9. 操作変数法 - GitHub...

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計量経済学応用 9. 操作変数法 矢内 勇生 2019年5月20日 高知工科大学 経済・マネジメント学群

Transcript of 9. 操作変数法 - GitHub...

Page 1: 9. 操作変数法 - GitHub Pagesyukiyanai.github.io/jp/classes/econometrics2/contents/...計量経済学応用 9. 操作変数法 矢内 勇生 2019年5月20日 高知工科大学

計量経済学応用

9. 操作変数法

矢内 勇生

2019年5月20日

高知工科大学 経済・マネジメント学群

Page 2: 9. 操作変数法 - GitHub Pagesyukiyanai.github.io/jp/classes/econometrics2/contents/...計量経済学応用 9. 操作変数法 矢内 勇生 2019年5月20日 高知工科大学

今日の目標

• 操作変数法の使い方を身につける

‣ 操作変数法とは?

‣ 操作変数法の具体的な実行方法は?

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Page 3: 9. 操作変数法 - GitHub Pagesyukiyanai.github.io/jp/classes/econometrics2/contents/...計量経済学応用 9. 操作変数法 矢内 勇生 2019年5月20日 高知工科大学

説明変数の内生性•回帰分析では、因果効果の推定にバイアスが生じることがある

‣ 交絡変数を統制し損ねる(欠落変数バイアス)

‣ 結果変数が説明変数に影響を与える(逆の因果関係)

‣ 測定誤差

‣ 交絡変数が測定・観察不能

• こういう状況下の説明変数を内生変数 (endogenous variable) と呼ぶ

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Page 4: 9. 操作変数法 - GitHub Pagesyukiyanai.github.io/jp/classes/econometrics2/contents/...計量経済学応用 9. 操作変数法 矢内 勇生 2019年5月20日 高知工科大学

例:教育が所得に及ぼす影響•結果変数:所得

• 説明変数:教育(修学年数)

• 交絡:能力

• 母回帰モデル(母集団における本当の関係):

• βを推定したい!

• 問題: が観察できない

• 妥協案:交絡変数を除いた回帰分析 (short regression)

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Ai<latexit sha1_base64="m8ETsR6bTWaZAvSRrLNJWuHBcNU=">AAAB6nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPFi8eK9gPaUDbbTbt0swm7E6GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvTKUw6HnfTmltfWNzq7xd2dnd2z9wD49aJsk0402WyER3Qmq4FIo3UaDknVRzGoeSt8Px7cxvP3FtRKIecZLyIKZDJSLBKFrp4aYv+m7Vq3lzkFXiF6QKBRp996s3SFgWc4VMUmO6vpdikFONgkk+rfQyw1PKxnTIu5YqGnMT5PNTp+TMKgMSJdqWQjJXf0/kNDZmEoe2M6Y4MsveTPzP62YYXQe5UGmGXLHFoiiTBBMy+5sMhOYM5cQSyrSwtxI2opoytOlUbAj+8surpHVR872af39ZrdeLOMpwAqdwDj5cQR3uoAFNYDCEZ3iFN0c6L86787FoLTnFzDH8gfP5AwwsjZ8=</latexit><latexit sha1_base64="m8ETsR6bTWaZAvSRrLNJWuHBcNU=">AAAB6nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPFi8eK9gPaUDbbTbt0swm7E6GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvTKUw6HnfTmltfWNzq7xd2dnd2z9wD49aJsk0402WyER3Qmq4FIo3UaDknVRzGoeSt8Px7cxvP3FtRKIecZLyIKZDJSLBKFrp4aYv+m7Vq3lzkFXiF6QKBRp996s3SFgWc4VMUmO6vpdikFONgkk+rfQyw1PKxnTIu5YqGnMT5PNTp+TMKgMSJdqWQjJXf0/kNDZmEoe2M6Y4MsveTPzP62YYXQe5UGmGXLHFoiiTBBMy+5sMhOYM5cQSyrSwtxI2opoytOlUbAj+8surpHVR872af39ZrdeLOMpwAqdwDj5cQR3uoAFNYDCEZ3iFN0c6L86787FoLTnFzDH8gfP5AwwsjZ8=</latexit><latexit sha1_base64="m8ETsR6bTWaZAvSRrLNJWuHBcNU=">AAAB6nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPFi8eK9gPaUDbbTbt0swm7E6GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvTKUw6HnfTmltfWNzq7xd2dnd2z9wD49aJsk0402WyER3Qmq4FIo3UaDknVRzGoeSt8Px7cxvP3FtRKIecZLyIKZDJSLBKFrp4aYv+m7Vq3lzkFXiF6QKBRp996s3SFgWc4VMUmO6vpdikFONgkk+rfQyw1PKxnTIu5YqGnMT5PNTp+TMKgMSJdqWQjJXf0/kNDZmEoe2M6Y4MsveTPzP62YYXQe5UGmGXLHFoiiTBBMy+5sMhOYM5cQSyrSwtxI2opoytOlUbAj+8surpHVR872af39ZrdeLOMpwAqdwDj5cQR3uoAFNYDCEZ3iFN0c6L86787FoLTnFzDH8gfP5AwwsjZ8=</latexit><latexit sha1_base64="m8ETsR6bTWaZAvSRrLNJWuHBcNU=">AAAB6nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPFi8eK9gPaUDbbTbt0swm7E6GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvTKUw6HnfTmltfWNzq7xd2dnd2z9wD49aJsk0402WyER3Qmq4FIo3UaDknVRzGoeSt8Px7cxvP3FtRKIecZLyIKZDJSLBKFrp4aYv+m7Vq3lzkFXiF6QKBRp996s3SFgWc4VMUmO6vpdikFONgkk+rfQyw1PKxnTIu5YqGnMT5PNTp+TMKgMSJdqWQjJXf0/kNDZmEoe2M6Y4MsveTPzP62YYXQe5UGmGXLHFoiiTBBMy+5sMhOYM5cQSyrSwtxI2opoytOlUbAj+8surpHVR872af39ZrdeLOMpwAqdwDj5cQR3uoAFNYDCEZ3iFN0c6L86787FoLTnFzDH8gfP5AwwsjZ8=</latexit>

