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Indirect Inference for Nonlinear Panel Models with Fixed Effects. (2022). Chen, Shuowen.
In: Papers.
RePEc:arx:papers:2203.10683.

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  1. 2𝜀E𝑀(𝑥𝑖𝑡) > 𝜂 i =𝑜(𝑇−2 ), where the last line follows as the first two terms on the right–hand side are 𝑜(𝑇−2) by Lemma 1 in Hahn and Kuersteiner (2011) and the last term is of probability zero by construction. Since 𝑛 = 𝑂(𝑇), 𝑃 h max 1≤𝑖≤𝑛 sup (𝛽,𝛾) e 𝐺(𝑖) (𝛽, 𝛾) − 𝐺(𝑖) (𝛽, 𝛾) ≥ 𝜂 i ≤ 𝑛 ∑︁ 𝑖=1 𝑚(𝜀) ∑︁ 𝑗=1 𝑃 h sup Γ𝑗 e 𝐺(𝑖) (𝛽, 𝛾) − 𝐺(𝑖) (𝛽, 𝛾) ≥ 𝜂 i = 𝑜(𝑇−1 ) A.2 Proof of Lemma 2 Lemma 2 (Pointwise Consistency of Auxiliary Estimator in the Simulation World). ∀𝜃 ∈ Θ, b 𝛽ℎ (𝜃, b 𝜶) 𝑝 − → 𝛽(𝜃, 𝜶0) = 𝜃.
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  11. Because 1 𝑇 Í𝑇 𝑡=1 ln 𝑓 (𝑦ℎ 𝑖𝑡 (𝜃, 𝜏) | 𝑥𝑖𝑡; 𝛽, 𝛾) is an additive and multiplicative combination of the two classes of components, its function class also satisfies the entropy condition (Andrews, 1994), which is the primitive condition for stochastic equicontinuity. More specifically, define the following empirical process: 𝜈𝑇 (𝜏) = 𝑇 𝑇 ∑︁ 𝑡=1 h ln 𝑓 (𝑦ℎ 𝑖𝑡 (𝜃, 𝜏) | 𝑥𝑖𝑡; 𝛽, 𝛾) − ∫ 𝑈 ln 𝑓 (𝑦ℎ 𝑖𝑡 (𝜃, 𝜏) | 𝑥𝑖𝑡; 𝛽, 𝛾)𝑑𝐹𝑢 i ,
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  55. Step 1: Given 𝜃 and a scalar 𝜏, note that 𝑇 𝑇 ∑︁ 𝑡=1 ln 𝑓 (𝑦ℎ 𝑖𝑡 (𝜃, 𝜏) | 𝑥𝑖𝑡; 𝛽, 𝛾) := 𝑇 𝑇 ∑︁ 𝑡=1 𝑦ℎ 𝑖𝑡 (𝜃, 𝜏) ln Φ(𝑥0 𝑖𝑡 𝛽 + 𝛾) + (1 − 𝑦ℎ 𝑖𝑡 (𝜃, 𝜏)) ln(1 − Φ(𝑥0 𝑖𝑡 𝛽 + 𝛾)) consists of two components: (1) an indicator function of scalar 𝜏 and (2) a smooth, bounded and monotone function of (𝛽, 𝛾). The indicator function 𝑦ℎ 𝑖𝑡 (𝜃, b 𝛼𝑖) belongs to type I class of Andrews (1994), which satisfies Pollard’s entropy condition. The second component belongs to a class of functions satisfying bracketing entropy condition (van der Vaart and Wellner, 1996, Section 2.7.2).
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  56. Step 4: By Theorem 1 in Newey (1991), b 𝛽ℎ(𝜃, b 𝜶) converges to 𝛽(𝜃, 𝜶0) uniformly over 𝜃 ∈ Θ. B.2 Proof of Theorem 1 Proof. Following the argument as in Appendix 1 of Gouriéroux et al. (1993), consistency of e 𝜃𝐻 requires the following three conditions to hold: 1. the function 𝛽(𝜃, 𝜶0) is invertible; 2. b 𝜃 converges to 𝛽(𝜃0, 𝜶0) in 𝜃0 ∈ Θ pointwise; 3. b 𝛽ℎ(𝜃, b 𝜶) converges to 𝛽(𝜃, 𝜶0) uniformly over 𝜃 ∈ Θ.
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