近年來，需求預測一直是經濟學中研究的經典主題。而隨機係數的離散選擇
模型被廣泛應用在不同產品的市場中來揭示消費者對產品的異質偏好。
本文建立了一個具有數學規劃均衡約束的最佳化模型，並使用Fox等人(2011)
年提出的無母數方法來估計純特徵需求模型中的隨機係數，然後利用這些隨機係
數來預測個體的決策。
Fox 等人（2011）提出了一種簡單且計算上具有優勢的混合方法來估計隨機
係數的分佈。該方法固定一個隨機係數的網格，並且僅估計網格點上的權重，通
過這些網格點的權重的線性組合來近似整個隨機係數的分佈。這種方法使我們能
夠估計隨機係數的聯合分佈，而不需要對分佈做出假設。
我們打算運用 AMPL 來進行數值實驗，以驗證我們提出的無母數均衡限制
式最佳化模型的準確性，透過觀察在不同問題規模下進行實驗的結果，我們能夠
更深入地瞭解該模型的效能。此外，我們也將比較在不同問題大小下使用隨機網
格和固定網格、不同工具變量時模型的表現。我們的目標是將來在處理真實世界
的大規模數據時，可以在保持執行效率的同時，取得格點數量和實際需求之間的
平衡。

Estimation of demand has been the subject of many classic studies in recent
economics. Recently, the discrete choice model with random coefficients has been
widely applied in discerning heterogeneous preferences over consumers in various
product markets.
This paper builds a constrained optimization model called the Mathematical
Program with Equilibrium Constraint (MPEC) to estimate the random coefficients in
the pure characteristic demand model using the nonparametric method proposed by Fox
et al. (2011), then we will utilize these coefficients to predict individual decisions.
Fox et al. (2011), hereafter FKRB, proposed a simple and computationally
attractive mixture approach to estimate the random coefficients’ distribution. The
estimator fixes a grid of heterogenous parameters and estimates only the weights on the
grid points, and then finds the proper mixture of those models that best approximates
the actual data. This approach allows us to estimate the joint distribution of random
coefficients without having to impose a specific distribution.
We intend to conduct numerical experiments using the AMPL, which is a
mathematical program language, to validate the accuracy of our proposed nonparametric equilibrium-constrained optimization model. By observing the results of
5
these experiments on different problem scales, we can gain further insights into the
performance of the model. Additionally, we will compare the performance of the model
when using random grids versus fixed grids, as well as when employing different tool
variables. Our objective is to achieve a balance between the number of grid points and
execution efficiency in future scenarios while working with real-world and large-scale
data.

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