Yi = ↵s + �sDi + ⌘i<latexit sha1_base64="+evE3mNS3OXxvhGi2BaI/v7hQ20=">AAACDnicbVDLSsNAFJ34rPUVdelmsBQEoSQi6EYo6MJlBfuQJoSb6bQdOpmEmYlQQr/Ajb/ixoUibl2782+ctFlo64GBc8+5lzv3hAlnSjvOt7W0vLK6tl7aKG9ube/s2nv7LRWnktAmiXksOyEoypmgTc00p51EUohCTtvh6Cr32w9UKhaLOz1OqB/BQLA+I6CNFNjV+4DhS+wBT4YQKHyCvZDqnF0bw1R5wQK74tScKfAicQtSQQUagf3l9WKSRlRowkGprusk2s9AakY4nZS9VNEEyAgGtGuogIgqP5ueM8FVo/RwP5bmCY2n6u+JDCKlxlFoOiPQQzXv5eJ/XjfV/Qs/YyJJNRVktqifcqxjnGeDe0xSovnYECCSmb9iMgQJRJsEyyYEd/7kRdI6rblOzb09q9TrRRwldIiO0DFy0TmqoxvUQE1E0CN6Rq/ozXqyXqx362PWumQVMwfoD6zPH9/5mhc=</latexit><latexit sha1_base64="+evE3mNS3OXxvhGi2BaI/v7hQ20=">AAACDnicbVDLSsNAFJ34rPUVdelmsBQEoSQi6EYo6MJlBfuQJoSb6bQdOpmEmYlQQr/Ajb/ixoUibl2782+ctFlo64GBc8+5lzv3hAlnSjvOt7W0vLK6tl7aKG9ube/s2nv7LRWnktAmiXksOyEoypmgTc00p51EUohCTtvh6Cr32w9UKhaLOz1OqB/BQLA+I6CNFNjV+4DhS+wBT4YQKHyCvZDqnF0bw1R5wQK74tScKfAicQtSQQUagf3l9WKSRlRowkGprusk2s9AakY4nZS9VNEEyAgGtGuogIgqP5ueM8FVo/RwP5bmCY2n6u+JDCKlxlFoOiPQQzXv5eJ/XjfV/Qs/YyJJNRVktqifcqxjnGeDe0xSovnYECCSmb9iMgQJRJsEyyYEd/7kRdI6rblOzb09q9TrRRwldIiO0DFy0TmqoxvUQE1E0CN6Rq/ozXqyXqx362PWumQVMwfoD6zPH9/5mhc=</latexit><latexit sha1_base64="+evE3mNS3OXxvhGi2BaI/v7hQ20=">AAACDnicbVDLSsNAFJ34rPUVdelmsBQEoSQi6EYo6MJlBfuQJoSb6bQdOpmEmYlQQr/Ajb/ixoUibl2782+ctFlo64GBc8+5lzv3hAlnSjvOt7W0vLK6tl7aKG9ube/s2nv7LRWnktAmiXksOyEoypmgTc00p51EUohCTtvh6Cr32w9UKhaLOz1OqB/BQLA+I6CNFNjV+4DhS+wBT4YQKHyCvZDqnF0bw1R5wQK74tScKfAicQtSQQUagf3l9WKSRlRowkGprusk2s9AakY4nZS9VNEEyAgGtGuogIgqP5ueM8FVo/RwP5bmCY2n6u+JDCKlxlFoOiPQQzXv5eJ/XjfV/Qs/YyJJNRVktqifcqxjnGeDe0xSovnYECCSmb9iMgQJRJsEyyYEd/7kRdI6rblOzb09q9TrRRwldIiO0DFy0TmqoxvUQE1E0CN6Rq/ozXqyXqx362PWumQVMwfoD6zPH9/5mhc=</latexit><latexit sha1_base64="+evE3mNS3OXxvhGi2BaI/v7hQ20=">AAACDnicbVDLSsNAFJ34rPUVdelmsBQEoSQi6EYo6MJlBfuQJoSb6bQdOpmEmYlQQr/Ajb/ixoUibl2782+ctFlo64GBc8+5lzv3hAlnSjvOt7W0vLK6tl7aKG9ube/s2nv7LRWnktAmiXksOyEoypmgTc00p51EUohCTtvh6Cr32w9UKhaLOz1OqB/BQLA+I6CNFNjV+4DhS+wBT4YQKHyCvZDqnF0bw1R5wQK74tScKfAicQtSQQUagf3l9WKSRlRowkGprusk2s9AakY4nZS9VNEEyAgGtGuogIgqP5ueM8FVo/RwP5bmCY2n6u+JDCKlxlFoOiPQQzXv5eJ/XjfV/Qs/YyJJNRVktqifcqxjnGeDe0xSovnYECCSmb9iMgQJRJsEyyYEd/7kRdI6rblOzb09q9TrRRwldIiO0DFy0TmqoxvUQE1E0CN6Rq/ozXqyXqx362PWumQVMwfoD6zPH9/5mhc=</latexit>

