From 4b23dda940d33df82e9b073723648749d0352cd5 Mon Sep 17 00:00:00 2001 From: spring-haru Date: Fri, 3 Jan 2025 01:10:48 +0000 Subject: [PATCH] deploy: 15c3bc16c87ba70cb43afbcee472fa4eb3210ca3 --- 11a_Okun_Phillips.html | 4 +- 11b_Money_Inflation.html | 78 ++++---- 13a_Random_Variables.html | 162 +++++++-------- 13b_Stochastic_Solow.html | 160 +++++++-------- 18_ADAS-1.html | 186 +++++++----------- 18_ADAS-2.html | 48 ++--- 6_Regression.html | 4 +- ...033bb79a7b2556d8c6b948fd2d88907d119694.png | Bin 0 -> 48914 bytes ...6d2daa28916eac8ee709f2bc748e6905a32ef5.png | Bin 48671 -> 0 bytes ...8bd110d73da89f6499a2901d7b56b40ee1405e.png | Bin 8145 -> 0 bytes ...7819532cf5a329146c16276f3c9bf0b578c6af.png | Bin 0 -> 31267 bytes ...191a32015580237665fb03aceb24bef9ec9b82.png | Bin 77272 -> 0 bytes ...d1bbdfe26879c56def140827d11178f34df266.png | Bin 0 -> 7681 bytes ...f4e82e65496fd8802afb62551b965d37350703.png | Bin 0 -> 50691 bytes ...f9ad545ac53c9215f9fe4e3e122c9cda75c9c6.png | Bin 0 -> 10931 bytes ...12655e17cf6a0ef440dfd634f97e664380461d.png | Bin 39078 -> 0 bytes ...2c4d1334a4e3cec7d38d137458bddf5e7c6f51.png | Bin 35947 -> 0 bytes ...442bc38c9aa3e06989a11e16c5531109d518e4.png | Bin 33961 -> 0 bytes ...ba8667f65818dea60b7c422bb878bac3b626d1.png | Bin 62259 -> 0 bytes ...5344f30db23ed5dfb49d77e8de59cab651ebe2.png | Bin 0 -> 45241 bytes ...e71663510cf32a2a74944b5d19c69a44064368.png | Bin 46018 -> 0 bytes ...78eb80fa112c2783f2c20012d2739d3f7dc7b4.png | Bin 0 -> 38962 bytes ...96a6c55c1d706e9fe6d07921bf0482ecf893e5.png | Bin 0 -> 11745 bytes ...3dcc1c06c58d78e8a7abbadd1e67ed6a0bb56e.png | Bin 0 -> 32787 bytes ...df5ad5f1ba761a733d35771f373c5985f9d5e0.png | Bin 0 -> 41069 bytes ...346e8225c291308c87fe441424b9fbb15b1c00.png | Bin 39039 -> 0 bytes ...74ae7627102d0a5def759cbfba09c07ee5e1a1.png | Bin 12322 -> 0 bytes ...e359335ed540ee903236ac61e7105d39c4777c.png | Bin 0 -> 76789 bytes ...7b77b18dc6e4b30d8ad2eb7bf2ebeadfba5857.png | Bin 45654 -> 0 bytes ...9b42a43efdd45dcd11760a2c34e82a46f786f9.png | Bin 0 -> 59715 bytes ...191b7935f7dba58fde8c994177812fce3fc954.png | Bin 11131 -> 0 bytes _sources/18_ADAS-1.ipynb | 154 ++++----------- _sources/index.ipynb | 12 +- index.html | 2 +- searchindex.js | 2 +- 35 files changed, 350 insertions(+), 462 deletions(-) create mode 100644 _images/0ccc371866a21987df211f7faf033bb79a7b2556d8c6b948fd2d88907d119694.png delete mode 100644 _images/0f87a1e45587004669fd494a206d2daa28916eac8ee709f2bc748e6905a32ef5.png delete mode 100644 _images/1127910289f646796190ad73eb8bd110d73da89f6499a2901d7b56b40ee1405e.png create mode 100644 _images/1fd4ed086eb2f122776e7a077c7819532cf5a329146c16276f3c9bf0b578c6af.png delete mode 100644 _images/3233f4c4b4057d6a06133d5076191a32015580237665fb03aceb24bef9ec9b82.png create mode 100644 _images/3e861834efdb7a49f614713195d1bbdfe26879c56def140827d11178f34df266.png create mode 100644 _images/5183e8799a5de10cda15034d5cf4e82e65496fd8802afb62551b965d37350703.png create mode 100644 _images/5f9e5a782f241d5dfd07e89165f9ad545ac53c9215f9fe4e3e122c9cda75c9c6.png delete mode 100644 _images/61c203d33a818510108a9ba42412655e17cf6a0ef440dfd634f97e664380461d.png delete mode 100644 _images/628f813bd1bd7d32db0e2a42682c4d1334a4e3cec7d38d137458bddf5e7c6f51.png delete mode 100644 _images/7ce521608dc04719eacc05c1de442bc38c9aa3e06989a11e16c5531109d518e4.png delete mode 100644 _images/85691adc4a79b256ab3d12427fba8667f65818dea60b7c422bb878bac3b626d1.png create mode 100644 _images/8ad4051082caff7b62c965a4565344f30db23ed5dfb49d77e8de59cab651ebe2.png delete mode 100644 _images/9b1abeccbef906d1750331c8eae71663510cf32a2a74944b5d19c69a44064368.png create mode 100644 _images/aa77d668f0bc4f08e4f1f60f7878eb80fa112c2783f2c20012d2739d3f7dc7b4.png create mode 100644 _images/aff93ba8ca0807c48213648c4496a6c55c1d706e9fe6d07921bf0482ecf893e5.png create mode 100644 _images/c0da3bcfc3afe5f377a47ab5473dcc1c06c58d78e8a7abbadd1e67ed6a0bb56e.png create mode 100644 _images/c69aff5a769726b55773075cc2df5ad5f1ba761a733d35771f373c5985f9d5e0.png delete mode 100644 _images/c74f150b652bf15e2797006068346e8225c291308c87fe441424b9fbb15b1c00.png delete mode 100644 _images/ca8bd43629a1a295f46644109c74ae7627102d0a5def759cbfba09c07ee5e1a1.png create mode 100644 _images/dd26fadedfae37bc7bcd5a2125e359335ed540ee903236ac61e7105d39c4777c.png delete mode 100644 _images/de6f01d3324a2c0a5c9995f9ce7b77b18dc6e4b30d8ad2eb7bf2ebeadfba5857.png create mode 100644 _images/ebe1fd1db8278d9e821af856199b42a43efdd45dcd11760a2c34e82a46f786f9.png delete mode 100644 _images/f17063b495b348064060413363191b7935f7dba58fde8c994177812fce3fc954.png diff --git a/11a_Okun_Phillips.html b/11a_Okun_Phillips.html index 7d1e0d3d..d351b54c 100644 --- a/11a_Okun_Phillips.html +++ b/11a_Okun_Phillips.html @@ -706,8 +706,8 @@

