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takahashi-r/public_2022_09_28_コブ・ダグラス生産関数.ipynb

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public_2022_09_28_コブ・ダグラス生産関数.ipynb
{
"cells": [
{
"cell_type": "markdown",
"id": "36b73660",
"metadata": {},
"source": [
"本ノートブックはQiitaに投稿した「○○」の執筆に先立って制作したものです。定義等は「参考」に記載されている記事・書籍等から引用しております。\n",
"\n",
"\n",
"使用環境のバージョンは下記に記しています。"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "ab5b9331",
"metadata": {
"scrolled": true
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"outputs": [
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"output_type": "stream",
"text": [
"Julia Version 1.7.0\n",
"Commit 3bf9d17731 (2021-11-30 12:12 UTC)\n",
"Platform Info:\n",
" OS: Windows (x86_64-w64-mingw32)\n",
" CPU: Intel(R) Core(TM) i5-10210U CPU @ 1.60GHz\n",
" WORD_SIZE: 64\n",
" LIBM: libopenlibm\n",
" LLVM: libLLVM-12.0.1 (ORCJIT, skylake)\n"
]
}
],
"source": [
"versioninfo()"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "0d8dc4d2",
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"C:\\Users\\ryuma takahashi\\.conda\\envs\\pymc_env\\lib\\inspect.py:469: FutureWarning: pandas.Float64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead.\r\n",
" value = getattr(object, key)\r\n",
"C:\\Users\\ryuma takahashi\\.conda\\envs\\pymc_env\\lib\\inspect.py:469: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead.\r\n",
" value = getattr(object, key)\r\n",
"C:\\Users\\ryuma takahashi\\.conda\\envs\\pymc_env\\lib\\inspect.py:469: FutureWarning: pandas.UInt64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead.\r\n",
" value = getattr(object, key)\r\n"
]
}
],
"source": [
"using Plots\n",
"gr(fontfamily=\"IPAPGothic\") # 日本語\n",
"\n",
"using DataFrames\n",
"using GLM\n",
"using HypothesisTests\n",
"using Pandas: Pandas\n",
"using PyCall\n",
"@pyimport pandas as pd\n",
"using RCall\n",
"R\"\"\"Sys.setenv(LANG = \"en\")\"\"\" # おまじない\n",
"@rimport lmtest"
]
},
{
"cell_type": "markdown",
"id": "40388a3b",
"metadata": {},
"source": [
"# コブ・ダグラス生産関数"
]
},
{
"cell_type": "markdown",
"id": "b6616459",
"metadata": {},
"source": [
"Y: 生産額、K：資本額、L：労働投入量 \n",
"\n",
"$$ \n",
"\\begin{align}\n",
"Y = \\alpha K^\\beta L^\\gamma \\tag{1} \\\\\n",
"\\log{Y} = \\alpha' + \\beta \\log{K} + \\gamma \\log{L} \\tag{2} \n",
"\\end{align}\n",
"$$\n",
"\n",
"生産設備と労働投下量によって、生産額があらかた決まる"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "d8fcc9c4",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Y (generic function with 1 method)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# 式(1)\n",
"Y( α, β, γ, K, L ) = α*K^β*L^γ"
]
},
{
"cell_type": "markdown",
"id": "4d8c9d63",
"metadata": {},
"source": [
"## 弾力性"
]
},
{
"cell_type": "markdown",
"id": "7d7b5e3c",
"metadata": {},
"source": [
"係数β、γは個々の生産要素が生産量Yに及ぼす「弾力性」という意味をもつ。 \n",
"これは例えば、係数βを考えたとすると、資本Kを1%増やしたときの生産量の増加%を表している。\n",
"\n",
"(1)より\n",
"$$ \n",
"\\begin{align}\n",
"\\frac{\\partial Y}{\\partial K} &= \\alpha \\beta K^{\\beta - 1} L^\\gamma \\\\\n",
"&= \\beta \\frac{\\alpha K^\\beta L^\\gamma}{K} \\\\\n",
