TAILIEUCHUNG - ECONOMETRICS phần 2
Nếu bạn có thể đếm trên 10% thị trường của bạn (nói, cô gái trẻ) để tải về một bảo vệ màn hình tại một đô la, doanh thu sẽ là $ 12 triệu một tháng. Bạn chỉ nhận được $ đó, nhưng bạn không phải làm bất cứ điều gì cho nó. | CHAPTER 2. MOMENT ESTIMATION 23 The mean is well defined and finite if E Z1 -1 y dF y 1. If this does not hold we evaluate I1 b ri ydF y Jo i ydF y J 1 If I1 1 and I2 1 then we define Ey 1. If I1 1 and I2 1 then we define Ey 1. If both I1 1 and I2 1 then Ey is undefined. If y. Ey is well defined we say that y is identified meaning that the parameter is uniquely determined by the distribution of the observed variables. The demonstration that the parameters of an econometric model are identified is an important precondition for estimation. Typically identification holds under a set of restrictions and an identification theorem carefully describes a set of such conditions which are sufficient for identification. In the case of the mean a sufficient condition for identification is E y 1. The mean of y is finite if E y 1. More generally y has a finite r th moment if E y r 1. It is common in econometric theory to assume that the variables or certain transformations of the variables have finite moments of a certain order. How should we interpret this assumption How restrictive is it One way to visualize the importance is to consider the class of Pareto densities given by f y ay-a-1 y 1. The parameter a of the Pareto distribution indexes the rate of decay of the tail of the density. Larger a means that the tail declines to zero more quickly. See the figure below where we show the Pareto density for a 1 and a 2. The parameter a also determines which moments are finite. We can calculate that a i 1 yr a 1 dy a if r a I r I a r E y r 1 if r a Thus to allow for stricter finite moments larger r we need to restrict the class of permissible densities require larger a . Pareto Densities a 1 and a 2 CHAPTER 2. MOMENT ESTIMATION 24 Thus broadly speaking the restriction that y has a finite r th moment means that the tail of y s density declines to zero faster than y r 1. The faster decline of the tail means that the probability of observing an extreme value of y is a more rare event.
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