固定効果モデル#
パネルデータ#
異なる時点で同じ個体についての情報を観測したデータをパネルデータ （panel data）という。
被説明変数\(Y\)と説明変数\(X = (X_1, \dots, X_K)\)に時点\(t\)と個体\(i\)の添字をつけて
\[
\{ (Y_{it}, X_{it}) \},
\hspace{2em} i = 1, \dots, N,
\hspace{1em} t = 1, \dots, T
\]
のような形で表記される
import statsmodels.api as sm
import statsmodels.formula.api as smf
data = sm.datasets.get_rdataset("Grunfeld", package="plm").data
data
# Grunfeldは1935～1954年にかけてのアメリカの10の企業のbalanced panelデータ
# firm: 企業ID
# inv: 投資総額
# value: 企業価値
# capital: 資本ストック
| firm | year | inv | value | capital | |
|---|---|---|---|---|---|
| 0 | 1 | 1935 | 317.60 | 3078.50 | 2.80 |
| 1 | 1 | 1936 | 391.80 | 4661.70 | 52.60 |
| 2 | 1 | 1937 | 410.60 | 5387.10 | 156.90 |
| 3 | 1 | 1938 | 257.70 | 2792.20 | 209.20 |
| 4 | 1 | 1939 | 330.80 | 4313.20 | 203.40 |
| ... | ... | ... | ... | ... | ... |
| 195 | 10 | 1950 | 3.42 | 69.05 | 8.74 |
| 196 | 10 | 1951 | 4.67 | 83.04 | 9.07 |
| 197 | 10 | 1952 | 6.00 | 74.42 | 9.93 |
| 198 | 10 | 1953 | 6.53 | 63.51 | 11.68 |
| 199 | 10 | 1954 | 5.12 | 58.12 | 14.33 |
200 rows × 5 columns
Pooled OLS#
パネルデータ分析において、個体や時間による固定効果（fixed effect）を特に考慮しないで（固定効果が無いと仮定して）通常の重回帰モデルを用いたモデルをPooled OLSと呼ぶ。
\[
Y_{it} = \boldsymbol{\beta} X_{it} + \varepsilon_{it}
\]
pooled_ols = sm.OLS.from_formula(formula="inv ~ value + capital", data=data).fit()
pooled_ols.summary()
| Dep. Variable: | inv | R-squared: | 0.812 |
|---|---|---|---|
| Model: | OLS | Adj. R-squared: | 0.811 |
| Method: | Least Squares | F-statistic: | 426.6 |
| Date: | Wed, 08 May 2024 | Prob (F-statistic): | 2.58e-72 |
| Time: | 23:28:52 | Log-Likelihood: | -1191.8 |
| No. Observations: | 200 | AIC: | 2390. |
| Df Residuals: | 197 | BIC: | 2399. |
| Df Model: | 2 | ||
| Covariance Type: | nonrobust |
| coef | std err | t | P>|t| | [0.025 | 0.975] | |
|---|---|---|---|---|---|---|
| Intercept | -42.7144 | 9.512 | -4.491 | 0.000 | -61.472 | -23.957 |
| value | 0.1156 | 0.006 | 19.803 | 0.000 | 0.104 | 0.127 |
| capital | 0.2307 | 0.025 | 9.055 | 0.000 | 0.180 | 0.281 |
| Omnibus: | 27.240 | Durbin-Watson: | 0.358 |
|---|---|---|---|
| Prob(Omnibus): | 0.000 | Jarque-Bera (JB): | 89.572 |
| Skew: | 0.469 | Prob(JB): | 3.55e-20 |
| Kurtosis: | 6.141 | Cond. No. | 2.46e+03 |
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 2.46e+03. This might indicate that there are
strong multicollinearity or other numerical problems.