Page 5: 9. 操作変数法 - GitHub Pagesyukiyanai.github.io/jp/classes/econometrics2/contents/...計量経済学応用 9. 操作変数法 矢内 勇生 2019年5月20日 高知工科大学

Short Regression のバイアス• 最小二乗推定量  の期待値:

• がバイアス

• かつ であれば、バイアスが生じる

• は、 を に回帰したときの傾きパラメタ!5

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�Cov(A,D)

Var(D)<latexit sha1_base64="aiEGUHNTsHXPGW7r3E1uRa2jvSM=">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</latexit><latexit sha1_base64="aiEGUHNTsHXPGW7r3E1uRa2jvSM=">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</latexit><latexit sha1_base64="aiEGUHNTsHXPGW7r3E1uRa2jvSM=">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</latexit><latexit sha1_base64="aiEGUHNTsHXPGW7r3E1uRa2jvSM=">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</latexit>

� 6= 0<latexit sha1_base64="qiFH0hmBYjXXLx4pB5lLuOrdMmQ=">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</latexit><latexit sha1_base64="qiFH0hmBYjXXLx4pB5lLuOrdMmQ=">AAAD6HiclVNbi9NAFJ4mXtZ66+qjL4OlUlFKIoK+CF12BR9XsN1dmhBOptN23MnFmclCCfkFvvigiK/+JN/8M+JJ2i69xAUHAt9853zfuTAJUym0cZzfDcu+dv3Gzb1bzdt37t6739p/MNRJphgfsEQm6jQEzaWI+cAII/lpqjhEoeQn4flhGT+54EqLJP5g5in3I5jGYiIYGKSC/cafzlkg6BvqgUxnQJ9RL+QG6BGSiKcQRUAPFheeaiFRJJobmkCvVIjWdXhdKTGGKm8GJq8yi0DjNQIzU1H+thhtRPxm5wmaTxSwfJVzmFwU3bPn9OhpcckNQRVdJDzvX4LtodZGWhvoCldab7twK3WlTX3F3TpXtF4ZXopqSh7UN4lL/G/JShHzT9QJWm2n51SH7gJ3CdpkeY6D1i9vnLAs4rFhErQeuU5q/ByUEUxydM80T4Gdw5SPEMYQce3n1UMtaAeZMZ0kCr/Y0IpdV+QQaT2PQswsu9bbsZKsi40yM3nt5yJOM8Njtig0ySQ1CS1fPR0LxZmRcwTAlMBeKZsBrszgv9HEJbjbI++C4Yue6/Tc9y/b/f5yHXvkEXlMusQlr0ifvCPHZECYxa3P1lfrm/3R/mJ/t38sUq3GUvOQbBz751+QEztQ</latexit><latexit sha1_base64="qiFH0hmBYjXXLx4pB5lLuOrdMmQ=">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</latexit><latexit sha1_base64="qiFH0hmBYjXXLx4pB5lLuOrdMmQ=">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</latexit>

Cov(A,D) 6= 0<latexit sha1_base64="VmqFkJ42gmtIcCgM6l8RPrHFfCU=">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</latexit><latexit sha1_base64="VmqFkJ42gmtIcCgM6l8RPrHFfCU=">AAAEBHicnVNLi9RAEO5NfKzxsbN600vjMDKiDIkIehFm2RU8ruDM7jIJodLTM9Ns52F3Z2EIOXjxr3jxoIhXf4Q3/42VeSzziHuwIPDVV/XVi3SUSaGN6/7Zsexr12/c3L3l3L5z995eY/9+X6e5YrzHUpmq0wg0lyLhPSOM5KeZ4hBHkp9E54dV/OSCKy3S5IOZZjyIYZyIkWBgkAr3rYets1DQN9QHmU2APqN+xA3QIyQRjyGOgR7MHZ5pIVEknDVNqJcqRKs6dJdKjKHKn4ApZpllqNGNwUxUXLwtB2uRwGk9weIjBaxY5hymF2X77Dk9elpecn1QZRsJ3/+XYHOplZVWFrqiKq0vO69W6aoy9R23+1wx+qzgpaim5UH9kHjE/5ck/CN1nW3NPELdsNF0O+7M6DbwFqBJFnYcNn77w5TlMU8Mk6D1wHMzExSgjGCSl46fa54BO4cxHyBMIOY6KGY/cUlbyAzpKFX4JYbO2FVFAbHW0zjCzGpivRmryLrYIDej10Ehkiw3PGHzRqNcUpPS6kXQoVCcGTlFAEwJnJWyCeA5Db4bB4/gba68DfovOp7b8d6/bHa7i3PskkfkMWkTj7wiXfKOHJMeYdYn64v1zfpuf7a/2j/sn/NUa2eheUDWzP71F7keQ+8=</latexit><latexit sha1_base64="VmqFkJ42gmtIcCgM6l8RPrHFfCU=">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</latexit><latexit sha1_base64="VmqFkJ42gmtIcCgM6l8RPrHFfCU=">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</latexit>

