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ECONOMETRICS phần 5

Mật độ lấy mẫu của ^ hội tụ trong xác suất để những khoảnh khắc dân số. Và thứ ba, các tiểu bang CMT có chức năng liên tục bảo vệ hội tụ trong xác suất. Bây giờ chúng ta giải thích từng bước ngắn gọn và sau đó chi tiết hơn. Đầu tiên, quan sát ước tính | CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 110 Figure 6.1 Sampling Density of 3 converge in probability to population moments. And third the CMT states that continuous functions preserve convergence in probability. We now explain each step in brief and then in greater detail. First observe that the OLS estimator i 1 3 XixỊ n ẾÍ 1 n 1 1 xiyi Qxx Qxy n y i i is a function of the sample moments Q 1 pn XiX0 and Q 1 pn XjVj. xx n Si 1 i xy n Si 1 Second by an application of the WLLN these sample moments converge in probability to the population moments. Specifically the fact that yi Xi are mutually independent and identically distributed Assumption 1.5.1 implies that any function of yi Xi is iid including Xixi and Xiyi. These variables also have finite expectations by Theorem 3.16.1.1. Under these conditions the WLLN Theorem 2.7.2 implies that as n 1 and Q_ X x-x . - E xãx 0 Q Qxx xixi xixi Qxx n i 1 b Qxy 1 A p m J xiyi E xiyi Qxy. n i 1 6.1 6.2 Third the CMT Theorem 2.9.1 allows us to combine these equations to show that 3 converges in probability to 3. Specifically as n 1 1 3 Q xx Qxy QxxQxy 3. 6.3 We have shown that 3 3 as n 1. In words the OLS estimator converges in probability to the projection coefficient vector 3 as the sample size n gets large. CHAPTER 6. ASYMPTOTIC THEORY FOR LEAST SQUARES 111 To fully understand the application of the CMT we walk through it in detail. We can write 3 g Q xx Q xy where g A b A 1 b is a function of A and b. The function g A b is a continuous function of A and b at all values of the arguments such that A 1 exists. Assumption 3.16.1 implies that QxX exists and thus g A b is continuous at A Qxx. This justifies the application of the CMT in 6.3 . For a slightly different demonstration of 6.3 recall that 5.6 implies that b fl b 1 b 3 3 Qxx Qxe 6.4 where The WLLN and 3.25 imply Therefore b Qxe b Qxe 1 1X n i 1 xiei . - E xiei 0. 6.5 b fl b 1 b 3 3 Qxx Qxe - Q x10 0 which is the same as 3 3. Theorem 6.2.1 Consistency of .