\[ y_{it} = a_{it} + \beta_{it}^Tx_{it} + u_{it} \\ y_{it} = \alpha + \beta^T x_{it} + \mu_i + \varepsilon_{it} \\ y_{it} = \alpha + \beta^T x_{it} + \mu_i + \varepsilon_{it} \] \[ \Delta y_{it} = \beta^T \Delta x_{it} + \Delta u_{it} \\ \Delta y_{it} = y_{it} - y_{i,t-1} = x_{it} - x_{i,t-1} \\ \Delta u_{it}=u_{it} - u_{i,t-1} = \Delta \varepsilon_{it} \\ \] \[ \hat{\beta} = (X^T V^{-1}X)^{-1} (X^TV^{-1}y) \] \[ y_{it} - \theta \bar{y_i}=(X_{it}-\theta \bar{X_i})\beta + (u_{it}-\theta \bar{u_i}) \\ \theta = 1-[\sigma_u^2/(\sigma_u^2+T\sigma_e^2)]^{1/2}, \quad v_{it} - \theta \bar{v_i} \] $$ \ , \ (R-r)T[RR^T]{-1}(R) \ = (X^T X)^{-1} _{i=1}^{n} X_i^R E_i X_i (XTX){-1}
\[ \] \ var({}) = {k=1}^{m} var() + (1+) {k=1}{m}(-{})2 \[ \] {x}=Mx ; P= I_n jj’ ;Q=I_{nT}-P ;D=I_n d \ d = \[\begin{pmatrix} 1 & -1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & -1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & -1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\] \[ \] (T-1,T) \[ \] = ({i=1}{n}(+(X_iT X_i){-1}))(+(X_iTX_i){-1}){-1} \[ \] = ({i=1}^{n} - {i=1}^{n} ) ( - {i=1}^{n} )^T - {i=1}^{n} (X_iTX_i){-1} \[ \] y{it} = y_{it-1} + ^Tx_{it} + i + {it} \ y_{it} = y_{it-1} + ^Tx_{it} + _{it} \[ \] W_i = \[\begin{pmatrix} y_1 & 0 & 0 & 0 & 0 & 0 & \cdots &0 &0 &0 &0 &x_{i3} \\ 0 &y_1 &y_2 &0 &0 &0 &\cdots &0 &0 &0 &0 &x_{i4} \\ 0 &0 &0 &y_1 &y_2 &y_3 &\cdots &0 &0 &0 &0 &x_{i5} \\ \vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\ 0 &0 &0 &0 &\cdots &\cdots &\cdots &y_1 &y_2 &\cdots &y_{t-2} &x_{iT-2} \end{pmatrix}\] \[ \] ({i=1}{n}e_i()TW_i) A ({i=1}^{n} W_i^Te_i ) \ {i=1}^{n} W_i^Te_i\ A^{(1)} = ({i=1}{n}W_iT H^{(1) W_i})^{-1} \[ \] H^{(1)} = d^Td = \[\begin{pmatrix} 2 &-1 & 0 &\cdots &0 \\ -1 &2 &-1 &\cdots &0 \\ 0 &-1 &2 &\cdots &0 \\ \vdots &\vdots &\vdots &\vdots &\vdots \\ 0 &0 &0 &-1 &2 \end{pmatrix}\] \[ \] H_i^{(2)} = {i=1}^{n} e_i^{(1)} e_i^{(1)T} \ y{it} = y_{it-1} + i + {it} \ y_{it-1} - y_{it-3} \ e_i^+ = (e_i,e_i) \ Z_i^+ = \[\begin{pmatrix} Z_i & 0 &0 &\cdots &0 \\ 0 &\Delta y_{i2} &0 &\cdots &0 \\ 0 &0 &\Delta y_{i3} &\cdots &0 \\ 0 &0 &0 &\cdots &\Delta y_{iT-1} \end{pmatrix}\] \[ \] (_{i=1}{n}Z_i{+T} { \[\begin{pmatrix} \bar{e_i}(\beta) \\ e_i{(\beta)} \end{pmatrix}\]} )^T = ({i=1}^{n}y{i1}{e_{i3}},{i=1}^{n}y{i1}{e_{i4}},{i=1}^{n}y{i2}{e_{i4}} ,{i=1}^{n}y{i1}{e_{iT}},\,{i=1}^{n}y{i2}{e_{iT}},,{i=1}{n}y_{iT-2}{e_{iT}},{i=1}^{n}{t=3}{T}x{it}{e_{it}},{i=1}^{n}e{i3}y_{i2},\,,{i=1}^{n}e{iT}y_{iT-1} ) \[ \] \ = _{i=1}^{n}
\[ \] \ n^{1/2}{i=1}{n}{t=1}^{T-1}{s=t+1}{T} \ W = \[ \] \ cor(,)= \ =++{it} \ = \[ \] e_{it} = u_{it}-u_{i,t-1};cor(e_{it},e_{i,t-1}) = -0.5 \ = + {i,t} \ u{it} = u_{i,t-1} + e_{it} \[ \] = \ LM = {i-1}^{n-1}{j=i+1}^{n}T_{ij} \ ^2_{n(n-1)/2} \ T ; n \[ \] SCLM = ({i=1}^{n-1}{j=i+1}^{n}T_{ij}-1 ) \[ \] BCSCLM = ({i=1}^{n-1}{j=i+1}^{n}T_{ij}-1 )- \[ \] CD = ({i=1}{n-1}{j=i+1}^{n} ) \[ \] CD=({i=1}{n-1}{j=i+1}^{n}[(p)]{ij} ) \[ \] y{it}=y_{it-1}+{L=1}^{p_i}i {y{it-L}}+{mi}d{mt}+{it} \ y{it} = y_{it-1}+{L=1}{p_i}iy{y_{it-L}}+{mi}d{mt}+{it} \ = \ \[ \] ={t=2}Ty{it}^2+2_{L=1}{{k}}{{k}L}\[ \] s_i = \ {S}= \ t{}{*} = }{_{mT{~}}{*}} \[ \] {}^2 \[ \] {t} = {i=1}^{n} t{i} \ z = \[ \] _i = \[\begin{pmatrix} \sigma_{i1}^2 &\cdots &\cdots &0 \\ 0 &\sigma_{i2}^2 &\cdots &\vdots \\ \vdots &0 &\ddots &0 \\ 0 &\cdots &\cdots &\sigma_{iT}^2 \end{pmatrix}\] \[ \] _i = \[\begin{pmatrix} \sigma_{it}^2 &\sigma_{i1,i2} &\cdots &\cdots &\sigma_{i1,iT} \\ \sigma_{i2,i1} &\sigma_{i2}^2 &\cdots &\cdots &\vdots \\ \vdots &\vdots &\ddots &\vdots &\vdots \\ \vdots &\vdots &\vdots &\sigma_{iT-1}^2 &\sigma_{iT-1,iT} \\ \sigma_{iT,i1} &\cdots &\cdots &\sigma_{iT,iT-1} &\sigma_{iT}^2 \end{pmatrix}\]\[ \] y_{it}=1x{1ij}++p x{pij} \ b_1 z_{1ij}++b_p z_{pij}+{ij}\ b{ik} N(0,k^2),Cov(b_k,b{k’})={kk’} \ {ij} N(0,^2 {ijj}),Cov({ij},{ij’})=^2 {ijj’} \ b_{iq}=0 i,q,{ijj} = ^2 for j=j’ \ z{1ij} = 1 i,j,z_{qij}=0 i,j,q {ij}=1 for i=j,0 \ u{ij} = 1b_{i1}+{ij} \[ \] z_q=x_q q,\ {kk’}=^2_{}I_N {ijj}=1,{ijj’}=0 for j j’ $$