計量経済学応用

10. 操作変数法 (2)

矢内　勇生

2019年5月23日

高知工科大学　経済・マネジメント学群

今日の目標

• 操作変数法の理解を深める

‣ 操作変数の選び方：操作変数の2条件

‣ 操作変数法を使った研究例

!2

モデル•理論：原因D が 結果Yに影響を与える

• 変数

‣ 結果変数：Y

‣ （主な）説明変数：D

‣ 交絡因子：A

• 「正しい」モデル

!3

✏i ⇠ Normal(0,�✏)AAACUXicbZHLThsxFIZPplAglDYty24sIiQqUDRDkegGKVJZsKqo1HBRJhqdcU4SC3s8sj2IaJRXZEFXvAcbFiCcC+J6JEuf/3Ox/TvNpbAuDK8rwYe5+Y8Li0vV5U8rn7/Uvn47srownFpcS21OUrQkRUYtJ5ykk9wQqlTScXr2e5w/Pidjhc7+uWFOHYX9TPQER+elpDZYP00E22MxynyAbJPFKTlk+170LP2gLrLWdEe5FdI3ieoTstgKxWKV6ovyjzYK5Wgj3BqrfYVJ+Vg4+pHU6mEjnAR7C9EM6jCLw6R2FXc1LxRljku0th2FueuUaJzgkkbVuLCUIz/DPrU9ZqjIdsqJIyO27pUu62njV+bYRH3eUaKydqhSX6nQDezr3Fh8L9cuXO9XpxRZXjjK+PSgXiGZ02xsL+sKQ9zJoQfkRvi7Mj5Ag9z5T6h6E6LXT34LR9uN6Gdj++9Ovdmc2bEI32ENNiCCXWjCARxCCzhcwg3cwX3lf+U2gCCYlgaVWc8qvIhg+QHst7HJ

Cor(D, ✏) = 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

Yi = ↵+ �Di + �Ai + ✏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

観測された変数 観測不能な （観測していない）変数

D Y

重回帰分析は可能？

A

観察不能な交絡因子• 問題：交絡因子 A が観測できない（観測されていない）

• 妥協案：交絡因子を無視した回帰分析 (short regression)

!5

Yi = ↵s + �sDi + ⌘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

Short Regression の誤差項

!6

Yi = ↵s + �sDi + ⌘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

⌘i = �Ai + ✏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

とすると

Yi = ↵s + �Di + ⌘iAAAD/nicnVNdi9NAFJ1N/Fjjx3bVN18GS6XFpaSroC9CZSv4JCvY7i5NCTfTaTvsTBJnJoslBPwrvvigiK/+Dt/8N9603a+2i+BAyMm599xz7yUTpVIY6/t/Nhz32vUbNzdvebfv3L23Vdm+3zNJphnvskQm+jACw6WIedcKK/lhqjmoSPKD6HivjB+ccG1EEn+w05QPFIxjMRIMLFLhtvOwdhQK+ooGINMJ0Kc0iLgF2kESscRKQ6Dd+RdPjZCoEl7tHNPACEUDFSWf8neJViCLur9TsmMFYX6aWDRQpMBOtMr3El3v7JzVaxTo73sX+wjNcicI577l+5+eFtb6WUCvIOYf0e60FFqOQSmgr1eHXOjfFP1gAjafNVSEZkC92hPUjTSw/NzkpKgf7dBOozjjeqCLOhJBcJVgee2LUf+vTKkqC1yKzOuuDV1pFFaqftOfHboKWgtQJYuzH1Z+B8OEZYrHlkkwpt/yUzvIQVvBJC+8IDM8BXYMY95HGIPiZpDPft+C1pAZ0lGi8YktnbEXFTkoY6YqwsyyT7McK8l1sX5mRy8HuYjTzPKYzY1GmaQ2oeVdoEOhObNyigCYFtgrZRPA5Vq8MR4uobU88iro7TZbz5q7759X2+3FOjbJI/KY1EmLvCBt8pbsky5hTu58cb45393P7lf3h/tznupsLDQPyKXj/voLYqFCtA==

