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.

Current restriction: for the time being we limit ourselves to stochastic trends integrated of order 1.

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

Specification of the trends

For each sochastic trend, the user lists the growth factor of the stochastic trend in parentheses and the list of variables following that trend. The growth factor is function of parameters and (stationary) variables of the model:

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

Note 1: one variable can't follow several trends (our trends are not the common trends of cointegration analysis).
Note 2: the preprocessor attaches its trend to each variable

Algorithm of transformation and test

The algorithm is performed after introduction of auxiliary variables to reduce leads and lags over multiple periods.
- for each equation, in a temporary copy of the tree of this equation:
  1. each variables is multiplied by its trend (this trend is not a declared variable).
    - keep a list of variables treated that way for the test below
    - If the variabe appears with a lead, in addition, it is multiplied by the growth factor of the trend pushed forward by one period
    - If the variable appears with a lag, in addition it is divided by the growth factor at the current period
  2. test: evaluate the derivative of the equation with respect to each of the trend variable introduced above. If the derivative is not null, the equation or the specification of trend for each variable is not compatible with balanced growth. Send an error message identifying the equation and the list of nonstationary variables affected by the faulty trend
  3. write the equation on the model tree, removing the trend variables.
- 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:

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

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(pie);

assuming that we have

trend (exp(pie)), P;