Cov(A,D)

Var(D)<latexit sha1_base64="v4sOY5yI/VhHKXD3mefnfi7o5nk=">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</latexit><latexit sha1_base64="v4sOY5yI/VhHKXD3mefnfi7o5nk=">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</latexit><latexit sha1_base64="v4sOY5yI/VhHKXD3mefnfi7o5nk=">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</latexit><latexit sha1_base64="v4sOY5yI/VhHKXD3mefnfi7o5nk=">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</latexit>

A<latexit sha1_base64="UYMYx1T9Tojxuc04ke6iT0LOapM=">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</latexit><latexit sha1_base64="UYMYx1T9Tojxuc04ke6iT0LOapM=">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</latexit><latexit sha1_base64="UYMYx1T9Tojxuc04ke6iT0LOapM=">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</latexit><latexit sha1_base64="UYMYx1T9Tojxuc04ke6iT0LOapM=">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</latexit>

D<latexit sha1_base64="0NCjiy35oPj96DUKbOphYu+MGiY=">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</latexit><latexit sha1_base64="0NCjiy35oPj96DUKbOphYu+MGiY=">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</latexit><latexit sha1_base64="0NCjiy35oPj96DUKbOphYu+MGiY=">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</latexit><latexit sha1_base64="0NCjiy35oPj96DUKbOphYu+MGiY=">AAAEM3icnVNbi9NAFJ5NvKz11tVHXwZLpaKURAR9EbpsBX1bwXZ3aUo4mU7bYScXZyYLJeQ/+eIf8UEQHxTx1f/gSdLu9uYiDgTOfN/5vnNJEiRSaOM4X3cs+8rVa9d3b9Ru3rp95259715fx6livMdiGavjADSXIuI9I4zkx4niEAaSHwWnBwV/dMaVFnH03swSPgxhEomxYGAQ8vest80TX9BX1AOZTIE+oV7ADdAughhPIAyB7lcXnmghUSRqKxpfL1QYLevwulAihypvCiYrM3Nf4zUEM1Vh9jofrDDDWvMRmo8VsGyRcxCf5a2Tp7T7OD/H+qDyFgKe9zfB+lBLIy0NdIkr3W5buRW6wmZ7xc06l7ReGp6LtpTc394kLvH/JRH/QJ2L13AhqqiS+3fbrl9vOG2nPHQzcOdBg8zPoV//7I1iloY8MkyC1gPXScwwA2UEkzyveanmCbBTmPABhhGEXA+z8pvPaRORER3HCp/I0BJdVmQQaj0LA8wsOtXrXAFu4wapGb8cZiJKUsMjVhUap5KamBY/EB0JxZmRMwyAKYG9UjYFXJPB36yGS3DXR94M+s/artN23z1vdDrzdeySB+QhaRGXvCAd8oYckh5h1kfri/Xd+mF/sr/ZP+1fVaq1M9fcJyvH/v0HjElWEQ==</latexit>

E[�̂s] =Cov(Y,D)

Var(D)

=Cov(↵+ �D + �A+ ✏, D)

Var(D)

=Cov(�D,D) + Cov(↵+ �A+ ✏, D)

Var(D)

= � + �Cov(A,D)

Var(D)<latexit sha1_base64="EEzgzFw+r9F+EJ9mS3IrCx8usYo=">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</latexit><latexit sha1_base64="EEzgzFw+r9F+EJ9mS3IrCx8usYo=">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</latexit><latexit sha1_base64="EEzgzFw+r9F+EJ9mS3IrCx8usYo=">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</latexit><latexit sha1_base64="EEzgzFw+r9F+EJ9mS3IrCx8usYo=">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</latexit>

Page 6: 9. 操作変数法 - GitHub Pagesyukiyanai.github.io/jp/classes/econometrics2/contents/...計量経済学応用 9. 操作変数法 矢内 勇生 2019年5月20日 高知工科大学

操作変数法 (instrumental variable [IV] method)

• バイアスなしにβを推定するために、IV法を使う

• 操作変数 (instrument):

• 操作変数は、以下の条件を満たす必要がある

‣ 原因となる説明変数(内生変数)と相関がある

‣ 結果変数には、説明変数(内生変数)を通してのみ影響を与える(除外制約 [exclusion restriction])

• 操作変数法の公式(内生変数も操作変数も1つずつのとき):