回帰分析# Dep. Variable: gdp_cycle R-squared: 0.391 Model: OLS Adj. R-squared: 0.387 Method: Least Squares F-statistic: 111.5 -Date: Thu, 02 Jan 2025 Prob (F-statistic): 1.84e-20 -Time: 12:42:48 Log-Likelihood: -275.03 +Date: Fri, 03 Jan 2025 Prob (F-statistic): 1.84e-20 +Time: 01:09:27 Log-Likelihood: -275.03 No. Observations: 176 AIC: 554.1 Df Residuals: 174 BIC: 560.4 Df Model: 1 diff --git a/11b_Money_Inflation.html b/11b_Money_Inflation.html index f9f3e57f..1ba211bc 100644 --- a/11b_Money_Inflation.html +++ b/11b_Money_Inflation.html @@ -623,122 +623,122 @@

resample() 2019-01-01 10 - 5.075150 + 7.068212 2019-02-01 20 - 4.916978 + 5.507801 2019-03-01 30 - 3.979215 + 4.659180 2019-04-01 40 - 3.371814 + 6.750383 2019-05-01 50 - 6.031563 + 5.770613 2019-06-01 60 - 6.865312 + 5.057649 2019-07-01 70 - 5.930032 + 5.291954 2019-08-01 80 - 5.707334 + 5.090206 2019-09-01 90 - 3.460428 + 5.229165 2019-10-01 100 - 5.461242 + 4.371847 2019-11-01 110 - 4.073966 + 5.835155 2019-12-01 120 - 5.085689 + 6.839144 2020-01-01 10 - 6.377875 + 3.745913 2020-02-01 20 - 5.752744 + 4.395061 2020-03-01 30 - 6.360929 + 6.024376 2020-04-01 40 - 4.177215 + 6.345998 2020-05-01 50 - 5.736020 + 3.855593 2020-06-01 60 - 5.724809 + 3.339227 2020-07-01 70 - 6.046016 + 5.439351 2020-08-01 80 - 1.678281 + 5.569649 2020-09-01 90 - 4.670443 + 6.225637 2020-10-01 100 - 5.501243 + 5.086193 2020-11-01 110 - 4.895184 + 3.244917 2020-12-01 120 - 4.984782 + 4.087981 @@ -766,7 +766,7 @@

resample()
-
<pandas.core.resample.DatetimeIndexResampler object at 0x110be1e20>
+
<pandas.core.resample.DatetimeIndexResampler object at 0x107bf6b40>
 
@@ -807,42 +807,42 @@

resample() 2019-01-01 20.0 - 4.657114 + 5.745064 2019-04-01 50.0 - 5.422896 + 5.859548 2019-07-01 80.0 - 5.032598 + 5.203775 2019-10-01 110.0 - 4.873632 + 5.682049 2020-01-01 20.0 - 6.163849 + 4.721784 2020-04-01 50.0 - 5.212681 + 4.513606 2020-07-01 80.0 - 4.131580 + 5.744879 2020-10-01 110.0 - 5.127070 + 4.139697 @@ -883,12 +883,12 @@

resample() 2019-01-01 65.0 - 4.996560 + 5.622609 2020-01-01 65.0 - 5.158795 + 4.779991 @@ -973,12 +973,12 @@

resample() 2019-01-01 0.5547 - 0.218092 + 0.154317 2020-01-01 0.5547 - 0.249735 + 0.237913 @@ -1638,7 +1638,7 @@

変化率の計算 -
<pandas.core.groupby.generic.DataFrameGroupBy object at 0x115f48ef0>
+
<pandas.core.groupby.generic.DataFrameGroupBy object at 0x1136ebfe0>
 
