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‎18_ADAS-2.ipynb‎

Lines changed: 28 additions & 12 deletions
Original file line numberDiff line numberDiff line change
@@ -1526,12 +1526,14 @@
15261526
"\n",
15271527
"$e_{pt}$の分散を次のように計算する。\n",
15281528
"\n",
1529+
"$$\n",
15291530
"\\begin{align}\n",
15301531
"\\text{Var}(e_{pt})\n",
15311532
"&=\\text{Var}(ahu_{t}+hv_{t})\\qquad\\cdots\\cdots [1]\\\\\n",
15321533
"&=\\text{Var}(ahu_{t})+\\text{Var}(hv_{t})\\qquad\\cdots [2]\\\\\n",
15331534
"&=(ah)^2\\text{Var}(u_{t})+h^2\\text{Var}(v_{t})\\qquad\\cdots [3]\n",
1534-
"\\end{align} (eq:18-eq-varep)\n",
1535+
"\\end{align}\n",
1536+
"$$ (eq:18-eq-varep)\n",
15351537
"\n",
15361538
"* [1]：$e_{pt}$に式[](eq:18-eq-ept)を代入。\n",
15371539
"* [2]：右辺を展開。\n",
@@ -1544,13 +1546,15 @@
15441546
"source": [
15451547
"最後に式[](eq:18-eq-varep)を式[](eq:18-eq-varp)に代入し，次のように整理する。\n",
15461548
"\n",
1549+
"$$\n",
15471550
"\\begin{align}\n",
15481551
"\\text{Var}(p_{t})\n",
15491552
"&=\\frac{(ah)^2\\text{Var}(u_{t})+h^2\\text{Var}(v_{t})}{1-h^2}\\\\\n",
15501553
"&=\\frac{(ah)^2}{1-h^2}\\text{Var}(u_{t})\n",
15511554
"+\\frac{h^2}{1-h^2}\\text{Var}(v_{t})\\\\\n",
15521555
"&=\\left(\\text{需要ショックに起因する変動}\\right)+\\left(\\text{供給ショックに起因する変動}\\right)\n",
1553-
"\\end{align} (eq:18-eq-varp2)"
1556+
"\\end{align}\n",
1557+
"$$ (eq:18-eq-varp2)"
15541558
]
15551559
},
15561560
{
@@ -1586,16 +1590,18 @@
15861590
"metadata": {},
15871591
"source": [
15881592
"$$\n",
1593+
"\\begin{align*}\n",
15891594
"\\left[\n",
15901595
"\\begin{split}\n",
15911596
"&p_t\\text{の変動に対する}\\\\\n",
15921597
"&u_t\\text{(需要ショック)}\\\\\n",
15931598
"&\\text{の寄与度}\n",
15941599
"\\end{split}\n",
15951600
"\\right]\n",
1596-
"=\\frac{(ah)^2}{1-h^2}\\cdot\\frac{\\text{Var}(u_{t})}{\\text{Var}(p_{t})}\n",
1601+
"&=\\frac{(ah)^2}{1-h^2}\\cdot\\frac{\\text{Var}(u_{t})}{\\text{Var}(p_{t})}\n",
15971602
"=\\frac{a^2\\text{Var}(u_{t})}{a^2\\text{Var}(u_{t})+\\text{Var}(v_{t})}\n",
15981603
"\\tag{u-p}\n",
1604+
"\\end{align*}\n",
15991605
"$$"
16001606
]
16011607
},
@@ -1604,16 +1610,18 @@
16041610
"metadata": {},
16051611
"source": [
16061612
"$$\n",
1613+
"\\begin{align*}\n",
16071614
"\\left[\n",
16081615
"\\begin{split}\n",
16091616
"&p_t\\text{の変動に対する}\\\\\n",
16101617
"&v_t\\text{(供給ショック)}\\\\\n",
16111618
"&\\text{の寄与度}\n",
16121619
"\\end{split}\n",
16131620
"\\right]\n",
1614-
"=\\frac{h^2}{1-h^2}\\cdot\\frac{\\text{Var}(v_{t})}{\\text{Var}(p_{t})}\n",
1621+
"&=\\frac{h^2}{1-h^2}\\cdot\\frac{\\text{Var}(v_{t})}{\\text{Var}(p_{t})}\n",
16151622
"=\\frac{\\text{Var}(v_{t})}{a^2\\text{Var}(u_{t})+\\text{Var}(v_{t})}\n",
16161623
"\\tag{v-p}\n",
1624+
"\\end{align*}\n",
16171625
"$$"
16181626
]
16191627
},
@@ -1627,9 +1635,7 @@
16271635
},
16281636
{
16291637
"cell_type": "markdown",
1630-
"metadata": {
1631-
"jp-MarkdownHeadingCollapsed": true
1632-
},
1638+
"metadata": {},
16331639
"source": [
16341640
"### $y_t$に対する影響：分散分解分析"
16351641
]
@@ -1648,14 +1654,16 @@
16481654
"\n",
16491655
"$y_{t}$の分散$\\text{Var}(y_{t})$を次のように書き換えよう。\n",
16501656
"\n",
1657+
"$$\n",
16511658
"\\begin{align}\n",
16521659
"\\text{Var}(y_{t})\n",
16531660