固定効果モデル#
パネルデータを分析する際に個体差による効果（個体固定効果）や時点ごとの固有の効果（時間固定効果）の影響を除くように作ったモデルのことを固定効果モデル（fixed effect model）という。
\[
Y_{it} = \boldsymbol{\beta} X_{it} +\theta_i + \pi_t + \varepsilon_{it}
\]
個体内の変動を使うため、固定効果推定量は「群内（within）推定量」と呼ばれたりもする。
one-way fixed effect model#
時点や個体など、1つの固定効果に対処するモデル。一元配置固定効果モデル（one-way fixed effect model）などと呼ばれる。その推定量は「群内（within）」推定量や「固定効果（fixed effect）」推定量などとも呼ばれる。
個体固定効果モデル#
以下のような個体固定効果モデルを考える。（単純化のため説明変数\(X\)は1つのみとする）
\[
Y_{it} = \beta_0 + \beta_1 X_{it} +\theta_i + \varepsilon_{it}
\hspace{2em} (i = 1, \dots, N, \ t=1,\dots, T)
\]
パネルデータを用いることができる場合、以下の3つの方法によって個体固定効果（entity fixed effects）\(\theta_i\)を除去することができる。
(1) “一回の階差モデル（first difference model）”によるOLS推定#
\[
(Y_{i,t+1} - Y_{it}) = (\beta_0 - \beta_0) + \beta_1 (X_{i,t+1}-X_{it}) + (\varepsilon_{i,t+1} - \varepsilon_{it})
\]
記号を置き換えて、
\[
\Delta Y_{it} = \beta_1 \Delta X_{it}+ \Delta \varepsilon_{it}
\]
推定方法：
説明変数、被説明変数それぞれ\(t+1\)期から\(t\)期を引く
上の式をOLS推定する
import pandas as pd
data = sm.datasets.get_rdataset("Grunfeld", package="plm").data
deltas = []
data = data.sort_values(["firm", "year"])
for firm in data["firm"].unique():
d = data.query(f"firm == {firm}").copy()
delta = d - d.shift(1)
delta["year"] = d["year"]
delta["firm"] = firm
deltas.append(delta)
delta = pd.concat(deltas).dropna().sort_values("firm").reset_index(drop=True)
delta
| firm | year | inv | value | capital | |
|---|---|---|---|---|---|
| 0 | 1 | 1936 | 74.20 | 1583.20 | 49.80 |
| 1 | 1 | 1954 | 182.30 | -648.10 | 449.00 |
| 2 | 1 | 1953 | 413.20 | 1316.80 | 346.80 |
| 3 | 1 | 1952 | 135.30 | 91.90 | 222.80 |
| 4 | 1 | 1951 | 113.00 | 1077.40 | 108.70 |
| ... | ... | ... | ... | ... | ... |
| 185 | 10 | 1937 | 0.19 | -5.74 | -0.14 |
| 186 | 10 | 1936 | -0.54 | 17.03 | 0.21 |
| 187 | 10 | 1953 | 0.53 | -10.91 | 1.75 |
| 188 | 10 | 1944 | 0.25 | -0.42 | -0.17 |
| 189 | 10 | 1954 | -1.41 | -5.39 | 2.65 |
190 rows × 5 columns
first_diff = sm.OLS.from_formula(formula="inv ~ -1 + value + capital", data=delta).fit()
first_diff.summary()
| Dep. Variable: | inv | R-squared (uncentered): | 0.429 |
|---|---|---|---|
| Model: | OLS | Adj. R-squared (uncentered): | 0.423 |
| Method: | Least Squares | F-statistic: | 70.58 |
| Date: | Wed, 08 May 2024 | Prob (F-statistic): | 1.36e-23 |
| Time: | 23:28:52 | Log-Likelihood: | -982.76 |
| No. Observations: | 190 | AIC: | 1970. |
| Df Residuals: | 188 | BIC: | 1976. |
| Df Model: | 2 | ||
| Covariance Type: | nonrobust |
| coef | std err | t | P>|t| | [0.025 | 0.975] | |
|---|---|---|---|---|---|---|
| value | 0.0891 | 0.008 | 10.816 | 0.000 | 0.073 | 0.105 |
| capital | 0.2787 | 0.047 | 5.910 | 0.000 | 0.186 | 0.372 |