@@ -1653,7 +1653,7 @@

変化率の計算 -
<pandas.core.groupby.generic.SeriesGroupBy object at 0x115f2e3c0>
+
<pandas.core.groupby.generic.SeriesGroupBy object at 0x113cbbd70>
 
diff --git a/13a_Random_Variables.html b/13a_Random_Variables.html index 9a5f9e3f..af71e3cd 100644 --- a/13a_Random_Variables.html +++ b/13a_Random_Variables.html @@ -659,53 +659,53 @@

.cov 0 - 10.532872 - 99.347011 + 9.226527 + 59.884129 1 - 10.328511 - 57.027872 + 8.447087 + 10.864864 2 - 9.253206 - 30.935320 + 9.089684 + 98.844441 3 - 10.043987 - 27.351686 + 9.307079 + 30.984169 4 - 11.121341 - 69.431080 + 10.704744 + 79.099151 5 - 9.996576 - 48.125961 + 9.598606 + 55.097851 6 - 9.400748 - 96.637544 + 9.372898 + 8.376976 7 - 9.302751 - 40.649557 + 8.954774 + 73.010548 8 - 9.819194 - 42.198536 + 10.789657 + 65.658884 9 - 11.177317 - 92.616570 + 9.109282 + 35.203200 @@ -746,13 +746,13 @@

.cov X - 0.486400 - 9.289946 + 0.551495 + 8.565577 Y - 9.289946 - 754.804638 + 8.565577 + 882.945539 @@ -795,11 +795,11 @@

.cov X 1.000000 - 0.484841 + 0.388167 Y - 0.484841 + 0.388167 1.000000 @@ -816,7 +816,7 @@

.cov

-
0.4848409755436408
+
0.38816740703052205
 
@@ -1019,53 +1019,53 @@

.cov() 0 - 12.159850 + 10.743490 NaN 1 - 9.587224 - 12.159850 + 11.515666 + 10.743490 2 - 10.607244 - 9.587224 + 11.058309 + 11.515666 3 - 12.467029 - 10.607244 + 9.699965 + 11.058309 4 - 8.569824 - 12.467029 + 10.317081 + 9.699965 5 - 11.024675 - 8.569824 + 8.613991 + 10.317081 6 - 10.528026 - 11.024675 + 11.934332 + 8.613991 7 - 9.172312 - 10.528026 + 11.172057 + 11.934332 8 - 8.896764 - 9.172312 + 10.105710 + 11.172057 9 - 8.134338 - 8.896764 + 10.341831 + 10.105710 @@ -1114,13 +1114,13 @@

.cov() X - 2.212862 - -0.152933 + 0.921925 + -0.137977 X_lag - -0.152933 - 1.944755 + -0.137977 + 1.031133 @@ -1165,11 +1165,11 @@

.cov() X 1.000000 - -0.079381 + -0.133756 X_lag - -0.079381 + -0.133756 1.000000 @@ -1186,7 +1186,7 @@

.cov()

自己共分散がゼロであれば,ランダムに散らばっているはずであり,何らかのパターンも確認できないはずである。

@@ -1431,7 +1431,7 @@

自己共分散 -_images/ca8bd43629a1a295f46644109c74ae7627102d0a5def759cbfba09c07ee5e1a1.png +_images/aff93ba8ca0807c48213648c4496a6c55c1d706e9fe6d07921bf0482ecf893e5.png

\(\rho\)の値が高くなると、定常状態の\(0\)に戻ろうとする作用が弱くなり、持続性が強くなることがわかる。

@@ -1689,7 +1689,7 @@

3つの例:持続性の違い -_images/628f813bd1bd7d32db0e2a42682c4d1334a4e3cec7d38d137458bddf5e7c6f51.png +_images/c0da3bcfc3afe5f377a47ab5473dcc1c06c58d78e8a7abbadd1e67ed6a0bb56e.png

\(\rho\)が非常に高いため、持続性も非常に強くなっている。即ち、今期の値は前期の値に対する依存度が大きい。

diff --git a/13b_Stochastic_Solow.html b/13b_Stochastic_Solow.html index 5fe61611..6990f05b 100644 --- a/13b_Stochastic_Solow.html +++ b/13b_Stochastic_Solow.html @@ -1094,10 +1094,10 @@

TFP#

-
TFPの自己相関係数:0.687
+
TFPの自己相関係数:0.663
 
-_images/7ce521608dc04719eacc05c1de442bc38c9aa3e06989a11e16c5531109d518e4.png +_images/1fd4ed086eb2f122776e7a077c7819532cf5a329146c16276f3c9bf0b578c6af.png

