Microsoft Word - 計量研修テキスト_第5版）.doc

(1)

Q3-1-1【テキスト P59】

Dependent Variable: LOG(TAXH) Method: Least Squares

Date: 10/26/05 Time: 15:42 Sample: 1975 1998

Included observations: 24

Variable Coefficient Std. Error t-Statistic Prob.

LOG(YNH) 1.264042 0.079310 15.93808 0.0000 C -5.594743 0.977449 -5.723819 0.0000

R-squared 0.920296 Mean dependent var 9.977606 Adjusted R-squared 0.916673 S.D. dependent var 0.472175 S.E. of regression 0.136300 Akaike info criterion -1.068265 Sum squared resid 0.408708 Schwarz criterion -0.970094 Log likelihood 14.81918 F-statistic 254.0223 Durbin-Watson stat 0.203893 Prob(F-statistic) 0.000000

-.3

-.2

-.1

.0

.1

.2

.3

8.8

9.2

9.6

10.0

10.4

10.8

76

78

80

82

84

86

88

90

92

94

96

98 R

e

s

i

d

u

al

A

c

tual

Fi

tte

d

誤差項に 1 階の自己相関が

発生している

（視覚的には上記図参照）

(2)

Q3-1-2【テキスト P59】

◆ Omitted Variable の追加：新たに財産所得［LOG(YAH)］を説明変数として加える。

Date: 10/26/05 Time: 15:47 Sample: 1975 1998

LOG(YNH) 0.916294 0.035528 25.79047 0.0000 LOG(YAH) 0.616127 0.044149 13.95566 0.0000 C -7.299619 0.335175 -21.77856 0.0000

-.10

-.05

.00

.05

.10

8.8

9.2

9.6

10.0

10.4

10.8

76

78

80

82

84

86

88

90

92

94

96

98 R

e

s

i

d

ual

A

c

tu

a

l

Fi

tted

DW 値が改善

Omitted Variable(=YAH)

(3)

Q3-1-3【テキスト P61】

◆ 誤差項に１階の自己相関があることを考慮した最尤法による推定

Date: 10/26/05 Time: 15:50 Sample (adjusted): 1976 1998

Included observations: 23 after adjustments Convergence achieved after 13 iterations

LOG(YNH) 0.925516 0.058818 15.73535 0.0000 LOG(YAH) 0.565030 0.078030 7.241160 0.0000 C -6.917079 0.839160 -8.242857 0.0000 AR(1) 0.423531 0.287921 1.470996 0.1577

Inverted AR Roots .42

-.12

-.08

-.04

.00

.04

.08

8.8

9.2

9.6

10.0

10.4

10.8

76

78

80

82

84

86

88

90

92

94

96

98 R

e

s

i

d

ual

A

c

tu

a

l

Fi

tted

誤差項の１階自己相関係数

(4)

Q3-2-1【テキスト P61】

◆ データの季節性を考慮しない推定

Dependent Variable: C95 Method: Least Squares Date: 10/26/05 Time: 15:57 Sample: 1990Q1 2002Q4 Included observations: 52

RYLE 0.225107 0.035320 6.373288 0.0000 RDEPO 0.022780 0.002511 9.070765 0.0000

C 37201.96 2197.414 16.92988 0.0000

R-squared 0.825474 Mean dependent var 67935.34 Adjusted R-squared 0.818350 S.D. dependent var 4380.313 S.E. of regression 1866.907 Akaike info criterion 17.95791 Sum squared resid 1.71E+08 Schwarz criterion 18.07049 Log likelihood -463.9058 F-statistic 115.8799 Durbin-Watson stat 1.999396 Prob(F-statistic) 0.000000

-4000

-2000

0 2000

4000

6000

56000

60000

64000

68000

72000

76000

90 91 92 93 94 95 96 97 98 99 00 01 02

R

es

i

dua

l

A

c

tu

a

l

Fi

tte

d

一見誤差項の自己相関の問題

はないように思えるが・・・

(5)

Q3-2-2【テキスト P63】

◆ Breusch-Godfrey検定

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 9.865135 Prob. F(8,41) 0.000000 Obs*R-squared 34.22164 Prob. Chi-Square(8) 0.000037

Test Equation:

Dependent Variable: RESID Method: Least Squares Date: 11/11/05 Time: 14:50

Presample missing value lagged residuals set to zero.

