Differences between revisions 3 and 4

Automatic removing of trends

Stationarizing a non-linear model by hand is a tedious process that is better done by the computer.

Computing the equilibrium growth rates of a balanced growth model is complicated and will not be attempted here. We limit ourselves to replace non-stationary variables by their stationary counterpart as specified by the user.

First case: exogenous nonstationary process

$\begin{eqnarray*} y_t &=& A_t k_{t-1}^\alpha\\ A_t &=& (1+g_t)A_{t-1}\\ g_t &=& \rho\, g_{t-1}+e_t \end{eqnarray*}$

Assuming that $y_t$ and $k_t$ have common trend $A_t$ , the stationarizing procedure calls for dividing first and second equation by $A_t$ . The second equation becomes meaningless and corresponds to the fact that $A_t$ doesn't belong to the stationary model.

Second case: endogenous nonstationary process

$\begin{eqnarray*} P_t\,C_t &=& W_t\, L_t\\ P_t &=& (1+\pi_t) P_{t-1}\\ r_t &=& \rho_1 (\pi_t - \bar \pi) \end{eqnarray*}$

In this case as well, the second equation becomes meaningless after stationarizing the model.

Interface specification and usage

the user lists only the variables to be stationarized, but not the trend in the VAR statement
for each sochastic trend, the user lists the deterministic growth factor and the list of corresponding variables:

trend (exp(gamma)), C, K;
trend (exp(m-gamma)), P;
trend (exp(m)), W, D;

algorithm of transformation, after introduction of auxiliary variables to reduce leads and lags over multiple periods:
- for each equation, the original variable becomes the stationarized variable, multiplied by the growth factor when this variable appears with a lead, divided by the growth factor when this variable appears with a lag.
- question: should we change the name of the variable when stationarized? It would be clearer, but more complicated as well. The user would have to use the modified name of the variable in all subsequent instructions and we would have to keep two lists of variable names. So I think that we should keep the same names.
- the transformation is triggered by the keyword stationarize_model;
- When the user wants to estimate the model in level, the nonstationary variables must be linked to the stationarized variable via (log-)linear relations:

$Pobs_t/Pobs_{t-1} = (P_t/P_{t-1})exp(\bar \pi)\\$

where P is the stationarized price level. In that case, variable Pobs shouldn't be listed in a trend expression, but only P and the original equation

Pobs/Pobs(-1)=P/P(-1);

shall be transformed in

Pobs/Pobs(-1)=P/P(-1)*exp(PIbar);

assuming that we have

trend (exp(PIbar)), P;

-  ⇤ ← Revision 3 as of 2010-04-08 15:13:57 → 
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+   ← Revision 4 as of 2010-04-15 11:00:43 → ⇥
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  Comment:
-Deletions are marked like this.
+Additions are marked like this.
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+== Interface specification and usage ==
 * the user lists only the variables to be stationarized, but not the trend in the VAR statement
 * for each sochastic trend, the user lists the deterministic growth factor and the list of corresponding variables:
{{{
trend (exp(gamma)), C, K;
trend (exp(m-gamma)), P;
trend (exp(m)), W, D;
}}}
 * algorithm of transformation, after introduction of auxiliary variables to reduce leads and lags over multiple periods:
   * for each equation, the original variable becomes the stationarized variable, multiplied by the growth factor when this variable appears with a lead, divided by the growth factor when this variable appears with a lag.
   * question: should we change the name of the variable when stationarized? It would be clearer, but more complicated as well. The user would have to use the modified name of the variable in all subsequent instructions and we would have to keep two lists of variable names. So I think that we should keep the same names.
   * the transformation is triggered by the keyword {{{stationarize_model;}}}
   *When the user wants to estimate the model in level, the nonstationary variables must be linked to the stationarized variable via (log-)linear relations:
{{{#!latex
\[
Pobs_t/Pobs_{t-1} = (P_t/P_{t-1})exp(\bar \pi)\\  
\]
}}}
where {{{P}}} is the stationarized price level. In that case, variable {{{Pobs}}} shouldn't be listed in a {{{trend}}} expression, but only {{{P}}} and the original equation
{{{
Pobs/Pobs(-1)=P/P(-1);
}}}
shall be transformed in 
{{{
Pobs/Pobs(-1)=P/P(-1)*exp(PIbar);
}}}
assuming that we have
{{{
trend (exp(PIbar)), P;
}}}