"&=\\text{Var}(-chp_{t-1} + e_{yt})\\qquad\\cdots\\cdots [1]\\\\\n",
16541661
"&=\\text{Var}(-chp_{t-1})+\\text{Var}(e_{yt})+2\\text{Cov}(-chp_{t-1},e_{yt})\\qquad\\cdots [2]\\\\\n",
16551662
"&=\\text{Var}(-chp_{t-1})+\\text{Var}(e_{yt})\\qquad\\cdots [3]\\\\\n",
16561663
"&=(-ch)^2\\text{Var}(p_{t-1})+\\text{Var}(e_{yt})\\qquad\\cdots [4]\\\\\n",
16571664
"&=(ch)^2\\text{Var}(p_{t})+\\text{Var}(e_{yt})\\qquad\\cdots [5]\\\\\n",
1658-
"\\end{align} (eq:18-eq-vary)"
1665+
"\\end{align}\n",
1666+
"$$ (eq:18-eq-vary)"
16591667
]
16601668
},
16611669
{
@@ -1690,13 +1698,15 @@
16901698
"\n",
16911699
"$e_{yt}$の分散を次のように計算する。\n",
16921700
"\n",
1701+
"$$\n",
16931702
"\\begin{align}\n",
16941703
"\\text{Var}(e_{yt})\n",
16951704
"&=\\text{Var}(hu_t-chv_t)\\qquad\\cdots\\cdots [1]\\\\\n",
16961705
"&=\\text{Var}(hu_t)+\\text{Var}(-chv_t)\\qquad\\cdots [2]\\\\\n",
16971706
"&=h^2\\text{Var}(u_{t})+(-ch)^2\\text{Var}(v_{t})\\qquad\\cdots [3]\\\\\n",
16981707
"&=h^2\\text{Var}(u_{t})+(ch)^2\\text{Var}(v_{t})\\qquad\\cdots [4]\n",
1699-
"\\end{align} (eq:18-eq-varyp)"
1708+
"\\end{align}\n",
1709+
"$$ (eq:18-eq-varyp)"
17001710
]
17011711
},
17021712
{
@@ -1715,12 +1725,14 @@
17151725
"source": [
17161726
"最後に，式[](eq:18-eq-varyp)と式[](eq:18-eq-varp2)を式[](eq:18-eq-vary)に代入すると次式を導出できる（詳細は[付録Ｂ](sec:18-appendix_B)を参照）。\n",
17171727
"\n",
1728+
"$$\n",
17181729
"\\begin{align}\n",
17191730
"\\text{Var}(y_{t})\n",
17201731
"&=\\frac{h^2}{1-h^2}2ach\\text{Var}(u_{t})\n",
17211732
"+\\frac{h^2}{1-h^2}c^2\\text{Var}(v_{t})\\\\\n",
17221733
"&=\\left(\\text{需要ショックに起因する変動}\\right)+\\left(\\text{供給ショックに起因する変動}\\right)\n",
1723-
"\\end{align} (eq:18-eq-vary2)"
1734+
"\\end{align}\n",
1735+
"$$ (eq:18-eq-vary2)"
17241736
]
17251737
},
17261738
{
@@ -1752,16 +1764,18 @@
17521764
"この結果を使い次のように定義する。\n",
17531765
"\n",
17541766
"$$\n",
1767+
"\\begin{align*}\n",
17551768
"\\left[\n",
17561769
"\\begin{split}\n",
17571770
"&y_t\\text{の変動に対する}\\\\\n",
17581771
"&u_t\\text{(需要ショック)}\\\\\n",
17591772
"&\\text{の寄与度}\n",
17601773
"\\end{split}\n",
17611774
"\\right]\n",
1762-
"=\\frac{h^2}{1-h^2}2ach\\frac{\\text{Var}(u_{t})}{\\text{Var}(y_{t})}\n",
1775+
"&=\\frac{h^2}{1-h^2}2ach\\frac{\\text{Var}(u_{t})}{\\text{Var}(y_{t})}\n",
17631776
"=\\frac{2a\\text{Var}(u_{t})}{2a\\text{Var}(u_{t})+c(1+ac)\\text{Var}(v_{t})}\n",
17641777
"\\tag{u-y}\n",
1778+
"\\end{align*}\n",
17651779
"$$"
17661780
]
17671781
},
@@ -1770,16 +1784,18 @@
17701784
"metadata": {},
17711785
"source": [
17721786
"$$\n",
1787+
"\\begin{align*}\n",
17731788
"\\left[\n",
17741789
"\\begin{split}\n",
17751790
"&p_t\\text{の変動に対する}\\\\\n",
17761791
"&v_t\\text{(供給ショック)}\\\\\n",
17771792
"&\\text{の寄与度}\n",
17781793
"\\end{split}\n",
17791794
"\\right]\n",
1780-
"=\\frac{h^2}{1-h^2}c^2\\frac{\\text{Var}(v_{t})}{\\text{Var}(y_{t})}\n",
1795+
"&=\\frac{h^2}{1-h^2}c^2\\frac{\\text{Var}(v_{t})}{\\text{Var}(y_{t})}\n",
17811796
"=\\frac{c(1+ac)\\text{Var}(v_{t})}{2a\\text{Var}(u_{t})+c(1+ac)\\text{Var}(v_{t})}\\\\\n",
17821797
"\\tag{v-y}\n",
1798+
"\\end{align*}\n",
17831799
"$$"
17841800
]
17851801
},

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