I have the following panel-data model:
$$ y_{it} = \alpha_i + \lambda_t + \beta_i X_{it} + \varepsilon_{it}. $$
It contains individual-specific intercept $\alpha_i$, time-specific intercept $\lambda_t$ and individual-specific slope $\beta_i$ (a vector). $X_{it}$ are exogenous variables.
If I got the panel data terminology right, it would be a fairly standard two-way fixed effects model, if not for the individual-specific slopes.
Questions:
- Does this model have a name? If so, how is it called?
- Where can I read more about this model and its estimation?
- What is a good estimator for the above model if
- $y_{i,\cdot}$ is integrated of order 1 (I(1)),
- $X_{i,\cdot}$ are I(1),
- $y_{i,\cdot}$ are cointegrated across individuals (i.e. across $i$), but
- there is no cointegration between $y_{i,\cdot}$ and $X_{i,\cdot}$?
- Is the model implemented in R? If not, is it implemented in some other software?
I have found something like this model in Stata's panel data manual, function xtxdpd
(see bottom of page 15); but I did not like that source too much.
Edit:
The model does not look good if $y_{i,\cdot}$ is not cointegrated with $X_{i,\cdot}$, because then the regressors diverge from the regressand. So a model in first differences would make more sense.
Best Answer
Here is one way of estimating $\lambda_t$ and $\beta_i$.
Take the original equation (but consider only one $x_{it}$ in place of a vector $X_{it}$, that will help save some space and typesetting later on)
$$ y_{it} = \alpha_i + \lambda_t + \beta_i x_{it} + \varepsilon_{it} $$
and difference it with respect to time to obtain
$$ \Delta y_{it} = \Delta \lambda_t + \beta_i \Delta x_{it} + \Delta \varepsilon_{it}. $$
If $y_{i,\cdot}$ and $x_{i,\cdot}$ are integrated but not cointegrated, we get a relatively nice representation in terms of their stationary transformations.
Construct a set of dummies corresponding to $\Delta \lambda_t$ and stack the equations to get
$$ \begin{pmatrix} \Delta y_{11} \\ \Delta y_{12} \\ \vdots \\ \Delta y_{1T} \\ \Delta y_{21} \\ \Delta y_{22} \\ \vdots \\ \Delta y_{2T} \\ \vdots \\ \Delta y_{m1} \\ \Delta y_{m2} \\ \vdots \\ \Delta y_{mT} \end{pmatrix} = \begin{pmatrix} 1 & 0 & \dotsb & 0 & \Delta x_{11} & 0 & \dotsb & 0 \\ 0 & 1 & \dotsb & 0 & \Delta x_{12} & 0 & \dotsb & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dotsb & 1 & \Delta x_{1T} & 0 & \dotsb & 0 \\ 1 & 0 & \dotsb & 0 & 0 & \Delta x_{21} & \dotsb & 0 \\ 0 & 1 & \dotsb & 0 & 0 & \Delta x_{22} & \dotsb & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dotsb & 1 & 0 & \Delta x_{2T} & \dotsb & 0 \\ \vdots \\ 1 & 0 & \dotsb & 0 & 0 & 0 & \dotsb & \Delta x_{m1} \\ 0 & 1 & \dotsb & 0 & 0 & 0 & \dotsb & \Delta x_{m2} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dotsb & 1 & 0 & 0 & \dotsb & \Delta x_{mT} \\ \end{pmatrix} \times \begin{pmatrix} \Delta\lambda_1 \\ \Delta\lambda_2 \\ \vdots \\ \Delta\lambda_T \\ \beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \\ \end{pmatrix} + \begin{pmatrix} \Delta\varepsilon_{11} \\ \Delta\varepsilon_{12} \\ \vdots \\ \Delta\varepsilon_{1T} \\ \Delta\varepsilon_{21} \\ \Delta\varepsilon_{22} \\ \vdots \\ \Delta\varepsilon_{2T} \\ \vdots \\ \Delta\varepsilon_{m1} \\ \Delta\varepsilon_{m2} \\ \vdots \\ \Delta\varepsilon_{mT} \end{pmatrix} $$
(for notational simplicity, I assumed the original observations at time $t=0$ are available).
This is a shape of ordinary linear regression, and the estimator of the coefficient vector is straightforward. I have not given much thought on how good such estimator is, though.