シミュレーションを行う度に図と自己相関係数は異なることになる。

@@ -1612,95 +1612,95 @@

実行#< 8.500577e+05 1.000000 0.366461 - 2.722106e+05 + 272210.617761 213095.442439 59115.175322 - 13.758567 - -10.550732 - 0.305866 - -30.586611 - 12.858180 - -34.384874 - 12.613344 - -34.384874 - 11.331092 - -34.384874 + 13.704888 + -5.182883 + 0.083737 + -8.373726 + 12.616727 + -10.239564 + 12.371891 + -10.239564 + 11.089639 + -10.239564 1 8.879214e+05 - 0.541715 + 0.956201 0.367346 - 1.501540e+05 - 117545.495100 - 32608.499138 - 13.801442 - -10.480357 - 0.244313 - -85.732918 - 12.814475 - -89.505847 - 12.569639 - -89.505847 - 11.287387 - -89.505847 + 265042.428091 + 207483.947333 + 57558.480757 + 13.760500 + -6.386177 + 0.086970 + -13.175700 + 12.642392 + -15.474724 + 12.397556 + -15.474724 + 11.115304 + -15.474724 2 - 8.983319e+05 - 1.381046 + 9.232819e+05 + 3.177727 0.368233 - 3.853405e+05 - 301657.237477 - 83683.256087 - 13.844251 - -13.595659 - 0.182570 - 14.027174 - 12.770555 - 9.132736 - 12.525719 - 9.132736 - 11.243467 - 9.132736 + 895439.064586 + 700979.209461 + 194459.855125 + 13.816080 + -8.039004 + 0.090150 + 106.601600 + 12.667994 + 103.707559 + 12.423158 + 103.707559 + 11.140905 + 103.707559 3 - 9.595568e+05 - 3.811484 + 1.094660e+06 + 3.142032 0.369123 - 1.091658e+06 - 854585.690388 - 237072.094723 - 13.886863 - -11.263657 - 0.119908 - 121.811102 - 12.725646 - 117.756185 - 12.480810 - 117.756185 - 11.198558 - 117.756185 + 943621.976175 + 738698.380549 + 204923.595627 + 13.871555 + 3.439931 + 0.093143 + 105.172659 + 12.693371 + 106.411034 + 12.448535 + 106.411034 + 11.166282 + 106.411034 4 - 1.172640e+06 - 0.651463 + 1.272217e+06 + 1.718926 0.370015 - 2.010405e+05 - 157381.110947 - 43659.366242 - 13.929062 - 4.570662 - 0.055688 - -48.422343 - 12.679031 - -46.776904 - 12.434195 - -46.776904 - 11.151942 - -46.776904 + 546252.979147 + 427624.834154 + 118628.144993 + 13.926802 + 12.946903 + 0.096481 + 44.521859 + 12.719010 + 49.182744 + 12.474174 + 49.182744 + 11.191922 + 49.182744 @@ -1716,7 +1716,7 @@

実行#<

-_images/de6f01d3324a2c0a5c9995f9ce7b77b18dc6e4b30d8ad2eb7bf2ebeadfba5857.png +_images/5183e8799a5de10cda15034d5cf4e82e65496fd8802afb62551b965d37350703.png

トレンドの傾きは成長率を表しているが,徐々に減少していることが分かる。最初は成長率に対する資本蓄積の効果が大きいが,定常状態に近づくにつれて資本蓄積の効果が減少しているためである。

@@ -1730,7 +1730,7 @@

実行#<
-_images/0f87a1e45587004669fd494a206d2daa28916eac8ee709f2bc748e6905a32ef5.png +_images/0ccc371866a21987df211f7faf033bb79a7b2556d8c6b948fd2d88907d119694.png

絶対値で概ね2%内に収まった変動となっている。データでは,リーマンショック時はマイナス約4%程乖離したが,それ以外は絶対値で2%以内に収まっており,シミュレーションは概ねデータと同じような特徴を持っていると言える。

@@ -1754,11 +1754,11 @@

結果1:自己相関
--- シミュレーション:変動の自己相関係数 ---
 
-Yの変動: 0.416
-Cの変動: 0.416
-Iの変動: 0.416
-Kの変動: 0.830
-Aの変動: 0.411
+Yの変動: 0.395
+Cの変動: 0.395
+Iの変動: 0.395
+Kの変動: 0.828
+Aの変動: 0.394
 
@@ -1842,7 +1842,7 @@

結果2:GDPとの相関\(h\)の推定値
  • \(e_{pt}\equiv h\left(a u_{t}+v_{t}\right)\)

-

説明変数である\(p_{t-1}\)は誤差項\(e_{pt}\)とは期間がズレているため\(\text{E}(e_{pt}|p_{t-1})=0\)が満たされ,推定値は一致性を満たすことになる(不偏性は満たさない)。

+

説明変数である\(p_{t-1}\)は誤差項\(e_{pt}\)とは期間がズレているため,仮定に基づくと\(\text{E}(e_{pt}|p_{t-1})=0\)が満たされ,推定値は一致性を満たすことになる(不偏性は満たさない)。

まず,dfのメソッド.shift()を使ってdeflator_cycleを1期ずらした列をdeflator_cycle_lagとしてdfに追加しよう。

@@ -1266,8 +1266,8 @@

準備:\(h\)の推定値 Dep. Variable: deflator_cycle R-squared: 0.689 Model: OLS Adj. R-squared: 0.688 Method: Least Squares F-statistic: 384.1 -Date: Thu, 02 Jan 2025 Prob (F-statistic): 8.56e-46 -Time: 12:43:26 Log-Likelihood: 716.66 +Date: Fri, 03 Jan 2025 Prob (F-statistic): 8.56e-46 +Time: 01:09:49 Log-Likelihood: 716.66 No. Observations: 175 AIC: -1429. Df Residuals: 173 BIC: -1423. Df Model: 1 @@ -1294,7 +1294,7 @@

準備:\(h\)の推定値 -
  • 不均一分散自己相関頑健推定(HAC)を使うと推定値ht検定は有効になるが,その場合のdeflator_cycle_lag\(p\)値は0.0であり、推定値の統計的優位性は高いことが確認できる。

  • +
  • 不均一分散自己相関頑健推定(HAC

    +
      +
    • この手法を使うと推定値ht検定は有効になる。その場合のdeflator_cycle_lag\(p\)値は0.0であり(付録Aを参照)、推定値の統計的優位性は高いことが確認できる。

    • +
    +
  • 定数項なしで推定しても結果は殆ど変わらない。(試してみよう!)