"&= \\beta \\frac{Y}{K} \\\\\n",
"\\Rightarrow \\beta &= \\frac{\\partial Y}{\\partial K} \\frac{K}{Y} \\\\\n",
"&= \\frac{\\frac{\\partial Y}{Y}}{ \\frac{\\partial K}{K}}\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "8e2edcf2",
"metadata": {},
"source": [
"## 収穫の不変性\n",
"\n",
"生産関数のインプットである資本と労働投入量を定数倍したときのアウトプットの増加倍数を問うのが「収穫の不変性」という\n",
"\n",
"コブ・ダグラス生産関数においてインプットをc倍してみる\n",
"\n",
"$$ \n",
"\\begin{align}\n",
"Y = \\alpha (cK)^\\beta (cL)^\\gamma = c^{\\beta + \\gamma} \\lbrace \\alpha K^\\beta L^\\gamma \\rbrace \\tag{3}\n",
"\\end{align}\n",
"$$\n",
"\n",
"$\\beta + \\gamma$ が1ならアウトプットも定数倍になる。このように資本と労働投入量をともにc倍したときに産出量が同じくc倍になることを「収穫不変」という。1以上なら「収穫逓増」、1以下なら「収穫逓減」となる。 \n",
"※製造業では収穫逓減が通常で、農業は収穫不変に近いとされる。通常収穫逓増は起こらないと考えられている。\n",
"\n",
"収穫不変が成立していれば式(2)は次のように書き直される\n",
"\n",
"$$ \n",
"\\begin{align}\n",
"\\log{Y} &= \\alpha' + \\beta \\log{K} + (1 - \\beta) \\log{L} \\tag{4} \\\\\n",
"\\log{Y/L} &= \\alpha' + \\beta \\log{K/L}\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "2a962c96",
"metadata": {},
"source": [
"- 定数項は技術水準である"
]
},
{
"cell_type": "markdown",
"id": "18129893",
"metadata": {},
"source": [
"## 生産規模と収穫の関係\n",
"\n",
"さて、では無限に資本と労働をつぎ込めばうまくいくことはなさそうだ。→通常収穫逓増は起こらないと考えられている\n",
"\n",
"ここで、収穫不変の状態を考えて、資本と労働投入量の値を動かして、生産量がどんな変化をするのか見てみよう。簡単のために技術水準αは取り除いた。\n",
"\n",
"$$\n",
"\\begin{align}\n",
"MP_k &= \\beta K^{(\\beta-1)} L^{(1-\\beta)} \\tag{5} \\\\\n",
"MP_l &= (1 - \\beta) K^\\beta L^{-\\beta} \\tag{6} \\\\\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "e2bb2717",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"MPₗ (generic function with 1 method)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# 生産関数を Y = K^β L^(1-β) とする. 資本:Kと労働投入量:Lでそれぞれ微分する\n",
"\n",
"# 資本の限界生産力 式(3)\n",
"MPₖ( β, K, L ) = Y( β, β-1, 1-β, K, L ) \n",
"# 労働投入量の限界生産力 式(4)\n",
"MPₗ( β, K, L ) = Y( 1-β, β, -β, K, L )"
]
},
{
"cell_type": "markdown",
"id": "d04ca381",
"metadata": {},
"source": [
"- 生産要素の投入量が増えるにつれて，限界生産力は逓減していくことがわかる→資本と労働投入量のバランスが大事だということ"
]
},
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},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"xs = range(0, 1, 100)\n",
"\n",
"p1 = plot(xs, MPₖ.( 0.5, 1, xs ), xlabel=\"L: 労働投入量\", ylabel=\"Y: 生産額\", label=\"MPₖ\")\n",
"p2 = plot(xs, MPₗ.( 0.5, xs, 1 ), xlabel=\"K: 資本\", ylabel=\"Y: 生産額\", label=\"MPₗ\")\n",
"plot(p1, p2, layout=(2, 1))"
]
},
{
"cell_type": "markdown",
"id": "775a00c2",
"metadata": {},
"source": [
"## 製造業の生産関数を求める\n",
"\n",
"技術水準一定として生産関数を求める"
]
},
{
"cell_type": "markdown",
"id": "6d0820b4",
"metadata": {},
"source": [
"- [内閣府](https://www.cao.go.jp/)\n",
" - [製造業GDP.xlsx](https://www.esri.cao.go.jp/jp/sna/data/data_list/kakuhou/files/2020/tables/2020ffm1n_jp.xlsx)\n",
" - [就業者数.xlsx](https://www.esri.cao.go.jp/jp/sna/data/data_list/kakuhou/files/2020/tables/2020s3_jp.xlsx)\n",
" - [資本ストック.xlsx](https://www.esri.cao.go.jp/jp/sna/data/data_list/minkan/files/tables/h17/h26y_stock_all.xls) \n",
"\n",
"にある公開データを利用する。いずれのデータもすべて名目値を扱っている。"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "50f7b71c",
"metadata": {},