!6

Zi<latexit sha1_base64="lmW8z/68Y4OChfheHr+CybFm20g=">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</latexit><latexit sha1_base64="lmW8z/68Y4OChfheHr+CybFm20g=">AAAENXicnVNbi9NAFJ5NvKz1sl199GWwVCpKSUTQF6HLriD4soLt7tqUcDKdtMNOLs5MFkrIn/LF/+GTPvigiK/+BU/Sdrc3F3EgcOb7zvedS5IglUIbx/m6ZdlXrl67vn2jdvPW7Ts79d27PZ1kivEuS2SijgPQXIqYd40wkh+nikMUSH4UnO6X/NEZV1ok8TszSfkgglEsQsHAIOTvWm+aJ76gL6kHMh0DfUy9gBugBwhiPIIoAro3vfBUC4kiUVvS+HquwmhRh9e5EjlUeWMweZVZ+BqvEZixivJXRX+JGdSaD9E8VMDyec5+cla0Tp7Qg0fFOdYDVbQQ8Ly/CVaHWhhpYaBLXOlm26lbqSttNldcr3NJ65XhuWhDyb3NTeIS/18S8w/UuXgNF6IpVXH/bvveF3694bSd6tD1wJ0FDTI7h379szdMWBbx2DAJWvddJzWDHJQRTPKi5mWap8BOYcT7GMYQcT3Iq6++oE1EhjRMFD6xoRW6qMgh0noSBZhZ9qpXuRLcxPUzE74Y5CJOM8NjNi0UZpKahJa/EB0KxZmREwyAKYG9UjYGXJTBH62GS3BXR14Pek/brtN23z5rdDqzdWyT++QBaRGXPCcd8pocki5h1kfri/Xd+mF/sr/ZP+1f01Rra6a5R5aO/fsPuWlXAw==</latexit><latexit sha1_base64="lmW8z/68Y4OChfheHr+CybFm20g=">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</latexit><latexit sha1_base64="lmW8z/68Y4OChfheHr+CybFm20g=">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</latexit>

�IV =Cov(Y, Z)

Cov(D,Z)<latexit sha1_base64="tW3W9G7XOGDx131R/fjdEqALOa8=">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</latexit><latexit sha1_base64="tW3W9G7XOGDx131R/fjdEqALOa8=">AAAIXXicnVVLb9NAEHYLbdJQSgsHDlxWVEVBoGBHSHCp1KpBAolDUWn6iIO12W7iVf3Cu6lUWf6T3ODCX2F2nfidULAUaf3NfN88djIeBQ7jQtd/rqzeu7+23mhutB5sPtx6tL3zuM/9aUjoKfEdPzwfYU4d5tFTwYRDz4OQYnfk0LPR9ZG0n93QkDPf+ypuAzp08cRjY0awAMjaWed7FxZD+8jETmBj9AqZIyow6gEI5wl2XYwOkxcacOYAibUKHIvPWXDK8+B1zgQbsEwbi0h5xhaHVxcLO3SjD/GgYBm29l6A+DjEJJr7HPk3cfviNeq9jFOsj8O4DYBpLiKUi8qVlCtoiSqql03UJE/K1EesxlmSuhJMSTUhD+uThCb+P8Wj35GeXUNGSkzKdnfZZAJS/FM/juu7B7d4mRNQWE9hpVwUmKVZ1V94QctCoAV3eqdRqRctjoqxMDA8pkPHYpDEqCk2G4FSyy+rk5ZOr2xSyCa2GKZdgu6g2ViVsX/r1mICeoNyM9AG7zqJolcs3bIkQtsHTmAzCNTLbxRDFgs4ulT7g1qRIX0KW0eXPqCQ+ej1On9V6qb7Ky2ge/L5BOZXraVePKd2JTXD9hN7ohInwRQgK5IBVTaVyU20l+y4WYTynyyFF+87SS53sCJTdSgKSrB2PsDwrVu5ziI5u9L8aiv9ba3tXb2jqwdVD8bssKvNnmNr+4d55ZOpSz1BHMz5wNADMYxwKBhxKOyeKacBJtd4Qgdw9LBL+TBSX8cY7QFyhcZ+CD9PIIXmGRF2Ob91R+Apk+RlmwTrbIOpGL8fRswLpoJ6JAk0njpI+Eh+atEVCykRzi0cMAkZ5IqIjaFNAj7IsglGueTqod/tGHrH+PJ29+Bg1o6m9kx7rrU1Q3unHWgftWPtVCPrvxpaY6PRavxurjU3m1uJ6+rKjPNEKzzNp38AVua74w==</latexit><latexit sha1_base64="tW3W9G7XOGDx131R/fjdEqALOa8=">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</latexit><latexit sha1_base64="tW3W9G7XOGDx131R/fjdEqALOa8=">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</latexit>

Page 7: 9. 操作変数法 - GitHub Pagesyukiyanai.github.io/jp/classes/econometrics2/contents/...計量経済学応用 9. 操作変数法 矢内 勇生 2019年5月20日 高知工科大学