RYLE 0.006128 0.047806 0.128181 0.8986 RDEPO -0.000306 0.002158 -0.141920 0.8878 C -207.4857 2373.977 -0.087400 0.9308 RESID(-1) 0.087539 0.156865 0.558052 0.5798 RESID(-2) 0.029607 0.163283 0.181321 0.8570 RESID(-3) 0.146358 0.185603 0.788553 0.4349 RESID(-4) 0.747395 0.155621 4.802669 0.0000 RESID(-5) -0.154662 0.159225 -0.971342 0.3371 RESID(-6) -0.050635 0.169050 -0.299525 0.7661 RESID(-7) -0.254810 0.164733 -1.546801 0.1296 RESID(-8) 0.086977 0.163270 0.532717 0.5971

R-squared 0.658108 Mean dependent var -4.31E-12 Adjusted R-squared 0.574720 S.D. dependent var 1829.935 S.E. of regression 1193.364 Akaike info criterion 17.19235 Sum squared resid 58388834 Schwarz criterion 17.60511 Log likelihood -436.0010 F-statistic 7.892108 Durbin-Watson stat 1.961748 Prob(F-statistic) 0.000001

4 期ラグとの相

関関係強い

(6)

Q3-2-3【テキスト P64】

◆ データの季節性を考慮した推定①：季節ダミー変数（Q1～Q3）導入

Dependent Variable: C95 Method: Least Squares Date: 10/26/05 Time: 16:01 Sample: 1990Q1 2002Q4 Included observations: 52

Variable Coefficient Std. Error t-Statistic Prob. RYLE 0.657721 0.102308 6.428806 0.0000 RDEPO 0.008269 0.003482 2.374602 0.0218 Q1 8719.269 2108.725 4.134853 0.0001 Q2 430.3446 1000.776 0.430011 0.6692 Q3 4517.505 1224.668 3.688760 0.0006 C 14050.98 5868.881 2.394149 0.0208

R-squared 0.954293 Mean dependent var 67935.34 Adjusted R-squared 0.949325 S.D. dependent var 4380.313 S.E. of regression 986.0591 Akaike info criterion 16.73348 Sum squared resid 44726378 Schwarz criterion 16.95862 Log likelihood -429.0704 F-statistic 192.0818 Durbin-Watson stat 1.789597 Prob(F-statistic) 0.000000

-3000

-2000

-1000

0 1000

2000

3000

56000

60000

64000

68000

72000

76000

90 91 92 93 94 95 96 97 98 99 00 01 02

R

es

i

dua

l

A

c

tu

a

l

Fi

tte

d

自由度修正済み決定係数の向上

(7)

Q3-2-3〈続き〉

◆ 高階自己相関の検定

Breusch-Godfrey Serial Correlation LM Test:

Test Equation:

Dependent Variable: RESID Method: Least Squares Date: 11/11/05 Time: 14:59

Presample missing value lagged residuals set to zero.

RYLE -0.002311 0.131009 -0.017644 0.9860 RDEPO 0.000190 0.004450 0.042659 0.9662 Q1 -25.90941 2683.408 -0.009655 0.9923 Q2 17.74427 1245.205 0.014250 0.9887 Q3 -36.32161 1536.495 -0.023639 0.9813 C 48.82814 7432.082 0.006570 0.9948 RESID(-1) 0.079197 0.162745 0.486633 0.6293 RESID(-2) 0.066109 0.167125 0.395565 0.6946 RESID(-3) 0.238338 0.167262 1.424935 0.1623 RESID(-4) 0.150854 0.175575 0.859199 0.3956 RESID(-5) -0.162026 0.184445 -0.878451 0.3852 RESID(-6) -0.084119 0.178913 -0.470166 0.6409 RESID(-7) 0.135445 0.172621 0.784640 0.4375 RESID(-8) -0.140330 0.174836 -0.802634 0.4272

R-squared 0.172357 Mean dependent var 9.79E-13 Adjusted R-squared -0.110784 S.D. dependent var 936.4763 S.E. of regression 986.9871 Akaike info criterion 16.85200 Sum squared resid 37017458 Schwarz criterion 17.37733 Log likelihood -424.1519 F-statistic 0.608733 Durbin-Watson stat 1.921601 Prob(F-statistic) 0.831502

帰無仮説：“高階の自己相関なし”が棄却されない⇒

高階自己相関なし

(8)

Q3-2-4【テキスト P65】

◆ 高階自己相関を考慮したＢＭ最尤法

Dependent Variable: C95 Method: Least Squares Date: 10/26/05 Time: 16:05 Sample (adjusted): 1991Q1 2002Q4

Included observations: 48 after adjustments Convergence achieved after 9 iterations

RYLE 0.121915 0.103215 1.181180 0.2439 RDEPO 0.012172 0.009471 1.285122 0.2055

C 54243.32 12646.04 4.289353 0.0001

AR(4) 0.829315 0.081224 10.21025 0.0000

Inverted AR Roots .95 .00-.95 i -.00+.95i -.95

-4000

-2000

0 2000

4000

56000

60000

64000

68000

72000

76000

91 92 93

94 95

96 97 98 99

00 01 02

R

es

i

dua

l

A

c

tu

a

l

Fi

tte

d

(9)