"outputs": [],
"source": [
"pydf = pd.read_csv(\"./製造業の生産関数.csv\");"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "f8f1b273",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"PyObject"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# julia上での型\n",
"typeof(pydf)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "d8701487",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"PyObject <class 'pandas.core.frame.DataFrame'>"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# python上での型\n",
"py\"type\"(pydf)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "6d3efda9",
"metadata": {},
"outputs": [
{
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"<div class=\"data-frame\"><p>21 rows × 5 columns</p><table class=\"data-frame\"><thead><tr><th></th><th>年次</th><th>製造業GDP(10億円)</th><th>就業者数(万人)</th><th>労働時間</th><th>資本ストック(10億円)</th></tr><tr><th></th><th title=\"Int64\">Int64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th></tr></thead><tbody><tr><th>1</th><td>1994</td><td>119908.0</td><td>1403.4</td><td>1961.3</td><td>3.06303e5</td></tr><tr><th>2</th><td>1995</td><td>1.22432e5</td><td>1363.8</td><td>1975.6</td><td>3.15492e5</td></tr><tr><th>3</th><td>1996</td><td>1.25114e5</td><td>1356.3</td><td>1975.2</td><td>3.22602e5</td></tr><tr><th>4</th><td>1997</td><td>1.2692e5</td><td>1334.3</td><td>1961.3</td><td>3.32636e5</td></tr><tr><th>5</th><td>1998</td><td>122518.0</td><td>1279.3</td><td>1939.3</td><td>3.41929e5</td></tr><tr><th>6</th><td>1999</td><td>1.18298e5</td><td>1248.0</td><td>1938.8</td><td>3.4479e5</td></tr><tr><th>7</th><td>2000</td><td>1.20213e5</td><td>1229.0</td><td>1944.6</td><td>3.49805e5</td></tr><tr><th>8</th><td>2001</td><td>1.12284e5</td><td>1185.2</td><td>1946.7</td><td>3.54669e5</td></tr><tr><th>9</th><td>2002</td><td>1.08644e5</td><td>1140.4</td><td>1961.3</td><td>3.52087e5</td></tr><tr><th>10</th><td>2003</td><td>1.09599e5</td><td>1123.6</td><td>1982.4</td><td>350463.0</td></tr><tr><th>11</th><td>2004</td><td>1.11813e5</td><td>1109.1</td><td>1992.0</td><td>3.53903e5</td></tr><tr><th>12</th><td>2005</td><td>1.14083e5</td><td>1096.5</td><td>1995.8</td><td>3.59879e5</td></tr><tr><th>13</th><td>2006</td><td>1.14523e5</td><td>1112.1</td><td>2003.3</td><td>369886.0</td></tr><tr><th>14</th><td>2007</td><td>1.17971e5</td><td>1117.1</td><td>1996.5</td><td>3.83308e5</td></tr><tr><th>15</th><td>2008</td><td>1.12289e5</td><td>1100.5</td><td>1926.5</td><td>3.94291e5</td></tr><tr><th>16</th><td>2009</td><td>94274.3</td><td>1029.9</td><td>1877.5</td><td>3.9625e5</td></tr><tr><th>17</th><td>2010</td><td>1.0498e5</td><td>1020.6</td><td>1944.0</td><td>4.00983e5</td></tr><tr><th>18</th><td>2011</td><td>97179.4</td><td>1024.8</td><td>1955.2</td><td>4.07542e5</td></tr><tr><th>19</th><td>2012</td><td>98426.9</td><td>1018.6</td><td>1946.2</td><td>4.12368e5</td></tr><tr><th>20</th><td>2013</td><td>98326.9</td><td>1032.1</td><td>1955.6</td><td>4.12936e5</td></tr><tr><th>21</th><td>2014</td><td>1.01654e5</td><td>1047.4</td><td>1955.6</td><td>416822.0</td></tr></tbody></table></div>"
