Estimating dynamic discrete choice models with aggregate data: Properties of the inclusive value approximation

Abstract

We investigate the use of the inclusive value based approach for estimating dynamic discrete choice models of demand with aggregate data. The inclusive value sufficiency (IVS) approach approximates a multi-dimensional state space with a single “sufficient statistic” in order to mitigate the curse of dimensionality and tractability estimate model primitives. Although in widespread use, the conditions under which IVS is appropriate have not been examined. Theoretically, we show that the estimator is biased and inconsistent. We then use Monte Carlo simulations (of a simple model of dynamic durable goods adoption) to demonstrate the degree of bias associated with the inclusive value approximation estimator under an array of parameterizations and data generating processes. In our examination, we show that the estimator performs better when the discount factor is smaller and/or when the price sensitivity of the consumer is larger. Examining how the bias impacts economic quantities of interest, we find that the IVS method under estimates the true long-run own-price elasticities and over estimates the change in profits as prices change. Theses findings highlight the importance of correctly specifying how consumers form expectations. As a result, researchers should consider how to empirically support their assumption for the underlying consumer belief structure.

Appendices

Appendix A: Short term elasticity

In addition to estimates of a long run own-price elasticities and profits to inform the reader of the difference between the data generating process and the results of the estimation methodologies, we also present short-term elasticities. Specifically, this short-term own-price elasticity is the % change in total quantity for good j for the first τ_short = 4 periods resulting from a 1% temporary decrease in the price of good j in period 1. We determine this measure for each good and average over the number of products as done above with the long-term elasticity measures.The profit measure is computed as the sum of discounted period profits and computed based on prices, marginal costs and sales in each period. We then determine the percent change in profit from a 1% temporary decrease in price in period 1, assuming an initial market size of 10,000,000 consumers and a discount factor of β_j = 0.975 for the firm.

We present the results of a short-term temporary price change in Tables 5 and 6 for the setting of identical price transitions and heterogeneous transitions, respectively. When analyzes the results we find the results from the own-price elasticity differs from the long term elasticity above. Specifically, the tables below indicate that % change from the short-term own-price DGP elasticity becomes more negative as J increases whereas the long-term own-price elasticity improves as J increases. Moreover, it appears that with respect to both elasticity measures that the IVS is competitive with the full solution when J = 2 or J = 3.

Table 5 Short term elasticities: Identical transition

Table 6 Short term elasticities: Different transition

Appendix B: Computational details

We use the following computational algorithm to estimate the model parameters. We employ a GMM procedure using mathematical programming with equilibrium constraints (MPEC). Model parameters are $\mathbf {\theta }=\left (\bar {\alpha }^{p},\alpha \right )$. Let W be the GMM weighting matrix. The constrained optimization formulation is

$$ \begin{array}{@{}rcl@{}} \min_{\mathbf{\theta,\xi}}\left[\mathbf{\xi^{\prime}}\mathbf{Z}\mathbf{WZ}^{\prime}\mathbf{\xi}\right],\\ st: \hat{s}_{jt}(\xi,\theta)=S_{jt} \end{array} $$

with the market share equations imposed as constraints to the optimization problem.

2.1 Overall procedure

• 1. Given a guess of $\mathbf {\theta }=\left (\bar {\alpha }^{p},\alpha \right )$ and ξ_jt determine the simulated market share for each product in each time period.

• 2. With the same guess of $\mathbf {\theta }=\left (\bar {\alpha }^{p},\alpha \right )$ and ξ_jt compute the GMM objective function defined in the equation above.

• 3. Search over $\mathbf {\theta }=\left (\bar {\alpha }^{p},\alpha \right )$ and ξ_jtto minimize the objective function given the constraint that the observed market share equals the simulated share.

2.2 Formation of the market share constraint

• 1. Given a guess of $\mathbf {\theta }=\left (\bar {\alpha }^{p},\alpha \right )$ and ξ_jt formulate $f_{k,t}\left ({x_{t}^{c}},\xi _{t}\right )$ for each product k, and for each period t.

• 2. Obtain $\delta _{kt}^{h}$ for each product k and period t, using the following equation:

  $$ \delta_{kt}=\frac{F_{k,t}}{\left( 1-\beta\right)}+\bar{\alpha}^{p}p_{k,t}\qquad k\in\mathbf{J_{t}} $$

• 3. Compute the inclusive value for each consumer:

  $$ \delta_{t}=\log\left( \sum\limits_{k}\exp\left( \delta_{kt}\right)\right) $$

• 4. Obtain the coefficients through estimation of an AR(1) regression of δ_it:

  • (a) estimate δ_t+ 1 = γ₀ + γ₁δ_t + ζ_t

  • (b) given estimates of γ₁ and the variance of ζ_t discretize (N = 30) formulate the transition matrix of δ_t using the Rouwenhorst method

• 5. Obtain consumer-specific expected value of not purchasing (and hence continuing):

  $$ EV\left( \delta\right)=\log\left( \exp\left( \delta\right)+\exp\left( \beta\mathbf{\ E}\left[EV(\delta^{\prime})|\delta\right]\right)\right) $$

  • (a) Given the discretized values of δ_t and the corresponding transition matrix perform a value function iteration to determine $EV\left (\delta \right )$

  • (b) Perfrom a linear interpolation of the expected value function back to the estimated values of δ_it

• 6. The model-predicted purchase probability or market share for each product k in each period t is then given as:

  $$ \hat{s}_{kt}=\frac{\exp\left( \delta_{t}\right)}{\left[\exp\left( EV\left( \delta_{t}\right)\right)\right]}\frac{\exp\left( \delta_{kt}\right)}{\exp\left( \delta_{t}\right)} $$

• 7. Determine the difference between the observed and simulated market shares at a given parameter set $s_{k,t}-\hat {s}_{kt}\left (\delta _{t}\right )$