Steady state & residuals + colinear relationships

Dynare help (legacy posts)
1

Hi!

I’m trying to implement the model from “TECHNOLOGY INNOVATION AND DIFFUSION AS SOURCES OF OUTPUT AND
ASSET PRICE FLUCTUATIONS” by Comin, Gertler and Santacreu. In the steady state I get non-zero residuals in 2 equations. I know what it means. Obviously, there is a mistake in my steady-state, but I can’t find it. Remark: some parameters are chosen in an another way than in the paper. I ask you to look at the mod-file and I would be very thankful if you could tell, whether there is and where is the mistake. Thanks in advance!!!
cgs.mod (6.5 KB)
w15029.pdf (539 KB)

2

Are you sure the shock processes are correct?

% stochastic processes log(chi) = rho*log(chi(-1)) + eps; %31 log(x) = eta*log(x(-1)) + sigma; %32 log(g) = nu*log(g(-1)) + varrho; %33 log(p_k_st) = rho_st*log(p_k_st(-1)) + eps_st; %34
means they will all have steady state 0. That is not what you use for x and g.

3

Thanks a lot! Setting the four parameters to 1, or, equivalently, using

% stochastic processes
log(chi) = log(chi(-1)) + eps; %31
log(x) = log(x(-1)) + sigma; %32
log(g) = log(g(-1)) + varrho; %33
log(p_k_st) = log(p_k_st(-1)) + eps_st; %34

makes all the residuals = 0.

4

I finally could derive the steady state equations to get all the residuals = 0. I’m getting 8 eigenvalues greater than one for 12 jump variables. So I use the model_diagnostics command. And I get the following output:

model_diagnostic: the Jacobian of the static model is singular there is 7 colinear relationships between the variables and the equations Relation 1 Colinear variables: y c g p_k j i z_k z_y v_k v_y n_y a_k a_y h_k h_y x k l o_k o_y pi_k pi_y p_k_et n_k j_y j_k i_s i_e lambda_y lambda_k p_k_bar Relation 2 Colinear variables: c g p_k j z_k z_y v_k v_y n_y a_k a_y h_k h_y x k l o_k o_y pi_k pi_y p_k_st p_k_et n_k j_y j_k lambda_y lambda_k p_k_bar Relation 3 Colinear variables: c g p_k j z_k z_y v_k v_y n_y a_k a_y h_k h_y x k l o_k o_y pi_k pi_y p_k_st p_k_et n_k j_y j_k lambda_y lambda_k p_k_bar Relation 4 Colinear variables: c g p_k j z_k z_y v_k v_y n_y a_k a_y h_k h_y x k l o_k o_y pi_k pi_y p_k_st p_k_et n_k j_y j_k lambda_y lambda_k p_k_bar Relation 5 Colinear variables: c g p_k j z_k z_y v_k v_y n_y a_k a_y h_k h_y x k l o_k o_y pi_k pi_y p_k_st p_k_et n_k j_y j_k lambda_y lambda_k p_k_bar Relation 6 Colinear variables: c g p_k j z_k z_y v_k v_y n_y a_k a_y h_k h_y x k l o_k o_y pi_k pi_y p_k_st p_k_et n_k j_y j_k lambda_y lambda_k p_k_bar Relation 7 Colinear variables: c g p_k j z_k z_y v_k v_y n_y a_k a_y h_k h_y x k l o_k o_y pi_k pi_y p_k_st p_k_et n_k j_y j_k lambda_y lambda_k p_k_bar Relation 1 Colinear equations 6 7 9 10 12 13 14 Relation 2 Colinear equations 6 7 9 10 12 13 14 Relation 3 Colinear equations 31 Relation 4 Colinear equations 32 Relation 5 Colinear equations 33 Relation 6 Colinear equations 34 Relation 7 Colinear equations 6 7 9 10 12 13 14 The presence of a singularity problem typically indicates that there is one redundant equation entered in the model block, while another non-redundant equation is missing. The problem often derives from Walras Law.

What does this colinearity mean? How could it be, that only one equation at a time (No. 31 - 34, stochastic processes) is colinear? Is the eigenvalues problem arising from the steady state or does it depend on the starting values for k, lambdas and other? Varying this I get from 5 to 8 eigenvalues > 1, bot not more.
cgs.mod (6.53 KB)

5

Your exogenous processes specify unit roots. That’s why there is collinearity in these equations. Given the many missing explosive eigenvalues, I would think that you are having a systematic timing issue in your model.

6

I rewrote three of four exogenous processes, such that they are sorting with the paper now. But in the paper one process looks like

log(X_t) = log(X_t-1) + sigma; %31

Can it be a problem?

Furthermore I get 3 colinear equations (number 6,7 and 30). In order to have the equation

log(Chi_t) = rho*log(Chi_t-1) + eps; %30

fulfilled in the steady state I choose Chi = 1. But then the equations 6 and 7

z_y(+1) = (chi_y_bar*(chi^ksi_y) + phi)*z_y; %6 z_k(+1) = (chi_k_bar*(chi^ksi_k) + phi)*z_k; %7
imply that in the steady state chi_k_bar + phi = 1 and chi_y_bar + phi = 1. But that contradicts the calibration of the model (chi_k_bar = 0.0304, chi_y_bar = 0.0202, phi = 0.99). What is the right way? Thanks in advance!
cgs_main_steady.mod (6.87 KB)

7

The first one will give you a unit root, which is a feature of the model as far as I can see. But i

I cannot work myself into the calibration of the model. So you need to figure out how to make the steady states consistent.

But I would like to point out that the last two equations are not conforming to Dynare’s timing convention. They are laws of motion that describe predetermined variables.

in Dynare means

I guess that z_y(+1) must be predetermined here like capital.

8

jpfeifer, thanks a lot!!!