面板数据变系数模型

前言：在这一篇文章中，我们将某些影响因素的作用范围扩大，这些因素不仅影响截距项的变动，而且也能影响到斜率项。因素的作用范围就可能有一下几种组合，单独影响截距，单独影响斜率，既影响截距又影响斜率，既不影响截距也不影响斜率（随机效应）。因素又区分为两类，时间因素与个体特质因素。推荐先阅读数据分析-面板数据变截距模型 再阅读本文。

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因素
时间因素
个体特质因素
随机因素
影响截距
影响斜率
影响截距与斜率
不影响截距,斜率
影响截距
影响斜率
影响截距与斜率
不影响截距,斜率
影响截距
影响斜率
影响截距与斜率
不影响截距,斜率

为了方便理解，我们将包含个体特质与时间因素的面板回归方程拆写为：
Yit=α0+αi+λ0+λt+Xit′βi+Xit′βt+Xit′βc+εitY_{it}=\alpha_0 +\alpha_i + \lambda_0 +\lambda_t + X_{it}' \beta_i+ X_{it}' \beta_t+ X_{it}' \beta_c + \varepsilon_{it}Yit​=α0​+αi​+λ0​+λt​+Xit′​βi​+Xit′​βt​+Xit′​βc​+εit​
β=βi+βt+βc\beta= \beta_i+ \beta_t + \beta_cβ=βi​+βt​+βc​
,i=1,2,3,...,N;t=1,2,3,...,T,i = 1,2,3,...,N;t=1,2,3,...,T,i=1,2,3,...,N;t=1,2,3,...,T
当然这里的βt与βi\beta_t与\beta_iβt​与βi​也可以像拆分 α和λ\alpha和\lambdaα和λ一样，拆分出均值和差异项

项目 含义
iii 个体标志序数
ttt 时间序数
XitX_{it}Xit​ 观测变量，K∗1K*1K∗1向量，(X1it,,X2it,..,Xkit)′(X_{1it,},X_{2it},..,X_{kit})'(X1it,​,X2it​,..,Xkit​)′
βi\beta_iβi​ 随个体特质而变动的参数，K∗1K*1K∗1向量, (0,0,...,βi,..0)′(0,0,...,\beta_i,..0)'(0,0,...,βi​,..0)′
βt\beta_tβt​ 随时间而变动的参数，K∗1K*1K∗1向量, (0,0,...,βt,..0)′(0,0,...,\beta_t,..0)'(0,0,...,βt​,..0)′
βc\beta_cβc​ 不变动的参数，K∗1K*1K∗1向量, (β1,β2,..0...,βk)′(\beta_{1},\beta_{2},..0...,\beta_{k})'(β1​,β2​,..0...,βk​)′
β\betaβ 总参数向量，K∗1K*1K∗1向量, (β1,β2,...,βi,...,βt,...,βk)′(\beta_{1},\beta_{2},...,\beta_i,...,\beta_t,...,\beta_{k})'(β1​,β2​,...,βi​,...,βt​,...,βk​)′
α0\alpha_0α0​ 个体效应在个体维度上的平均值
αi\alpha_iαi​ 个体效应在个体维度上差异
α0+αi\alpha_0+\alpha_iα0​+αi​ 个体效应引起的截距项
λ0\lambda_0λ0​ 时间效应在时间维度上的平均值
λt\lambda_tλt​ 时间效应在时间维度上差异
λ0+λt\lambda_0 +\lambda_tλ0​+λt​ 时间效应引起的截距项
εit\varepsilon_{it}εit​ 随机扰动项

固定系数模型

模型

以截距项为个体固定效应，系数为个体固定效应：
Yit=α0+αi+Xit′βi+Xit′βc+εitY_{it}=\alpha_0 +\alpha_i +X_{it}' \beta_i + X_{it}' \beta_c + \varepsilon_{it}Yit​=α0​+αi​+Xit′​βi​+Xit′​βc​+εit​
以截距项为个体固定效应，系数为时间固定效应：
Yit=α0+αi+Xit′βt+Xit′βc+εitY_{it}=\alpha_0 +\alpha_i +X_{it}' \beta_t + X_{it}' \beta_c + \varepsilon_{it}Yit​=α0​+αi​+Xit′​βt​+Xit′​βc​+εit​

