Model specification

Dolo Model Classification

Dynare models

A Dynare model is specified by a set of equations captured by one vector function $f$ so that the vector of endogeneous states $y_t$ satisfies

$0 = E_t f(y_{t|1}, y_t, y_{t-1}, \epsilon_t)$

where $\epsilon_t$ is a vector of exogenous variables following an i.i.d. series of shocks.

In this formulation, any variable appearing at date $t-1$ in the system (a predetermined variable) is part of the state-space as well as all shocks. Hence, the solution of the system is a function $g$ such that:

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

Discrete Time - Continuous States - Continuous Controls models (DTCSCC)

Solution

State-space is characterized by a vector $s$ of continuous variables. The unknown vector of controls $x$ is expressed by a function $\varphi$ such that:

$x = \varphi(s)$

Transitions

- name: `transition`
- short name: `g`

Transitions are given by a function $g$ such that at all times:

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

where $\epsilon_t$ is a vector of i.i.d. shocks.

Value equation

- name: `value`
- short name: `v`

The (separable) value equation defines a value $v_t$ as:

$v_t = U(s_t,x_t) | \beta E_t \left[ v_{t|1} \right]$

In general the Bellman equations are completely characterized by the reward function $U$ and the discount rate $\beta$.

Note that several values can be computed at once, if $U$ is a vector function and $\beta$ a vector of discount factors.

Generalized Value Equation

- name: `value_2`
- short name: `v_2`

A more general updating equation can be useful to express non-separable utilities or prices. In that case, we define a function $U^{*}$ such that

$v^{}_t = U^{}(s_t,x_t,v^{}t,s{t|1},x_{t|1},v^{}_{t|1})$

This equation defines the vector of (generalized) values $v^{*}$

Boundaries

- name: `controls_lb` and `controls_ub`
- short name: `lb` and `ub`

The optimal controls must also satisfy bounds that are function of states. There are two functions $\underline{b}()$ and $\overline{b}()$ such that:

$\underline{b}(s_t) \leq x_t \leq \overline{b}(s_t)$

Euler equation

- name: `arbitrage` (`equilibrium`)
- short name: `f`

A general formulation of the Euler equation is:

$0 = E_t [ f(s_t, x_t, s_{t|1}, x_{t|1}) ]$

Note that the Euler equation and the boundaries interact via "complentarity equations". Evaluated at one given state, with the vector of controls $x=(x_1, ..., x_i, ..., x_{n_x})$, the Euler equation gives us the residuals $r=(f_1, ..., f_i, ..., f_{n_x})$. Suppose that the $i$-th control $x_i$ is supposed to lie in the interval $[ \underline{b}_i, \overline{b}_i ]$. Then one of the following conditions must be true:

• $f_i$ = 0
• $f_i>0$ and $x_i=\underline{b}_i$
• $f_i<0$ and $x_i=\overline{b}_i$

By definition, this set of conditions is denoted by:

• $f_i = 0 \perp \underline{b}_i \leq x_i \leq \overline{b}_i$

These notations extend to a vector setting so that the Euler equations can also be written:

$0 = E_t [ f(s_t, x_t, s_{t|1}, x_{t|1}) ] \perp \underline{b}(s_t) \leq x_t \leq \overline{b}(s_t) $

Specifying the boundaries together with Euler equation, or specifying them as independent equations is (read: should be) equivalent. In any case, when the boundaries are finite and occasionally binding, some attention should be devoted to write the Euler equations in a consistent manner. In particular, note, that the Euler equations are order-sensitive.

The Euler conditions, together with the complementarity conditions typically come from the Kuhn-Tucker conditions associated with the maximization problem, but that is not true in general.

