Linear Regression: single regressor

Undergraduate class slides

Transcript

1. . . ܭྔܦࡁֶɿ୯ճؼ Keiichi Shima Mie University K.Shima (Mie University) ܭྔܦࡁֶ 1 / 13
2. ࠷খೋ৐๏ ਅͷճؼํఔࣜ yi = α + βxi + ui (1) yiɺxi ͸؍ଌ͞ΕΔม਺ɺඪຊ਺͸ n ޡ߲ࠩ͸ະ஌ɺui ∼ N ( 0, σ2 ) ΛԾఆ α, β ͸ະ஌܎਺ɺOLS ʹΑΔਪఆ஋Λ α, β ͱදΘ͢ ޡ߲ࠩͷਪఆ஋Λ࢒ࠩͱݺͿɿui = yi − α − βxi J = ∑ u2 i ͷ࠷খԽʹΑΓɺα, β ͷਖ਼نํఔ͕ࣜಘΒΕΔ K.Shima (Mie University) ܭྔܦࡁֶ 2 / 13
3. ࠷খೋ৐๏ ਅͷճؼํఔࣜ yi = α + βxi + ui (1) yiɺxi ͸؍ଌ͞ΕΔม਺ɺඪຊ਺͸ n ޡ߲ࠩ͸ະ஌ɺui ∼ N ( 0, σ2 ) ΛԾఆ α, β ͸ະ஌܎਺ɺOLS ʹΑΔਪఆ஋Λ α, β ͱදΘ͢ ޡ߲ࠩͷਪఆ஋Λ࢒ࠩͱݺͿɿui = yi − α − βxi J = ∑ u2 i ͷ࠷খԽʹΑΓɺα, β ͷਖ਼نํఔ͕ࣜಘΒΕΔ K.Shima (Mie University) ܭྔܦࡁֶ 2 / 13
4. ࠷খೋ৐๏ ਅͷճؼํఔࣜ yi = α + βxi + ui (1) yiɺxi ͸؍ଌ͞ΕΔม਺ɺඪຊ਺͸ n ޡ߲ࠩ͸ະ஌ɺui ∼ N ( 0, σ2 ) ΛԾఆ α, β ͸ະ஌܎਺ɺOLS ʹΑΔਪఆ஋Λ α, β ͱදΘ͢ ޡ߲ࠩͷਪఆ஋Λ࢒ࠩͱݺͿɿui = yi − α − βxi J = ∑ u2 i ͷ࠷খԽʹΑΓɺα, β ͷਖ਼نํఔ͕ࣜಘΒΕΔ K.Shima (Mie University) ܭྔܦࡁֶ 2 / 13
5. ࠷খೋ৐๏ ਅͷճؼํఔࣜ yi = α + βxi + ui (1) yiɺxi ͸؍ଌ͞ΕΔม਺ɺඪຊ਺͸ n ޡ߲ࠩ͸ະ஌ɺui ∼ N ( 0, σ2 ) ΛԾఆ α, β ͸ະ஌܎਺ɺOLS ʹΑΔਪఆ஋Λ α, β ͱදΘ͢ ޡ߲ࠩͷਪఆ஋Λ࢒ࠩͱݺͿɿui = yi − α − βxi J = ∑ u2 i ͷ࠷খԽʹΑΓɺα, β ͷਖ਼نํఔ͕ࣜಘΒΕΔ K.Shima (Mie University) ܭྔܦࡁֶ 2 / 13
6. ࠷খೋ৐๏ ਅͷճؼํఔࣜ yi = α + βxi + ui (1) yiɺxi ͸؍ଌ͞ΕΔม਺ɺඪຊ਺͸ n ޡ߲ࠩ͸ະ஌ɺui ∼ N ( 0, σ2 ) ΛԾఆ α, β ͸ະ஌܎਺ɺOLS ʹΑΔਪఆ஋Λ α, β ͱදΘ͢ ޡ߲ࠩͷਪఆ஋Λ࢒ࠩͱݺͿɿui = yi − α − βxi J = ∑ u2 i ͷ࠷খԽʹΑΓɺα, β ͷਖ਼نํఔ͕ࣜಘΒΕΔ K.Shima (Mie University) ܭྔܦࡁֶ 2 / 13
