(x\).
compute_elasticities
([name, market_id])Estimate matrices of elasticities of demand, \(\varepsilon\), with respect to a variable, \(x\).
compute_hhi
([firm_ids, shares, market_id])Estimate HerfindahlHirschman Indices, \(\text{HHI}\).
compute_long_run_diversion_ratios
([market_id])Estimate matrices of longrun diversion ratios, \(\bar{\mathscr{D}}\).
compute_markups
([prices, costs, market_id])Estimate markups, \(\mathscr{M}\).
compute_optimal_instruments
([method, draws, …])Estimate feasible optimal or efficient instruments, \(Z_D^\text{opt}\) and \(Z_S^\text{opt}\).
compute_prices
([firm_ids, ownership, costs, …])Estimate equilibrium prices after firm or cost changes, \(p^*\).
compute_probabilities
([market_id])Estimate matrices of choice probabilities.
compute_profits
([prices, shares, costs, …])Estimate populationnormalized gross expected profits, \(\pi\).
compute_shares
([prices, delta, agent_data, …])Estimate shares.
extract_diagonal_means
(matrices)Extract means of diagonals from stacked \(J_t \times J_t\) matrices for each market \(t\).
extract_diagonals
(matrices)Extract diagonals from stacked \(J_t \times J_t\) matrices for each market \(t\).
importance_sampling
(draws[, ar_constant, …])Use importance sampling to construct nodes and weights for integration.
run_distance_test
(unrestricted)Test the validity of model restrictions with the distance test.
Test the validity of overidentifying restrictions with the Hansen \(J\) test.
Test the validity of model restrictions with the Lagrange multiplier test.
run_wald_test
(restrictions, …)Test the validity of model restrictions with the Wald test.
to_dict
([attributes])Convert these results into a dictionary that maps attribute names to values.