操作変数を使った推定量• 真のモデルを操作変数の公式に代入してみると

• 除外制約があるので、 かつ

• 説明変数と操作変数には相関があるので

• したがって、!7

Cov(A,Z) = 0<latexit sha1_base64="J1k3GeA0fmxrIST2hnSKGfpl1Ao=">AAAEhHicnVPbbtNAEN00ppRwS+GRlxVRIBVVZHMRvIBSNUjwViSSXuLIGm82yarrC7vrSpHlH+GzeONvGDtJm4upBCtZmj0z58xlPX4shTa2/buyU7Xu7N7du1e7/+Dho8f1/Sd9HSWK8R6LZKTOfNBcipD3jDCSn8WKQ+BLfupfHuf+0yuutIjC72YW82EAk1CMBQODkLe/87N57gn6kbog4ynQV9T1uQHaRRDtCQQB0KP5hcdaSCSJ2hrH00sWWqs8vC6Z6EOWOwWTFpGZp/EagJmqIP2cDdY8w1rzBYqPFbB0GXMcXWWt80PaPciusT6orIWA6/6NsNnUSksrDd2iSstl52o5L5cpz7id55bSC8FrUknKo/IicYj/Twn5D2rfPMMNae4qfP8iW/wC6dd+Vj41fL2LFWKBdQustpng4gAlbK/esNt2cei24SyMBlmcE6/+yx1FLAl4aJgErQeOHZthCsoIJjmmSTSPgV3ChA/QDCHgepgWS5TRJiIjOo4UfqGhBbrKSCHQehb4GJlXqzd9OVjmGyRm/GGYijBODA/ZPNE4kdRENN9IOhKKMyNnaABTAmulbAo4PoN7W8MhOJstbxv9123Hbjvf3jY6ncU49sgz8py0iEPekw75Qk5Ij7Bqpfqyalcda9c6tN5Y7+ahO5UF5ylZO9anP1DMbAE=</latexit><latexit sha1_base64="J1k3GeA0fmxrIST2hnSKGfpl1Ao=">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</latexit><latexit sha1_base64="J1k3GeA0fmxrIST2hnSKGfpl1Ao=">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</latexit><latexit sha1_base64="J1k3GeA0fmxrIST2hnSKGfpl1Ao=">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</latexit>

Cov(✏, Z) = 0<latexit sha1_base64="tQQfwsu6/kZ0DIqWUoyTtOBzJ+I=">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</latexit><latexit sha1_base64="tQQfwsu6/kZ0DIqWUoyTtOBzJ+I=">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</latexit><latexit sha1_base64="tQQfwsu6/kZ0DIqWUoyTtOBzJ+I=">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</latexit><latexit sha1_base64="tQQfwsu6/kZ0DIqWUoyTtOBzJ+I=">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</latexit>

Cov(D,Z) 6= 0<latexit sha1_base64="irJo5gOjylnF0xn4HCmD6/bAqZY=">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</latexit><latexit sha1_base64="irJo5gOjylnF0xn4HCmD6/bAqZY=">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</latexit><latexit sha1_base64="irJo5gOjylnF0xn4HCmD6/bAqZY=">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</latexit><latexit sha1_base64="irJo5gOjylnF0xn4HCmD6/bAqZY=">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</latexit>

�IV = �<latexit sha1_base64="nqanr5dtIhyYatqVY+B+o38kFKA=">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</latexit><latexit sha1_base64="nqanr5dtIhyYatqVY+B+o38kFKA=">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</latexit><latexit sha1_base64="nqanr5dtIhyYatqVY+B+o38kFKA=">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</latexit><latexit sha1_base64="nqanr5dtIhyYatqVY+B+o38kFKA=">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</latexit>

�IV =Cov(Y, Z)

Cov(D,Z)=

Cov(↵+ �D + �A+ ✏, Z)

Cov(D,Z)

=1

Cov(D,Z)[�Cov(D,Z) + �Cov(A,Z) + Cov(✏, Z)]

<latexit sha1_base64="2Hrbih+iUQRu6fpRDY235MDu154=">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</latexit><latexit sha1_base64="2Hrbih+iUQRu6fpRDY235MDu154=">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</latexit><latexit sha1_base64="2Hrbih+iUQRu6fpRDY235MDu154=">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</latexit><latexit sha1_base64="2Hrbih+iUQRu6fpRDY235MDu154=">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</latexit>

Page 8: 9. 操作変数法 - GitHub Pagesyukiyanai.github.io/jp/classes/econometrics2/contents/...計量経済学応用 9. 操作変数法 矢内 勇生 2019年5月20日 高知工科大学

操作変数法の推定量は2つの回帰係数の比

• 操作変数法で得られる推定量は、2つの回帰係数の比である

• ρとφは以下の回帰モデルで推定できる

‣ 第1段階の回帰:

‣ 誘導型回帰:

!8

Di = ↵1 + �Zi + e1i<latexit sha1_base64="0WVI6v4saoPA8x4Mo6KsYwFnBu4=">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</latexit><latexit sha1_base64="0WVI6v4saoPA8x4Mo6KsYwFnBu4=">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</latexit><latexit sha1_base64="0WVI6v4saoPA8x4Mo6KsYwFnBu4=">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</latexit><latexit sha1_base64="0WVI6v4saoPA8x4Mo6KsYwFnBu4=">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</latexit>

Yi = ↵0 + ⇢Zi + e0i<latexit sha1_base64="PrxAzuqT9Y60yRcUilrpO+sV2Oc=">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</latexit><latexit sha1_base64="PrxAzuqT9Y60yRcUilrpO+sV2Oc=">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</latexit><latexit sha1_base64="PrxAzuqT9Y60yRcUilrpO+sV2Oc=">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</latexit><latexit sha1_base64="PrxAzuqT9Y60yRcUilrpO+sV2Oc=">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</latexit>