| Omnibus: | 41.981 | Durbin-Watson: | 1.782 |
|---|---|---|---|
| Prob(Omnibus): | 0.000 | Jarque-Bera (JB): | 403.556 |
| Skew: | 0.398 | Prob(JB): | 2.34e-88 |
| Kurtosis: | 10.095 | Cond. No. | 5.75 |
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
(2) “\(N-1\)個のダミー説明変数”を用いたOLS推定#
最小二乗ダミー変数推定（Least Squares Dummy Variables (LSDV) 推定）とも呼ばれる。
\[\begin{split}
Y_{it} = \beta_0 + \beta_1 X_{it} + \gamma_2 D^{(2)}_i + \cdots + \gamma_N D^{(N)}_i + \varepsilon_{it} \\
\text{where } D^{(2)}_i =
\begin{cases}
1 & \text{for } i = 2\\
0 & \text{otherwise}
\end{cases} \text{, etc.}
\end{split}\]
推定方法：
個体ダミー変数（個体\(i\)に該当する場合に1、それ以外は0となるダミー変数）\(D^{(2)}_i, \cdots, D^{(N)}_i\)を作成する
上の式をOLS推定する
data = sm.datasets.get_rdataset("Grunfeld", package="plm").data
lsdv = sm.OLS.from_formula(
formula="inv ~ value + capital + firm",
data=data.assign(firm = data["firm"].astype("category")) # category型にすれば自動でダミー変数にしてくれる
).fit()
lsdv.summary()
| Dep. Variable: | inv | R-squared: | 0.944 |
|---|---|---|---|
| Model: | OLS | Adj. R-squared: | 0.941 |
| Method: | Least Squares | F-statistic: | 288.5 |
| Date: | Wed, 08 May 2024 | Prob (F-statistic): | 2.41e-111 |
| Time: | 23:28:52 | Log-Likelihood: | -1070.8 |
| No. Observations: | 200 | AIC: | 2166. |
| Df Residuals: | 188 | BIC: | 2205. |
| Df Model: | 11 | ||
| Covariance Type: | nonrobust |
| coef | std err | t | P>|t| | [0.025 | 0.975] | |
|---|---|---|---|---|---|---|
| Intercept | -70.2967 | 49.708 | -1.414 | 0.159 | -168.354 | 27.760 |
| firm[T.2] | 172.2025 | 31.161 | 5.526 | 0.000 | 110.732 | 233.673 |
| firm[T.3] | -165.2751 | 31.776 | -5.201 | 0.000 | -227.958 | -102.593 |
| firm[T.4] | 42.4874 | 43.910 | 0.968 | 0.334 | -44.132 | 129.107 |
| firm[T.5] | -44.3201 | 50.492 | -0.878 | 0.381 | -143.924 | 55.284 |
| firm[T.6] | 47.1354 | 46.811 | 1.007 | 0.315 | -45.206 | 139.477 |
| firm[T.7] | 3.7432 | 50.565 | 0.074 | 0.941 | -96.004 | 103.491 |
| firm[T.8] | 12.7511 | 44.053 | 0.289 | 0.773 | -74.150 | 99.652 |
| firm[T.9] | -16.9256 | 48.453 | -0.349 | 0.727 | -112.508 | 78.656 |
| firm[T.10] | 63.7289 | 50.330 | 1.266 | 0.207 | -35.556 | 163.013 |
| value | 0.1101 | 0.012 | 9.288 | 0.000 | 0.087 | 0.134 |
| capital | 0.3101 | 0.017 | 17.867 | 0.000 | 0.276 | 0.344 |
| Omnibus: | 29.604 | Durbin-Watson: | 1.079 |
|---|---|---|---|
| Prob(Omnibus): | 0.000 | Jarque-Bera (JB): | 162.947 |
| Skew: | 0.283 | Prob(JB): | 4.13e-36 |
| Kurtosis: | 7.386 | Cond. No. | 6.42e+04 |
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 6.42e+04. This might indicate that there are
strong multicollinearity or other numerical problems.
