# pyblp.Problem.solve¶

Problem.solve(sigma=None, pi=None, rho=None, beta=None, gamma=None, sigma_bounds=None, pi_bounds=None, rho_bounds=None, beta_bounds=None, gamma_bounds=None, delta=None, method='2s', initial_update=False, optimization=None, scale_objective=True, check_optimality='both', finite_differences=False, error_behavior='revert', error_punishment=1, delta_behavior='first', iteration=None, fp_type='safe_linear', shares_bounds=(1e-300, None), costs_bounds=None, W=None, center_moments=True, W_type='robust', se_type='robust', micro_moments=(), extra_micro_covariances=None)

Solve the problem.

The problem is solved in one or more GMM steps. During each step, any parameters in $$\theta$$ are optimized to minimize the GMM objective value, giving the estimated $$\hat{\theta}$$. If there are no parameters in $$\theta$$ (for example, in the logit model there are no nonlinear parameters and all linear parameters can be concentrated out), the objective is evaluated once during the step.

If there are nonlinear parameters, the mean utility, $$\delta(\theta)$$ is computed market-by-market with fixed point iteration. Otherwise, it is computed analytically according to the solution of the logit model. If a supply side is to be estimated, marginal costs, $$c(\theta)$$, are also computed market-by-market. Linear parameters are then estimated, which are used to recover structural error terms, which in turn are used to form the objective value. By default, the objective gradient is computed as well.

Note

This method supports parallel() processing. If multiprocessing is used, market-by-market computation of $$\delta(\theta)$$ (and $$\tilde{c}(\theta)$$ if a supply side is estimated), along with associated Jacobians, will be distributed among the processes.

Parameters
• sigma (array-like, optional) –

Configuration for which elements in the lower-triangular Cholesky root of the covariance matrix for unobserved taste heterogeneity, $$\Sigma$$, are fixed at zero and starting values for the other elements, which, if not fixed by sigma_bounds, are in the vector of unknown elements, $$\theta$$.

Rows and columns correspond to columns in $$X_2$$, which is formulated according product_formulations in Problem. If $$X_2$$ was not formulated, this should not be specified, since the logit model will be estimated.

Values above the diagonal are ignored. Zeros are assumed to be zero throughout estimation and nonzeros are, if not fixed by sigma_bounds, starting values for unknown elements in $$\theta$$. If any columns are fixed at zero, only the first few columns of integration nodes (specified in Problem) will be used.

To have nonzero covariances for only a subset of the random coefficients, the characteristics for those random coefficients with zero covariances should come first in $$X_2$$. This can be seen by looking at the expression for $$\Sigma\Sigma'$$, the actual covariance matrix of the random coefficients.

• pi (array-like, optional) –

Configuration for which elements in the matrix of parameters that measures how agent tastes vary with demographics, $$\Pi$$, are fixed at zero and starting values for the other elements, which, if not fixed by pi_bounds, are in the vector of unknown elements, $$\theta$$.

Rows correspond to the same product characteristics as in sigma. Columns correspond to columns in $$d$$, which is formulated according to agent_formulation in Problem. If $$d$$ was not formulated, this should not be specified.

Zeros are assumed to be zero throughout estimation and nonzeros are, if not fixed by pi_bounds, starting values for unknown elements in $$\theta$$.

• rho (array-like, optional) –

Configuration for which elements in the vector of parameters that measure within nesting group correlation, $$\rho$$, are fixed at zero and starting values for the other elements, which, if not fixed by rho_bounds, are in the vector of unknown elements, $$\theta$$.

If this is a scalar, it corresponds to all groups defined by the nesting_ids field of product_data in Problem. If this is a vector, it must have $$H$$ elements, one for each nesting group. Elements correspond to group IDs in the sorted order of Problem.unique_nesting_ids. If nesting IDs were not specified, this should not be specified either.

Zeros are assumed to be zero throughout estimation and nonzeros are, if not fixed by rho_bounds, starting values for unknown elements in $$\theta$$.