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"text/latex": [
"\\begin{tabular}{r|ccccc}\n",
"\t& 年次 & 製造業GDP(10億円) & 就業者数(万人) & 労働時間 & 資本ストック(10億円)\\\\\n",
"\t\\hline\n",
"\t& Int64 & Float64 & Float64 & Float64 & Float64\\\\\n",
"\t\\hline\n",
"\t1 & 1994 & 119908.0 & 1403.4 & 1961.3 & 3.06303e5 \\\\\n",
"\t2 & 1995 & 1.22432e5 & 1363.8 & 1975.6 & 3.15492e5 \\\\\n",
"\t3 & 1996 & 1.25114e5 & 1356.3 & 1975.2 & 3.22602e5 \\\\\n",
"\t4 & 1997 & 1.2692e5 & 1334.3 & 1961.3 & 3.32636e5 \\\\\n",
"\t5 & 1998 & 122518.0 & 1279.3 & 1939.3 & 3.41929e5 \\\\\n",
"\t6 & 1999 & 1.18298e5 & 1248.0 & 1938.8 & 3.4479e5 \\\\\n",
"\t7 & 2000 & 1.20213e5 & 1229.0 & 1944.6 & 3.49805e5 \\\\\n",
"\t8 & 2001 & 1.12284e5 & 1185.2 & 1946.7 & 3.54669e5 \\\\\n",
"\t9 & 2002 & 1.08644e5 & 1140.4 & 1961.3 & 3.52087e5 \\\\\n",
"\t10 & 2003 & 1.09599e5 & 1123.6 & 1982.4 & 350463.0 \\\\\n",
"\t11 & 2004 & 1.11813e5 & 1109.1 & 1992.0 & 3.53903e5 \\\\\n",
"\t12 & 2005 & 1.14083e5 & 1096.5 & 1995.8 & 3.59879e5 \\\\\n",
"\t13 & 2006 & 1.14523e5 & 1112.1 & 2003.3 & 369886.0 \\\\\n",
"\t14 & 2007 & 1.17971e5 & 1117.1 & 1996.5 & 3.83308e5 \\\\\n",
"\t15 & 2008 & 1.12289e5 & 1100.5 & 1926.5 & 3.94291e5 \\\\\n",
"\t16 & 2009 & 94274.3 & 1029.9 & 1877.5 & 3.9625e5 \\\\\n",
"\t17 & 2010 & 1.0498e5 & 1020.6 & 1944.0 & 4.00983e5 \\\\\n",
"\t18 & 2011 & 97179.4 & 1024.8 & 1955.2 & 4.07542e5 \\\\\n",
"\t19 & 2012 & 98426.9 & 1018.6 & 1946.2 & 4.12368e5 \\\\\n",
"\t20 & 2013 & 98326.9 & 1032.1 & 1955.6 & 4.12936e5 \\\\\n",
"\t21 & 2014 & 1.01654e5 & 1047.4 & 1955.6 & 416822.0 \\\\\n",
"\\end{tabular}\n"
],
"text/plain": [
"\u001b[1m21×5 DataFrame\u001b[0m\n",
"\u001b[1m Row \u001b[0m│\u001b[1m 年次 \u001b[0m\u001b[1m 製造業GDP(10億円) \u001b[0m\u001b[1m 就業者数(万人) \u001b[0m\u001b[1m 労働時間 \u001b[0m\u001b[1m 資本ストック(10億円\u001b[0m ⋯\n",
"\u001b[1m \u001b[0m│\u001b[90m Int64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m ⋯\n",
"─────┼──────────────────────────────────────────────────────────────────────────\n",
" 1 │ 1994 119908.0 1403.4 1961.3 3.06303e ⋯\n",
" 2 │ 1995 1.22432e5 1363.8 1975.6 3.15492e\n",
" 3 │ 1996 1.25114e5 1356.3 1975.2 3.22602e\n",
" 4 │ 1997 1.2692e5 1334.3 1961.3 3.32636e\n",
" 5 │ 1998 122518.0 1279.3 1939.3 3.41929e ⋯\n",
" 6 │ 1999 1.18298e5 1248.0 1938.8 3.4479e5\n",
" 7 │ 2000 1.20213e5 1229.0 1944.6 3.49805e\n",
" 8 │ 2001 1.12284e5 1185.2 1946.7 3.54669e\n",
" 9 │ 2002 1.08644e5 1140.4 1961.3 3.52087e ⋯\n",
" 10 │ 2003 1.09599e5 1123.6 1982.4 350463.0\n",
" 11 │ 2004 1.11813e5 1109.1 1992.0 3.53903e\n",
" 12 │ 2005 1.14083e5 1096.5 1995.8 3.59879e\n",
" 13 │ 2006 1.14523e5 1112.1 2003.3 369886.0 ⋯\n",
" 14 │ 2007 1.17971e5 1117.1 1996.5 3.83308e\n",
" 15 │ 2008 1.12289e5 1100.5 1926.5 3.94291e\n",
" 16 │ 2009 94274.3 1029.9 1877.5 3.9625e5\n",
" 17 │ 2010 1.0498e5 1020.6 1944.0 4.00983e ⋯\n",
" 18 │ 2011 97179.4 1024.8 1955.2 4.07542e\n",
" 19 │ 2012 98426.9 1018.6 1946.2 4.12368e\n",
" 20 │ 2013 98326.9 1032.1 1955.6 4.12936e\n",
" 21 │ 2014 1.01654e5 1047.4 1955.6 416822.0 ⋯\n",
"\u001b[36m 1 column omitted\u001b[0m"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# py to jl\n",