(3) ”平均差分法（Entity-demeaned）”を用いたOLS推定#
\[\begin{split}
\begin{align}
\tilde{Y}_{it} &= \beta_1 \tilde{X}_{it} + \tilde{\varepsilon}_{it}, \\
\text{where }
\tilde{Y}_{it} &= Y_{it} - \bar{Y}_i, \hspace{1em} \bar{Y}_i = \frac{1}{T} \sum^T_{t=1} Y_{it}\\
\tilde{X}_{it} &= X_{it} - \bar{X}_i, \hspace{1em} \bar{X}_i = \frac{1}{T} \sum^T_{t=1} X_{it}\\
\tilde{\varepsilon}_{it} &= \varepsilon_{it}- \bar{\varepsilon}_i, \hspace{1em} \bar{\varepsilon}_i = \frac{1}{T} \sum_{t=1}^T \varepsilon_{it}
\end{align}
\end{split}\]
推定方法：
説明変数・被説明変数について、変数から期間平均を引く
上の式をOLS推定する
\(N-1\)個の個体ダミー説明変数による推定と同じ推定値が得られる
統計ソフトでは通常は平均差分法による推定が行われる
data = sm.datasets.get_rdataset("Grunfeld", package="plm").data
group = "firm"
rows = []
for _, d in data.groupby(group):
for col in ["value", "inv", "capital"]:
d[col] = (d[col] - d[col].mean())
rows.append(d)
df = pd.concat(rows)
entity_demeaned = sm.OLS.from_formula(formula="inv ~ -1 + value + capital", data=df).fit()
entity_demeaned.summary()
| Dep. Variable: | inv | R-squared (uncentered): | 0.767 |
|---|---|---|---|
| Model: | OLS | Adj. R-squared (uncentered): | 0.764 |
| Method: | Least Squares | F-statistic: | 325.5 |
| Date: | Wed, 08 May 2024 | Prob (F-statistic): | 2.59e-63 |
| Time: | 23:28:53 | Log-Likelihood: | -1070.8 |
| No. Observations: | 200 | AIC: | 2146. |
| Df Residuals: | 198 | BIC: | 2152. |
| Df Model: | 2 | ||
| Covariance Type: | nonrobust |
| coef | std err | t | P>|t| | [0.025 | 0.975] | |
|---|---|---|---|---|---|---|
| value | 0.1101 | 0.012 | 9.532 | 0.000 | 0.087 | 0.133 |
| capital | 0.3101 | 0.017 | 18.336 | 0.000 | 0.277 | 0.343 |
| Omnibus: | 29.604 | Durbin-Watson: | 1.079 |
|---|---|---|---|
| Prob(Omnibus): | 0.000 | Jarque-Bera (JB): | 162.947 |
| Skew: | 0.283 | Prob(JB): | 4.13e-36 |
| Kurtosis: | 7.386 | Cond. No. | 1.74 |
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
md = smf.mixedlm("inv ~ value + capital", data, groups=data["year"])
mdf = md.fit()
print(mdf.summary())
/usr/local/lib/python3.10/site-packages/statsmodels/base/model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
warnings.warn("Maximum Likelihood optimization failed to "
/usr/local/lib/python3.10/site-packages/statsmodels/regression/mixed_linear_model.py:2200: ConvergenceWarning: Retrying MixedLM optimization with lbfgs
warnings.warn(
/usr/local/lib/python3.10/site-packages/statsmodels/base/model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
warnings.warn("Maximum Likelihood optimization failed to "
/usr/local/lib/python3.10/site-packages/statsmodels/regression/mixed_linear_model.py:2200: ConvergenceWarning: Retrying MixedLM optimization with cg
warnings.warn(
Mixed Linear Model Regression Results
========================================================
Model: MixedLM Dependent Variable: inv
No. Observations: 200 Method: REML
No. Groups: 20 Scale: 8912.8075
Min. group size: 10 Log-Likelihood: -1196.1133
Max. group size: 10 Converged: No
Mean group size: 10.0
--------------------------------------------------------
Coef. Std.Err. z P>|z| [0.025 0.975]
--------------------------------------------------------
Intercept -42.714 9.501 -4.496 0.000 -61.335 -24.092
value 0.116 0.006 19.854 0.000 0.104 0.127
capital 0.231 0.025 9.136 0.000 0.181 0.280
Group Var 0.585
========================================================
/usr/local/lib/python3.10/site-packages/statsmodels/base/model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
warnings.warn("Maximum Likelihood optimization failed to "
/usr/local/lib/python3.10/site-packages/statsmodels/regression/mixed_linear_model.py:2206: ConvergenceWarning: MixedLM optimization failed, trying a different optimizer may help.
