họ gọi một người đàn ông một sô-cô-la "vua" hay một "vua bông" hoặc một "ông vua ô tô." Sử dụng của họ như vậy thuật ngữ ngụ ý rằng họ nhìn thấy thực tế không có sự khác biệt giữa người đứng đầu của ngành công nghiệp hiện đại và những vị vua phong kiến , | CHAPTER 14. UNIVARIATE TIME SERIES 226 The vector xt is strictly stationary and ergodic and by Theorem so is xtx t. Thus by the Ergodic Theorem 1 p T E xtxt E xtxt Q. t 1 Combined with and the continuous mapping theorem we see that 1 T .y1 Í 1 y 3 3 i E xt xd E XteA y Q-10 0. We have shown the following Theorem If the AR k process yt is strictly stationary and ergodic and Ey2 1 then 3 - 3 as T 1. Asymptotic Distribution Theorem MDS CLT. If ut is a strictly stationary and ergodic MDS and E ut ut ft 1 then as T 1 PT E ut - N 0 ft . T t 1 Since Xtet is a MDS we can apply Theorem to see that 1 r . P E xtet - N 0 ft T t 1 where ft E xtx te2 . Theorem If the AR k process yt is strictly stationary and ergodic and Ey4 1 then as T 1 PT p - 0 - N 0 Q-1ftQ-1 . This is identical in form to the asymptotic distribution of OLS in cross-section regression. The implication is that asymptotic inference is the same. In particular the asymptotic covariance matrix is estimated just as in the cross-section case. CHAPTER 14. UNIVARIATE TIME SERIES 227 Bootstrap for Autoregressions In the non-parametric bootstrap we constructed the bootstrap sample by randomly resampling from the data values yt xtg. This creates an iid bootstrap sample. Clearly this cannot work in a time-series application as this imposes inappropriate independence. Briefly there are two popular methods to implement bootstrap resampling for time-series data. Method 1 Model-Based Parametric Bootstrap. 1. Estimate ft and residuals et. 2. Fix an initial condition y k 1 y-k 2 . yo . 3. Simulate iid draws e from the empirical distribution of the residuals êi . êyg. 4. Create the bootstrap series y by the recursive formula y â P1yt-1 P2Vt-2 pkyĩ-k et. This construction imposes homoskedasticity on the errors et which may be different than the properties of the actual ei. It also presumes that the AR k structure is the truth. Method 2 Block Resampling 1. Divide the sample into T m
