預測產品需求是近年的經濟學研究核心之一，而隨機係數的離散選擇模型已經廣泛應用於各種異質消費者偏好的產品市場。我們希望藉由逆最佳化的方式建立一個具有均衡限制式的數學最佳化模型，並引入Fox等人在2011年提出的無母數方法估計純特徵需求模型中的隨機變數，接著未來再以此係數進行個人選擇的預測。在這篇研究中，首先將會說明我們為何需要工具變數(instrumental variables)，再來呈現如何套用至現有的估計方法。係數估計主要由兩種方法組成並利用數學規劃程式AMPL搭配非線性規畫 solver “knirtro” 進行求解，透過利用經濟學中常用的GMM(General Method of Moment)作為估計市場的消費者隨機係數之基準值，再以非線性規劃求解器Knitro求解無母數特徵需求模型，估計無母數模型中的係數。我們在未來的目標是能夠將其應用於真實世界規模問題的數值實驗，並在真實規模下找出此模型有效性與計算效率間的平衡點。

Estimation of demand is always a core issue in recent economic studies. At the time, discrete choice model with random coefficient have been widely applied in discriminating heterogeneous preferences over consumers in differentiated products market. We aim to estimate the coefficients of pure characteristics demand model by building a mathematical program with equilibrium constraints (MPEC) adopting nonparametric inverse optimization method proposed by Fox et al.(2011). After the estimation, the next step will hope to extend and utilize the coefficients we obtain to predict decision of the individuals. In this paper, we first explain why we should introduce the instrumental variables, which is important in the model, and then we will show the derivation of how to fit the generalized method of moment (GMM) estimation. We will utilize AMPL, which is a mathematical program for the experiment and Knitro as the solver. We will implement GMM which is a common method in economic research with MPEC to find the base value of random coefficient, the other is the use of Knitro to acquire the coefficient in nonparametric pure characteristic model. We aim to carry out these experiments with real-world data and to tradeoff between the number of grid point and execution efficiency for such problem in the future.

摘要---------------------------------------------------- i
Abstract------------------------------------------------ ii
Chapter 1 Introduction----------------------------------- 1
Chapter 2 Literature Review----------------------------------- 4
2.1 Pure characteristics demand model-------------------------- 4
2.2 Utility estimation of random-coefficient functions--------- 5
2.3 General Method of Moment (GMM)---------------------------- 7
2.4 Adjusting weighting matrix to enhance the estimation accuracy-------------- 11
2.5 Nonparametric estimation.---------------------------------- 11
2.5.1 Random grid for selected characteristics----------------- 12
2.5.2 Concept for proving the consistency---------------------- 13
Chapter 3 Experiments----------------------------------- 16
3.1 The constructive approach to reformulate the problem------- 16
3.2 Consistency for the nonparametric GMM estimator------------ 21
3.3 The environment of experiments------------------------ 26
3.4 The data description------------------------------------ 27
3.5 Design of experiments------------------------------------ 28
3.5.1 Utility function estimation with constraints------------ 28
3.5.2 Estimation of the grids’ weight------------------------ 31
3.5.3 How to validate the effectiveness------------------------ 32
3.5.4 The efficiency and computing limit----------------------- 33
Chapter 4 Numerical Result and Analysis------------------------ 34
4.1 Validating the effectiveness of in-sample prediction------ 34
4.2 Validating the efficiency and computing limit for single machine----- 40
4.3 Additional experiment for model validation------------ 42
Chapter 5 Conclusion and Future direction---------------------- 47
References------------------------------------------------ 49

Su, C. L., & Judd, K. L. (2012). Constrained optimization approaches to estimation of structural models.

Fox, J. T., Kim, K.I., K., & Yang, C. (2016). A simple nonparametric approach to estimating the distribution of random coefficients in structural models.

Fox, Jeremy T., et al.(2011). A simple estimator for the distribution of random coefficients. Quantitative Economics 2.3, 381-418.

Holtz Eakin, D., Newey, W., & Rosen, H. S. (1988). Estimating vector autoregressions with panel data. Econometrica: Journal of the Econometric Society, 1371-1395.

Nevo, A. (2000). A practitioner's guide to estimation of random coefficients logit models of demand. Journal of economics & management strategy, 9(4), 513-548.

Pang, J. S., Su, C. L., & Lee, Y. C. (2015). A constructive approach to estimating pure characteristics demand models with pricing. Operations Research, 63(3), 639-659.

Berry, S., & Pakes, A. (2007). The pure characteristics demand model. International Economic Review, 48(4), 1193-1225.

Berry, S., Levinsohn, J., & Pakes, A. (1995). Automobile prices in market equilibrium. Econometrica: Journal of the Econometric Society, 841-890.

Fox, J. T., Kim, K. I., Ryan, S. P., & Bajari, P. (2011). A simple estimator for the distribution of random coefficients. Quantitative Economics, 2(3), 381-418.

Reynaert, M., & Verboven, F. (2014). Improving the performance of random coefficients demand models: the role of optimal instruments.

Newey, W. K., & Powell, J. L. (2003). Instrumental variable estimation of nonparametric models.

Ahn, S. C., & Schmidt, P. (1997). Efficient estimation of dynamic panel data models: Alternative assumptions and simplified estimation. Journal of econometrics, 76(1 2), 309 321.

Chaussé, P. (2010). Computing generalized method of moments and generalized empirical likelihood with R. Journal of Statistical Software, 34(11), 1 35.

Bajari, P., Fox, J. T., & Ryan, S. P. (2007). Linear regression estimation of discrete choice models with nonparametric distributions of random coefficients. American Economic Review, 97(2), 459-463.

Doran, H. E., & Schmidt, P. (2006). GMM estimators with improved finite sample properties using principal components of the weighting matrix, with an application to the dynamic panel data model. Journal of econometrics, 133(1), 387-409.

Gautier, E., & Kitamura, Y. (2013). Nonparametric estimation in random coefficients binary choice models. Econometrica, 81(2), 581-607.

Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica: Journal of the Econometric Society, 1029-1054.

Hoderlein, S., Klemelä, J., & Mammen, E. (2010). Analyzing the random coefficient model nonparametrically. Econometric Theory, 26(3), 804-837.

Ichimura, H., & Thompson, T. S. (1998). Maximum likelihood estimation of a binary choice model with random coefficients of unknown distribution. Journal of Econometrics, 86(2), 269-295.

Luo, Z. Q., Pang, J. S., & Ralph, D. (1996). Mathematical programs with equilibrium constraints. Cambridge University Press.

McFadden, D. (1981). Econometric models of probabilistic choice. Structural analysis
of discrete data with econometric applications, 198-272.

Newey, W. K., & Powell, J. L. (2003). Instrumental variable estimation of nonparametric models.

Skrainka, B. S. (2012). A large scale study of the small sample performance of random coefficient models of demand.