warnings.warn(msg, ConvergenceWarning)
/usr/local/lib/python3.10/site-packages/statsmodels/regression/mixed_linear_model.py:2218: ConvergenceWarning: Gradient optimization failed, |grad| = 1.131047
warnings.warn(msg, ConvergenceWarning)
/usr/local/lib/python3.10/site-packages/statsmodels/regression/mixed_linear_model.py:2261: ConvergenceWarning: The Hessian matrix at the estimated parameter values is not positive definite.
warnings.warn(msg, ConvergenceWarning)
導出#
固定効果のある
\[
Y_{it} = \beta_0 + \beta_1 X_{it} +\theta_i + \varepsilon_{it}
\]
というモデルを考えるとき、その推定量は
\[
(\hat{\beta}_0, \hat{\beta}_1, \hat{\theta}_1, \hat{\theta}_2, \dots, \hat{\theta}_N)
\newcommand{\argmin}{\mathop{\rm arg~min}\limits}
= \argmin_{\beta_0, \beta_1, \theta_1, \dots, \theta_N}
\sum^N_{i=1} \sum^T_{t=1} (Y_{it} - \beta_0 - \beta_1 X_{it} - \theta_i)^2
\]
より、この最小化問題の一階条件は
\[
\sum^N_{i=1} \sum^T_{t=1} X_{it} (Y_{it} - \hat{\beta}_0 - \hat{\beta}_1 X_{it} - \hat{\theta}_i) = 0
\]
となり、各ユニット\(i\)について
\[\begin{split}
\sum^T_{t=1} (Y_{it} - \hat{\beta}_0 - \hat{\beta}_1 X_{it} - \hat{\theta}_i) = 0\\
\to \sum^T_{t=1} (Y_{it} - \hat{\beta}_0 - \hat{\beta}_1 X_{it}) = T \hat{\theta}_i\\
\begin{align}
\to \hat{\theta}_i &= \frac{1}{T} \sum^T_{t=1} (Y_{it} - \hat{\beta}_0 - \hat{\beta}_1 X_{it})\\
&= \frac{1}{T} \sum^T_{t=1} Y_{it} - \hat{\beta}_0 - \hat{\beta}_1 \frac{1}{T} \sum^T_{t=1} X_{it}\\
&= \bar{Y}_{i} - \hat{\beta}_0 - \hat{\beta}_1 \bar{X}_{i},
\hspace{1em} \left(\bar{Y}_{i} := \frac{1}{T} \sum^T_{t=1} Y_{it}, \bar{X}_{i} := \frac{1}{T} \sum^T_{t=1} X_{it}\right)\\
\end{align}
\end{split}\]
となる。これを一階条件に代入すると
\[\begin{split}
\begin{align}
\sum^N_{i=1} \sum^T_{t=1} X_{it}
(Y_{it} - \hat{\beta}_0 - \hat{\beta}_1 X_{it} - \bar{Y}_{i} + \hat{\beta}_0 + \hat{\beta}_1 \bar{X}_{i}) &= 0
\\
\sum^N_{i=1} \sum^T_{t=1} X_{it}
(Y_{it} - \bar{Y}_{i} - \hat{\beta}_1 X_{it} + \hat{\beta}_1 \bar{X}_{i}) &= 0
\\
\sum^N_{i=1} \sum^T_{t=1} X_{it} (Y_{it} - \bar{Y}_{i} - \hat{\beta}_1 (X_{it} - \bar{X}_{i})) &= 0
\\
\sum^N_{i=1} \sum^T_{t=1} X_{it} (Y_{it} - \bar{Y}_{i})
- \sum^N_{i=1} \sum^T_{t=1} X_{it} \hat{\beta}_1 (X_{it} - \bar{X}_{i})
&= 0
\end{align}
\end{split}\]
\[
\sum^N_{i=1} \sum^T_{t=1} X_{it} \hat{\beta}_1 (X_{it} - \bar{X}_{i})