"jldf = DataFrame(Pandas.DataFrame(pydf))"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "fa9c467f",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"DataFrame"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"typeof(jldf)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "8f542a47",
"metadata": {},
"outputs": [
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" 2328.93,631.512 \n",
" \"/>\n",
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" \"/>\n",
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" \"/>\n",
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" \"/>\n",
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" 310.802,1163.84 329.699,1163.84 \n",
" \"/>\n",
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" \"/>\n",
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" 1128.93,1140.08 \n",
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" 1510.8,1487.47 1510.8,921.675 \n",
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" \"/>\n",
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" 1510.8,1139.68 1529.7,1139.68 \n",
" \"/>\n",
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"</g>\n",
"<g clip-path=\"url(#clip210)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:38px; text-anchor:start;\" transform=\"rotate(0, 1453.06, 1377.48)\" x=\"1453.06\" y=\"1377.48\">5</text>\n",
"</g>\n",
"<g clip-path=\"url(#clip210)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 1250.99, 1284.14)\" x=\"1250.99\" y=\"1284.14\">3.500×10</text>\n",
"</g>\n",
"<g clip-path=\"url(#clip210)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:38px; text-anchor:start;\" transform=\"rotate(0, 1453.06, 1256.73)\" x=\"1453.06\" y=\"1256.73\">5</text>\n",
"</g>\n",
"<g clip-path=\"url(#clip210)\">\n",
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"</g>\n",
"<g clip-path=\"url(#clip210)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:38px; text-anchor:start;\" transform=\"rotate(0, 1453.06, 1135.99)\" x=\"1453.06\" y=\"1135.99\">5</text>\n",
"</g>\n",
"<g clip-path=\"url(#clip210)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 1250.99, 1042.66)\" x=\"1250.99\" y=\"1042.66\">4.000×10</text>\n",
"</g>\n",
"<g clip-path=\"url(#clip210)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:38px; text-anchor:start;\" transform=\"rotate(0, 1453.06, 1015.25)\" x=\"1453.06\" y=\"1015.25\">5</text>\n",
"</g>\n",
"<g clip-path=\"url(#clip210)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:84px; text-anchor:middle;\" transform=\"rotate(0, 1931.78, 873.2)\" x=\"1931.78\" y=\"873.2\">資本ストック(10億円)</text>\n",
"</g>\n",
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" 1534.63,1471.46 1574.35,1427.08 1614.06,1392.74 1653.78,1344.28 1693.49,1299.4 1733.2,1285.58 1772.92,1261.36 1812.63,1237.87 1852.35,1250.34 1892.06,1258.18 \n",
" 1931.78,1241.57 1971.49,1212.7 2011.21,1164.37 2050.92,1099.55 2090.64,1046.5 2130.35,1037.05 2170.07,1014.19 2209.78,982.508 2249.5,959.201 2289.21,956.455 \n",
" 2328.93,937.688 \n",
" \"/>\n",
