Panel data model with two-way fixed effects and individual-specific slopes 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.
 A: 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.