�IV =Cov(Y, Z)

Cov(D,Z)=

Cov(Y, Z)/Var(Z)

Cov(D,Z)/Var(Z)=

�<latexit sha1_base64="D1PAHlfEqkOkMKygi9UmIAJIhX4=">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</latexit><latexit sha1_base64="D1PAHlfEqkOkMKygi9UmIAJIhX4=">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</latexit><latexit sha1_base64="D1PAHlfEqkOkMKygi9UmIAJIhX4=">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</latexit><latexit sha1_base64="D1PAHlfEqkOkMKygi9UmIAJIhX4=">AAAIbXicrVXZbtNAFHULNCFsKYgHFqERVSEIVOwICV4qtWqQQOKhqDRd4mBNnEk8qjc8k0qV5Sf+kDd+gRd+gTvjxLvDIixFGp97z7nL3FyPfJsyrqrfV1YvXb6y1mhebV27fuPmrfb67T7zZoFJDk3P9oLjEWbEpi455JTb5NgPCHZGNjkane0J+9E5CRj13E/8widDB09dOqEm5gAZ62tfN08MiraRjm3fwug50keEY9QDEM5T7DgY7cYvxGfUBhJt5TgGW7DglOXB64IJNmDpFuah9IwMBq8O5lbghG+jQc4ybG0+AfFJgM1w4bPnnUedkxeo9yxKsD4Oog4Aul5HKBaVKSlT0BJVVC0bqwmekKmOWI6zJHUpmJAqQu5WJwlN/HeKS74gNb2GlBSbpO1vZOUIJIb3/Siqbh9c42lGQWI9iRWSkWCaZ1m/9oaWhUA1l/pHs1Itmp8VrTYwPLpNJnwQx6goNp2BQs9Py6OWjK9oUkCnFh/WXYJAW//jgur90UuUmYkOOFcp5L0i4ZbmEFgecHyLQrN62Q2jidoBR6dynxAj1IRPbgupwgcUUh+1Wue3St1knyUFdA8+HEC35JrqRQtqV1BTbDu2xypRHEwCoiIRUGZTGuRYe8nOm0co/ukSuH7/CXKxgyWZskNeUICV4wGGz93SdebJ6ZVmV13hX2y0N9QtVT6ofNDmhw1l/uwb7W/62DNnDnG5aWPGBprq82GIA05Nm0QtfcaIj80zPCUDOLrYIWwYyq9lhDYBGaOJF8DP5UiiWUaIHcYunBF4iiRZ0SbAKttgxidvhiF1/RknrhkHmsxsxD0kPr1oTANicvsCDtgMKOSKTAtDmzh8oEUTtGLJ5UO/u6WpW9rHVxs7O/N2NJUHymOlo2jKa2VHeafsK4eKufaj0W7ca9xv/GzebT5sPopdV1fmnDtK7mk+/QXifsQO</latexit>

Page 9: 9. 操作変数法 - GitHub Pagesyukiyanai.github.io/jp/classes/econometrics2/contents/...計量経済学応用 9. 操作変数法 矢内 勇生 2019年5月20日 高知工科大学

2段階回帰分析法• 2段階回帰 (two-stage least squares, 2SLS, TSLS)

• 第1段階の回帰

• 第2段階の回帰

‣ ただし、  は第1段階で得られた  の予測値

!9

Di = ↵1 + �Zi + e1i<latexit sha1_base64="+4+O98fK+LFgq8IFsT/DZNRAMlE=">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</latexit><latexit sha1_base64="+4+O98fK+LFgq8IFsT/DZNRAMlE=">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</latexit><latexit sha1_base64="+4+O98fK+LFgq8IFsT/DZNRAMlE=">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</latexit><latexit sha1_base64="+4+O98fK+LFgq8IFsT/DZNRAMlE=">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</latexit>

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Di<latexit sha1_base64="LkBTExc/JR7zfT0aJNPqhFRhIuo=">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</latexit><latexit sha1_base64="LkBTExc/JR7zfT0aJNPqhFRhIuo=">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</latexit><latexit sha1_base64="LkBTExc/JR7zfT0aJNPqhFRhIuo=">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</latexit><latexit sha1_base64="LkBTExc/JR7zfT0aJNPqhFRhIuo=">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</latexit>

Page 10: 9. 操作変数法 - GitHub Pagesyukiyanai.github.io/jp/classes/econometrics2/contents/...計量経済学応用 9. 操作変数法 矢内 勇生 2019年5月20日 高知工科大学

2段階回帰の推定量と操作変数法の推定量

• 操作変数の数と内生変数の数が一致するとき、2段階回帰の推定量と操作変数法の推定量は一致する

!10

�2SLS =Cov(Y, D̂)

Var(D̂)

=Cov(Y,↵1 + �Z)

Var(↵1 + �Z)

=�Cov(Y, Z)

�2Var(Z)