  • (127)AR(1)であるため,\(\hat{h}\)\(p_t\)の自己相関係数でもあり,非常に高い持続性が特徴となっている。

    @@ -1329,7 +1333,7 @@

    最初のスイッチ:\(c\)

    \(d\equiv -ch\)

  • \(e_{yt}\equiv hu_t-chv_t\)

  • -

    説明変数である\(p_{t-1}\)は誤差項\(e_{yt}\)とは期間がズレているため\(\text{E}(e_{yt}|p_{t-1})=0\)となり,推定値は一致性を満たすことになる(不偏性は満たさない)。

    +

    説明変数である\(p_{t-1}\)は誤差項\(e_{yt}\)とは期間がズレているため,仮定に基づくと\(\text{E}(e_{yt}|p_{t-1})=0\)となり,推定値は一致性を満たすことになる(不偏性は満たさない)。

    res_d = smf.ols('gdp_cycle ~ deflator_cycle_lag', data=df).fit()
    @@ -1343,8 +1347,8 @@ 

    最初のスイッチ:\(c\)最初のスイッチ:\(c\)
  • Durbin-Watson検定量は0.633であり残差の正の系列相関が疑われる。

      -
    • \(t\)検定は無効の可能性がある。

    • +
    • \(t\)検定は無効の可能性を拭えない。

  • 残差の均一分散(付録Bを参照)

      -
    • ブルーシュペーガン検定の\(p\)値は0.03149であり、帰無仮説(均一分散)は有意水準5%で棄却される。一方、ホワイト検定の\(p\)値は0.0569847となり、帰無仮説(均一分散)は有意水準5%で棄却できないが、10%では棄却される。従って、不均一分散の可能性も残っている。

    • +
    • ブルーシュペーガン検定の\(p\)値は0.03149であり、帰無仮説(均一分散)は有意水準5%で棄却される。一方、ホワイト検定の\(p\)値は0.0569847となり、帰無仮説(均一分散)は有意水準5%で棄却できないが、10%では棄却される。従って、不均一分散の可能性も残っている。従って,この理由からも\(t\)検定は無効になるかも知れない。

  • -
  • 不均一分散自己相関頑健推定(HAC)を使うと推定値ht検定は有効になるが,その場合のdeflator_cycle_lag\(p\)値は0.0406244147であり、推定値の統計的優位性は高いことが確認できる(付録Bを参照)。この結果に基づきdの推定値を採用する。

  • +
  • 不均一分散自己相関頑健推定(HAC)を使うと推定値ht検定は有効になる。その場合のdeflator_cycle_lag\(p\)値は0.0406244147であり、推定値の統計的優位性は高いことが確認できる(付録Bを参照)。この結果に基づきdの推定値を採用する。

  • 定数項なしで推定しても結果は殆ど変わらない。

  • \(d\)の推定値である\(\hat{d}\)は負であり,モデルに沿った結果である。 @@ -1545,127 +1549,89 @@

    付録#<

    付録A#

    残差の自己相関の検定#

    -

    (129)は自己回帰モデルとなるため,(通常の)ダービン・ワトソン検定(\(d\)検定)は無効となる。その代わりに,次の統計量で与えられるダービンの\(H\)検定を使う。

    +

    (129)は自己回帰モデルとなるため,(通常の)ダービン・ワトソン検定(\(d\)検定)は無効となる。その代わりに,Breusch-Godfrey検定を使う。

    +

    <帰無仮説>

    \[ -H = -\left( -1-\frac{d}{2} -\right) -\sqrt{ - \frac{T} - { - 1-T\times\widehat{\text{Var}} - \left( - \hat{\beta} - \right) - } -} -\sim{\cal N}(0,1) +e_{pt}=\rho_{p}e_{pt-1}+e_{pt} \]
    -

    ここで

      -
    • \(H\):ダービン\(H\)検定統計量

    • -
    • \(d\):ダービン・ワトソン(\(d\))検定統計量

    • -
    • \(T\):標本の大きさ

    • -
    • \(\widehat{\text{Var}}\left(\hat{\beta}\right)\)\(p_{t-1}\)の係数hの分散の推定値

    • -
    • 仮定:\(1>T\times\widehat{\text{Var}}\left(\hat{\beta}\right)\)

    • +
    • \(H_0\)\(\rho_{p}=0\)

    • +
    • \(H_A\)\(\rho_{p}\ne0\)

