One-Way Random-Effects Model :: SAS/ETS(R) 13.1 User's Guide: High-Performance Procedures

One-Way Random-Effects Model

The specification for the one-way random-effects model is

$u_{it}={\nu }_{i}+{\epsilon }_{it}$

Let $\mb {Z} _{0}=\mr {diag}(\mb {J} _{T_{i}}$ ), ${\mb {P} _{0}=\mr {diag}({\bar{\mb {J}}}_{T_{i}})}$ , and $\mb {Q} _{0}=\mr {diag}(\mb {E} _{T_{i}})$ , with ${\bar{\mb {J}}}_{T_{i}}=\mb {J} _{T_{i}}/\mi {T} _{i}$ and ${\mb {E} _{T_{i}}=\mb {I} _{T_{i}}-{\bar{\mb {J}}}_{T_{i}} }$ . Define ${\tilde{\mb {X} }_{s}=\mb {Q} _{0}\mb {X} _{s} }$ . Also define ${\tilde{\mb {y} }=\mb {Q} _{0}\mb {y} }$ and $\mb {J}$ as a vector of 1s whose length is ${T_{i}}$ .

In the one-way model, estimation proceeds in a two-step fashion. First, you obtain estimates of the variance of the ${ {\sigma }^{2}_{{\epsilon } } }$ and ${{\sigma }^2_{{\nu }} }$ . There are multiple ways to derive these estimates; PROC HPPANEL provides four options. For more information, see the section One-Way Random-Effects Model in SAS/ETS User's Guide.

After the variance components are calculated from any method, the next task is to estimate the regression model of interest. For each individual, you form a weight ( $\theta _\mi {i}$ ),

$\theta _\mi {i} = 1 - \sigma _{\epsilon } / w_\mi {i}$

$w_{i}^{2} = T_{i}{\sigma }^{2}_{{\nu }} + {\sigma }^{2}_{{\epsilon }}$

where $T_{i}$ is the $\emph{i}$ th cross section’s time observations.

Taking the $\theta _\mi {i}$ , you form the partial deviations,

$\tilde{y}_\mi {it} = y_\mi {it}- \theta _\mi {i} \bar{y}_\mi {i \cdot }$

$\tilde{x}_\mi {it} = x_\mi {it}- \theta _\mi {i} \bar{x}_\mi {i \cdot }$

where $\bar{y}_\mi {i \cdot }$ and $\bar{x}_\mi {i \cdot }$ are cross section means of the dependent variable and independent variables (including the constant if any), respectively.

The random-effects $\beta$ is then the result of simple OLS on the transformed data.

The HPPANEL Procedure (Experimental)

One-Way Random-Effects Model