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By Jean-Pierre Florens, Velayoudom Marimoutou, Anne Peguin-Feissolle, Josef Perktold, Marine Carrasco

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The random variables xi are independent but their distributions differ. d. conditionally on λ such that n ⊥⊥i=1 xi |λ and xi |λ ∼ N a + ξ, σ 2 . Now, suppose ξ is an unobservable random variable generated by a normal distribution with mean 0 and variance 1. Then, the parameter of interest is θ = (a, σ 2 ). We integrate out ξ to obtain the distribution of the sample conditional on θ. It follows that x|θ ∼ N (a, V ) with V = σ 2 I n + 1n 1n . 13:56 P1: OSO/OVY P2: OSO/OVY QC: OSO/OVY T1: OSO Ecoeok CUFX117-Florens 0521876407 8 April 22, 2007 Econometric Modeling and Inference Then, the observations xi have the same marginal distributions but are not independent.

It can be sustained only to some extent (for example, study of the conditional properties of estimators in small samples) but some assumptions on the generating process of z i are needed to analyze the asymptotic properties of the estimators and tests. This will appear clearly in the following chapters. d. for instance, without assuming a specific distribution). We will see later that when the conditional model is wrong (meaning that the true data-generating process does not belong to the model), then it is necessary, in order to study its behavior, to explore the generating process of z i .

D. models. d. model {X ∞ , , P∞ } and ϕ : X → Rk (or X × k θ θ → R ) such that E (ϕ) = μ and Var (ϕ) = . Then √ n 1 n n θ ϕ (xi ) − μ → N (0, ) in P∞ -distribution. i=1 This theorem is called the Lindeberg-Levy theorem. One can extend this theorem by keeping independence but dropping the condition of identical distributions. We then obtain the Lindeberg-Feller theorem or its extensions which we will not present here. We will present only one central limit theorem that is useful for mixing stationary models.

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