"</svg>\n"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ps = []\n",
"for col in names(jldf)[2:end]\n",
" push!(ps, plot(jldf[!, \"年次\"], jldf[!, col], title=\"$col\", label=false))\n",
"end\n",
"plot(ps...)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "0b9b42b5",
"metadata": {},
"outputs": [],
"source": [
"# カラム名の変更 単位が邪魔くさい\n",
"rename!(jldf, Symbol(\"製造業GDP(10億円)\") => :製造業GDP);\n",
"rename!(jldf, Symbol(\"就業者数(万人)\") => :就業者数);\n",
"rename!(jldf, Symbol(\"資本ストック(10億円)\") => :資本ストック);"
]
},
{
"cell_type": "markdown",
"id": "b5bf8e79",
"metadata": {},
"source": [
"係数推定値を確認する。 \n",
"β: 0.857002, γ: 0.154591, β+γ: 1.011593 \n",
"β+γ > 1 ではあるので、収穫逓増である。ただ、+0.011593がどれほどの差になるかはよくわからない。ただ、今回使用したデータでは収穫逓減にはなっていないことがわかった。\n",
"\n",
"係数推定値を弾力性で見ると、1%労働投入量を増やせば生産量が85%増え、1%資本を増やせば生産量が15%増えるということになる。"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "2a4ec72b",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}}}}, Matrix{Float64}}\n",
"\n",
":(log(製造業GDP)) ~ 1 + :(log(就業者数 * 労働時間)) + :(log(資本ストック))\n",
"\n",
"Coefficients:\n",
"───────────────────────────────────────────────────────────────────────────────\n",
" Coef. Std. Error t Pr(>|t|) Lower 95% Upper 95%\n",
"───────────────────────────────────────────────────────────────────────────────\n",
"(Intercept) -2.89628 7.44976 -0.39 0.7020 -18.5477 12.7551\n",
"log(就業者数 * 労働時間) 0.857002 0.254899 3.36 0.0035 0.32148 1.39252\n",
"log(資本ストック) 0.154591 0.298641 0.52 0.6110 -0.472831 0.782012\n",
"───────────────────────────────────────────────────────────────────────────────"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ols = lm(\n",
" @formula(log(製造業GDP) ~ log(就業者数*労働時間) + log(資本ストック)),\n",
" jldf\n",
")"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "e4a446b8",
"metadata": {},
"outputs": [
{
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},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(residuals(ols), title=\"残差\", legend=false)"
]
},
{
"cell_type": "markdown",
"id": "0e4bad6a",
"metadata": {},
"source": [
"## DW検定"
]
},
{
"cell_type": "markdown",
"id": "214a09ae",
"metadata": {},
"source": [
"回帰分析後の残差の自己相関を確認する。自己相関がある場合GLSを用いて推定を行わなければならない→自己相関なし"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "9cd200d5",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"RObject{ClosSxp}\n",
"function (formula, order.by = NULL, alternative = c(\"greater\", \n",
" \"two.sided\", \"less\"), iterations = 15, exact = NULL, \n",
" tol = 1e-10, data = list()) \n",
"{\n",
" dname <- paste(deparse(substitute(formula)))\n",
" alternative <- match.arg(alternative)\n",
" if (!inherits(formula, \"formula\")) {\n",
" if (!is.null(w <- weights(formula))) {\n",
" if (!isTRUE(all.equal(as.vector(w), rep(1L, length(w))))) \n",
" stop(\"weighted regressions are not supported\")\n",
" }\n",
" X <- if (is.matrix(formula$x)) \n",
" formula$x\n",
" else model.matrix(terms(formula), model.frame(formula))\n",
" y <- if (is.vector(formula$y)) \n",
" formula$y\n",
" else model.response(model.frame(formula))\n",
" }\n",
" else {\n",
" mf <- model.frame(formula, data = data)\n",