Q3-2-5【テキスト P65】

◆ 季節調整をしたデータによる推定

400000 500000 600000 700000 800000 900000 90 91 92 93 94 95 96 97 98 99 00 01 02 RDEPO RDEPO_SA 56000 60000 64000 68000 72000 76000 90 91 92 93 94 95 96 97 98 99 00 01 02 C9 5 C9 5 _ S A 48000 52000 56000 60000 64000 68000 72000 76000 80000 84000 90 91 92 93 94 95 96 97 98 99 00 01 02 R Y L E R Y L E _ S A

(10)

Dependent Variable: C95_SA Method: Least Squares Date: 10/26/05 Time: 16:08 Sample: 1990Q1 2002Q4 Included observations: 52

RYLE_SA 0.717351 0.095644 7.500220 0.0000 RDEPO_SA 0.006412 0.003253 1.971218 0.0544

C 14652.88 4502.330 3.254510 0.0021

-2000

-1000

0 1000

2000

3000

56000

60000

64000

68000

72000

76000

90 91 92 93 94 95 96 97 98 99 00 01 02

R

es

i

dua

l

A

c

tu

a

l

Fi

tte

d

(11)

Q3-3-1【テキスト P67】

◆ 誤差項の不均一分散

Dependent Variable: C95 Method: Least Squares Date: 10/26/05 Time: 16:11 Sample: 1 47

Y95 0.402304 0.016823 23.91344 0.0000

C 895493.3 302864.5 2.956746 0.0049

R-squared 0.927049 Mean dependent var 5354219. Adjusted R-squared 0.925428 S.D. dependent var 5991758. S.E. of regression 1636223. Akaike info criterion 31.49530 Sum squared resid 1.20E+14 Schwarz criterion 31.57403 Log likelihood -738.1396 F-statistic 571.8524 Durbin-Watson stat 2.456389 Prob(F-statistic) 0.000000

-8000000

-4000000

0 4000000

8000000

0.0E+00

1.0E+07

2.0E+07

3.0E+07

4.0E+07

5

10

15

20

25

30

35

40

45 R

e

s

i

d

u

a

l

A

c

tu

al

Fi

t

e

d

(12)

Q3-3-2【テキスト P68】

◆ 不均一分散の有無を確認する検定

White Heteroskedasticity Test:

Test Equation:

Dependent Variable: RESID^2 Method: Least Squares Date: 10/26/05 Time: 16:13 Sample: 1 47

C -1.77E+12 1.21E+12 -1.464800 0.1501 Y95 405573.2 132664.8 3.057128 0.0038 Y95^2 -0.000501 0.001674 -0.299661 0.7658

R-squared 0.562360 Mean dependent var 2.56E+12 Adjusted R-squared 0.542467 S.D. dependent var 7.05E+12 S.E. of regression 4.77E+12 Akaike info criterion 61.28715 Sum squared resid 1.00E+27 Schwarz criterion 61.40524 Log likelihood -1437.248 F-statistic 28.26965 Durbin-Watson stat 1.657246 Prob(F-statistic) 0.000000

帰無仮説： H

０

が棄却される ⇒ 誤差項の不均一分散がある

H

0

:不均一分散なし

(13)

Q3-3-3【テキスト P70】

◆ 残差の２乗系列と POP の２乗系列の相関

Dependent Variable: RES^2 Method: Least Squares Date: 10/26/05 Time: 16:17 Sample: 1 47

POP^2 0.214883 0.018736 11.46915 0.0000 R-squared 0.705958 Mean dependent var 2.56E+12 Adjusted R-squared 0.705958 S.D. dependent var 7.05E+12 S.E. of regression 3.83E+12 Akaike info criterion 60.80437 Sum squared resid 6.73E+26 Schwarz criterion 60.84373 Log likelihood -1427.903 Durbin-Watson stat 1.672270

◆ 加重最小２乗法による推定

Included observations: 47 Weighting series: POP

Y95 0.306956 0.013314 23.05541 0.0000

C 5017366. 637411.4 7.871472 0.0000

Weighted Statistics

R-squared 0.978550 Mean dependent var 10778244 Adjusted R-squared 0.978073 S.D. dependent var 23941334 S.E. of regression 3545171. Akaike info criterion 33.04169 Sum squared resid 5.66E+14 Schwarz criterion 33.12042 Log likelihood -774.4798 F-statistic 531.5521 Durbin-Watson stat 1.997821 Prob(F-statistic) 0.000000

Unweighted Statistics

R-squared 0.607595 Mean dependent var 5354219. Adjusted R-squared 0.598875 S.D. dependent var 5991758. S.E. of regression 3794846. Sum squared resid 6.48E+14 Durbin-Watson stat 0.417828