7. ਖ਼نํఔࣜ        ∑ yi − nα − β ∑ xi = 0, ∑ yixi − α ∑ xi − β ∑ x2 i = 0. (2) (2) ࣜͷҰஈ໨ΑΓɺ α = y − βx (3) ͕ಘΒΕΔɻͨͩ͠ɺxɺy ͸ฏۉΛද͢ɻ(3) ࣜΛ (2) ࣜͷೋஈ໨ʹ୅ೖ ͢Ε͹ɺ͕࣍ࣜಘΒΕΔ ∑ yixi − ( y − βx ) nx − β ∑ x2 i = 0, ∑ yixi − nyx − β (∑ x2 i − nx2 ) = 0. (4) K.Shima (Mie University) ܭྔܦࡁֶ 3 / 13
8. ਖ਼نํఔࣜ        ∑ yi − nα − β ∑ xi = 0, ∑ yixi − α ∑ xi − β ∑ x2 i = 0. (2) (2) ࣜͷҰஈ໨ΑΓɺ α = y − βx (3) ͕ಘΒΕΔɻͨͩ͠ɺxɺy ͸ฏۉΛද͢ɻ(3) ࣜΛ (2) ࣜͷೋஈ໨ʹ୅ೖ ͢Ε͹ɺ͕࣍ࣜಘΒΕΔ ∑ yixi − ( y − βx ) nx − β ∑ x2 i = 0, ∑ yixi − nyx − β (∑ x2 i − nx2 ) = 0. (4) K.Shima (Mie University) ܭྔܦࡁֶ 3 / 13
9. ਖ਼نํఔࣜ        ∑ yi − nα − β ∑ xi = 0, ∑ yixi − α ∑ xi − β ∑ x2 i = 0. (2) (2) ࣜͷҰஈ໨ΑΓɺ α = y − βx (3) ͕ಘΒΕΔɻͨͩ͠ɺxɺy ͸ฏۉΛද͢ɻ(3) ࣜΛ (2) ࣜͷೋஈ໨ʹ୅ೖ ͢Ε͹ɺ͕࣍ࣜಘΒΕΔ ∑ yixi − ( y − βx ) nx − β ∑ x2 i = 0, ∑ yixi − nyx − β (∑ x2 i − nx2 ) = 0. (4) K.Shima (Mie University) ܭྔܦࡁֶ 3 / 13
10. ਖ਼نํఔࣜ ͜͜ͰɺSyx = ∑ yixi − nyxɺSxx = ∑ x2 i − nx2 ͱஔ͘ɻҎ্Λ༻͍ͯ (4) ࣜΛॻ͖௚ͤ͹͕࣍ಘΒΕΔ Syx = Sxx β ͜ΕΛղ͚͹ɺβ ͸࣍ͷΑ͏ʹٻΊΒΕΔ β = Syx Sxx (5) β ͷਪఆ஋Λ (3) ࣜʹ୅ೖ͢Ε͹͕࣍ٻ·Δ α = y − Syx Sxx x (6) K.Shima (Mie University) ܭྔܦࡁֶ 4 / 13
11. ਖ਼نํఔࣜ ͜͜ͰɺSyx = ∑ yixi − nyxɺSxx = ∑ x2 i − nx2 ͱஔ͘ɻҎ্Λ༻͍ͯ (4) ࣜΛॻ͖௚ͤ͹͕࣍ಘΒΕΔ Syx = Sxx β ͜ΕΛղ͚͹ɺβ ͸࣍ͷΑ͏ʹٻΊΒΕΔ β = Syx Sxx (5) β ͷਪఆ஋Λ (3) ࣜʹ୅ೖ͢Ε͹͕࣍ٻ·Δ α = y − Syx Sxx x (6) K.Shima (Mie University) ܭྔܦࡁֶ 4 / 13