ここで、

説明変数と誤差項の間に相関がある！ Dは内生変数

( )Cor(D, ⌘) 6= 0AAACBnicbVBNS8NAEN3Ur1q/qh5FWCxCBSlJFfRYqAePFewHNKFstpN26WYTdzdCCT158a948aCIV3+DN/+Nm7YHrT4YeLw3w8w8P+ZMadv+snJLyyura/n1wsbm1vZOcXevpaJEUmjSiEey4xMFnAloaqY5dGIJJPQ5tP1RPfPb9yAVi8StHsfghWQgWMAo0UbqFQ/dkOihDNN6JCflq1PsgiYn2BVwh23cK5bsij0F/kucOSmhORq94qfbj2gSgtCUE6W6jh1rLyVSM8phUnATBTGhIzKArqGChKC8dPrGBB8bpY+DSJoSGk/VnxMpCZUah77pzI5Wi14m/ud1Ex1ceikTcaJB0NmiIOFYRzjLBPeZBKr52BBCJTO3YjokklBtkiuYEJzFl/+SVrXinFWqN+elWm0eRx4doCNURg66QDV0jRqoiSh6QE/oBb1aj9az9Wa9z1pz1nxmH/2C9fEN7viXdw==

* Cor(D,A) 6= 0AAACDHicbVDLSsNAFJ34rPVVdelmsAgVpCRV0GWlLlxWsA9oSplMb9qhk0mcmQgl9APc+CtuXCji1g9w5984abPQ1gMXDufcy733eBFnStv2t7W0vLK6tp7byG9ube/sFvb2myqMJYUGDXko2x5RwJmAhmaaQzuSQAKPQ8sb1VK/9QBSsVDc6XEE3YAMBPMZJdpIvULR9YCSWAF2A6KHMkhqoZyUrk/x1Ql2BdxjG5suu2xPgReJk5EiylDvFb7cfkjjAISmnCjVcexIdxMiNaMcJnnX7IsIHZEBdAwVJADVTabPTPCxUfrYD6UpofFU/T2RkECpceCZzvRiNe+l4n9eJ9b+ZTdhIoo1CDpb5Mcc6xCnyeA+k0A1HxtCqGTmVkyHRBKqTX55E4Iz//IiaVbKzlm5cnterFazOHLoEB2hEnLQBaqiG1RHDUTRI3pGr+jNerJerHfrY9a6ZGUzB+gPrM8fGlaZsg==

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

!7

�̂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

E[�̂s] =Cov(Y,D)

Var(D)

=Cov(↵+ �D + ⌘, D)

Var(D)

=Cov(↵, D) + Cov(�D,D) + Cov(⌘, D)

Var(D)

= � +Cov(⌘, D)

Var(D)

= � +Cor(D, ⌘)

SD(D)SD(⌘)

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

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

!8

Di = ↵1 + �Zi + 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

D̂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

D̂i = ↵1 + �ZiAAAE3nicnVRdb9MwFM3aACN8dfDIi0VV1GrVlAwkeEHqtCLxhIag3UZThRvXba3ZSbDdiSryAy88gBCv/C7e+B/8AJw0rJ9o0ixFOr7X59zjazthwqhUrvt7q1S2r12/sX3TuXX7zt17lZ37XRlPBCYdHLNYnIQgCaMR6SiqGDlJBAEeMnIcnh1m+eNzIiSNo3dqmpA+h1FEhxSDMqFgp/SndhpQ9AL5wJIxoF3kh0QBapugwcwoDQB1ZjOSSMoMizq1OUa+pBz5PIw/pa9jwYHputvMoiMOQfpvoW4YEgc1Fjw9jEW93bzQa2hT33WWjARy1YqBeeG5hp6JKMikQ4JhIglazR9kyZx7qVEFG00afY38iHzMPBZSxuYIOAd0sN6Zgv9S9/wxqDTfhA5kHzm1x4Y3FIDTeZFzXT9tonZDX8S6kDlvaN//H2H1rIr2XE0mY2UCS5mZ7sbUpYVy7u7GglflLhz1nPu2nVPR4jxfYsTyzpuLo+cXyst0kzFF7wMaVKrunpsPtA68AlStYhwFlV/+IMYTTiKFGUjZ89xE9VMQimJGtOObq5cAPoMR6RkYASeyn+bPU6OaiQzQMBbmixTKo4uMFLiUUx6aldlW5GouC27K9SZq+Lyf0iiZKBLhWaHhhCEVo+ytowEVBCs2NQCwoMYrwmMwrVXmj+CYJnirW14H3f0978ne/pun1VaraMe29dB6ZNUtz3pmtaxX1pHVsXC5V/5c/lr+Zn+wv9jf7R+zpaWtgvPAWhr2z7+gK46E

