Compare commits
2 Commits
f5a5ebb910
...
c33b53b045
Author | SHA1 | Date |
---|---|---|
Stéphane Adjemian (Argos) | c33b53b045 | |
Stéphane Adjemian (Argos) | 50beb8000d |
|
@ -14299,7 +14299,7 @@ a trend target to which the endogenous variables may be attracted in the long-ru
|
||||||
:math:`n\times 1` vector of parameters, :math:`A_i` (:math:`i=0,\ldots,p`)
|
:math:`n\times 1` vector of parameters, :math:`A_i` (:math:`i=0,\ldots,p`)
|
||||||
are :math:`n\times n` matrices of parameters, and :math:`A_0` is non
|
are :math:`n\times n` matrices of parameters, and :math:`A_0` is non
|
||||||
singular square matrix. Vector :math:`\mathbf{c}` and matrices :math:`A_i`
|
singular square matrix. Vector :math:`\mathbf{c}` and matrices :math:`A_i`
|
||||||
(:math:`i=0,\ldots,p`) are set by Dynare by parsing the equations in the
|
(:math:`i=0,\ldots,p`) are set by parsing the equations in the
|
||||||
``model`` block. Then, Dynare builds a VAR(1)-companion form model for
|
``model`` block. Then, Dynare builds a VAR(1)-companion form model for
|
||||||
:math:`\mathcal{Y}_t = (1, Y_t, \ldots, Y_{t-p+1})'` as:
|
:math:`\mathcal{Y}_t = (1, Y_t, \ldots, Y_{t-p+1})'` as:
|
||||||
|
|
||||||
|
@ -14510,7 +14510,7 @@ up to time :math:`t-\tau`, :math:`\mathcal{Y}_{\underline{t-\tau}}`) is:
|
||||||
|
|
||||||
In a semi-structural model, variables appearing in :math:`t+h` (*e.g.*
|
In a semi-structural model, variables appearing in :math:`t+h` (*e.g.*
|
||||||
the expected output gap in a dynamic IS curve or expected inflation in a
|
the expected output gap in a dynamic IS curve or expected inflation in a
|
||||||
(New Keynesian) Phillips curve) will be replaced by the expectation implied by an auxiliary VAR
|
New Keynesian Phillips curve) will be replaced by the expectation implied by an auxiliary VAR
|
||||||
model. Another use case is for the computation of permanent
|
model. Another use case is for the computation of permanent
|
||||||
incomes. Typically, consumption will depend on something like:
|
incomes. Typically, consumption will depend on something like:
|
||||||
|
|
||||||
|
@ -14518,13 +14518,13 @@ incomes. Typically, consumption will depend on something like:
|
||||||
|
|
||||||
\sum_{h=0}^{\infty} \beta^h y_{t+h|t-\tau}
|
\sum_{h=0}^{\infty} \beta^h y_{t+h|t-\tau}
|
||||||
|
|
||||||
Assuming that $0<\beta<1$ and knowing the limit of geometric series, the conditional expectation of this variable can be evaluated based on the same auxiliary model:
|
Assuming that :math:`0<\beta<1` and knowing the limit of geometric series, the conditional expectation of this variable can be evaluated based on the same auxiliary model:
|
||||||
|
|
||||||
.. math ::
|
.. math ::
|
||||||
|
|
||||||
\mathbb E \left[\sum_{h=0}^{\infty} \beta^h y_{t+h}\Biggl| \mathcal{Y}_{\underline{t-\tau}}\right] = \alpha \mathcal{C}^\tau(I-\beta\mathcal{C})^{-1}\mathcal{Y}_{t-\tau}
|
\mathbb E \left[\sum_{h=0}^{\infty} \beta^h y_{t+h}\Biggl| \mathcal{Y}_{\underline{t-\tau}}\right] = \alpha \mathcal{C}^\tau(I-\beta\mathcal{C})^{-1}\mathcal{Y}_{t-\tau}
|
||||||
|
|
||||||
More generally, it is possible to consider finite discounted sums.
|
Finite discounted sums can also be considered.
|
||||||
|
|
||||||
.. command:: var_expectation_model (OPTIONS...);
|
.. command:: var_expectation_model (OPTIONS...);
|
||||||
|
|
||||||
|
|
|
@ -18,7 +18,7 @@ function [h, lrcp] = hVectors(params, H, auxmodel, kind, id)
|
||||||
% params(2:end-1) ⟶ Autoregressive parameters.
|
% params(2:end-1) ⟶ Autoregressive parameters.
|
||||||
% params(end) ⟶ Discount factor.
|
% params(end) ⟶ Discount factor.
|
||||||
|
|
||||||
% Copyright © 2018-2021 Dynare Team
|
% Copyright © 2018-2024 Dynare Team
|
||||||
%
|
%
|
||||||
% This file is part of Dynare.
|
% This file is part of Dynare.
|
||||||
%
|
%
|
||||||
|
@ -52,21 +52,21 @@ n = length(H);
|
||||||
tmp = eye(n*m)-kron(G, transpose(H)); % inv(W2)
|
tmp = eye(n*m)-kron(G, transpose(H)); % inv(W2)
|
||||||
|
|
||||||
switch kind
|
switch kind
|
||||||
case 'll'
|
case 'll' % (A.84), page 28 in Brayton, Davis and Tulip (2000)
|
||||||
h = A_1*A_b*((kron(iota(m, m), H))'*(tmp\kron(iota(m, m), iota(n, id))));
|
h = A_1*A_b*((kron(iota(m, m), H))'*(tmp\kron(iota(m, m), iota(n, id))));
|
||||||
case 'dd'
|
case 'dd' % (A.79), page 26 in Brayton, Davis and Tulip (2000)
|
||||||
h = A_1*A_b*(kron(iota(m, m)'*inv(eye(m)-G), H')*(tmp\kron(iota(m, m), iota(n, id))));
|
h = A_1*A_b*(kron(iota(m, m)'*inv(eye(m)-G), H')*(tmp\kron(iota(m, m), iota(n, id))));
|
||||||
case 'dl'
|
case 'dl' % (A.74), page 24 in Brayton, Davis and Tulip (2000)
|
||||||
h = A_1*A_b*(kron(iota(m, m)'*inv(eye(m)-G), (H'-eye(length(H))))*(tmp\kron(iota(m, m), iota(n, id))));
|
h = A_1*A_b*(kron(iota(m, m)'*inv(eye(m)-G), (H'-eye(length(H))))*(tmp\kron(iota(m, m), iota(n, id))));
|
||||||
otherwise
|
otherwise
|
||||||
error('Unknown kind value in PAC model.')
|
error('Unknown kind value in PAC model.')
|
||||||
end
|
end
|
||||||
|
|
||||||
if nargin>1
|
if nargout>1
|
||||||
if isequal(kind, 'll')
|
if isequal(kind, 'll')
|
||||||
lrcp = NaN;
|
lrcp = NaN;
|
||||||
else
|
else
|
||||||
d = A_1*A_b*(iota(m, m)'*inv((eye(m)-G)*(eye(m)-G))*iota(m, m));
|
d = A_1*A_b*(iota(m, m)'*inv((eye(m)-G)*(eye(m)-G))*iota(m, m));
|
||||||
lrcp = (1-sum(params(2:end-1))-d);
|
lrcp = (1-sum(params(2:end-1))-d);
|
||||||
end
|
end
|
||||||
end
|
end
|
||||||
|
|
Loading…
Reference in New Issue