  • 以截距项为个体固定效应，系数为个体固定效应，仅考虑第3个参数随个体变化，举例理解：
    Yit=α0+αi+β1x1it+β2x2it+β3ix3it+εitY_{it}=\alpha_0 +\alpha_i + \beta_1 x_{1it}+\beta_2x_{2it}+ \beta_{3i}x_{3it} + \varepsilon_{it}Yit​=α0​+αi​+β1​x1it​+β2​x2it​+β3i​x3it​+εit​
    其中x1it表示第i个个体在t时刻的第1个变量值，β1表示第1个变量前的参数x_{1it} 表示第i个个体在t时刻的第1个变量值， \beta_1表示第1个变量前的参数x1it​表示第i个个体在t时刻的第1个变量值，β1​表示第1个变量前的参数
    其中x2it表示第i个个体在t时刻的第2个变量值，β2表示第2个变量前的参数x_{2it} 表示第i个个体在t时刻的第2个变量值， \beta_2表示第2个变量前的参数x2it​表示第i个个体在t时刻的第2个变量值，β2​表示第2个变量前的参数
    其中x3it表示第i个个体在t时刻的第3个变量值，β3i表示依赖于第i个个体特质(第i个个体特质是个体分类的类别，表示个体差异影响x3的斜率)、第3个变量前的参数x_{3it} 表示第i个个体在t时刻的第3个变量值， \beta_{3i}表示依赖于第i个个体特质(第i个个体特质是个体分类的类别，表示个体差异影响x_3的斜率)、第3个变量前的参数x3it​表示第i个个体在t时刻的第3个变量值，β3i​表示依赖于第i个个体特质(第i个个体特质是个体分类的类别，表示个体差异影响x3​的斜率)、第3个变量前的参数

  • 以截距项为个体固定效应，系数为时间固定效应，仅考虑第3个参数随时间变化，举例理解：
    Yit=α0+αi+β1x1it+β2x2it+β3tx3it+εitY_{it}=\alpha_0 +\alpha_i + \beta_1 x_{1it}+\beta_2x_{2it}+ \beta_{3t}x_{3it} + \varepsilon_{it}Yit​=α0​+αi​+β1​x1it​+β2​x2it​+β3t​x3it​+εit​
    其中x1it表示第i个个体在t时刻的第1个变量值，β1表示第1个变量前的参数x_{1it} 表示第i个个体在t时刻的第1个变量值， \beta_1表示第1个变量前的参数x1it​表示第i个个体在t时刻的第1个变量值，β1​表示第1个变量前的参数
    其中x2it表示第i个个体在t时刻的第2个变量值，β2表示第2个变量前的参数x_{2it} 表示第i个个体在t时刻的第2个变量值， \beta_2表示第2个变量前的参数x2it​表示第i个个体在t时刻的第2个变量值，β2​表示第2个变量前的参数
    其中x3it表示第i个个体在t时刻的第3个变量值，β3t表示依赖于第t个时段特质(第t个时段是依据时间段分类的类别，表示时间段变动影响x3的斜率)、第3个变量前的参数x_{3it} 表示第i个个体在t时刻的第3个变量值， \beta_{3t}表示依赖于第t个时段特质(第t个时段是依据时间段分类的类别，表示时间段变动影响x_3的斜率)、第3个变量前的参数x3it​表示第i个个体在t时刻的第3个变量值，β3t​表示依赖于第t个时段特质(第t个时段是依据时间段分类的类别，表示时间段变动影响x3​的斜率)、第3个变量前的参数

