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### 4.3 Expressions

Dynare distinguishes between two types of mathematical expressions: those that are used to describe the model, and those that are used outside the model block (e.g. for initializing parameters or variables, or as command options). In this manual, those two types of expressions are respectively denoted by MODEL_EXPRESSION and EXPRESSION.

Unlike MATLAB or Octave expressions, Dynare expressions are necessarily scalar ones: they cannot contain matrices or evaluate to matrices1.

Expressions can be constructed using integers (INTEGER), floating point numbers (DOUBLE), parameter names (PARAMETER_NAME), variable names (VARIABLE_NAME), operators and functions.

The following special constants are also accepted in some contexts:

Constant: inf

Represents infinity.

Constant: nan

“Not a number”: represents an undefined or unrepresentable value.

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#### 4.3.1 Parameters and variables

Parameters and variables can be introduced in expressions by simply typing their names. The semantics of parameters and variables is quite different whether they are used inside or outside the model block.

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#### 4.3.1.1 Inside the model

Parameters used inside the model refer to the value given through parameter initialization (see Parameter initialization) or `homotopy_setup` when doing a simulation, or are the estimated variables when doing an estimation.

Variables used in a MODEL_EXPRESSION denote current period values when neither a lead or a lag is given. A lead or a lag can be given by enclosing an integer between parenthesis just after the variable name: a positive integer means a lead, a negative one means a lag. Leads or lags of more than one period are allowed. For example, if `c` is an endogenous variable, then `c(+1)` is the variable one period ahead, and `c(-2)` is the variable two periods before.

When specifying the leads and lags of endogenous variables, it is important to respect the following convention: in Dynare, the timing of a variable reflects when that variable is decided. A control variable — which by definition is decided in the current period — must have no lead. A predetermined variable — which by definition has been decided in a previous period — must have a lag. A consequence of this is that all stock variables must use the “stock at the end of the period” convention. Please refer to Mancini-Griffoli (2007) for more details and concrete examples.

Leads and lags are primarily used for endogenous variables, but can be used for exogenous variables. They have no effect on parameters and are forbidden for local model variables (see Model declaration).

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#### 4.3.1.2 Outside the model

When used in an expression outside the model block, a parameter or a variable simply refers to the last value given to that variable. More precisely, for a parameter it refers to the value given in the corresponding parameter initialization (see Parameter initialization); for an endogenous or exogenous variable, it refers to the value given in the most recent `initval` or `endval` block.

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#### 4.3.2 Operators

The following operators are allowed in both MODEL_EXPRESSION and EXPRESSION:

• binary arithmetic operators: `+`, `-`, `*`, `/`, `^`
• unary arithmetic operators: `+`, `-`
• binary comparison operators (which evaluate to either `0` or `1`): `<`, `>`, `<=`, `>=`, `==`, `!=`

Note that these operators are differentiable everywhere except on a line of the 2-dimensional real plane. However for facilitating convergence of Newton-type methods, Dynare assumes that, at the points of non-differentiability, the partial derivatives of these operators with respect to both arguments is equal to (since this is the value of the partial derivatives everywhere else).

The following special operators are accepted in MODEL_EXPRESSION (but not in EXPRESSION):

This operator is used to take the value of the enclosed expression at the steady state. A typical usage is in the Taylor rule, where you may want to use the value of GDP at steady state to compute the output gap.

Operator: EXPECTATION (INTEGER) (MODEL_EXPRESSION)

This operator is used to take the expectation of some expression using a different information set than the information available at current period. For example, `EXPECTATION(-1)(x(+1))` is equal to the expected value of variable `x` at next period, using the information set available at the previous period. See Auxiliary variables, for an explanation of how this operator is handled internally and how this affects the output.

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#### 4.3.3 Functions

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#### 4.3.3.1 Built-in Functions

The following standard functions are supported internally for both MODEL_EXPRESSION and EXPRESSION:

Function: exp (x)

Natural exponential.

Function: log (x)
Function: ln (x)

Natural logarithm.

Function: log10 (x)

Base 10 logarithm.

Function: sqrt (x)

Square root.

Function: abs (x)

Absolute value.