" y <- model.response(mf)\n",
" X <- model.matrix(formula, data = data)\n",
" }\n",
" if (!is.null(order.by)) {\n",
" if (inherits(order.by, \"formula\")) {\n",
" z <- model.matrix(order.by, data = data)\n",
" z <- as.vector(z[, ncol(z)])\n",
" }\n",
" else {\n",
" z <- order.by\n",
" }\n",
" X <- as.matrix(X[order(z), ])\n",
" y <- y[order(z)]\n",
" }\n",
" n <- nrow(X)\n",
" if (is.null(exact)) \n",
" exact <- (n < 100)\n",
" k <- ncol(X)\n",
" res <- lm.fit(X, y)$residuals\n",
" dw <- sum(diff(res)^2)/sum(res^2)\n",
" Q1 <- chol2inv(qr.R(qr(X)))\n",
" if (n < 3) {\n",
" warning(\"not enough observations for computing a p value, set to 1\")\n",
" pval <- 1\n",
" }\n",
" else {\n",
" if (exact) {\n",
" A <- diag(c(1, rep(2, n - 2), 1))\n",
" A[abs(row(A) - col(A)) == 1] <- -1\n",
" MA <- diag(rep(1, n)) - X %*% Q1 %*% t(X)\n",
" MA <- MA %*% A\n",
" ev <- eigen(MA)$values[1:(n - k)]\n",
" if (any(Im(ev) > tol)) \n",
" warning(\"imaginary parts of eigenvalues discarded\")\n",
" ev <- Re(ev)\n",
" ev <- ev[ev > tol]\n",
" pdw <- function(dw) .Fortran(\"pan\", as.double(c(dw, \n",
" ev)), as.integer(length(ev)), as.double(0), as.integer(iterations), \n",
" x = double(1), PACKAGE = \"lmtest\")$x\n",
" pval <- switch(alternative, two.sided = (2 * min(pdw(dw), \n",
" 1 - pdw(dw))), less = (1 - pdw(dw)), greater = pdw(dw))\n",
" if (is.na(pval) || ((pval > 1) | (pval < 0))) {\n",
" warning(\"exact p value cannot be computed (not in [0,1]), approximate p value will be used\")\n",
" exact <- FALSE\n",
" }\n",
" }\n",
" if (!exact) {\n",
" if (n < max(5, k)) {\n",
" warning(\"not enough observations for computing an approximate p value, set to 1\")\n",
" pval <- 1\n",
" }\n",
" else {\n",
" AX <- matrix(as.vector(filter(X, c(-1, 2, -1))), \n",
" ncol = k)\n",
" AX[1, ] <- X[1, ] - X[2, ]\n",
" AX[n, ] <- X[n, ] - X[(n - 1), ]\n",
" XAXQ <- t(X) %*% AX %*% Q1\n",
" P <- 2 * (n - 1) - sum(diag(XAXQ))\n",
" Q <- 2 * (3 * n - 4) - 2 * sum(diag(crossprod(AX) %*% \n",
" Q1)) + sum(diag(XAXQ %*% XAXQ))\n",
" dmean <- P/(n - k)\n",
" dvar <- 2/((n - k) * (n - k + 2)) * (Q - P * \n",
" dmean)\n",
" pval <- switch(alternative, two.sided = (2 * \n",
" pnorm(abs(dw - dmean), sd = sqrt(dvar), lower.tail = FALSE)), \n",
" less = pnorm(dw, mean = dmean, sd = sqrt(dvar), \n",
" lower.tail = FALSE), greater = pnorm(dw, \n",
" mean = dmean, sd = sqrt(dvar)))\n",
" }\n",
" }\n",
" }\n",
" alternative <- switch(alternative, two.sided = \"true autocorrelation is not 0\", \n",
" less = \"true autocorrelation is less than 0\", greater = \"true autocorrelation is greater than 0\")\n",
" names(dw) <- \"DW\"\n",
" RVAL <- list(statistic = dw, method = \"Durbin-Watson test\", \n",
" alternative = alternative, p.value = pval, data.name = dname)\n",
" class(RVAL) <- \"htest\"\n",
" return(RVAL)\n",
"}\n",
"<bytecode: 0x0000000097558ad8>\n",
"<environment: namespace:lmtest>\n"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# formulaじゃないとだめなのか、、、\n",
"lmtest.dwtest"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "9615e124",
"metadata": {},