12. ਖ਼نํఔࣜ ͜͜ͰɺSyx = ∑ yixi − nyxɺSxx = ∑ x2 i − nx2 ͱஔ͘ɻҎ্Λ༻͍ͯ (4) ࣜΛॻ͖௚ͤ͹͕࣍ಘΒΕΔ Syx = Sxx β ͜ΕΛղ͚͹ɺβ ͸࣍ͷΑ͏ʹٻΊΒΕΔ β = Syx Sxx (5) β ͷਪఆ஋Λ (3) ࣜʹ୅ೖ͢Ε͹͕࣍ٻ·Δ α = y − Syx Sxx x (6) K.Shima (Mie University) ܭྔܦࡁֶ 4 / 13
13. β ͷظ଴஋ (5) ࣜΑΓɺβ ͷӈลͷ෼ࢠ Syx ͸ Syx = (∑ yixi − nyx ) = ∑ ( yi − y ) ( xi − x ) (7) ਅͷճؼํఔࣜ (1) ʹ͍ͭͯɺฏۉ͸ҎԼΛຬͨ͢ɿ y = α + βx + u (8) ͨͩ͠ɺu = ∑ ui /n ͸ະ஌Ͱ͋Δ K.Shima (Mie University) ܭྔܦࡁֶ 5 / 13
14. β ͷظ଴஋ (5) ࣜΑΓɺβ ͷӈลͷ෼ࢠ Syx ͸ Syx = (∑ yixi − nyx ) = ∑ ( yi − y ) ( xi − x ) (7) ਅͷճؼํఔࣜ (1) ʹ͍ͭͯɺฏۉ͸ҎԼΛຬͨ͢ɿ y = α + βx + u (8) ͨͩ͠ɺu = ∑ ui /n ͸ະ஌Ͱ͋Δ K.Shima (Mie University) ܭྔܦࡁֶ 5 / 13
15. β ͷظ଴஋ ਅͷճؼํఔࣜ (1) ͱͦͷฏۉͰ͋Δ (8) ࣜΛ (7) ࣜʹ୅ೖɺ ∑ ( yi − y ) ( xi − x ) = ∑ [ β ( xi − x ) + ui ] ( xi − x ) = β ∑ ( xi − x )2 + ∑ ui ( xi − x ) = βSxx + ∑ ui ( xi − x ) (9) (9) ࣜΛ༻͍ͯ (5) ࣜͷ෼ࢠΛॻ͖׵͑Δ β = β + ∑ ui ( xi − x ) Sxx (10) K.Shima (Mie University) ܭྔܦࡁֶ 6 / 13
16. β ͷظ଴஋ ਅͷճؼํఔࣜ (1) ͱͦͷฏۉͰ͋Δ (8) ࣜΛ (7) ࣜʹ୅ೖɺ ∑ ( yi − y ) ( xi − x ) = ∑ [ β ( xi − x ) + ui ] ( xi − x ) = β ∑ ( xi − x )2 + ∑ ui ( xi − x ) = βSxx + ∑ ui ( xi − x ) (9) (9) ࣜΛ༻͍ͯ (5) ࣜͷ෼ࢠΛॻ͖׵͑Δ β = β + ∑ ui ( xi − x ) Sxx (10) K.Shima (Mie University) ܭྔܦࡁֶ 6 / 13
17. β ͷෆภੑ OLS ͷԾఆɿ ਅͷޡ߲ࠩͱઆ໌ม਺ͱ͕ಠཱʢແ૬ؔʣͰ͋Δ͜ͱɺ͢ͳΘͪ E [uixi] = 0 ҟͳΔ࣌఺ i j ͷޡ߲ࠩ΋ಠཱͰ͋ΓɺE [ uiuj ] = 0 ͜ͷͱ͖ɺ E ( β ) = β ͕੒Γཱͭ OLS ʹΑΓٻΊΒΕͨ β ͷظ଴஋͕ԿͷภΓ΋ͳ͘ਅͷ β ʹҰக͢Δੑ ࣭ΛʮෆภੑʯͱݺͿ K.Shima (Mie University) ܭྔܦࡁֶ 7 / 13