Yi = ↵2 + �2SLSD̂i + 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

2段階回帰で得られる係数

!9

* � = Cov(D,Z)Var(Z)

AAACI3icbVDLSsNAFJ3UV62vqEs3g0WoICWpgiIIhbpwWcE+sCllMp20QyeZMDMplJB/ceOvuHGhFDcu/BcnbRBtPTBwOOde5tzjhoxKZVmfRm5ldW19I79Z2Nre2d0z9w+akkcCkwbmjIu2iyRhNCANRRUj7VAQ5LuMtNxRLfVbYyIk5cGDmoSk66NBQD2KkdJSz7x2XIJRJAl0wiGFN9DxBMKx4yM1FH5c4+OkdHsGH0+TH62JRFLSQs8sWmVrBrhM7IwUQYZ6z5w6fY4jnwQKMyRlx7ZC1Y2RUBQzkhQcHSNEeIQGpKNpgHwiu/HsxgSeaKUPPS70CxScqb83YuRLOfFdPZnmlIteKv7ndSLlXXVjGoSRIgGef+RFDCoO08JgnwqCFZtogrCgOivEQ6RLUrrWgi7BXjx5mTQrZfu8XLm/KFarWR15cASOQQnY4BJUwR2ogwbA4Am8gDfwbjwbr8bU+JiP5oxs5xD8gfH1DY9hpDg=

( )

E[�̂2SLS] =Cov(Y, D̂)

Var(D̂)

=Cov(↵+ �D + ⌘,↵1 + �Z)

Var(↵1 + �Z)

=��Cov(D,Z) + �Cov(⌘, Z)

�2Var(Z)

= � +1

�

Cov(⌘, Z)

Var(Z)

= � +Cor(Z, ⌘)

Cor(Z,D)

SD(⌘)

SD(D)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

操作変数の条件1. 操作変数 Z と内生変数 D（主な説明変数、理論における「原因」）の間に相関がある

2. 操作変数 Z は、説明変数 D を通してのみ結果変数 Y に影響を与える（除外制約、唯一経路）

!10

Cor(Z, ⌘) = 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

Cor(Z,D) 6= 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

条件1： 操作変数と内生変数の相関• 操作変数 Z と内生変数 D には相関がなければならない

‣ 操作変数の値を変えれば、内生変数の値も変わる

!11

Cor(Z,D) 6= 0 , Cov(Z,D) 6= 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

相関：操作変数 Z の値が変化すると、内生変数 D 　　の値も変化する

D Y

Z

操作変数と内生変数の相関 A

相関：操作変数 Z の値が変化すると、内生変数 D 　　の値も変化する

D Y

Z

操作変数と内生変数の相関 A

相関：操作変数 Z の値が変化すると、内生変数 D 　　の値も変化する

D Y

Z

操作変数と内生変数の相関A

V

相関の確認方法• 「ZとDの相関はゼロ」という帰無仮説を棄却すればよい

‣ 操作変数が1つのとき：t 検定

‣ 操作変数が複数あるとき: F 検定

• 実践的には「強い」相関が求められる（理由は後述）

‣ t 検定：t 値が3.2超（p 値が 0.0016未満）

‣ F 検定：F 値が10超!15

条件2： 除外制約 (exclusion restriction)• 操作変数 Z は、内生変数 D を通じてのみ結果変数 Y に影響を与える

‣ 唯一経路条件 (“only through” condition) とも呼ばれる

‣ 2段階回帰の2段階目の式

からZ を除外できる：

!16

Cor(Z, ⌘) = 0 , Cov(Z, ⌘) = 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

Yi = ↵2 + �2SLSD̂i + 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

唯一経路：操作変数 Z は、内生変数 D を通してのみ 結果変数 Y に影響を与える(Z affects Y only through D)