"outputs": [],
"source": [
"# 説明変数の行列\n",
"X = hcat(Matrix(jldf)[:,3] .* Matrix(jldf)[:,4], Matrix(jldf)[:,end]);"
]
},
{
"cell_type": "markdown",
"id": "d44fc676",
"metadata": {},
"source": [
"[【docs】HypothesisTests.jl > Time series tests > Durbin-Watson test](https://juliastats.org/HypothesisTests.jl/stable/time_series/#HypothesisTests.DurbinWatsonTest)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "044d1b3e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Durbin-Watson autocorrelation test\n",
"----------------------------------\n",
"Population details:\n",
" parameter of interest: sample autocorrelation parameter\n",
" value under h_0: \"0\"\n",
" point estimate: 0.450282\n",
"\n",
"Test summary:\n",
" outcome with 95% confidence: reject h_0\n",
" two-sided p-value: 0.0127\n",
"\n",
"Details:\n",
" number of observations: 21\n",
" DW statistic: 1.09944\n"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"DurbinWatsonTest(X, residuals(ols), p_compute=:exact)"
]
},
{
"cell_type": "markdown",
"id": "a7c35707",
"metadata": {},
"source": [
"# 参考\n",
"- [基礎コース 計量経済学 森棟 広夫](amazon.co.jp/基礎コース-計量経済学-基礎コース経済学-森棟-公夫/dp/4883840816)\n",
"- [なぜ生産関数はCobb-Douglas型\n",
"になるのか？\n",
"ー実データによる生産関数の直接観測－](http://www2.kanazawa-gu.ac.jp/~ishikawa/20190914_,meiji.pdf)\n",
"- [「生産者行動の理論」のポイント](http://www2.toyo.ac.jp/~kawano/lec-semi/mat/mic3.pdf)\n",
"- [生産関数と潜在GDP](http://www2.toyo.ac.jp/~mihira/keizaitoukei2014/08_production/08_production_slides.pdf)\n",
"\n",
"\n",
"\n",
"- [terasakisatoshi/Pandas.jl_example.ipynb Gist](https://gist.github.com/terasakisatoshi/23b7b69af2893691a992fb7263e185c9)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "784d2db1",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.7.0",
"language": "julia",
"name": "julia-1.7"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.7.0"
},
"toc": {
"base_numbering": 1,
"nav_menu": {},
"number_sections": true,
"sideBar": true,
"skip_h1_title": false,
"title_cell": "Table of Contents",
"title_sidebar": "Contents",
"toc_cell": false,
"toc_position": {},
"toc_section_display": true,
"toc_window_display": true
}
},
"nbformat": 4,
"nbformat_minor": 5
}
Raw
製造業の生産関数.csv
年次 製造業GDP(10億円) 就業者数(万人) 労働時間 資本ストック(10億円)
1994 119908 1403.4 1961.3 306302.677
1995 122431.9 1363.8 1975.6 315492.465
1996 125113.5 1356.3 1975.2 322602.08
1997 126920.5 1334.3 1961.3 332635.868
1998 122518 1279.3 1939.3 341929.404
1999 118297.5 1248 1938.8 344789.539
2000 120213.1 1229 1944.6 349805.344
2001 112283.8 1185.2 1946.7 354668.893
2002 108644.2 1140.4 1961.3 352087.06
2003 109599.2 1123.6 1982.4 350462.978
2004 111812.6 1109.1 1992 353902.75
2005 114082.8 1096.5 1995.8 359879.447
2006 114522.6 1112.1 2003.3 369886.017
2007 117970.6 1117.1 1996.5 383308.096
2008 112289.4 1100.5 1926.5 394291.39
2009 94274.3 1029.9 1877.5 396249.684
2010 104979.5 1020.6 1944 400983.145
2011 97179.4 1024.8 1955.2 407542.074
2012 98426.9 1018.6 1946.2 412367.725
2013 98326.9 1032.1 1955.6 412936.362
2014 101653.8 1047.4 1955.6 416822.02
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