• beta (array-like, optional) –

Configuration for which elements in the vector of demand-side linear parameters, $$\beta$$, are concentrated out of the problem. Usually, this is left unspecified, unless there is a supply side, in which case parameters on endogenous product characteristics cannot be concentrated out of the problem. Values specify which elements are fixed at zero and starting values for the other elements, which, if not fixed by beta_bounds, are in the vector of unknown elements, $$\theta$$.

Elements correspond to columns in $$X_1$$, which is formulated according to product_formulations in Problem.

Both None and numpy.nan indicate that the parameter should be concentrated out of the problem. That is, it will be estimated, but does not have to be included in $$\theta$$. Zeros are assumed to be zero throughout estimation and nonzeros are, if not fixed by beta_bounds, starting values for unknown elements in $$\theta$$.

• gamma (array-like, optional) –

Configuration for which elements in the vector of supply-side linear parameters, $$\gamma$$, are concentrated out of the problem. Usually, this is left unspecified. Values specify which elements are fixed at zero and starting values for the other elements, which, if not fixed by gamma_bounds, are in the vector of unknown elements, $$\theta$$.

Elements correspond to columns in $$X_3$$, which is formulated according to product_formulations in Problem. If $$X_3$$ was not formulated, this should not be specified.

Both None and numpy.nan indicate that the parameter should be concentrated out of the problem. That is, it will be estimated, but does not have to be included in $$\theta$$. Zeros are assumed to be zero throughout estimation and nonzeros are, if not fixed by gamma_bounds, starting values for unknown elements in $$\theta$$.

• sigma_bounds (tuple, optional) –

Configuration for $$\Sigma$$ bounds of the form (lb, ub), in which both lb and ub are of the same size as sigma. Each element in lb and ub determines the lower and upper bound for its counterpart in sigma. If optimization does not support bounds, these will be ignored. If bounds are supported, the diagonal of sigma is by default bounded from below by zero.

Values above the diagonal are ignored. Lower and upper bounds corresponding to zeros in sigma are set to zero. Setting a lower bound equal to an upper bound fixes the corresponding element, removing it from $$\theta$$. Both None and numpy.nan are converted to -numpy.inf in lb and to numpy.inf in ub.

• pi_bounds (tuple, optional) –

Configuration for $$\Pi$$ bounds of the form (lb, ub), in which both lb and ub are of the same size as pi. Each element in lb and ub determines the lower and upper bound for its counterpart in pi. If optimization does not support bounds, these will be ignored. By default, pi is unbounded.

Lower and upper bounds corresponding to zeros in pi are set to zero. Setting a lower bound equal to an upper bound fixes the corresponding element, removing it from $$\theta$$. Both None and numpy.nan are converted to -numpy.inf in lb and to numpy.inf in ub.

• rho_bounds (tuple, optional) –

Configuration for $$\rho$$ bounds of the form (lb, ub), in which both lb and ub are of the same size as rho. Each element in lb and ub determines the lower and upper bound for its counterpart in rho. If optimization does not support bounds, these will be ignored.

If bounds are supported, rho is by default bounded from below by 0, which corresponds to the simple logit model, and bounded from above by 0.99 because values greater than 1 are inconsistent with utility maximization.

Lower and upper bounds corresponding to zeros in rho are set to zero. Setting a lower bound equal to an upper bound fixes the corresponding element, removing it from $$\theta$$. Both None and numpy.nan are converted to -numpy.inf in lb and to numpy.inf in ub.

• beta_bounds (tuple, optional) –

Configuration for $$\beta$$ bounds of the form (lb, ub), in which both lb and ub are of the same size as beta. Each element in lb and ub determines the lower and upper bound for its counterpart in beta. If optimization does not support bounds, these will be ignored.

Usually, this is left unspecified unless there is a supply side, in which case parameters on endogenous product characteristics cannot be concentrated out of the problem. It is generally a good idea to constrain such parameters to be nonzero so that the intra-firm Jacobian of shares with respect to prices does not become singular.