    -

    であり,\(H\)は標準正規分布に従う。

    -

    <帰無仮説>

    -

    \(H_0\):自己相関はない

    -

    <両側検定>

    -
    -
    -
    import scipy
    -cv_10 = scipy.stats.norm.ppf(0.05)
    -cv_025 = scipy.stats.norm.ppf(0.025)
    -cv_01 = scipy.stats.norm.ppf(0.005)
    -print(f'有意水準10%:Hの絶対値が {abs(cv_10):.3f} よりも大きい場合,帰無仮説を棄却する。')
    -print(f'有意水準5%:Hの絶対値が {abs(cv_025):.3f} よりも大きい場合,帰無仮説を棄却する。')
    -print(f'有意水準1%:Hの絶対値が {abs(cv_01):.3f} よりも大きい場合,帰無仮説を棄却する。')
    -
    -
    -
    -
    -
    有意水準10%:Hの絶対値が 1.645 よりも大きい場合,帰無仮説を棄却する。
    -有意水準5%:Hの絶対値が 1.960 よりも大きい場合,帰無仮説を棄却する。
    -有意水準1%:Hの絶対値が 2.576 よりも大きい場合,帰無仮説を棄却する。
    -
    -
    -
    -
    -

    \(H\)検定統計量の計算>

    -

    \(d\)の計算

    -
    -
    -
    from statsmodels.stats.stattools import durbin_watson
    -d = durbin_watson(res_h.resid)
    -d
    -
    -
    -
    -
    -
    1.6164501184887314
    -
    -
    -
    -
    -

    標本の大きさ

    -
    -
    -
    T = res_h.nobs
    -T
    -
    -
    -
    -
    -
    175.0
    -
    -
    -
    -
    -

    \(\widehat{\text{Var}}\left(\hat{\beta}\right)\)の計算

    -
    -
    -
    var_h = res_h.cov_params().loc['deflator_cycle_lag','deflator_cycle_lag']
    -var_h
    -
    -
    -
    -
    -
    0.001748183392678259
    -
    -
    -
    -
    -

    \(H\)検定統計量の計算

    -
    H = ( 1-d/2 ) * ( T /(1-T*var_h) )**(1/2)
    -H
    +
    from statsmodels.stats.diagnostic import acorr_breusch_godfrey
    +
    +acorr_breusch_godfrey(res_h, nlags=1)
     
    -
    3.045158482774858
    +
    (5.427205281450245,
    + 0.019825381990804594,
    + 5.504888387071769,
    + 0.020102813690939046)
     
    -

    \(p\)値の計算

    +

    上から次の値となる。

    +
      +
    • \(LM\)検定統計量

    • +
    • \(LM\)検定統計量に関する\(p\)

    • +
    • \(F\)検定統計量

    • +
    • \(F\)検定統計量に関する\(p\)

    • +
    +

    5%の有意水準で帰無仮説を棄却できる。即ち,残差の自己相関があるようだ。

    +

    次に,Ljung-Box検定もおこなってみよう。

    +

    <帰無仮説>

    +
      +
    • \(H_0\):残差は独立分布(無相関)

    • +
    • \(H_A\)\(\rho_{p}\ne0\)

    • +
    -
    1-scipy.stats.norm.cdf(H)
    +
    from statsmodels.stats.diagnostic import acorr_ljungbox
    +
    +acorr_ljungbox(res_h.resid, lags=1)
     
    -
    0.001162788645233337
    -
    -
    -
    +
    + + + + + + + + + + + + + + + + +
    lb_statlb_pvalue
    13.9227610.047637
    +
    +

    左の値が検定統計量であり,右の値が\(p\)値となる。5%の有意水準で帰無仮説を棄却される。自己相関が疑われる結果である。

    均一分散の検定#

    diff --git a/18_ADAS-2.html b/18_ADAS-2.html index 68ca57fd..72e601b4 100644 --- a/18_ADAS-2.html +++ b/18_ADAS-2.html @@ -1076,7 +1076,7 @@

    プロット

    -_images/85691adc4a79b256ab3d12427fba8667f65818dea60b7c422bb878bac3b626d1.png +_images/ebe1fd1db8278d9e821af856199b42a43efdd45dcd11760a2c34e82a46f786f9.png

    \(u_t\)\(v_t\)が毎期毎期変動しており,その結果として\(p_t\)\(y_t\)が変動していることになる。 @@ -1095,7 +1095,7 @@

    プロット

    -_images/3233f4c4b4057d6a06133d5076191a32015580237665fb03aceb24bef9ec9b82.png +_images/dd26fadedfae37bc7bcd5a2125e359335ed540ee903236ac61e7105d39c4777c.png

    このプロットから次の点を確認できる。

    @@ -1149,8 +1149,8 @@

    GDPと価格水準の乖離率の標準偏差 -
    -
    -0.11532550507310885
    +
    -0.10826051696903893
     
    @@ -1278,11 +1278,11 @@

    結果のまとめと拡張-0.15298

    WNショック

    -

    0.01369

    -

    0.01433

    -

    0.05058

    -

    0.82254

    -

    -0.11533

    +

    0.01375

    +

    0.01415

    +

    0.04270

    +

    0.81852

    +

    -0.10826

    AR(1)ショック

    0.01252

    @@ -1471,8 +1471,8 @@

    \(p_t\)に対する影響

    需要ショックと供給ショックに起因する\(p_t\)の変動の割合を計算するために,式(144)の両辺を\(\text{Var}(p_{t})\)で割ると次式となる。

    -
    -(145)#\[\begin{align} +
    +(145)#\[\begin{align} 1&= \frac{(ah)^2}{1-h^2}\cdot\frac{\text{Var}(u_{t})}{\text{Var}(p_{t})} +\frac{h^2}{1-h^2}\cdot\frac{\text{Var}(v_{t})}{\text{Var}(p_{t})} @@ -1580,8 +1580,8 @@

