Dynare's use of auxiliary variables

The Dynare preprocessor modifies the model given by the user, under certain circumstances, by adding auxiliary variables to it. The aim is to bring the model into a canonical form and to simplify computations,

Auxiliary variables are created in the following cases:

• for endogenous variables with a lag or a lead of two or more (only in stochastic contexts in Dynare 4.1, in all cases in Dynare ≥ 4.2)
• for exogenous variables with a lag or a lead (only in stochastic contexts in Dynare ≤ 4.6, in all cases in Dynare ≥ 4.7)
• when the EXPECTATION() operator is used
• when the ramsey_model-statement is used
• if the diff-operator is used.

Auxiliary variables are always endogenous variables. Internally, the indices of the original endogenous (as declared by the user) are in the range 1:M_.orig_endo_nbr. Auxiliary variables have indices in the range M_.orig_endo_nbr+1:M_.endo_nbr.

In most cases, auxiliary equations are appended along with auxiliary variables, so that the number of equations and endogenous variables stays constant.

In order to get information about auxiliary variables and equations added by the preprocessor, the write_latex_dynamic_model and write_latex_static_model commands can be useful tools.

The M_.aux_vars-structure

The matrix structure M_.aux_vars also contains information about auxiliary variables. For each auxiliary variable, the following fields are declared:

• M_.aux_vars(i).endo_index: the index of the auxiliary variable, in the range M_.orig_endo_nbr+1:M_.endo_nbr
• M_.aux_vars(i).type: an integer representing the type of the auxiliary variable (see below for the list of types).

The structure can also contain other information, depending on the type of the aux var:

• M_.aux_vars(i).orig_index contains the index of the endogenous or exogenous variable for whose leads/lags the auxiliary variables is a substitute
• M_.aux_vars(i).orig_lead_lag contains the lag or lead number of the original variable (as a negative value for lags and a positive one for leads; it will be empty for no lag).
• M_.aux_vars(i).eq_nbr contains the associated equation in case of Lagrange multipliers
• M_.aux_vars(i).orig_expr contains the original expression to which the auxiliary variable is attached, for example in case of a lead of two on a Lagrange multiplier, there will be an auxiliary variable of type 0 for the lead on an auxiliary variable of type 6 (the multiplier). In this case, orig_expr may contain MULT_1(1) to indicate that the auxiliary variable for the lead refers to that leaded multiplier.

Types of auxiliary variables

Type 0: Leads of endogenous >=2

This kind of aux var has M_.aux_vars(i).type = 0 and substitutes for an expression in the original model where a lead of 2 or more appears on an endogenous variable. The name of such an aux var begins with AUX_ENDO_LEAD_.

Type 1: Lags of endogenous >=2

This kind of aux var has M_.aux_vars(i).type = 1 and substitutes for an expression in the original model where a lag of 2 or more appears on an endogenous variable. The name of such an aux var begins with AUX_ENDO_LAG_.

Type 2: Lead of exogenous

This kind of aux var has M_.aux_vars(i).type = 2 and substitutes for an expression in the original model where a lead on an exogenous variable appears. The name of such an aux var begins with AUX_EXO_LEAD_.

Type 3: Lag of exogenous

This kind of aux var has M_.aux_vars(i).type = 3 and substitutes for an expression in the original model where a lag on an exogenous variable appears. The name of such an aux var begins with AUX_EXO_LAG_.

Type 4: EXPECTATION()-operator

This kind of aux var has M_.aux_vars(i).type = 4 and results from the EXPECTATION()-operator employing the (full) information set of another period. It substitutes for an expression like EXPECTATION(-1)(...). The name of such an aux var begins with AUX_EXPECT_LEAD_ or AUX_EXPECT_LAG_.

Type 5: differentiate_forward_vars-option (since Dynare 4.4)

This kind of aux var has M_.aux_vars(i).type = 5 and results from differentiate_forward_vars-option of model. In this case, forward variables are differentiated and and the difference replaced by an aux var, whose name begins with AUX_DIFF_FWRD_.

M_.aux_vars(i).orig_index contains the index of the endogenous variable of which it is the differentiate.

Type 6: Lagrange multipliers of the ramsey_model (since Dynare 4.3)

This kind of aux var has M_.aux_vars(i).type = 6 and corresponds to the Lagrange multipliers attached to the first order conditions of the Ramsey problem. They will appear whenever ramsey_model is used. Their name begins is MULT_i, where i is the equation number with which the multiplier is associated.

M_.aux_vars(i).eq_nbr contains the number of the equation (constraint) with which the multiplier is associated.

Type 7: Unused

Type 8: diff operator (since Dynare 4.6)

This kind of aux var has M_.aux_vars(i).type = 8 and substitutes for the diff operator. The name of such an auxiliary variable begins with AUX_DIFF_.

The substitution will only happen if there is no lead in the argument to the diff operator.

• M_.aux_vars(i).orig_index contains the index of the variable contained in the diff operator. This field exists if the diff was taken of a variable, not of an expression.
• M_.aux_vars(i).orig_lead_lag contains the lag of the variable contained in the diff operator. This field exists if the diff was taken of a variable, not of an expression.

Type 9: lag within diff operator (since Dynare 4.6)

This kind of aux var has M_.aux_vars(i).type = 9 and accounts for applying the difference operator to a lag. If a diff operator is applied to a variable with lagged values (e.g. diff(x(-3)), and if there is also a diff operator with the same arguments shifted in the future (e.g. diff(x)), then a diff auxiliary variable (of type 8) is created for the diff operator applied to the variable with lag equal to zero (e.g. diff(x)) and auxiliary variables (of type 9) are created for the following sequence diff(x(-1), diff(x(-2)), and diff(x(-3)). The name of such auxiliary variables begins with AUX_DIFF_LAG_.

• M_.aux_vars(i).orig_index contains the index of the preceding auxiliary variable for diff lag or of the auxiliary variable for diff itself (types 9 or 8)
• M_.aux_vars(i).orig_lead_lag contains 0 (not needed because we know the lag is equal to 1 compared to the previous aux variable in the sequence)

Type 10: unary operations (since Dynare 4.7)

This kind of aux var has M_.aux_vars(i).type = 10 is substituted for an expression consisting of a unary operator (namely exp, log, log10, cos, sin, tan, acos, asin, atan, cosh, sinh, tanh, acosh, asinh, atanh, sqrt, cbrt, abs, sign and erf). Such an auxiliary variable is typically created for an expression of the form log(x) in the context of a var_model or a trend_component_model. The name of such an auxiliary variable begins with AUX_UOP_.

If the transform_unary_ops option has been passed to dynare, then the substitution is done everywhere in the model. Otherwise, by default, it is only performed in the equations appearing in a var_model or a trend_component_model.

Type 11: lead within diff operator (since Dynare 4.7)

This kind of aux var has M_.aux_vars(i).type = 11 and accounts for applying the difference operator to a lead. Similarly to Type 9, it will create a sequence of auxiliary variables, but for leads.

Such aux vars only appear in the context of PAC models with model-consistent expectations (when constructing the expectation term).

• M_.aux_vars(i).orig_index contains the index of the preceding auxiliary variable for diff lead or of the auxiliary variable for diff itself (types 11 or 8)
• M_.aux_vars(i).orig_lead_lag contains 0 (not needed because we know the lead is equal to 1 compared to the previous aux var in the sequence)