Calibration Procedure#

EKW models are calibrated to data on observed individual decisions and experiences, given the assumption that individuals behave according to the behavioral model. For example, the Keane and Wolpin (1997, [13]) model is calibrated to a subsample of young men from the National Longitudinal Survey 1979 (NLSY79). The goal is to back out information on utility functions, preference parameters, and transition probabilities. This requires to fully parameterize the model. We denote \(\theta\) the vector of structural parameters out of an admissible parameter space \(\Theta\).

The discount factor is pre-defined in respy and mandatory. It is located in params data frame under the key delta.

Under the paradigm of revealed preferences (Samuelson, 1938, [21]) structural preference parameters are estimable with microdata on individual decisions. Generally, we have access to information for \(i = 1, \dots, N\) individuals at each point in time \(1, \dots, T_i\). For each observation \((i,t)\) in the data we observe that action taken \(a_{it}\) some components of the utility \(\bar{u}_{it}\), and the observable state space \(\bar{s}_t\). At time \(t\) both the individual and we as a researcher observe \(\bar{s}_t\) but the stochastic component \(\epsilon_t\) is only observed by the individual. [1] In summary, the data structure is given by

(1)#\[\mathcal{D} = \big\{ a_{it}, \bar{s}_{it}, \bar{u}_{it}: i = 1, \dots, N; t = 1, \dots, T_i \big\}.\]

Given the present data structure numerous calibraion procedures exist (Davidson & MacKinnon, 2003, [6]; Gourieroux & Monfort, 1996, [9]). We will outline likelihood-based and simulation-based calibration. Independent of the calibration criterion, it is necessary to solve for the optimal policy \(\pi^*\) at each candidate parameterization of the model.

respy supports the calibration via simulated maximum-likelihood and the method of simulated moments. Both can be called with respy.likelihood and respy.method_of_simulated_moments, respectively.

Likelihood-based calibration seeks to find the parameterization \(\theta\) that maximizes the likelihood function \(\mathcal{L}(\theta \,|\, \mathcal{D})\), i.e. the probability of observing the given data as a function of \(\theta\). As we only observe a subset \(\bar{s}_{it}\) of the state, we can determine the probability \(p_{it}(a_{it}, \bar{u}_{it} \,|\, \bar{s}_{it}, \theta)\) of individual \(i\) at time \(t\) choosing \(a_{it}\) and receiving \(u_{it}\) given a parametric assumption about the distribution of \(\epsilon_{it}\). [2] The objective function takes the following form:

(2)#\[\hat{\theta} \equiv \underset{\theta \in \Theta}{{\arg \max}} \underbrace{\prod_{i=1}^N \prod_{t = 1}^{T_i} p_{it}(a_{it}, \bar{u}_{it} \,|\, \bar{s}_{it}, \theta)}_{\mathcal{L} (\theta \,|\, \mathcal{D})}.\]

The implementation in respy minimizes the simulated negative log-likelihood of the observed sample.

Simulation-based calibration seeks to find the parameterization \(\hat{\theta}\) that yields a simulated data set \(M_S(\theta)\) from the model that closest resembles the observed data. More precisely, the goal is to minimize the weighted squared distance between a set of moments \(M_{\mathcal{D}}\) computed on the observed data and the same set of moments computed on the simulated data \(M_{\mathcal{S}}(\theta)\). The objective function takes the following form:

(3)#\[\hat{\theta} \equiv \underset{\theta \in \Theta}{{\arg \min}} \big( M_{\mathcal{D}} - M_{\mathcal{S}}(\theta) \big)' \, W \, \big( M_{\mathcal{D}} - M_{\mathcal{S}}(\theta) \big).\]

The work by Eisenhauer, Heckman, and Mosso (2015, [7]) compares the performance of the MSM estimator against a standard maximum likelihood estimator in a simplified dynamic discrete choice model of schooling. Different to Keane and Wolpin (1994, [15]; 1997, [13]) their restriction to binary choices of agents allows to solve for the likelihood analytically and so dispenses the need for simulation or interpolation. Their maximum likelihood estimates are close to the ‘’true’’ structural objects of interest while MSM fails to recover some of them. At p.351 the authors provide a comparison of alternative weighting matrices.

To How-to guide

The implementation of MSM estimation in respy is extensively described in the guide onMethods of Simulated Moments (MSM) and the guide onHow to Estimate Model Parameters with MSM

We have explained the economic model, its solution, one particular specification, and the calibration procedure. The Robinson Crusoe tutorial provides a great applied resource to familiarize with the main functionalities of respy. Reading through will help you to set-up and calibrate your own DCDP model.

To Tutorials

We have explained the economic model, its solution, one particular specification, and the calibration procedure. TheRobinson Crusoe tutorial provides a great applied resource to familiarize with the main functionalities of respy. Reading through will help you to set-up and calibrate your own DCDP model.

Footnotes