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Introduction to Econometrics 3rd Edition by James Stock, Mark Watson
Edition 3ISBN: 978-9352863501Introduction to Econometrics 3rd Edition by James Stock, Mark Watson
Edition 3ISBN: 978-9352863501 Exercise 11
This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model
![This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model where all variables are scalars and the constant term/intercept is omitted for convenience. a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent. b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that are i.i.d. i. Show that the OLS estimator can be written as ii. Suppose that data are missing completely at random in the sense that where p is a constant. Show that is unbiased and consistent. iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is, Show that is unbiased and consistent. iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is, Is unbiased Is consistent Explain. c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is unbiased Is consistent Explain.](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_781b_aeb2_84e6_45e8bfc9e6da_SM2685_11.jpg)
where all variables are scalars and the constant term/intercept is omitted for convenience.
a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent.
b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that
![This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model where all variables are scalars and the constant term/intercept is omitted for convenience. a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent. b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that are i.i.d. i. Show that the OLS estimator can be written as ii. Suppose that data are missing completely at random in the sense that where p is a constant. Show that is unbiased and consistent. iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is, Show that is unbiased and consistent. iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is, Is unbiased Is consistent Explain. c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is unbiased Is consistent Explain.](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_781b_aeb3_84e6_1b2ffeacc0c0_SM2685_00.jpg)
are i.i.d.
i. Show that the OLS estimator can be written as
![This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model where all variables are scalars and the constant term/intercept is omitted for convenience. a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent. b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that are i.i.d. i. Show that the OLS estimator can be written as ii. Suppose that data are missing completely at random in the sense that where p is a constant. Show that is unbiased and consistent. iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is, Show that is unbiased and consistent. iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is, Is unbiased Is consistent Explain. c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is unbiased Is consistent Explain.](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_781b_aeb4_84e6_352acfa5cf12_SM2685_00.jpg)
ii. Suppose that data are "missing completely at random" in the sense that
![This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model where all variables are scalars and the constant term/intercept is omitted for convenience. a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent. b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that are i.i.d. i. Show that the OLS estimator can be written as ii. Suppose that data are missing completely at random in the sense that where p is a constant. Show that is unbiased and consistent. iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is, Show that is unbiased and consistent. iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is, Is unbiased Is consistent Explain. c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is unbiased Is consistent Explain.](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_781b_d5c5_84e6_efd087fff819_SM2685_00.jpg)
where p is a constant. Show that
![This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model where all variables are scalars and the constant term/intercept is omitted for convenience. a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent. b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that are i.i.d. i. Show that the OLS estimator can be written as ii. Suppose that data are missing completely at random in the sense that where p is a constant. Show that is unbiased and consistent. iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is, Show that is unbiased and consistent. iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is, Is unbiased Is consistent Explain. c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is unbiased Is consistent Explain.](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_781b_d5c6_84e6_2d52340a42ec_SM2685_11.jpg)
is unbiased and consistent.
iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is,
![This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model where all variables are scalars and the constant term/intercept is omitted for convenience. a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent. b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that are i.i.d. i. Show that the OLS estimator can be written as ii. Suppose that data are missing completely at random in the sense that where p is a constant. Show that is unbiased and consistent. iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is, Show that is unbiased and consistent. iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is, Is unbiased Is consistent Explain. c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is unbiased Is consistent Explain.](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_781b_d5c7_84e6_0faecf992bad_SM2685_11.jpg)
Show that
![This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model where all variables are scalars and the constant term/intercept is omitted for convenience. a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent. b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that are i.i.d. i. Show that the OLS estimator can be written as ii. Suppose that data are missing completely at random in the sense that where p is a constant. Show that is unbiased and consistent. iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is, Show that is unbiased and consistent. iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is, Is unbiased Is consistent Explain. c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is unbiased Is consistent Explain.](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_781b_fcd8_84e6_ede4c7503f6e_SM2685_11.jpg)
is unbiased and consistent.
iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is,
![This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model where all variables are scalars and the constant term/intercept is omitted for convenience. a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent. b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that are i.i.d. i. Show that the OLS estimator can be written as ii. Suppose that data are missing completely at random in the sense that where p is a constant. Show that is unbiased and consistent. iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is, Show that is unbiased and consistent. iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is, Is unbiased Is consistent Explain. c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is unbiased Is consistent Explain.](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_781b_fcd9_84e6_4d02cc02c3c7_SM2685_11.jpg)
Is
![This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model where all variables are scalars and the constant term/intercept is omitted for convenience. a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent. b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that are i.i.d. i. Show that the OLS estimator can be written as ii. Suppose that data are missing completely at random in the sense that where p is a constant. Show that is unbiased and consistent. iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is, Show that is unbiased and consistent. iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is, Is unbiased Is consistent Explain. c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is unbiased Is consistent Explain.](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_781b_fcda_84e6_63659b1791aa_SM2685_11.jpg)
unbiased Is
![This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model where all variables are scalars and the constant term/intercept is omitted for convenience. a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent. b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that are i.i.d. i. Show that the OLS estimator can be written as ii. Suppose that data are missing completely at random in the sense that where p is a constant. Show that is unbiased and consistent. iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is, Show that is unbiased and consistent. iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is, Is unbiased Is consistent Explain. c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is unbiased Is consistent Explain.](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_781b_fcdb_84e6_f9ba083b55b5_SM2685_11.jpg)
consistent Explain.
c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is
![This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model where all variables are scalars and the constant term/intercept is omitted for convenience. a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent. b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that are i.i.d. i. Show that the OLS estimator can be written as ii. Suppose that data are missing completely at random in the sense that where p is a constant. Show that is unbiased and consistent. iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is, Show that is unbiased and consistent. iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is, Is unbiased Is consistent Explain. c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is unbiased Is consistent Explain.](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_781b_fcdc_84e6_07c018606e0b_SM2685_11.jpg)
unbiased Is
![This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model where all variables are scalars and the constant term/intercept is omitted for convenience. a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent. b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that are i.i.d. i. Show that the OLS estimator can be written as ii. Suppose that data are missing completely at random in the sense that where p is a constant. Show that is unbiased and consistent. iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is, Show that is unbiased and consistent. iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is, Is unbiased Is consistent Explain. c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is unbiased Is consistent Explain.](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_781b_fcdd_84e6_1bca904d7516_SM2685_11.jpg)
consistent Explain.
where all variables are scalars and the constant term/intercept is omitted for convenience.a. Suppose that the least assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of ß is unbiased and consistent.
b. Now suppose that some of the observations are missing. Let I i , denote a binary random variable that indicates the nonmissing observations; that is, I i = 1 if observation i is not missing and I i = 0 if observation i is missing. Assume that
are i.i.d.i. Show that the OLS estimator can be written as
ii. Suppose that data are "missing completely at random" in the sense that
where p is a constant. Show that is unbiased and consistent.iii. Suppose that the probability that the i th observation is missing depends of X i but not on u i ; that is,
Show that is unbiased and consistent.iv. Suppose that the probability that the i th observation is missing depends on both X i and u i ; that is,
Is unbiased Is consistent Explain.c. Suppose that ß = 1 and that X i and u i are mutually independent standard normal random variables [so that both X t and iq are distributed N (0,1)]. Suppose that I i = 1 when Y i 0, but I i = 0 when Y i 0. Is
unbiased Is consistent Explain.Explanation
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a) If the least squares assumptions are ...
Introduction to Econometrics 3rd Edition by James Stock, Mark Watson
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