    \(y_t\)に対する影響
  • \(p_t\)の総変動の大きさ(水準)を決定する上で,\(h\) が重要な役割を果たしている。\(h\)はは約0.8であり\(p_t\)の高い持続性を意味する。持続性が高い場合,一旦総供給曲線が定常状態の位置から離れると元の位置に戻るには時間が掛かることになる。即ち,定常状態から離れた状態が長くなる。そのような状況下で,更にショックが発生すると,定常状態から更に乖離する結果につながることになり,変動が増幅されることになる。その効果が\(\dfrac{h^2}{1-h^2}\)で捉えられている。

  • 需要ショックと供給ショックに起因する\(y_t\)の変動の割合を計算するために,式(150)の両辺を\(\text{Var}(y_{t})\)で割ると次式となる。

    -
    -(151)#\[\begin{align} +
    +(151)#\[\begin{align} 1 &=\frac{h^2}{1-h^2}2ach\frac{\text{Var}(u_{t})}{\text{Var}(y_{t})} +\frac{h^2}{1-h^2}c^2\frac{\text{Var}(v_{t})}{\text{Var}(y_{t})}\\ @@ -1814,8 +1814,8 @@

    付録#<

    付録B:式(150)の導出#

    (144)と式(149)を式(147)に代入し整理する。

    -
    -(152)#\[\begin{align} +
    +(152)#\[\begin{align} \text{Var}(y_{t}) &=(ch)^2\text{Var}(p_{t})+\text{Var}(e_{yt})\\ &=(ch)^2\left[ @@ -1839,8 +1839,8 @@

    付録#< +\frac{(ch)^2}{1-h^2}\text{Var}(v_{t}) \end{align}\]

    ここで

    -
    -(153)#\[\begin{align} +
    +(153)#\[\begin{align} 1+(ach)^2-h^2 &=1+h^2[(ac)^2-1]=1+\frac{(ac)^2-1}{(1+ac)^2}\\ &=1+\frac{(ac+1)(ac-1)}{(1+ac)^2}=1+\frac{ac-1}{1+ac}\\ @@ -1852,8 +1852,8 @@

    付録#<

    付録C:式(u-y)と(v-y)の導出#

    式(u-y)に式(150)を代入して整理する。

    -
    -(154)#\[\begin{align} +
    +(154)#\[\begin{align} \left[ \begin{split} &y_t\text{の変動に対する}\\ @@ -1880,8 +1880,8 @@

    付録#< \tag{u-y} \end{align}\]

    同様に式(v-y)に式(150)を代入して整理する。

    -
    -(155)#\[\begin{align} +
    +(155)#\[\begin{align} \left[ \begin{split} &p_t\text{の変動に対する}\\ diff --git a/6_Regression.html b/6_Regression.html index 0b23d6a4..bf5f3ada 100644 --- a/6_Regression.html +++ b/6_Regression.html @@ -610,8 +610,8 @@