By default, all non-concentrated out parameters are unbounded. Bounds should only be specified for parameters that are included in $$\theta$$; that is, those with initial values specified in beta.

Lower and upper bounds corresponding to zeros in beta are set to zero. Setting a lower bound equal to an upper bound fixes the corresponding element, removing it from $$\theta$$. Both None and numpy.nan are converted to -numpy.inf in lb and to numpy.inf in ub.

• gamma_bounds (tuple, optional) –

Configuration for $$\gamma$$ bounds of the form (lb, ub), in which both lb and ub are of the same size as gamma. Each element in lb and ub determines the lower and upper bound for its counterpart in gamma. If optimization does not support bounds, these will be ignored.

By default, all non-concentrated out parameters are unbounded. Bounds should only be specified for parameters that are included in $$\theta$$; that is, those with initial values specified in gamma.

Lower and upper bounds corresponding to zeros in gamma are set to zero. Setting a lower bound equal to an upper bound fixes the corresponding element, removing it from $$\theta$$. Both None and numpy.nan are converted to -numpy.inf in lb and to numpy.inf in ub.

• delta (array-like, optional) – Initial values for the mean utility, $$\delta$$. If there are any nonlinear parameters, these are the values at which the fixed point iteration routine will start during the first objective evaluation. By default, the solution to the logit model in (40) is used. If $$\rho$$ is specified, the solution to the nested logit model in (41) under the initial rho is used instead.

• method (str, optional) –

The estimation routine that will be used. The following methods are supported:

• '1s' - One-step GMM.

• '2s' (default) - Two-step GMM.

Iterated GMM can be manually implemented by executing single GMM steps in a loop, in which after the first iteration, nonlinear parameters and weighting matrices from the last ProblemResults are passed as arguments.

• initial_update (bool, optional) – Whether to update starting values for the mean utility $$\delta$$, and the weighting matrix, $$W$$, at the initial parameter values before the first GMM step (this initial update will be called a zeroth step). By default, initial values are not updated because this requires an additional objective evaluation.

• optimization (Optimization, optional) – Optimization configuration for how to solve the optimization problem in each GMM step, which is only used if there are unfixed nonlinear parameters over which to optimize. By default, Optimization('l-bfgs-b', {'ftol': 0, 'gtol': 1e-8}) is used. If available, Optimization('knitro') may be preferable. Generally, it is recommended to consider a number of different optimization routines and starting values, verifying that $$\hat{\theta}$$ satisfies both the first and second order conditions. Choosing a routine that supports bounds (and configuring bounds) is typically a good idea. Choosing a routine that does not use analytic gradients will often down estimation.

• scale_objective (bool, optional) –

Whether to scale the objective in (10) by $$N$$, the number of observations, in which case the objective after two GMM steps is equal to the $$J$$ statistic from Hansen (1982). By default, the objective is scaled by $$N$$.

In theory the scale of the objective should not matter, but in practice having similar objective values for different problem sizes is helpful because similar optimization tolerances can be used.

• check_optimality (str, optional) –

How to check for optimality (first and second order conditions) after the optimization routine finishes. The following configurations are supported:

• 'gradient' - Analytically compute the gradient after optimization finishes, but do not compute the Hessian. Since Jacobians needed to compute standard errors will already be computed, gradient computation will not take a long time. This option may be useful if Hessian computation takes a long time when, for example, there are a large number of parameters.

• 'both' (default) - Also compute the Hessian with central finite differences after optimization finishes.

• finite_differences (bool, optional) –

Whether to use finite differences to compute Jacobians and the gradient instead of analytic expressions. Since finite differences comes with numerical approximation error and is typically slower, analytic expressions are used by default.

One situation in which finite differences may be preferable is when there are a sufficiently large number of products and integration nodes in individual markets so as to make computing analytic Jacobians infeasible because of large memory requirements. Note that an analytic expression for the Hessian has not been implemented, so when computed it is always approximated with finite differences.