18. β ͷෆภੑ OLS ͷԾఆɿ ਅͷޡ߲ࠩͱઆ໌ม਺ͱ͕ಠཱʢແ૬ؔʣͰ͋Δ͜ͱɺ͢ͳΘͪ E [uixi] = 0 ҟͳΔ࣌఺ i j ͷޡ߲ࠩ΋ಠཱͰ͋ΓɺE [ uiuj ] = 0 ͜ͷͱ͖ɺ E ( β ) = β ͕੒Γཱͭ OLS ʹΑΓٻΊΒΕͨ β ͷظ଴஋͕ԿͷภΓ΋ͳ͘ਅͷ β ʹҰக͢Δੑ ࣭ΛʮෆภੑʯͱݺͿ K.Shima (Mie University) ܭྔܦࡁֶ 7 / 13
19. β ͷෆภੑ OLS ͷԾఆɿ ਅͷޡ߲ࠩͱઆ໌ม਺ͱ͕ಠཱʢແ૬ؔʣͰ͋Δ͜ͱɺ͢ͳΘͪ E [uixi] = 0 ҟͳΔ࣌఺ i j ͷޡ߲ࠩ΋ಠཱͰ͋ΓɺE [ uiuj ] = 0 ͜ͷͱ͖ɺ E ( β ) = β ͕੒Γཱͭ OLS ʹΑΓٻΊΒΕͨ β ͷظ଴஋͕ԿͷภΓ΋ͳ͘ਅͷ β ʹҰக͢Δੑ ࣭ΛʮෆภੑʯͱݺͿ K.Shima (Mie University) ܭྔܦࡁֶ 7 / 13
20. β ͷෆภੑ OLS ͷԾఆɿ ਅͷޡ߲ࠩͱઆ໌ม਺ͱ͕ಠཱʢແ૬ؔʣͰ͋Δ͜ͱɺ͢ͳΘͪ E [uixi] = 0 ҟͳΔ࣌఺ i j ͷޡ߲ࠩ΋ಠཱͰ͋ΓɺE [ uiuj ] = 0 ͜ͷͱ͖ɺ E ( β ) = β ͕੒Γཱͭ OLS ʹΑΓٻΊΒΕͨ β ͷظ଴஋͕ԿͷภΓ΋ͳ͘ਅͷ β ʹҰக͢Δੑ ࣭ΛʮෆภੑʯͱݺͿ K.Shima (Mie University) ܭྔܦࡁֶ 7 / 13
21. β ͷෆภੑ OLS ͷԾఆɿ ਅͷޡ߲ࠩͱઆ໌ม਺ͱ͕ಠཱʢແ૬ؔʣͰ͋Δ͜ͱɺ͢ͳΘͪ E [uixi] = 0 ҟͳΔ࣌఺ i j ͷޡ߲ࠩ΋ಠཱͰ͋ΓɺE [ uiuj ] = 0 ͜ͷͱ͖ɺ E ( β ) = β ͕੒Γཱͭ OLS ʹΑΓٻΊΒΕͨ β ͷظ଴஋͕ԿͷภΓ΋ͳ͘ਅͷ β ʹҰக͢Δੑ ࣭ΛʮෆภੑʯͱݺͿ K.Shima (Mie University) ܭྔܦࡁֶ 7 / 13
22. β ͷෆภੑ OLS ͷԾఆɿ ਅͷޡ߲ࠩͱઆ໌ม਺ͱ͕ಠཱʢແ૬ؔʣͰ͋Δ͜ͱɺ͢ͳΘͪ E [uixi] = 0 ҟͳΔ࣌఺ i j ͷޡ߲ࠩ΋ಠཱͰ͋ΓɺE [ uiuj ] = 0 ͜ͷͱ͖ɺ E ( β ) = β ͕੒Γཱͭ OLS ʹΑΓٻΊΒΕͨ β ͷظ଴஋͕ԿͷภΓ΋ͳ͘ਅͷ β ʹҰக͢Δੑ ࣭ΛʮෆภੑʯͱݺͿ K.Shima (Mie University) ܭྔܦࡁֶ 7 / 13
23. β ͷ෼ࢄ (10) ࣜΛ༻͍Ε͹ɺ෼ࢄ͸࣍ࣜͰදͤΔ E ( β − β )2 = E           ∑ ui ( xi − x ) Sxx           2 (11) (11) ࣜͷ෼ࢠΛల։͢Ε͹ɺ͕࣍ಘΒΕΔ E [∑ ui ( xi − x )]2 = E          ∑ i ∑ j uiuj ( xi − x ) ( xj − x )          (12) K.Shima (Mie University) ܭྔܦࡁֶ 8 / 13