=
\sum^N_{i=1} \sum^T_{t=1} X_{it} (Y_{it} - \bar{Y}_{i})
\]
こう移行できるか？
\[\begin{split}
\frac{1}{NT} \sum^N_{i=1} \sum^T_{t=1} X_{it} \hat{\beta}_1 (X_{it} - \bar{X}_{i})
= \frac{1}{NT} \sum^N_{i=1} \sum^T_{t=1} X_{it} (Y_{it} - \bar{Y}_{i})
\\
\hat{\beta}_1 \frac{1}{NT} \sum^N_{i=1} \sum^T_{t=1} X_{it} (X_{it} - \bar{X}_{i})
= \frac{1}{NT} \sum^N_{i=1} \sum^T_{t=1} X_{it} (Y_{it} - \bar{Y}_{i})
\end{split}\]
\[
\hat{\beta}_1 =
\frac{
\sum^N_{i=1} \sum^T_{t=1} X_{it} (Y_{it} - \bar{Y}_{i})
}{
\sum^N_{i=1} \sum^T_{t=1} X_{it} (X_{it} - \bar{X}_{i})
}
\]
TODO: このへん整理
y = data.value
x = data.inv
sum(x * (y - y.mean())) / sum(x * (x - x.mean()))
5.193824955195333
two-way effect model#
時点効果＋個体効果 といった2つの効果を同時に固定する
\[
Y_{it} =\beta_1 X_{it} +\theta_i + \pi_t + \varepsilon_{it}
\]
個体の固定効果\(\theta_i\)と時間の固定効果\(\pi_t\)の両方を除去したい場合は、それぞれの推定方法の組み合わせになる。
\(N-1\)個の個体ダミー変数と\(T-1\)個の時間ダミー変数を用いたOLS推定
entity demeaningと\(T-1\)個の時間ダミー変数を用いたOLS推定
time demeaningと\(N-1\)個の個体ダミー変数を用いたOLS推定
entity & time demeaningを用いたOLS推定
説明変数と被説明変数について、個体と時間両方の平均を引いてOLS推定
なお、パネルデータを活用した計量経済分析では、時間固定効果がないと仮定できるケースはまれであるため、通常はone-way固定効果モデルではなくtwo-way固定効果モデルを用いる。
data = sm.datasets.get_rdataset("Grunfeld", package="plm").data
rows = []
for group in ["year", "firm"]:
for _, d in data.groupby(group):
for col in ["value", "inv", "capital"]:
d[col] = (d[col] - d[col].mean())
rows.append(d)
df = pd.concat(rows)
two_way_entity_demeaned = sm.OLS.from_formula(formula="inv ~ -1 + value + capital", data=df).fit()
two_way_entity_demeaned.summary()
| Dep. Variable: | inv | R-squared (uncentered): | 0.898 |
|---|---|---|---|
| Model: | OLS | Adj. R-squared (uncentered): | 0.891 |
| Method: | Least Squares | F-statistic: | 123.8 |
| Date: | Wed, 08 May 2024 | Prob (F-statistic): | 1.24e-14 |
| Time: | 23:28:53 | Log-Likelihood: | -173.11 |
| No. Observations: | 30 | AIC: | 350.2 |
| Df Residuals: | 28 | BIC: | 353.0 |
| Df Model: | 2 | ||
| Covariance Type: | nonrobust |
| coef | std err | t | P>|t| | [0.025 | 0.975] | |
|---|---|---|---|---|---|---|
| value | 0.1345 | 0.035 | 3.818 | 0.001 | 0.062 | 0.207 |
| capital | 0.3315 | 0.097 | 3.431 | 0.002 | 0.134 | 0.529 |
| Omnibus: | 39.780 | Durbin-Watson: | 2.574 |