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R-squared: 0.848 Method: Least Squares F-statistic: 336.7 -Date: Thu, 02 Jan 2025 Prob (F-statistic): 4.64e-26 -Time: 12:43:49 Log-Likelihood: 49.767 +Date: Fri, 03 Jan 2025 Prob (F-statistic): 4.64e-26 +Time: 01:10:19 Log-Likelihood: 49.767 No. Observations: 61 AIC: -95.53 Df Residuals: 59 BIC: -91.31 Df Model: 1 diff --git a/_images/0ccc371866a21987df211f7faf033bb79a7b2556d8c6b948fd2d88907d119694.png b/_images/0ccc371866a21987df211f7faf033bb79a7b2556d8c6b948fd2d88907d119694.png new file mode 100644 index 0000000000000000000000000000000000000000..5afe0deeb135e0c4c9f18ca2095088d47696e4b5 GIT binary patch literal 48914 zcmbUIbx>6AA3qE)T}v$>3rOk05(+FKC9s6ZA}k%!E{K94-Q6sqz=9wpARr~((kv;8 zL3gM~hjiYD&-XXaJ@Y)z+<)A|Fr3*rd(IW_c*O-)8~u=)l9dt!0#T#XRCGZgLIemz zP(@A(e1el)sRI7UxIZv<*K@LU_j=;`6r}mY-Pys(-N6pS<@war&Ccnim=HolNaPOJ zb9Z-VHyL4J$N$eALQbyFgp*y3U4f@iIIBH&1A%CtT>lajDSWU4f!YF5DvJ8vnVSJ* zEjJAgt^!(rx2Ddqanq?OQK^v;P_S3!oHG&A_k>0>7DNY=s}?IL>Lk$QiIXIM^Gihj z&USYLn#3MLKf 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z#gF5K^W@iFZ1`;Q(X??OR2da7+R_w=y5DedIOQw9f_B|8;-`r6?#&(~wv3_ky0I=G zI9V71&s{9c%RZdS7bL6<=MhU2wRm>jOH=@~)zWgJvA9#x-syUAjJ<(#O7dHcJ66EG z`~IFCAO;3a_S-VPws!Mpf0~M`nkrumFrwz0HEioD z3Y@0uyowvI{z0rR1Lg7=ejJ>=)p+KZB`7>N}uZRCM%4rE1=X$ztXEm65az|X=ZNy2L5`31}CT3 z#s1KWe{Y4Q)#+bdWatX!zgqf#_LcoR;DgHq02dI(}&?103 z>rHupwiaoImeZ}eKfb4d`LO*6<8x^=fC1EsWw_i9bKPC)nk+6P9|=8p#s$f@vQVv6BStMv-&hgO{gfFDuazBhXjd;K=8p1;)m1D`{RoHmhcf z64UukX8r`QIMwfj)&p{J_d6Z~a=tZe8Jy8q{%D4aRrymFig)M;gk7g-|KblJ%o}}T z8W^%Tk^K@^26#yW+4Ae1BauiY?uHy|8?9pn(RBO1)Q4#36XTEXYI{q*-1}hmJt23! zdjAG!V^aO)vqEovlJ(9>&5=L%`=4U(SK*S~1qs7plboA zJJ*@y6nuPYIE)+XJ*%ER*4pej-dS8u_uJ|c-pjVN8$NW@Hc2(OKMm?hN#g#s=O~?~ zcaAf)T>#zgDJ61!Flz8UIevgcmSB~2kpazJNrw`!vnk&-aQ=?7+wjL))CJL15*&{6Q!8A{~Tx-BpbToHk=cX(W6~2^<4=o@mH6f%GI!%T\\times\\widehat{\\text{Var}}\\left(\\hat{\\beta}\\right)$\n", - "\n", - "であり,$H$は標準正規分布に従う。\n", + "式[](eq:18-regression-h)は自己回帰モデルとなるため,(通常の)ダービン・ワトソン検定($d$検定)は無効となる。その代わりに,Breusch-Godfrey検定を使う。\n", "\n", "<帰無仮説>\n", "\n", - "$H_0$:自己相関はない" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "<両側検定>" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "import scipy\n", - "cv_10 = scipy.stats.norm.ppf(0.05)\n", - "cv_025 = scipy.stats.norm.ppf(0.025)\n", - "cv_01 = scipy.stats.norm.ppf(0.005)\n", - "print(f'有意水準10%:Hの絶対値が {abs(cv_10):.3f} よりも大きい場合,帰無仮説を棄却する。')\n", - "print(f'有意水準5%:Hの絶対値が {abs(cv_025):.3f} よりも大きい場合,帰無仮説を棄却する。')\n", - "print(f'有意水準1%:Hの絶対値が {abs(cv_01):.3f} よりも大きい場合,帰無仮説を棄却する。')" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "<$H$検定統計量の計算>\n", + "$$\n", + "e_{pt}=\\rho_{p}e_{pt-1}+e_{pt}\n", + "$$\n", "\n", - "$d$の計算" + "* $H_0$:$\\rho_{p}=0$\n", + "* $H_A$:$\\rho_{p}\\ne0$" ] }, { @@ -1657,50 +1603,34 @@ "metadata": {}, "outputs": [], "source": [ - "from statsmodels.stats.stattools import durbin_watson\n", - "d = durbin_watson(res_h.resid)\n", - "d" + "from statsmodels.stats.diagnostic import acorr_breusch_godfrey\n", + "\n", + "acorr_breusch_godfrey(res_h, nlags=1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "標本の大きさ" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "T = res_h.nobs\n", - "T" + "上から次の値となる。\n", + "* $LM$検定統計量\n", + "* $LM$検定統計量に関する$p$値\n", + "* $F$検定統計量\n", + "* $F$検定統計量に関する$p$値" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "$\\widehat{\\text{Var}}\\left(\\hat{\\beta}\\right)$の計算" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "var_h = res_h.cov_params().loc['deflator_cycle_lag','deflator_cycle_lag']\n", - "var_h" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$H$検定統計量の計算" + "5%の有意水準で帰無仮説を棄却できる。即ち,残差の自己相関があるようだ。\n", + "\n", + "次に,Ljung-Box検定もおこなってみよう。\n", + "\n", + "<帰無仮説>\n", + "\n", + "* $H_0$:残差は独立分布(無相関)\n", + "* $H_A$:$\\rho_{p}\\ne0$" ] }, { @@ -1709,24 +1639,16 @@ "metadata": {}, "outputs": [], "source": [ - "H = ( 1-d/2 ) * ( T /(1-T*var_h) )**(1/2)\n", - "H" + "from statsmodels.stats.diagnostic import acorr_ljungbox\n", + "\n", + "acorr_ljungbox(res_h.resid, lags=1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "$p$値の計算" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "1-scipy.stats.norm.cdf(H)" + "左の値が検定統計量であり,右の値が$p$値となる。5%の有意水準で帰無仮説を棄却される。自己相関が疑われる結果である。" ] }, { diff --git a/_sources/index.ipynb b/_sources/index.ipynb index c97af675..37b33d95 100644 --- a/_sources/index.ipynb +++ b/_sources/index.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "633a7b68", + "id": "9515b5c2", "metadata": {}, "source": [ "# Pythonで学ぶマクロ経済学 (中級+レベル)\n", @@ -22,14 +22,14 @@ { "cell_type": "code", "execution_count": 1, - "id": "1ce9b561", + "id": "f0488bae", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "Version: 2025年01月02日\n" + "Version: 2025年01月03日\n" ] } ], @@ -41,7 +41,7 @@ }, { "cell_type": "markdown", - "id": "a234a2c7", + "id": "bf66ad96", "metadata": {}, "source": [ "