24. β ͷ෼ࢄ (10) ࣜΛ༻͍Ε͹ɺ෼ࢄ͸࣍ࣜͰදͤΔ E ( β − β )2 = E           ∑ ui ( xi − x ) Sxx           2 (11) (11) ࣜͷ෼ࢠΛల։͢Ε͹ɺ͕࣍ಘΒΕΔ E [∑ ui ( xi − x )]2 = E          ∑ i ∑ j uiuj ( xi − x ) ( xj − x )          (12) K.Shima (Mie University) ܭྔܦࡁֶ 8 / 13
25. OLS ͷԾఆɺE ( uiuj ) = 0ɺE ( u2 j ) = σ2 Λ༻͍Ε͹ɺҎ্͸ E          ∑ i ∑ j uiuj ( xi − x ) ( xj − x )          = ∑ i ∑ j ( xi − x ) ( xj − x ) E ( uiuj ) = ∑ j ( xj − x ) ( xj − x ) E ( u2 j ) = Sxx σ2 (13) ͱॻ͖௚ͤΔɻΑͬͯɺ(11)-(13) ࣜʹΑΓɺβ ͷ෼ࢄ͸࣍ͷΑ͏ʹٻΊΒ ΕΔ E ( β − β )2 = Sxx σ2 S2 xx = σ2 Sxx K.Shima (Mie University) ܭྔܦࡁֶ 9 / 13
26. OLS ͷԾఆɺE ( uiuj ) = 0ɺE ( u2 j ) = σ2 Λ༻͍Ε͹ɺҎ্͸ E          ∑ i ∑ j uiuj ( xi − x ) ( xj − x )          = ∑ i ∑ j ( xi − x ) ( xj − x ) E ( uiuj ) = ∑ j ( xj − x ) ( xj − x ) E ( u2 j ) = Sxx σ2 (13) ͱॻ͖௚ͤΔɻΑͬͯɺ(11)-(13) ࣜʹΑΓɺβ ͷ෼ࢄ͸࣍ͷΑ͏ʹٻΊΒ ΕΔ E ( β − β )2 = Sxx σ2 S2 xx = σ2 Sxx K.Shima (Mie University) ܭྔܦࡁֶ 9 / 13
27. α ͷظ଴஋ͱ෼ࢄ ࣍ʹɺα ͷظ଴஋ͱ෼ࢄΛٻΊΔ (8) ࣜͷ୅ೖʹΑΓɺ(3) ࣜ͸ α = α + βx + u − βx = α − ( β − β ) x + u (14) ͱॻ͚Δɻैͬͯɺα ͷظ଴஋͸࣍ͷΑ͏ʹදݱͰ͖Δ E ( α ) = E [ α − ( β − β ) x + u ] = α − xE ( β − β ) + E ( u ) K.Shima (Mie University) ܭྔܦࡁֶ 10 / 13
28. α ͷظ଴஋ͱ෼ࢄ ࣍ʹɺα ͷظ଴஋ͱ෼ࢄΛٻΊΔ (8) ࣜͷ୅ೖʹΑΓɺ(3) ࣜ͸ α = α + βx + u − βx = α − ( β − β ) x + u (14) ͱॻ͚Δɻैͬͯɺα ͷظ଴஋͸࣍ͷΑ͏ʹදݱͰ͖Δ E ( α ) = E [ α − ( β − β ) x + u ] = α − xE ( β − β ) + E ( u ) K.Shima (Mie University) ܭྔܦࡁֶ 10 / 13