|---|---|---|---|
| Prob(Omnibus): | 0.000 | Jarque-Bera (JB): | 167.640 |
| Skew: | -2.559 | Prob(JB): | 3.96e-37 |
| Kurtosis: | 13.389 | Cond. No. | 6.79 |
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
比較#
from stargazer.stargazer import Stargazer
sg = Stargazer([pooled_ols, first_diff, lsdv, entity_demeaned])
sg.custom_columns(labels=["pooled", "first_diff", "lsdv", "entity_demeaned"], separators=[1] * 4)
sg
| Dependent variable: inv | ||||
| pooled | first_diff | lsdv | entity_demeaned | |
| (1) | (2) | (3) | (4) | |
| Intercept | -42.714*** | -70.297 | ||
| (9.512) | (49.708) | |||
| capital | 0.231*** | 0.279*** | 0.310*** | 0.310*** |
| (0.025) | (0.047) | (0.017) | (0.017) | |
| firm[T.10] | 63.729 | |||
| (50.330) | ||||
| firm[T.2] | 172.203*** | |||
| (31.161) | ||||
| firm[T.3] | -165.275*** | |||
| (31.776) | ||||
| firm[T.4] | 42.487 | |||
| (43.910) | ||||
| firm[T.5] | -44.320 | |||
| (50.492) | ||||
| firm[T.6] | 47.135 | |||
| (46.811) | ||||
| firm[T.7] | 3.743 | |||
| (50.565) | ||||
| firm[T.8] | 12.751 | |||
| (44.053) | ||||
| firm[T.9] | -16.926 | |||
| (48.453) | ||||
| value | 0.116*** | 0.089*** | 0.110*** | 0.110*** |
| (0.006) | (0.008) | (0.012) | (0.012) | |
| Observations | 200 | 190 | 200 | 200 |
| R2 | 0.812 | 0.429 | 0.944 | 0.767 |
| Adjusted R2 | 0.811 | 0.423 | 0.941 | 0.764 |
| Residual Std. Error | 94.408 (df=197) | 42.896 (df=188) | 52.768 (df=188) | 51.418 (df=198) |
| F Statistic | 426.576*** (df=2; 197) | 70.578*** (df=2; 188) | 288.500*** (df=11; 188) | 325.451*** (df=2; 198) |
| Note: | *p<0.1; **p<0.05; ***p<0.01 | |||
変量効果モデル#
変量効果という概念もある
個体に固有の効果が説明変数と相関する場合、その効果を**固定効果（fixed effect）**と呼ぶ。
個体に固有の効果が説明変数と相関しない場合、その効果を**変量効果（random effect）**と呼ぶ。
Pooled OLSでも推定できる#
\(\theta_i\)を個体に固有の効果として
\[
Y_{it} = \boldsymbol{\beta} X_{it} + \theta_i + \varepsilon_{it}
\]
という線形モデルを考えたとき、変量効果モデルでは\(\theta_i\)は\(X_{it}\)と無相関であるため\(\theta_i + \varepsilon_{it}\)をひとまとめに誤差項\(\epsilon_{it}\)として捉えて
\[
Y_{it} = \boldsymbol{\beta} X_{it} + \epsilon_{it}
\]
のように扱いPooled OLSとして推定すると、通常の最小二乗法の仮定を満たすためPooled OLS推定量は一致性をもつ。
変量効果モデル#
通常、変量効果モデルと呼ぶ場合は\(\theta_i\)と\(X_{it}\)の独立を仮定し、FGLSで推定を行うらしい
参考文献#
mixtape
西山ほか（2019）『計量経済学』有斐閣