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xây dựng 0,95 khoảng tin cậy cho và bằng cách sử dụng các ma trận hiệp phương sai OLS thông thường và . Khoảng OLS nên được dựa trên phân phối t của sinh viên với 47 bậc tự do, và những người khác phải dựa trên sự phân bố . | 5.8 Exercises 211 For each of the two DGPs and each of the N simulated data sets construct .95 confidence intervals for 1 and 2 using the usual OLS covariance matrix and the HCCMEs HC0 HC1 HC2 and HC3. The OLS interval should be based on the Student s t distribution with 47 degrees of freedom and the others should be based on the N 0 1 distribution. Report the proportion of the time that each of these confidence intervals included the true values of the parameters. On the basis of these results which covariance matrix estimator would you recommend using in practice 5.13 Write down a second-order Taylor expansion of the nonlinear function g 0 around 00 where 0 is an OLS estimator and 00 is the true value of the parameter 0. Explain why the last term is asymptotically negligible relative to the second term. 5.14 Using a multivariate first-order Taylor expansion show that if 7 g 0 the asymptotic covariance matrix of the l-vector n1 2 7 70 is given by the l l matrix G0V1 0 G0 . Here 0 is a k-vector with k l G0 is an l k matrix with typical element dattOi dO evaluated at Ớ0 and V1 0 is the 1 j 1 2 0 k k asymptotic covariance matrix of n- 0 o0 . 5.15 Suppose that exp and f3 1.324 with a standard error of 0.2432. Calculate 3 exp and its standard error. Construct two different .99 confidence intervals for . One should be based on 5.51 and the other should be based on 5.52 . 5.16 Construct two .95 bootstrap confidence intervals for the log of the mean income not the mean of the log of income of group 3 individuals from the data in earnings.data. These intervals should be based on 5.53 and 5.54 . Verify that these two intervals are different. 5.17 Use the DGP yt 0.8yt_i ut ut NID 0 1 to generate a sample of 30 observations. Using these simulated data obtain estimates of p and CT2 for the model yt pyt_i ut E ut 0 E utus ơ2ỗts where ts is the Kronecker delta introduced in Section 1.4. By use of the parametric bootstrap with the assumption of normal errors obtain two .95 .