估计方法

  • 最小二乘虚拟变量法（LSDV）
    引入虚拟变量进行回归
    举例，以截距项为个体固定效应，系数为个体固定效应：
    考虑β2与β3\beta_2 与 \beta_3β2​与β3​受到性别的影响
    Yit=α0+αi+β1x1it+β2ix2it+β3ix3it+εitY_{it}=\alpha_0 +\alpha_i + \beta_1 x_{1it}+\beta_{2i}x_{2it}+ \beta_{3i}x_{3it} + \varepsilon_{it}Yit​=α0​+αi​+β1​x1it​+β2i​x2it​+β3i​x3it​+εit​
    =α0+αi+β1x1it+(β2x2it+γ1x2it∗D1+γ2x2it∗D2)+(β3x3it+η1x3it∗D1+η2x3it∗D2)+εit=\alpha_0 +\alpha_i +\beta_1 x_{1it}+(\beta_{2}x_{2it}+ \gamma_1 x_{2it}*D_1+ \gamma_2 x_{2it}*D_2)+( \beta_{3}x_{3it} + \eta_1 x_{3it}*D_1+\eta_2 x_{3it}*D_2)+ \varepsilon_{it}=α0​+αi​+β1​x1it​+(β2​x2it​+γ1​x2it​∗D1​+γ2​x2it​∗D2​)+(β3​x3it​+η1​x3it​∗D1​+η2​x3it​∗D2​)+εit​
    =α0+αi+β1x1it+(γ3x2it∗D3+γ1x2it∗D1+γ2x2it∗D2)+(η3x3it∗D3+η1x3it∗D1+η2x3it∗D2)+εit=\alpha_0 +\alpha_i +\beta_1 x_{1it}+( \gamma_{3}x_{2it}*D_3+ \gamma_1 x_{2it}*D_1+ \gamma_2 x_{2it}*D_2)+ (\eta_{3}x_{3it}*D_3 + \eta_1 x_{3it}*D_1+\eta_2 x_{3it}*D_2)+ \varepsilon_{it}=α0​+αi​+β1​x1it​+(γ3​x2it​∗D3​+γ1​x2it​∗D1​+γ2​x2it​∗D2​)+(η3​x3it​∗D3​+η1​x3it​∗D1​+η2​x3it​∗D2​)+εit​
    设置虚拟变量：
    D1={1if 第i个个体性别为男性0if 第i个个体性别为其他D_1=\begin{cases} 1 &\text{if } 第i个个体性别为男性 \\ 0 &\text{if } 第i个个体性别为其他 \end{cases}D1​={10​if 第i个个体性别为男性if 第i个个体性别为其他​
    D2={1if 第i个个体性别为女性0if 第i个个体性别为其他D_2=\begin{cases} 1 &\text{if } 第i个个体性别为女性 \\ 0 &\text{if } 第i个个体性别为其他 \end{cases}D2​={10​if 第i个个体性别为女性if 第i个个体性别为其他​
    D3={1if 第i个个体性别为中性0if 第i个个体性别为其他D_3=\begin{cases} 1 &\text{if } 第i个个体性别为中性 \\ 0 &\text{if } 第i个个体性别为其他 \end{cases}D3​={10​if 第i个个体性别为中性if 第i个个体性别为其他​
    注意：这里引入m-1个虚拟变量与m个虚拟变量的两种方式等价。

随机系数模型

这个模型是有局限性的：模型多多少少会忽略一些解释变量，因此会导致截距项与解释变量相关。所以说模型设置为个体固定效应的模型很正常。随机变系数效应模型的截距项也应该是随机的，截距项如果不是随机的最好不要用随机变系数效应模型。
模型举例：
Swamy随机模型：
Yi=Xiβi~+εi,i=1,2,...,NY_i=X_i\tilde{\beta_i}+\varepsilon_i,i=1,2,...,NYi​=Xi​βi​~​+εi​,i=1,2,...,N
βi~=β0+βi\tilde{\beta_i}=\beta_0+\beta_iβi​~​=β0​+βi​
E(βi)=0k∗1,E(\beta_i)=0_{k *1},E(βi​)=0k∗1​,

E(βiβj′)={Δii=j0i≠jE(\beta_i\beta_j')=\begin{cases} \Delta_i &\text{ }i=j \\ 0 &\text{ } i \neq j \end{cases}E(βi​βj′​)={Δi​0​ i=j i​=j​;

E(Xit′βi)=0E(X_{it}'\beta_i)=0E(Xit′​βi​)=0;

E(εiεj′)={σii=j0i≠jE(\varepsilon_i\varepsilon_j')=\begin{cases} \sigma_i &\text{ }i=j \\ 0 &\text{ } i \neq j \end{cases}E(εi​εj′​)={σi​0​ i=j i​=j​;