(14)

Q3-3-4【テキスト P71】

White Heteroskedasticity-Consistent Standard Errors & Covariance

Y95 0.402304 0.048281 8.332499 0.0000

C 895493.3 402135.4 2.226845 0.0310

R-squared 0.927049 Mean dependent var 5354219. Adjusted R-squared 0.925428 S.D. dependent var 5991758. S.E. of regression 1636223. Akaike info criterion 31.49530 Sum squared resid 1.20E+14 Schwarz criterion 31.57403 Log likelihood -738.1396 F-statistic 571.8524 Durbin-Watson stat 2.456389 Prob(F-statistic) 0.000000

b

s

b

t

ˆ

=

値

が Q3-3-1 より低く算出されている係数推定値は Q3-3-1 と不変

(15)

演習 3【テキスト P72】

Dependent Variable: I90 Method: Least Squares Date: 10/26/05 Time: 16:43 Sample: 1 46

YF90 0.189574 0.052880 3.585007 0.0009 KP90 0.075103 0.003093 24.27810 0.0000 KG90 0.064635 0.005678 11.38287 0.0000

C 31.66028 42.89274 0.738127 0.4645

R-squared 0.997026 Mean dependent var 2704.154 Adjusted R-squared 0.996814 S.D. dependent var 2854.576 S.E. of regression 161.1305 Akaike info criterion 13.08525 Sum squared resid 1090448. Schwarz criterion 13.24426 Log likelihood -296.9607 F-statistic 4693.810 Durbin-Watson stat 2.292439 Prob(F-statistic) 0.000000

(16)

◆ 不均一分散検定（クロス項を考慮した場合）

White Heteroskedasticity Test:

Test Equation:

C 16992.09 18222.33 0.932487 0.3573 YF90 31.46843 36.63880 0.858883 0.3961 YF90^2 0.052062 0.018870 2.758903 0.0091 YF90*KP90 -0.000374 0.001887 -0.198314 0.8439 YF90*KG90 -0.018688 0.007469 -2.502186 0.0170 KP90 -3.472903 3.345424 -1.038106 0.3061 KP90^2 -0.000230 0.000105 -2.183258 0.0356 KP90*KG90 0.001338 0.000439 3.044910 0.0043 KG90 -0.434575 4.673830 -0.092981 0.9264 KG90^2 0.000191 0.000161 1.186791 0.2431

R-squared 0.618270 Mean dependent var 23705.39 Adjusted R-squared 0.522838 S.D. dependent var 40041.43 S.E. of regression 27659.40 Akaike info criterion 23.48298 Sum squared resid 2.75E+10 Schwarz criterion 23.88051 Log likelihood -530.1085 F-statistic 6.478611 Durbin-Watson stat 2.418430 Prob(F-statistic) 0.000020

(17)

◆ 不均一分散の問題を解決する方法の一例

『対数変換』

Dependent Variable: LOG(I90) Method: Least Squares Date: 10/26/05 Time: 16:44 Sample: 1 46

LOG(YF90) 0.129030 0.051881 2.487053 0.0169 LOG(KP90) 0.471989 0.052387 9.009568 0.0000 LOG(KG90) 0.415954 0.048082 8.650974 0.0000 C -1.640984 0.196237 -8.362252 0.0000

(18)

White Heteroskedasticity Test:（対数変換したもの）

F-statistic 0.536864 Prob. F(9,36) 0.837797

Obs*R-squared 5.443347 Prob. Chi-Square(9) 0.794076

Test Equation:

C 0.099511 0.237257 0.419425 0.6774 LOG(YF90) 0.065050 0.111033 0.585862 0.5616 (LOG(YF90))^2 0.016748 0.015752 1.063247 0.2948 (LOG(YF90))*(LOG(KP90)) 0.003019 0.026549 0.113696 0.9101 (LOG(YF90))*(LOG(KG90)) -0.037112 0.026797 -1.384943 0.1746 LOG(KP90) -0.020220 0.086660 -0.233331 0.8168 (LOG(KP90))^2 -0.007940 0.017536 -0.452806 0.6534 (LOG(KP90))*(LOG(KG90)) 0.016769 0.025401 0.660176 0.5133 LOG(KG90) -0.050493 0.084294 -0.599012 0.5529 (LOG(KG90))^2 0.008580 0.010766 0.796970 0.4307

R-squared 0.118334 Mean dependent var 0.004678 Adjusted R-squared -0.102083 S.D. dependent var 0.005723 S.E. of regression 0.006008 Akaike info criterion -7.201660 Sum squared resid 0.001300 Schwarz criterion -6.804129

Log likelihood 175.6382 F-statistic 0.536864

Durbin-Watson stat 2.014904 Prob(F-statistic) 0.837797