预测量

具体形式

第1类

OLS

${\hat{y}}_{i, T + 1} = {\hat{α}}_{O L S} + x_{i, T + 1}^{T} {\hat{β}}_{O L S}$

其中 ${\hat{α}}_{O L S} = {(τ_{N T}^{T} τ_{N T})}^{- 1} τ_{N T}^{T} y$ ， ${\hat{β}}_{O L S} = {(X^{T} X)}^{- 1} X^{T} y$

第2类

$\begin{array}{l} σ_{b}^{2} = 0 \\ AR (1) : ϕ_{1} = 0 \\ AR (2) : ϕ_{1} = ϕ_{2} = 0 \end{array}$

${\hat{y}}_{i, T + 1} = \hat{α} + {\hat{μ}}_{i} + x_{i, T + 1}^{T} \hat{β}$

其中 $\hat{α} = {(τ_{N T}^{T} Ω^{- 1} τ_{N T})}^{- 1} τ_{N T}^{T} Ω^{- 1} y$

$\hat{β} = {(X^{T} Ω^{- 1} X)}^{- 1} X^{T} Ω^{- 1} y$

${\hat{μ}}_{i} = σ_{μ}^{2} (l_{i}^{T} \otimes τ_{T}^{T}) Ω^{- 1} (y - X \hat{β} - τ_{N T} \hat{α})$

$Ω = σ_{μ}^{2} (I_{N} \otimes J_{T})$

第3类

$\begin{array}{l} σ_{b}^{2} = 0 \\ AR (2) : ϕ_{2} = 0 \end{array}$

${\hat{y}}_{i, T + 1} = \hat{α} + {\hat{μ}}_{i} + x_{i, T + 1}^{T} \hat{β} + {\hat{ϕ}}_{1} {\hat{u}}_{i T}$

其中 $\hat{α} = {(τ_{N T}^{T} Ω^{- 1} τ_{N T})}^{- 1} τ_{N T}^{T} Ω^{- 1} y$

$\hat{β} = {(X^{T} Ω^{- 1} X)}^{- 1} X^{T} Ω^{- 1} y$

${\hat{μ}}_{i} = σ_{μ}^{2} (l_{i}^{T} \otimes τ_{T}^{T}) Ω^{- 1} (y - X \hat{β} - τ_{N T} \hat{α})$

$Ω = σ_{μ}^{2} (I_{N} \otimes J_{T}) + σ_{u}^{2} [I_{N} \otimes Γ_{1}]$ ，式中 $Γ_{1}$ 为AR(1)相关系数矩阵

第4类

$σ_{b}^{2} = 0$

${\hat{y}}_{i, T + 1} = \hat{α} + {\hat{μ}}_{i} + x_{i, T + 1}^{T} \hat{β} + {\hat{ϕ}}_{1} {\hat{u}}_{i T} + {\hat{ϕ}}_{2} {\hat{u}}_{i, T - 1}$

其中 $\hat{α} = {(τ_{N T}^{T} Ω^{- 1} τ_{N T})}^{- 1} τ_{N T}^{T} Ω^{- 1} y$

$\hat{β} = {(X^{T} Ω^{- 1} X)}^{- 1} X^{T} Ω^{- 1} y$

${\hat{μ}}_{i} = σ_{μ}^{2} (l_{i}^{T} \otimes τ_{T}^{T}) Ω^{- 1} (y - X \hat{β} - τ_{N T} \hat{α})$

$Ω = σ_{μ}^{2} (I_{N} \otimes J_{T}) + σ_{u}^{2} [I_{N} \otimes Γ_{2}]$ ，式中 $Γ_{2}$ 为AR(2)相关系数矩阵

第5类

$\begin{array}{l} AR (1) : ϕ_{1} = 0 \\ AR (2) : ϕ_{1} = ϕ_{2} = 0 \end{array}$

${\hat{y}}_{i, T + 1} = \hat{α} + {\hat{μ}}_{i} + x_{i, T + 1}^{T} \hat{β} + x_{i, T + 1}^{T} {\hat{b}}_{i}$

其中 $\hat{α} = {(τ_{N T}^{T} Ω^{- 1} τ_{N T})}^{- 1} τ_{N T}^{T} Ω^{- 1} y$

$\hat{β} = {(X^{T} Ω^{- 1} X)}^{- 1} X^{T} Ω^{- 1} y$

${\hat{μ}}_{i} = σ_{μ}^{2} (l_{i}^{T} \otimes τ_{T}^{T}) Ω^{- 1} (y - X \hat{β} - τ_{N T} \hat{α})$

${\hat{b}}_{i} = σ_{b}^{2} (l_{i}^{T} \otimes X_{i}^{T}) Ω^{- 1} (y - X \hat{β} - τ_{N T} \hat{α})$

$Ω = σ_{μ}^{2} (I_{N} \otimes J_{T}) + σ_{b}^{2} Z Z^{T}$

第6类

$AR (2) : ϕ_{2} = 0$

${\hat{y}}_{i, T + 1} = \hat{α} + {\hat{μ}}_{i} + x_{i, T + 1}^{T} \hat{β} + x_{i, T + 1}^{T} {\hat{b}}_{i} + {\hat{ϕ}}_{1} {\hat{u}}_{i T}$

其中 $\hat{α} = {(τ_{N T}^{T} Ω^{- 1} τ_{N T})}^{- 1} τ_{N T}^{T} Ω^{- 1} y$

$\hat{β} = {(X^{T} Ω^{- 1} X)}^{- 1} X^{T} Ω^{- 1} y$

${\hat{μ}}_{i} = σ_{μ}^{2} (l_{i}^{T} \otimes τ_{T}^{T}) Ω^{- 1} (y - X \hat{β} - τ_{N T} \hat{α})$

${\hat{b}}_{i} = σ_{b}^{2} (l_{i}^{T} \otimes X_{i}^{T}) Ω^{- 1} (y - X \hat{β} - τ_{N T} \hat{α})$

$Ω = σ_{μ}^{2} (I_{N} \otimes J_{T}) + σ_{b}^{2} Z Z^{T} + σ_{u}^{2} [I_{N} \otimes Γ_{1}]$ ，式中 $Γ_{1}$ 为AR(1)相关系数矩阵

第7类

本文所提方法

${\hat{y}}_{i, T + 1} = \hat{α} + {\hat{μ}}_{i} + x_{i, T + 1}^{T} \hat{β} + x_{i, T + 1}^{T} {\hat{b}}_{i} + {\hat{ϕ}}_{1} {\hat{u}}_{i T} + {\hat{ϕ}}_{2} {\hat{u}}_{i, T - 1}$

其中 $\hat{α} = {(τ_{N T}^{T} Ω^{- 1} τ_{N T})}^{- 1} τ_{N T}^{T} Ω^{- 1} y$

$\hat{β} = {(X^{T} Ω^{- 1} X)}^{- 1} X^{T} Ω^{- 1} y$

${\hat{μ}}_{i} = σ_{μ}^{2} (l_{i}^{T} \otimes τ_{T}^{T}) Ω^{- 1} (y - X \hat{β} - τ_{N T} \hat{α})$

${\hat{b}}_{i} = σ_{b}^{2} (l_{i}^{T} \otimes X_{i}^{T}) Ω^{- 1} (y - X \hat{β} - τ_{N T} \hat{α})$

$Ω = σ_{μ}^{2} (I_{N} \otimes J_{T}) + σ_{b}^{2} Z Z^{T} + σ_{u}^{2} [I_{N} \otimes Γ_{2}]$ ，式中 $Γ_{2}$ 为AR(2)相关系数矩阵