模型设定检验

由于我们不知道模型中哪些变量的系数是变动的，所以需要依据检验是否某个变量的系数是变动的

  • 数据量很大，可以考虑全部变量系数变化
  • 依次从全部变量系数不同，m-1个系数不同，m-2个系数不同，…,1个系数不同逐个检验（此方法用于变量个数很多或者虚拟变量个数很多的情形）

LR检验

Yit=α0+αi+β1x1it+(β2x2it+γ1x2it∗D1+γ2x2it∗D2)+(β3x3it+η1x3it∗D1+η2x3it∗D2)+εitY_{it}=\alpha_0 +\alpha_i +\beta_1 x_{1it}+(\beta_{2}x_{2it}+ \gamma_1 x_{2it}*D_1+ \gamma_2 x_{2it}*D_2)+( \beta_{3}x_{3it} + \eta_1 x_{3it}*D_1+\eta_2 x_{3it}*D_2)+ \varepsilon_{it}Yit​=α0​+αi​+β1​x1it​+(β2​x2it​+γ1​x2it​∗D1​+γ2​x2it​∗D2​)+(β3​x3it​+η1​x3it​∗D1​+η2​x3it​∗D2​)+εit​
原假设：γ1=γ2=η1=η2=0\gamma_1=\gamma_2=\eta_1=\eta_2=0γ1​=γ2​=η1​=η2​=0；（变量的系数不变动）
备择假设：γ1,γ2,η1,η2\gamma_1,\gamma_2,\eta_1,\eta_2γ1​,γ2​,η1​,η2​不全为0；（变系数模型）

LR检验的无约束回归方程（备择假设成立）：
Yit=α0+αi+β1x1it+(β2x2it+γ1x2it∗D1+γ2x2it∗D2)+(β3x3it+η1x3it∗D1+η2x3it∗D2)+εitY_{it}=\alpha_0 +\alpha_i +\beta_1 x_{1it}+(\beta_{2}x_{2it}+ \gamma_1 x_{2it}*D_1+ \gamma_2 x_{2it}*D_2)+( \beta_{3}x_{3it} + \eta_1 x_{3it}*D_1+\eta_2 x_{3it}*D_2)+ \varepsilon_{it}Yit​=α0​+αi​+β1​x1it​+(β2​x2it​+γ1​x2it​∗D1​+γ2​x2it​∗D2​)+(β3​x3it​+η1​x3it​∗D1​+η2​x3it​∗D2​)+εit​
计算lnLulnL_ulnLu​
LR检验的约束回归方程（原假设成立）：
Yit=α0+αi+β1x1it+β2x2it+β3x3it+εitY_{it}=\alpha_0 +\alpha_i + \beta_1 x_{1it}+\beta_{2}x_{2it}+ \beta_{3}x_{3it} + \varepsilon_{it}Yit​=α0​+αi​+β1​x1it​+β2​x2it​+β3​x3it​+εit​
计算lnLrlnL_rlnLr​

Swamy检验

Yi=Xiβi~+εi,i=1,2,...,NY_i=X_i\tilde{\beta_i}+\varepsilon_i,i=1,2,...,NYi​=Xi​βi​~​+εi​,i=1,2,...,N
βi~=β0+βi\tilde{\beta_i}=\beta_0+\beta_iβi​~​=β0​+βi​
E(βi)=0k∗1,E(\beta_i)=0_{k *1},E(βi​)=0k∗1​,
原假设：β0=β1=β2=β3=...=βN\beta_0=\beta_1=\beta_2=\beta_3=...=\beta_Nβ0​=β1​=β2​=β3​=...=βN​ （不变系数）
备择假设：β0,β1,β2,β3,...,βN\beta_0,\beta_1,\beta_2,\beta_3,...,\beta_Nβ0​,β1​,β2​,β3​,...,βN​不全相等（变系数）

