First-order perturbation
=========================

The state vector
-----------------

.. math ::

	z_{t}\equiv\left[
	\begin{array}
	[c]{ccc}%
	x_{t-1}^{\prime} & \sigma & \varepsilon_{t}^{\prime}%
	\end{array}
	\right]  ^{\prime}

where :math:`\sigma` is the perturbation parameter.

In some applications, we can assume agents have information on shocks :math:`k\geq 0` periods ahead into the future. In that case,
the state variables under anticipated shocks (see e.g. :cite:`Maih2010` and :cite:`JuillardMaih2010`) is

.. math ::

	z_{t}\equiv\left[
	\begin{array}
	[c]{cccccc}%
	x_{t-1}^{\prime} & \sigma & \varepsilon_{t}^{\prime} & \color{red}{\varepsilon
	_{t+1}^{\prime}} & \color{red}{\cdots} & \color{red}{\varepsilon_{t+k}^{\prime}}%
	\end{array}
	\right]  ^{\prime}

Solvers for the quadratic matrix polynomial
----------------------------------------------

The constant-parameter case (and the Maih-Waggoner perturbation)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The algorithms implemented in RISE for solving linear rational expectations models are

- *rise_1* : A Klein-like algorithm (Default).

- *sims* : This is the Gensys algorithm by :cite:`Sims2002`

- *Klein* : An algorithm by :cite:`Klein2000`

- *AIM* : See e.g. :cite:`Anderson2008`

- *cyclic_reduction* : the cyclic reduction algorithm 

.. note:: If there is a unique and stable solution in the Maih-Waggoner perturbation, these algorithms will also work in the
	regime-switching case.


The regime-switching case
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~	

The algorithms implemented in RISE for solving regime-switching linear rational expectations models are

- *mnk* : Newton algorithm by :cite:`Maih2015`, which explicitly forms and store kronecker products.

- *mn* :  Newton algorithm by :cite:`Maih2015`, which avoids the creation of kronecker products and is thereby more efficient
  for solving large systems.

- *fwz* :  Newton algorithm by :cite:`FarmerEtAl2011`

- *dsge_udc* :	Undetermined coefficients method by :cite:`MaihWaggoner2018`. This algorithm
  can only be applied with the Maih-Waggoner perturbation.

- *dsge_schur* : Schur-based method by :cite:`MaihWaggoner2018`. This algorithm returns the most stable solution. Therefore it gives
  the same answer as *rise_1* (and other constant-parameter solvers) when there is a unique stable solution. This algorithm
  can only be applied with the Maih-Waggoner perturbation.

- *dsge_groebner* : Groebner-basis based solver from Computational algebraic geometry by :cite:`MaihWaggoner2018`.
  When it works, this algorithm finds all possible solutions to the problem.

- *mfi* : Functional iteration algorithm by :cite:`Maih2015`

.. note:: All these algorithms for solving regime-switching linear rational expectations can solve constant-parameter models
	even when there is no unique and stable solution.

User-defined solver
~~~~~~~~~~~~~~~~~~~~~~~~~

A user-defined solver should be a function with inputs and outputs as follows ::

	[Tz_pb,Tsig,Te,retcode]=user_solver(Gplus01,A0,Aminus,B,Q,stst_,vargs{:});

where the inputs are :

	- *Gplus01* is an :math:`nd\times nd\times h\times h` array of coefficient matrices on forward-looking terms
	  multiplied by the transition probabilities
	  
	- *A0* is an :math:`nd\times nd\times h` array of coefficient matrices on contemporaneous terms
	
	- *Aminus* is an :math:`nd\times nd\times h` array of coefficient matrices on backward-looking terms
	
	- *B* is an :math:`nd\times nx\times h` array of coefficient matrices on shock terms
	
	- *Q* is an :math:`h\times h` transition matrix with Qij the probability of going from i to j
	
	- *stst_* is an :math:`nd\times h` matrix whose columns represent the steady state in each regime
	
	- *vargs* represents potential additional arguments entering the user-defined function.

And the outputs are:

	- *Tz_pb* is an :math:`nd\times nd\times h` array of the impact of autoregressive terms
	
	- *Tsig* is an :math:`nd\times nd` matrix whose columns represent the impact of the perturbation parameter in each regime
	
	- *Te* is an :math:`nd\times nx\times h` array representing the impact of shock terms
	
	- *retcode* is a flag which takes on value 0 if there is no problem or a number different from 0 if there is a problem
	  finding the solution.