Merge pull request #484 from JohannesPfeifer/manual_DR
Clarify manual on different/inconsistent ordering of variables used in d...time-shift
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b207477c66
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@ -3735,7 +3735,7 @@ declaration order. Conversely, k-th declared variable is numbered
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@vindex M_.nsfwrd
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@vindex M_.ndynamic
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Finally, the state variables of the model are the purely backward variables
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and the mixed variables. They are orderer in DR-order when they appear in
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and the mixed variables. They are ordered in DR-order when they appear in
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decision rules elements. There are @code{M_.nspred = M_.npred + M_.nboth} such
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variables. Similarly, one has @code{M_.nsfwrd = M_.nfwrd + M_.nboth},
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and @code{M_.ndynamic = M_.nfwrd+M_.nboth+M_.npred}.
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@ -3743,7 +3743,7 @@ and @code{M_.ndynamic = M_.nfwrd+M_.nboth+M_.npred}.
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@node First order approximation
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@subsection First order approximation
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The approximation has the form:
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The approximation has the stylized form:
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@math{y_t = y^s + A y^h_{t-1} + B u_t}
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@ -3772,6 +3772,12 @@ endogenous in DR-order. The matrix columns correspond to exogenous
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variables in declaration order.
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@end itemize
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Of course, the shown form of the approximation is only stylized, because it neglects the required different ordering in @math{y^s} and @math{y^h_t}. The precise form of the approximation that shows the way Dynare deals with differences between declaration and DR-order, is
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@math{y_t(oo_.dr.order_var) = y^s(oo_.dr.order_var) + A (y_{t-1}(oo_.dr.order_var(k2))-y^s(oo_.dr.order_var(k2))) + B u_t}
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where @math{k2} selects the state variables, @math{y_t} and @math{y^s} are in declaration order and the coefficient matrices are in DR-order. Effectively, all variables on the right hand side are brought into DR order for computations and then assigned to @math{y_t} in declaration order.
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@node Second order approximation
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@subsection Second order approximation
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@ -3785,7 +3791,7 @@ A y^h_{t-1} + B u_t + 0.5 C
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where @math{y^s} is the steady state value of @math{y},
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@math{y^h_t=y_t-y^s}, and @math{\Delta^2} is the shift effect of the
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variance of future shocks.
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variance of future shocks. For the reordering required due to differences in declaration and DR order, see the first order approximation.
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The coefficients of the decision rules are stored in the variables
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described for first order approximation, plus the following variables:
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