Expectations

- name: `expectation`
- short name: `h`

The vector of explicit expectations $z_t$ is defined by a function $h$ such that:

$z_t = E_t \left[ h(s_{t|1},x_{t|1}) \right]$

Generalized expectations

- name: `expectation_2`
- short name: `h_2`

The vector of generalized explicit expectations $z_t$ is defined by a function $h^{\star}$ such that:

$z_t = E_t \left[ h^{\star}(s_t,x_t,e_{t|1},s_{t|1},x_{t|1}) \right]$

Euler equation with explicit equations

- name: `arbitrage_2` (`equilibrium_2`)
- short name: `f_2`

If expectations are defined using one of the two preceding definitions, the Euler equation can be rewritten as:

$0 = f(s_t, x_t, z_t) \perp \underline{b}(s_t) \leq x_t \leq \overline{b}(s_t) $

Direct response function

- name: `direct_response`
- short name: `d`

In some simple cases, there a function $d()$ giving an explicit definition of the controls:

$x_t = d(s_t, z_t)$

Compared to the preceding Euler equation, this formulation saves computational time by removing to solve a nonlinear to get the controls implicitly defined by the Euler equation.

Terminal conditions

- name: `terminal_control`
- short name: `f_T`

When solving a model over a finite number $T$ of periods, there must be a terminal condition defining the controls for the last period. This is a function $f^T$ such that:

$0 = f^T(s_T, x_T)$

Terminal conditions

- name: `terminal_control_2`
- short name: `f_T_2`

When solving a model over a finite number $T$ of periods, there must be a terminal condition defining the controls for the last period. This is a function $f^{T,\star}$ such that:

$x_T = f^{T,\star}(s_T)$

Auxiliary variables

- name: `auxiliary`
- short name: `a`

In order to reduce the number of variables, it is useful to define auxiliary variables $y_t$ using a function $a$ such that:

$y_t = a(s_t, x_t)$

When they appear in an equation they are automatically substituted by the corresponding expression in $s_t$ and $x_t$.

Discrete Time - Mixed States - Continuous Controls models (DTMSCC)

The definitions for this class of models differ from the former ones by the fact that states are split into exogenous and discrete markov states, and endogenous continous states as before. Most of the definition can be readily transposed by replacing only the state variables.

State-space and solution

For this kind of problem, the state-space, is the cartesian product of a vector of "markov states" $m_t$ that can take a finite number of values and a vector of "continuous states" $s_t$ which takes continuous values.

The unknown controls $x_t$ is a function $\varphi$ such that:

$x_t =\varphi (m_t, s_t)$

Transitions

- name: `transition`
- short name: `g`

$(m_t)$ follows an exogenous and discrete markov chain. The whole markov chain is specified by two matrices $P,Q$ where each line of $P$ is one admissible value for $m_t$ and where each element $Q(i,j)$ is the conditional probability to go from state $i$ to state $j$.

The continuous states $s_t$ evolve after the law of motion:

$s_t = g(s_{t-1}, x_{t-1}, \epsilon_t)$ where $\epsilon_t$

where $\epsilon_t$ is a vector of i.i.d. shocks.

Boundaries

- name: `controls_lb`, `controls_ub`
- short name: `lb`, `ub`

The optimal controls must satisfy bounds that are function of states. There are two functions $\underline{b}()$ and $\overline{b}()$ such that:

$\underline{b}(m_t, s_t) \leq x_t \leq \overline{b}(m_t, s_t)$

Value Equation

- name: `value`
- short name: `v`

The (separable) Bellman equation defines a value $v_t$ as:

$v_t = U(m_t,s_t,x_t) | \beta*E_t [ v_{t|1}]$

It is completely characterized by the reward function $U$ and the discount rate $\beta$.

Generalized Value Equation

- name: `value_2`
- short name: `v_2`

The generalized value equation defines a value $v^{\star}_t$ as:

$v^{\star}t = U^{\star}(m_t,s_t,x_t,v^{\star}t,m{t|1},s{t|1},x_{t|1})$

Euler equation

- name: `arbitrage` (`equilibrium`)
- short name: `f`

Many Euler equations can be defined a function $f$ such that:

$0 = E_t \left( f(m_t,s_t,x_t,m_{t|1},s_{t|1},x_{t|1}) \right) \perp \underline{b}(m_t, s_t) \leq x_t \leq \overline{b}(m_t, s_t)$

See discussion about complementarity equations in the Continuous States - Continuous Controls section.