• error_behavior (str, optional) –

How to handle any errors. For example, there can sometimes be overflow or underflow when computing $$\delta(\theta)$$ at a large $$\hat{\theta}$$. The following behaviors are supported:

• 'revert' (default) - Revert problematic values to their last computed values. If there are problematic values during the first objective evaluation, revert values in $$\delta(\theta)$$ to their starting values; in $$\tilde{c}(\hat{\theta})$$, to prices; in the objective, to 1e10; and in other matrices such as Jacobians, to zeros.

• 'punish' - Set the objective to 1 and its gradient to all zeros. This option along with a large error_punishment can be helpful for routines that do not use analytic gradients.

• 'raise' - Raise an exception.

• error_punishment (float, optional) – How to scale the GMM objective value after an error. By default, the objective value is not scaled.

• delta_behavior (str, optional) –

Configuration for the values at which the fixed point computation of $$\delta(\theta)$$ in each market will start. This configuration is only relevant if there are unfixed nonlinear parameters over which to optimize. The following behaviors are supported:

• 'first' (default) - Start at the values configured by delta during the first GMM step, and at the values computed by the last GMM step for each subsequent step.

• 'last' - Start at the values of $$\delta(\theta)$$ computed during the last objective evaluation, or, if this is the first evaluation, at the values configured by delta. This behavior tends to speed up computation but may introduce some instability into estimation.

• iteration (Iteration, optional) – Iteration configuration for how to solve the fixed point problem used to compute $$\delta(\theta)$$ in each market. This configuration is only relevant if there are nonlinear parameters, since $$\delta$$ can be estimated analytically in the logit model. By default, Iteration('squarem', {'atol': 1e-14}) is used. Newton-based routines such as Iteration('lm') that compute the Jacobian can often be faster (especially when there are nesting parameters), but the Jacobian-free SQUAREM routine is used by default because it speed is often comparable and in practice it can be slightly more stable.

• fp_type (str, optional) –

Configuration for the type of contraction mapping used to compute $$\delta(\theta)$$. The following types are supported:

• 'safe_linear' (default) - The standard linear contraction mapping in (13) (or (39) when there is nesting) with safeguards against numerical overflow. Specifically, $$\max_j V_{ijt}$$ (or $$\max_j V_{ijt} / (1 - \rho_{h(j)})$$ when there is nesting) is subtracted from $$V_{ijt}$$ and the logit expression for choice probabilities in (5) (or (37)) is re-scaled accordingly. Such re-scaling is known as the log-sum-exp trick.

• 'linear' - The standard linear contraction mapping without safeguards against numerical overflow. This option may be preferable to 'safe_linear' if utilities are reasonably small and unlikely to create overflow problems.

• 'nonlinear' - Iteration over $$\exp \delta_{jt}$$ instead of $$\delta_{jt}$$. This can be faster than 'linear' because it involves fewer logarithms. Also, following Brunner, Heiss, Romahn, and Weiser (2017), the $$\exp \delta_{jt}$$ term can be cancelled out of the expression because it also appears in the numerator of (5) in the definition of $$s_{jt}(\delta, \theta)$$. This second trick only works when there are no nesting parameters.

• 'safe_nonlinear' - Exponentiated version with minimal safeguards against numerical overflow. Specifically, $$\max_j \mu_{ijt}$$ is subtracted from $$\mu_{ijt}$$. This helps with stability but is less helpful than subtracting from the full $$V_{ijt}$$, so this version is less stable than 'safe_linear'.

This option is only relevant if sigma or pi are specified because $$\delta$$ can be estimated analytically in the logit model with (40) and in the nested logit model with (41).

• shares_bounds (tuple, optional) –

Configuration for $$s_{jt}(\delta, \theta)$$ bounds in the contraction in (13) of the form (lb, ub), in which both lb and ub are floats or None. By default, simulated shares are bounded from below by 1e-300. This is only relevant if fp_type is 'safe_linear' or 'linear'. Bounding shares in the contraction does nothing with a nonlinear fixed point.