  • 同方差var(εi)=σε2var(\varepsilon_i)=\sigma_\varepsilon^2var(εi​)=σε2​
    服从F分布
  • 异方差var(εi)=σi2var(\varepsilon_i)=\sigma_i^2var(εi​)=σi2​
    检验统计量为 Sw=∑i=1N(β^i−β^0∗)′Xi′Xi(β^i−β^0∗)σ^i2→dχ2((N−1)k)(给定N;T→∞时)Sw=\displaystyle\sum_{i=1}^N\frac{(\hat\beta_i-\hat\beta_0^*)'X_i'X_i(\hat\beta_i-\hat\beta_0^*)}{\hat\sigma_i^2}\xrightarrow[]{d}\chi^2((N-1)k)(给定N;T\xrightarrow{} \infty时 )Sw=i=1∑N​σ^i2​(β^​i​−β^​0∗​)′Xi′​Xi​(β^​i​−β^​0∗​)​d​χ2((N−1)k)(给定N;T​∞时)
    β^0∗=(∑i=1Nσ^i2Xi′Xi)−1(∑i=1Nσ^i2Xi′Yi)\hat\beta_0^*=(\displaystyle\sum_{i=1}^N\hat\sigma_i^2X_i'X_i)^{-1}(\displaystyle\sum_{i=1}^N\hat\sigma_i^2X_i'Y_i)β^​0∗​=(i=1∑N​σ^i2​Xi′​Xi​)−1(i=1∑N​σ^i2​Xi′​Yi​)

模型检验步骤

固定效应

LR逐次检验：

  1. 原假设：混合回归模型（截距与斜率都不变）
    备择假设：截距项与斜率项（k个变量）发生变化
    此时：不拒绝原假设，建立混合回归模型，检验结束；拒绝原假设，截距项与斜率项之中至少有一项在变化，因此进入下一步检验。

  2. 引入截距项的约束函数，验证是否成立
    原假设：变量的斜率变化 (约束条件成立)
    备择假设：截距项、变量的斜率变化（约束条件不成立）
    此时：不拒绝原假设，认为截距项不变。接下来要检验哪些变量的斜率发生变化；拒绝原假设，认为截距项变化，接下来需要检验截距项随时点变化、个体变化、个体时点变化，以及哪些变量的斜率发生变化。

  3. 在上一步原假设的基础上在引入任意k-1个关于变量系数的约束条件，有1个变量系数自由另外的k-1个约束条件的，认为这1个变量系数为模型唯一变动的变量系数，否则认为至少有2个变量系数变动。
    原假设：个体FX变截距，考察其中一个变量变化，另外k-1个变量不发生变化。
    备择假设：个体FX变截距，至少有两个变量系数变化。
    此时：不拒绝原假设，我们认为个体FE变截距，且只有一个变量斜率发生变动。检验结束。
    拒绝原假设，认为截距项发生变动，并且k-1个变量的斜率中至少有一个会变。继续检验。

  4. 减少1个约束条件个数，重复第三步检验。

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原假设
备择假设
原假设
原假设
备择假设
备择假设
原假设
备择假设
模型检验,约束全部系数与截距项
混合回归模型 : 截距与斜率都不变
结束检验,建立混合回归模型
截距项和斜率项的k个变量斜率之中至少有一个发生变化,约束截距项
截距不变,变量的斜率变化
引入k-1个约束,检测那个斜率变化
约束条件成立,检测出,检验结束
认为有两个斜率变动,因此i减少约束条件个数
截距项与变量的斜率变化
截距项变化,一个变量斜率变化,其余k-1个变量斜率不变化
检验结束,建立变截距,1个变量系数变化的模型
截距项变化,所有的变量系数都发生变化,下一步检验减少约束条件个数

随机效应

原假设：混合模型
备择假设：截距项、所有变量（k个变量）的斜率都是随机效应。
此时：若不拒绝原假设，表明建立混合（pool）模型，检验到此结束。
若拒绝原假设，建立随机系数模型。
注意：随机系数模型的截距项也应该是随机。

建模步骤

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数据非平稳
数据平稳
数据平稳
不拒绝原假设
拒绝原假设
不拒绝原假设
拒绝原假设
不拒绝原假设,意味着截距项不变动
拒绝原假设,意味着截距项变动
不拒绝原假设
拒绝原假设
输入数据
描述性统计分析
面板单位根检验
面板协整分析
F检验 or LR检验
变系数检验
固定系数检验
系数不变
系数变动
随机系数检验
系数不变
系数变动
使用混合回归
豪斯曼检验
选择个体随机效应模型
选择个体固定效应模型

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