Model conventions

This page describes various conventions used in dolo.

Model types

There are several ways to represent a DSGE model. Here we list some of them:

state-free approach

This approach is taken in Dynare and in most perturbation softwares. The model is specified by a vectorial function

\(f(y_{t+1},y_t,y_{t-1},\epsilon_t) = 0\)

with the restriction that \(\epsilon_t\) and \(y_{t+1}\) cannot appear in the same equations. For this kind of models, the solver finds a solution \(g\) such that the law of motion of \((y_t)\) is given by: \(y_t = g \left( y_{t-1}, \epsilon_t \right)\)

controlled process variants

We define several categories of models.

fg model

With these versions, the state-space is chosen by the user. A law of motion for the state-space must be specified (depending on the controls and on the shocks). And optimality conditions must be given to pin down all the controls. The model is specified by giving \(g\) and \(f\) such that:

\(s_t = g \left( s_{t-1}, x_{t-1}, \epsilon_t \right)\)

\(E_t \left[ f \left( s_t, x_t, s_{t+1}, x_{t+1} \right) \right]=0\)

The solution is a function \(\varphi\) such that \(x_t=\varphi(s_t)\).

fga model

In some cases, some variables can be directly expressed as a function of other variables. We call them auxiliary variables. Auxiliary variables are restricted to depend only on contemporaneous variables (controls or states). The model can be rewritten:

\(a_t = a\left(s_t, x_t\right)\)

\(s_t = g \left( s_{t-1}, x_{t-1}, a_{t-1}, \epsilon_t \right)\)

\(E_t \left[ f \left( s_t, x_t, a_t, s_{t+1}, x_{t+1}, a_{t+1} \right) \right]\)

Clearly, by substituting the variables a everywhere, this type of model can be seen as an fg model. Hence when some algorithm is applicable to an fg model, it can be also be applied to an fga model.

not implemented ; fgh model

A sub-variant of this specification let the user choose equations to define expectations. This is useful for PEA approaches. The model is specified by giving \(g\), \(f\) and \(h\) such that:

\(s_t = g \left( s_{t-1}, x_{t-1}, \epsilon_t \right)\)

\(f \left( s_t, x_t, z_t \right)\)

\(z_t = E_t h \left( s_{t+1}, x_{t+1} \right)\)

Same remark as for fga model: is needed, fgh models can behave exactly as an fg model.

not implemented ; fgah model

A sub-variant of this specification let the user choose equations to define expectations. This is useful for PEA approaches. The model is specified by giving \(g\), \(f\) and \(h\) such that:

\(a_t = a \left(s_t, x_t\right)\)

\(s_t = g \left( s_{t-1}, x_{t-1}, a_{t-1}, \epsilon_t \right)\)

\(f \left( s_t, x_t,a_t, z_t \right)\)

\(z_t = E_t h \left( s_{t+1}, x_{t+1} , a_{t+1} \right)\)

fgah models can behave exactly as an fg model, or an fga model

Numerical Conventions

• A list of N points in a d-dimensional space is represented by a d x N matrix. In other words the last index always corresponds to the successive observations. This is consistent with default ordering in numpy: last index varies faster.

Note

When dolo is used to compile a model for matlab the convention is reversed: first dimension denotes the successive points. This is adapted to matlab’s storage order (Fortran order).

• In order to evaluate many times a function \(f: R^s \rightarrow R^x\) for a list of N points we consider the vectorized version:

\(f: R^s x R^N \rightarrow R^x x R^N\) . If X is the matrix of inputs and F the matrix of output, then for any i

F[i,:] is the image of X[i,:] by f.

• In particular, the implementation of the model definition follow this convention, as well as the solution of the model.