Cosmetic changes.
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@ -14299,7 +14299,7 @@ a trend target to which the endogenous variables may be attracted in the long-ru
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:math:`n\times 1` vector of parameters, :math:`A_i` (:math:`i=0,\ldots,p`)
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are :math:`n\times n` matrices of parameters, and :math:`A_0` is non
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singular square matrix. Vector :math:`\mathbf{c}` and matrices :math:`A_i`
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(:math:`i=0,\ldots,p`) are set by Dynare by parsing the equations in the
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(:math:`i=0,\ldots,p`) are set by parsing the equations in the
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``model`` block. Then, Dynare builds a VAR(1)-companion form model for
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:math:`\mathcal{Y}_t = (1, Y_t, \ldots, Y_{t-p+1})'` as:
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@ -14510,7 +14510,7 @@ up to time :math:`t-\tau`, :math:`\mathcal{Y}_{\underline{t-\tau}}`) is:
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In a semi-structural model, variables appearing in :math:`t+h` (*e.g.*
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the expected output gap in a dynamic IS curve or expected inflation in a
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(New Keynesian) Phillips curve) will be replaced by the expectation implied by an auxiliary VAR
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New Keynesian Phillips curve) will be replaced by the expectation implied by an auxiliary VAR
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model. Another use case is for the computation of permanent
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incomes. Typically, consumption will depend on something like:
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@ -14518,13 +14518,13 @@ incomes. Typically, consumption will depend on something like:
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\sum_{h=0}^{\infty} \beta^h y_{t+h|t-\tau}
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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:
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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:
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.. math ::
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\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}
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More generally, it is possible to consider finite discounted sums.
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Finite discounted sums can also be considered.
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.. command:: var_expectation_model (OPTIONS...);
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