State space of a rotating panels error structure
Error Specification in a Rotating Panel
The error term \(e^{(i)}_{t}\) has a particular structure that is given by the rotating survey design:
\[\begin{eqnarray} e^{(i)}_{t} = \underbrace{ \color{red} b^{(i)}_{t}}_{\color{red} bias} + \underbrace{\varepsilon^{(i)}_{t}}_{autocorrelation} \end{eqnarray}\]Here, we describe the \(\color{red} bias\) component, which follows a random walk process. The sum of the bias terms accross waves is equal to zero:
\(b^{(i)}_{t} = b^{(i)}_{t-1} + w^{(i)}_{t}\),
Most of the times, this component cannot be identified together with additional trend components. Thus, one identification restriction is to assume that the bias terms for all weights is equal to zero (cite example):
\[b^{(1)}_{t} = -\sum_{i=2}^{W} b^{(i)}_{t}\]Alternatively, one could assume that there is no bias in the first wave (cite example).
State vectors:
The block in the state vector corresponding to the bias component takes the following form
\[\alpha_t= \begin{pmatrix} b^{(2)}_t \\ b^{(3)}_t\\ \vdots\\ b^{(W)}_t \end{pmatrix}\]Individally, each term is defined as a local level model
Dynamics
The identity matrix of size \(W\) defines the motion equation of the bias component of each wave:
\[T_t = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 1 \end{pmatrix}\] \[S_t = \begin{pmatrix} \sigma_{1} & 0 & \cdots & 0 \\ 0 & \sigma_{2} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & \sigma_{W} \end{pmatrix}\] \[V_t = S S'\]Diffuse initialization
\[a_0 = \mathrm{0_{W \times 1}}\] \[P_*= \mathrm{0_{W \times W}}\] \[B= \mathrm{I_{W \times W}}\] \[P_\infty= \mathrm{I_{W \times W}}\]Measurement
The measurement equation for each one of the $W$ waves incorporates the restriction that the bias component in the first wave plus the sum of the remaining ones equals zero, i.e. \(b^{(1)}_{t} = -\sum_{i=2}^{W} b^{(i)}_{t}\). The resulting $W \times (W-1)$ matrix takes the following form:
\[Z_t = \begin{pmatrix} -1 & -1 & \cdots & -1 \\ 1 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 1 \end{pmatrix}\]One could define alternative measurement equations. For example, assuming that the bias term for the first wave is equal to zero is possible by fixing to zero the first row of \(Z_t\)
\[H_t = 0\]Implementation
[check this]
ARMA models are implemented in the class demetra.arima.ssf.SsfArima