=⇢

= �IV<latexit sha1_base64="n+ARl41PRbBFtyGxJAdzo34d4U8=">AAAIRnicpVVJb9NAFHZZkmK2Fo5cRlRFQaBiR0hwqdSqQQKJQ1FpusQhmriTeFRveCaVKmt+HRfO3PgJXDiAEFfejBPvLttIkWa+977vbZPxOHQp44bxeenS5StXW+3la/r1Gzdv3V5ZvdNnwSyyyb4duEF0OMaMuNQn+5xylxyGEcHe2CUH49MdaT84IxGjgf+Wn4dk6OGpTyfUxhyg0WrLWj8aUbSJLOyGDkaPkDUmHKMegLCfYs/DaDs5kJBRF0hUL3BGbMGCXZ4HxwUTbMCyHMxj5SlGDI4e5k7kxS/EoGAZ6usPQHwSYTte+OwEZ6Jz9Bj1HooU6+NIdACwrCZCuahcSbmCLlBF9bKJmuRJmfqI1TgXpK4EU1JNyO36JKGJ/07xyXtkZGPISIlJ2f5GVl2B+FVf1HcNpnecIyqsp7BSDgrM0ktlG+dxkTJqGOEf3Yx60eLNMBsDw7JcMuGDJEZNjdnESx0+rl6s9LLK3kR06vBhuefy8H9zaCagJyg38g5410kUvYR0y5KInAA4oUMhUC//gJiyWMDRsXouyCg2pU/hkTGkDyhkPka9zm+VuulzlRbQ3Xu9JwRSr1BPLKhdSc2wzcSeqIgkmAJkRTKgykavldYbH7S5fvkflcJw2Zq55fZVVKoOBT2J1d4NMLzrVkZZ4GbTnMPFuuH6CX20smZsGGqh6sacb9a0+dodrXyyTgJ75hGf2y5mbGAaIR/GOOLUdonQrRkjIbZP8ZQMYOtjj7BhrD6DAq0DcoImQQQ/nyOF5hkx9hg798bgKZNkZZsE62yDGZ88H8bUD2ec+HYSaDJzEQ+Q/KaiExoRm7vnsMF2RCFXZDsYusThyyubYJZLrm763Q3T2DDfPF3b2pq3Y1m7p93XOpqpPdO2tJfarrav2a0PrS+tb63v7Y/tr+0f7Z+J66WlOeeuVljL2i8PsLiE</latexit><latexit sha1_base64="n+ARl41PRbBFtyGxJAdzo34d4U8=">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</latexit><latexit sha1_base64="n+ARl41PRbBFtyGxJAdzo34d4U8=">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</latexit><latexit sha1_base64="n+ARl41PRbBFtyGxJAdzo34d4U8=">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</latexit>

Page 11: 9. 操作変数法 - GitHub Pagesyukiyanai.github.io/jp/classes/econometrics2/contents/...計量経済学応用 9. 操作変数法 矢内 勇生 2019年5月20日 高知工科大学

共変量を伴う操作変数法

• 2段階回帰を行うなら、両方の回帰に共変量(統制変数)を含める(共変量を X とする)

‣ 第1段階:

‣ 第2段階:

!11

Di = ↵1 + �Zi + �1Xi + e1i<latexit sha1_base64="k9LgD+cULbqKTbCMMISN+8CMJuk=">AAACG3icbVBNS8MwGE7n15xfVY9egkMQhNFOQS/CQA8eJ7gPXEtJ03QLS9uQpMIo+x9e/CtePCjiSfDgvzHtetDNBwJPnud93+R9fM6oVJb1bVSWlldW16rrtY3Nre0dc3evK5NUYNLBCUtE30eSMBqTjqKKkT4XBEU+Iz1/fJX7vQciJE3iOzXhxI3QMKYhxUhpyTOb1x6Fl9BBjI+QZ8MT6PARhfda1ZTpQUEu94s78TKbTj2zbjWsAnCR2CWpgxJtz/x0ggSnEYkVZkjKgW1x5WZIKIoZmdacVBKO8BgNyUDTGEVEulmx2xQeaSWAYSL0iRUs1N8dGYqknES+royQGsl5Lxf/8wapCi/cjMY8VSTGs4fClEGVwDwoGFBBsGITTRAWVP8V4hESCCsdZ02HYM+vvEi6zYZ92mjentVbrTKOKjgAh+AY2OActMANaIMOwOARPINX8GY8GS/Gu/ExK60YZc8++APj6we36Z4f</latexit>

Yi = ↵2 + �2SLSD̂i + �2Xi + e2i<latexit sha1_base64="Vep7ahXI4lWD9DnVsDa2sjcwijo=">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</latexit>

Page 12: 9. 操作変数法 - GitHub Pagesyukiyanai.github.io/jp/classes/econometrics2/contents/...計量経済学応用 9. 操作変数法 矢内 勇生 2019年5月20日 高知工科大学

Rで推定する場合• 2つの回帰を別々には実行しない

‣ ただし、第1段階だけは別の目的で実行する

• 2段階回帰を別々に実行すると、

‣ 正しい(バイアスのない)回帰係数の推定値は得られる

‣ 標準誤差が正しくない!

• 操作変数法用に用意されたコマンドを使う!12

Page 13: 9. 操作変数法 - GitHub Pagesyukiyanai.github.io/jp/classes/econometrics2/contents/...計量経済学応用 9. 操作変数法 矢内 勇生 2019年5月20日 高知工科大学

AER::ivreg()

• AER パッケージの ivreg() 関数を使う

• 使い方

fit <- ivreg(Y ~ D + X | X + Z, data = …)

!13