The HPPANEL Procedure

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 14.1 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.