Note that this function is not differentiable at . However, for facilitating convergence of Newton-type methods, Dynare assumes that the derivative at is equal to (this assumption comes from the observation that the derivative of is equal to for and from the convention for the derivative of at ).

Function: sign (x)

Signum function.

Note that this function is not differentiable at . However, for facilitating convergence of Newton-type methods, Dynare assumes that the derivative at is equal to (this assumption comes from the observation that both the right- and left-derivatives at this point exist and are equal to ).

Function: sin (x)
Function: cos (x)
Function: tan (x)
Function: asin (x)
Function: acos (x)
Function: atan (x)

Trigonometric functions.

Function: max (a, b)
Function: min (a, b)

Maximum and minimum of two reals.

Note that these functions are differentiable everywhere except on a line of the 2-dimensional real plane defined by . However for facilitating convergence of Newton-type methods, Dynare assumes that, at the points of non-differentiability, the partial derivative of these functions with respect to the first (resp. the second) argument is equal to (resp. to ) (i.e. the derivatives at the kink are equal to the derivatives observed on the half-plane where the function is equal to its first argument).

Function: normcdf (x)
Function: normcdf (x, mu, sigma)

Gaussian cumulative density function, with mean mu and standard deviation sigma. Note that `normcdf(x)` is equivalent to `normcdf(x,0,1)`.

Function: normpdf (x)
Function: normpdf (x, mu, sigma)

Gaussian probability density function, with mean mu and standard deviation sigma. Note that `normpdf(x)` is equivalent to `normpdf(x,0,1)`.

Function: erf (x)

Gauss error function.

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#### 4.3.3.2 External Functions

Any other user-defined (or built-in) MATLAB or Octave function may be used in both a MODEL_EXPRESSION and an EXPRESSION, provided that this function has a scalar argument as a return value.

To use an external function in a MODEL_EXPRESSION, one must declare the function using the `external_function` statement. This is not necessary for external functions used in an EXPRESSION.

Command: external_function (OPTIONS…);

Description

This command declares the external functions used in the model block. It is required for every unique function used in the model block.

`external_function` commands can appear several times in the file and must come before the model block.

Options

`name = NAME`

The name of the function, which must also be the name of the M-/MEX file implementing it. This option is mandatory.

`nargs = INTEGER`

The number of arguments of the function. If this option is not provided, Dynare assumes `nargs = 1`.

`first_deriv_provided [= NAME]`

If NAME is provided, this tells Dynare that the Jacobian is provided as the only output of the M-/MEX file given as the option argument. If NAME is not provided, this tells Dynare that the M-/MEX file specified by the argument passed to `name` returns the Jacobian as its second output argument.

`second_deriv_provided [= NAME]`

If NAME is provided, this tells Dynare that the Hessian is provided as the only output of the M-/MEX file given as the option argument. If NAME is not provided, this tells Dynare that the M-/MEX file specified by the argument passed to `name` returns the Hessian as its third output argument. NB: This option can only be used if the `first_deriv_provided` option is used in the same `external_function` command.

Example

```external_function(name = funcname);
external_function(name = otherfuncname, nargs = 2,
first_deriv_provided, second_deriv_provided);
external_function(name = yetotherfuncname, nargs = 3,
first_deriv_provided = funcname_deriv);
```

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#### 4.3.4 A few words of warning in stochastic context

The use of the following functions and operators is strongly discouraged in a stochastic context: `max`, `min`, `abs`, `sign`, `<`, `>`, `<=`, `>=`, `==`, `!=`.

The reason is that the local approximation used by `stoch_simul` or `estimation` will by nature ignore the non-linearities introduced by these functions if the steady state is away from the kink. And, if the steady state is exactly at the kink, then the approximation will be bogus because the derivative of these functions at the kink is bogus (as explained in the respective documentations of these functions and operators).

Note that `extended_path` is not affected by this problem, because it does not rely on a local approximation of the model.

### (1)

Note that arbitrary MATLAB or Octave expressions can be put in a .mod file, but those expressions have to be on separate lines, generally at the end of the file for post-processing purposes. They are not interpreted by Dynare, and are simply passed on unmodified to MATLAB or Octave. Those constructions are not addresses in this section.

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