D Y

Z

唯一経路（除外制約）A

D Y

除外制約が満たされないときA

Z

ZはDとYの交絡因子なので、2段階目の回帰式から除外できない

D Y

Z

除外制約が満たされないとき

A

Cor(Z,A) 6= 0 ) Cor(Z, ⌘) 6= 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

Aが観測されないので、Zが交絡因子に

D Y

Z

除外制約が満たされないときA

X

Xが観測されているので、Xを統制（コントロール）すれば、除外制約は満たされる

除外制約の確認法•データでは確認できない！！！

‣ 「関係がないこと（値がゼロ）」は、統計的検定では示すことができないから

• では、どうするか？

‣ 操作変数が結果変数に影響を与えるとすれば、内生変数を通じた効果しか考えられないことを、データを使わずに説得する！

‣ 操作変数法を使った論文の成否は、この「説得力」に依存する!21

除外制約が疑われるとき• 除外制約は検定できないので、少しくらいは「他の経路」がありそう：

• 操作変数法（2段階回帰）の推定にバイアスが生じる

• バイアス：

!22

E[�̂2SLS] = � +Cor(Z, ⌘)

Cor(Z,D)

SD(⌘)

SD(Z)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

Cor(Z, ⌘) 6= 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

Cor(Z, ⌘)

Cor(Z,D)

SD(⌘)

SD(D)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

「弱い」操作変数

• 内生変数との相関が弱い操作変数：弱い操作変数 (weak instrument[s])

• 除外制約が完全に満たされていないとき、弱い操作変数を使うと推定がうまくいかない

‣ 弱すぎると、操作変数を使わない方がマシな場合も

!23

弱い操作変数とバイアス• Short regression（内生性に対処しない単回帰）のバイアス：

• 除外制約が完全でないときの、操作変数法のバイアス：

‣ 弱い操作変数を使うとバイアスが大きくなってしまう：弱い変数を使うくらいなら、short regressionのほうがマシかも

!24

Cor(Z, ⌘)

Cor(Z,D)

SD(⌘)

SD(D)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

Cor(D, ⌘)SD(⌘)

SD(D)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

操作変数の2条件：まとめ• 操作変数と内生変数の相関：データを使って確認できる

‣ 「相関がない」という帰無仮説が棄却されるかどうか

‣ 相関が十分強いか

• 除外制約（唯一経路）：データからは確かめられない

‣理論的に説得する必要がある：操作変数法を使う論文で、最も重要なパートの一つ！

‣ 自分が扱っている問題についての知識が求められる

★ 2つの条件をしっかり確認していない研究は信用できない！!25

操作変数の利用例• 研究課題：冷戦期、西側の文化は東側の体制崩壊に影響を与えたか？

• （かつての）一般的な見解：テレビやラジオで西側の文化に触れた東側の人間が、ソ連体制への不満を募らせ、それが体制崩壊を促進した

• 実験されたわけではない

- 本当に因果関係があったか？

- エビデンス（科学的方法で明らかにされた因果関係）がない

26

何が問題になり得るか• 例：反体制だから西側のテレビを見るのでは？（逆の因果関係 [内生性] の可能性）

• 最良の解決策：実験する

• 問題：実験できない

- もう冷戦は終わった！

• 自然実験（操作変数法）を利用する 27

28Source: Kern & Hainmueller (2009, Fig. 1: p.382)

東ドイツにおける自然実験

地形によってテレビ電波の強弱に違いがある

ことを利用

Kern, H. L., and J. Hainmueller. 2009. “Opium for the Masses: How Foreign Media Can Stabilize Authoritarian Regimes.” Political Analysis 17: 377-399.

冷戦当時 (1988, 1989) の世論調査を分析

分析結果（一部）

•西ドイツのテレビ番組を見ているほど、東ドイツ政府に好意的

- これまでの常識に反する結果

結果変数：東ドイツ政府に対する態度

内生変数：テレビの視聴

操作変数：西ドイツのテレビの電波 29

単純な差視聴者 - 非視聴者

操作変数を利用した分析西側テレビの影響

マルクス・レーニン主義を受け入れるか

-0.079 0.204

東ドイツに親近感を持っているか

-0.013 0.251