It can be particularly helpful to bound shares in the contraction from below by a small number to prevent the contraction from failing when there are issues with zero or negative simulated shares. Zero shares can occur when there are underflow issues and negative shares can occur when there are issues with the numerical integration routine having negative integration weights (e.g., for sparse grid integration).

The idea is that a small lower bound will allow the contraction to converge even when it encounters some issues with small or negative shares. However, if these issues are unlikely, disabling this behavior can speed up the iteration routine because fewer checks will be done.

Both None and numpy.nan are converted to -numpy.inf in lb and to numpy.inf in ub.

• costs_bounds (tuple, optional) –

Configuration for $$c_{jt}(\theta)$$ bounds of the form (lb, ub), in which both lb and ub are floats or None. This is only relevant if $$X_3$$ was formulated by product_formulations in Problem. By default, marginal costs are unbounded.

When costs_type in Problem is 'log', nonpositive $$c(\theta)$$ values can create problems when computing $$\tilde{c}(\theta) = \log c(\theta)$$. One solution is to set lb to a small number. Rows in Jacobians associated with clipped marginal costs will be zero.

Both None and numpy.nan are converted to -numpy.inf in lb and to numpy.inf in ub.

• W (array-like, optional) –

Starting values for the weighting matrix, $$W$$. By default, the 2SLS weighting matrix in (23) is used.

If there are any micro_moments, the initial weighting matrix will by default be block-diagonal with an identity matrix for the micro moment block. This micro moment block should usually be replaced by a matrix that better reflects micro moment covariances and the size of the micro dataset relative to $$N$$.

• center_moments (bool, optional) – Whether to center each column of the demand- and supply-side moments $$g$$ before updating the weighting matrix $$W$$ according to (24). By default, the moments are centered. This has no effect if W_type is 'unadjusted'.

• W_type (str, optional) –

How to update the weighting matrix. This has no effect if method is '1s'. Usually, se_type should be the same. The following types are supported:

• 'robust' (default) - Heteroscedasticity robust weighting matrix defined in (24) and (25).

• 'clustered' - Clustered weighting matrix defined in (24) and (26). Clusters must be defined by the clustering_ids field of product_data in Problem.

• 'unadjusted' - Homoskedastic weighting matrix defined in (24) and (28).

This only affects the standard demand- and supply-side block of the updated weighting matrix. If there are micro moments, this matrix will be block-diagonal with a micro moment block equal to the inverse of the covariance matrix defined in (35) plus any extra_micro_covariances.

• se_type (str, optional) –

How to compute parameter covariances and standard errors. Usually, W_type should be the same. The following types are supported:

• 'robust' (default) - Heteroscedasticity robust covariances defined in (30) and (25).

• 'clustered' - Clustered covariances defined in (30) and (26). Clusters must be defined by the clustering_ids field of product_data in Problem.

• 'unadjusted' - Homoskedastic covariances defined in (31), which are computed under the assumption that the weighting matrix is optimal.

This only affects the standard demand- and supply-side block of the matrix of averaged moment covariances. If there are micro moments, the $$S$$ matrix defined in the expressions referenced above will be block-diagonal with a micro moment block equal to the covariance matrix defined in (35) plus any extra_micro_covariances.

• micro_moments (sequence of Moment, optional) –

Configurations for the $$M_M$$ micro moments that will be added to the standard set of moments. For a list of supported micro moments, refer to Micro Moment Classes. By default, no micro moments are used, so $$M_M = 0$$.

If micro moments are specified, the micro moment block in W should usually be replaced by a matrix that better reflects micro moment covariances and the size of the micro dataset relative to $$N$$. If micro moments were computed with substantial sampling error, extra_micro_covariances can be specified to account for this additional source of error.

• extra_micro_covariances (array-like, optional) – Covariance matrix that is added on to the $$M_M \times M_M$$ matrix of micro moments covariances defined in (35), which is used to update the weighting matrix and compute standard errors. By default, this matrix is assumed to be zero. It should be specified if, for example, micro moments were computed with substantial sampling error.

Returns

ProblemResults` of the solved problem.

Return type

ProblemResults

Examples