29. α ͷظ଴஋ͱ෼ࢄ ࣍ʹɺα ͷظ଴஋ͱ෼ࢄΛٻΊΔ (8) ࣜͷ୅ೖʹΑΓɺ(3) ࣜ͸ α = α + βx + u − βx = α − ( β − β ) x + u (14) ͱॻ͚Δɻैͬͯɺα ͷظ଴஋͸࣍ͷΑ͏ʹදݱͰ͖Δ E ( α ) = E [ α − ( β − β ) x + u ] = α − xE ( β − β ) + E ( u ) K.Shima (Mie University) ܭྔܦࡁֶ 10 / 13
30. OLS ͷԾఆΑΓɺE [ u ] = 0 ͕੒Γཱͭɻߋʹ E ( β ) = β ΑΓɺ E ( α ) = α ͕ಘΒΕΔɻ෼ࢄ͸ (14) ࣜΑΓɺ࣍ͷΑ͏ʹද͢͜ͱ͕Ͱ͖Δ E ( α − α )2 = E [ − ( β − β ) x + 1 n ∑ ui ]2 (15) OLS ͷԾఆΑΓ E ( uiuj ) = 0ɺE ( u2 j ) = σ2 Λ༻͍Ε͹ɺ(15) ࣜΛ࣍ͷΑ ͏ʹॻ͖௚͢͜ͱ͕Ͱ͖Δ E [( β − β )2 x2 + 1 n2 ∑ ui ∑ uj − 2 n ( β − β ) x ∑ ui ]2 = V ( β ) x2 + 1 n σ2 − 2x n E         ( β − β ) ∑ i ui         (16) K.Shima (Mie University) ܭྔܦࡁֶ 11 / 13
31. OLS ͷԾఆΑΓɺE [ u ] = 0 ͕੒Γཱͭɻߋʹ E ( β ) = β ΑΓɺ E ( α ) = α ͕ಘΒΕΔɻ෼ࢄ͸ (14) ࣜΑΓɺ࣍ͷΑ͏ʹද͢͜ͱ͕Ͱ͖Δ E ( α − α )2 = E [ − ( β − β ) x + 1 n ∑ ui ]2 (15) OLS ͷԾఆΑΓ E ( uiuj ) = 0ɺE ( u2 j ) = σ2 Λ༻͍Ε͹ɺ(15) ࣜΛ࣍ͷΑ ͏ʹॻ͖௚͢͜ͱ͕Ͱ͖Δ E [( β − β )2 x2 + 1 n2 ∑ ui ∑ uj − 2 n ( β − β ) x ∑ ui ]2 = V ( β ) x2 + 1 n σ2 − 2x n E         ( β − β ) ∑ i ui         (16) K.Shima (Mie University) ܭྔܦࡁֶ 11 / 13
32. OLS ͷԾఆΑΓɺE [ u ] = 0 ͕੒Γཱͭɻߋʹ E ( β ) = β ΑΓɺ E ( α ) = α ͕ಘΒΕΔɻ෼ࢄ͸ (14) ࣜΑΓɺ࣍ͷΑ͏ʹද͢͜ͱ͕Ͱ͖Δ E ( α − α )2 = E [ − ( β − β ) x + 1 n ∑ ui ]2 (15) OLS ͷԾఆΑΓ E ( uiuj ) = 0ɺE ( u2 j ) = σ2 Λ༻͍Ε͹ɺ(15) ࣜΛ࣍ͷΑ ͏ʹॻ͖௚͢͜ͱ͕Ͱ͖Δ E [( β − β )2 x2 + 1 n2 ∑ ui ∑ uj − 2 n ( β − β ) x ∑ ui ]2 = V ( β ) x2 + 1 n σ2 − 2x n E         ( β − β ) ∑ i ui         (16) K.Shima (Mie University) ܭྔܦࡁֶ 11 / 13