Expectations

- name: `expectation`
- short name: `h`

The vector of explicit expectations $z_t$ is defined by a function $h$ such that:

$z_t = E_t \left[ h(m_{t|1},s_{t|1},x_{t|1}) \right]$

Generalized expectations

- name: `expectation_2`
- short name: `h_2`

The vector of generalized explicit expectations $z_t$ is defined by a function $h^{\star}$ such that:

$z_t = E_t \left[ h^{\star}(m_t,s_t,x_t,m_{t|1},s_{t|1},x_{t|1}) \right]$

Euler equation with explicit equations

- name: `arbitrage_2` (`equilibrium_2`)
- short name: `f_2`

If expectations are defined using one of the two preceding definitions, the Euler equation can be rewritten as:

$0 = f(m_t, s_t, x_t, z_t) \perp \underline{b}(s_t) \leq x_t \leq \overline{b}(s_t) $

Direct response function

- name: `direct_response`
- short name: `d`

In some simple cases, there a function $d()$ giving an explicit definition of the controls:

$x_t = d(s_t, z_t)$

Compared to the preceding Euler equation, this formulation saves computational time by removing to solve a nonlinear to get the controls implicitly defined by the Euler equation.

Direct states function

- name: `direct_states`
- short name: `d_s`

For some applications, it is also useful to have a function $d{\star}$ which gives the endogenous states as a function of the controls and the exogenous markov states:

$s_t = d^{\star}(m_t, x_t)$

Auxiliary variables

- name: `auxiliary`
- short name: `a`

In order to reduce the number of variables, it is useful to define auxiliary variables $y_t$ using a function $a$ such that:

$y_t = a(m_t,s_t, x_t)$

Terminal conditions

- name: `terminal_control`
- short name: `f_T`

When solving a model over a finite number $T$ of periods, there must be a terminal condition defining the controls for the last period. This is a function $f^T$ such that:

$x_T = f^T(m_T, s_T)$

Terminal conditions (explicit)

- name: `terminal_control`
- short name: `f_T_2`

When solving a model over a finite number $T$ of periods, there must be a terminal condition defining the controls for the last period. This is a function $f^{T,\star}$ such that:

$f^{T,\star}(m_T, s_T, x_T)$

Misc

Variables

For DTCSCC and DTMSCC models, the following list variable types can be used (abbreviation in parenthesis): Required:

• states (s)
• controls (x) For DTCSCC only:
• shocks (e) For DTMSCC only:
• markov_states (m) Optional:
• auxiliaries (y)
• values (v)
• values_2 (v_2)
• expectations (z)
• expectations_2 (z_2)

Algorithms

Several algorithm are available to solve a model, depending no the functions that are specified.

	Dynare model	DTCSCC	DTMSCC
Perturbations	yes	(f,g)	no
Perturbations (higher order)	yes	(f,g)	no
Value function iteration		(v,g)	(v,g)
Time iteration		(f,g),(f,g,h)	(f,g),(f,g,h)
Parameterized expectations		(f,g,h)	(f,g,h)
Parameterized expectations (2)		(f_2,g,h_2)	(f_2,g,h_2)
Parameterized expectations (3)		(d,g,h)	(d,g,h)
Endogeneous gridpoints			(d,d_s,g,h)

Additional informations

calibration

In general, the models will depend on a series of scalar parameters. A reference value for the endogeneous variables is also used, for instance to define the steady-state. We call a "calibration" a list of values for all parameters and steady-state.

state-space

When a global solution is computed, continuous states need to be bounded. This can be done by specifying an n-dimensional box for them.

Usually one also want to specify a finite grid, included in this grid and the interpolation method used to evaluate between the grid points.

specification of the shocks

For DTCSCC models, the shocks follow an i.i.d. series of random variables. If the shock is normal, this one is characterized by a covariance matrix.

For DTMSCC models, exogenous shocks are specified by a two matrices P and Q, containing respectively a list of nodes and the transition probabilities.

Remarks

Some autodetection is possible. For instance, some equations appearing in f fonctions, can be promoted (or downgraded) to expectational equation, based on incidence analysis.