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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 18
Consider the regression model in matrix form, Y = X + W + U , where X is an n × k 1 matrix of regressors and W is an n × k 2 matrix of regressors. Then, as shown in Exercise 18.17, the OLS estimator ß can be expressed
![Consider the regression model in matrix form, Y = X + W + U , where X is an n × k 1 matrix of regressors and W is an n × k 2 matrix of regressors. Then, as shown in Exercise 18.17, the OLS estimator ß can be expressed Now let be the binary variable fixed effects estimator computed by estimating Equation (10.11) by OLS and let be the de-meaning fixed effects estimator computed by estimating Equation (10.14) by OLS, in which the entity-specific sample means have been subtracted from X and Y. Use the expression for given above to prove that . [ Hint : Write Equation (10.11) using a full set of fixed effects, D 1 i , D 2 i , …, Dn i and no constant term. Include all of the fixed effects in W. Write out the matrix M W X.]](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_7816_f40b_84e6_f332daa20235_SM2685_11.jpg)
Now let
![Consider the regression model in matrix form, Y = X + W + U , where X is an n × k 1 matrix of regressors and W is an n × k 2 matrix of regressors. Then, as shown in Exercise 18.17, the OLS estimator ß can be expressed Now let be the binary variable fixed effects estimator computed by estimating Equation (10.11) by OLS and let be the de-meaning fixed effects estimator computed by estimating Equation (10.14) by OLS, in which the entity-specific sample means have been subtracted from X and Y. Use the expression for given above to prove that . [ Hint : Write Equation (10.11) using a full set of fixed effects, D 1 i , D 2 i , …, Dn i and no constant term. Include all of the fixed effects in W. Write out the matrix M W X.]](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_7817_1b1c_84e6_6b7d95a38b28_SM2685_11.jpg)
be the "binary variable" fixed effects estimator computed by estimating Equation (10.11) by OLS and let
![Consider the regression model in matrix form, Y = X + W + U , where X is an n × k 1 matrix of regressors and W is an n × k 2 matrix of regressors. Then, as shown in Exercise 18.17, the OLS estimator ß can be expressed Now let be the binary variable fixed effects estimator computed by estimating Equation (10.11) by OLS and let be the de-meaning fixed effects estimator computed by estimating Equation (10.14) by OLS, in which the entity-specific sample means have been subtracted from X and Y. Use the expression for given above to prove that . [ Hint : Write Equation (10.11) using a full set of fixed effects, D 1 i , D 2 i , …, Dn i and no constant term. Include all of the fixed effects in W. Write out the matrix M W X.]](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_7817_1b1d_84e6_5922ae13c395_SM2685_11.jpg)
be the "de-meaning" fixed effects estimator computed by estimating Equation (10.14) by OLS, in which the entity-specific sample means have been subtracted from X and Y. Use the expression for
![Consider the regression model in matrix form, Y = X + W + U , where X is an n × k 1 matrix of regressors and W is an n × k 2 matrix of regressors. Then, as shown in Exercise 18.17, the OLS estimator ß can be expressed Now let be the binary variable fixed effects estimator computed by estimating Equation (10.11) by OLS and let be the de-meaning fixed effects estimator computed by estimating Equation (10.14) by OLS, in which the entity-specific sample means have been subtracted from X and Y. Use the expression for given above to prove that . [ Hint : Write Equation (10.11) using a full set of fixed effects, D 1 i , D 2 i , …, Dn i and no constant term. Include all of the fixed effects in W. Write out the matrix M W X.]](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_7817_1b1e_84e6_c387499b93be_SM2685_11.jpg)
given above to prove that
![Consider the regression model in matrix form, Y = X + W + U , where X is an n × k 1 matrix of regressors and W is an n × k 2 matrix of regressors. Then, as shown in Exercise 18.17, the OLS estimator ß can be expressed Now let be the binary variable fixed effects estimator computed by estimating Equation (10.11) by OLS and let be the de-meaning fixed effects estimator computed by estimating Equation (10.14) by OLS, in which the entity-specific sample means have been subtracted from X and Y. Use the expression for given above to prove that . [ Hint : Write Equation (10.11) using a full set of fixed effects, D 1 i , D 2 i , …, Dn i and no constant term. Include all of the fixed effects in W. Write out the matrix M W X.]](https://d2lvgg3v3hfg70.cloudfront.net/SM2685/11eb817c_7817_1b1f_84e6_c3ba01f39b66_SM2685_11.jpg)
. [ Hint : Write Equation (10.11) using a full set of fixed effects, D 1 i , D 2 i , …, Dn i and no constant term. Include all of the fixed effects in W. Write out the matrix M W X.]
Now let
be the "binary variable" fixed effects estimator computed by estimating Equation (10.11) by OLS and let be the "de-meaning" fixed effects estimator computed by estimating Equation (10.14) by OLS, in which the entity-specific sample means have been subtracted from X and Y. Use the expression for given above to prove that . [ Hint : Write Equation (10.11) using a full set of fixed effects, D 1 i , D 2 i , …, Dn i and no constant term. Include all of the fixed effects in W. Write out the matrix M W X.]Explanation
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Verified
The given regression equation is
For a...
Introduction to Econometrics 3rd Edition by James Stock, Mark Watson
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