33. OLS ͷԾఆΑΓɺE [ u ] = 0 ͕੒Γཱͭɻߋʹ E ( β ) = β ΑΓɺ E ( α ) = α ͕ಘΒΕΔɻ෼ࢄ͸ (14) ࣜΑΓɺ࣍ͷΑ͏ʹද͢͜ͱ͕Ͱ͖Δ E ( α − α )2 = E [ − ( β − β ) x + 1 n ∑ ui ]2 (15) OLS ͷԾఆΑΓ E ( uiuj ) = 0ɺE ( u2 j ) = σ2 Λ༻͍Ε͹ɺ(15) ࣜΛ࣍ͷΑ ͏ʹॻ͖௚͢͜ͱ͕Ͱ͖Δ E [( β − β )2 x2 + 1 n2 ∑ ui ∑ uj − 2 n ( β − β ) x ∑ ui ]2 = V ( β ) x2 + 1 n σ2 − 2x n E         ( β − β ) ∑ i ui         (16) K.Shima (Mie University) ܭྔܦࡁֶ 11 / 13
34. OLS ͷԾఆΑΓɺE [ u ] = 0 ͕੒Γཱͭɻߋʹ E ( β ) = β ΑΓɺ E ( α ) = α ͕ಘΒΕΔɻ෼ࢄ͸ (14) ࣜΑΓɺ࣍ͷΑ͏ʹද͢͜ͱ͕Ͱ͖Δ E ( α − α )2 = E [ − ( β − β ) x + 1 n ∑ ui ]2 (15) OLS ͷԾఆΑΓ E ( uiuj ) = 0ɺE ( u2 j ) = σ2 Λ༻͍Ε͹ɺ(15) ࣜΛ࣍ͷΑ ͏ʹॻ͖௚͢͜ͱ͕Ͱ͖Δ E [( β − β )2 x2 + 1 n2 ∑ ui ∑ uj − 2 n ( β − β ) x ∑ ui ]2 = V ( β ) x2 + 1 n σ2 − 2x n E         ( β − β ) ∑ i ui         (16) K.Shima (Mie University) ܭྔܦࡁֶ 11 / 13
35. (16) ࣜͷୈ 3 ߲Λ੔ཧɺ E         ( β − β ) ∑ i ui         = E           ∑ uj ( xj − x ) Sxx ∑ i ui           = 1 Sxx E         ∑ uj ( xj − x ) ∑ i ui         = 1 Sxx E [∑ u2 j ( xj − x )] = σ2 Sxx ∑ ( xj − x ) = 0 (17) ͱͳΔ K.Shima (Mie University) ܭྔܦࡁֶ 12 / 13
36. Αͬͯɺ(15)-(17) ࣜʹΑΓɺα ͷ෼ࢄ͸࣍ͷΑ͏ʹٻΊΒΕΔ E ( α − α )2 = V ( β ) x2 + 1 n σ2 =         x2 Sxx + 1 n         σ2 = nx2 + Sxx nSxx σ2 = ∑ x2 i nSxx σ2 K.Shima (Mie University) ܭྔܦࡁֶ 13 / 13