Regression of a simple policy function

[Daniel Bendel]

Dear all,

I want to find a simple linear optimal policy rule for a Ramsey problem. Therefore I calculated the FOC of the Ramsey problem and simulated the model for 100000 periods. Then I want to do a regression where the dependent variable is the policy instrument "government debt" and the explained variables are the state variables of the model.
Now I am confused regarding the timing I should use:
Government debt is a predeterminded variable and evaluates as following:

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G + b  = b(+1)/(1+interest)*pi(+1) + T; (T= transfers, G = government consumption, b= government debt, pi = inflation)

I use the "predetermined_variables" command for "b". Now how do I have to interpret dynare's simulation of b. I know dynare converts variables like capital K(+1) into K. Does that mean that every b_t+1 of the simulation is in fact a b_t or every b_t is in fact a b_t+1???? How does my regression then look:

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b_t+1 = b_t + some state variables;

or

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b_t = b_t-1 + some state variables;

????

I hope you unterstand my problem...

[jpfeifer]

See the manual

Note that Dynare internally always uses the “stock at the end of the period” concept, even
when the model has been entered using the predetermined_variables-command. Thus, when
plotting, computing or simulating variables, Dynare will follow the convention to use variables
that are decided in the current period. For example, when generating impulse response functions
for capital, Dynare will plot k, which is the capital stock decided upon by investment today (and
which will be used in tomorrow’s production function). This is the reason that capital is shown
to be moving on impact, because it is k and not the predetermined k(-1) that is displayed. It
is important to remember that this also affects simulated time series and output from smoother
routines for predetermined variables. Compared to non-predetermined variables they might
otherwise appear to be falsely shifted to the future by one period.

That means: if you start at time t=0, the first simulated value for b is the value chosen at time 0 and which will be predetermined for period t=1

[Daniel Bendel]

Thank you. This clearifies my problem a little bit. But I was thinking about the problem again, and was wondering if there might be an easier way. I think, instead doing a regression, one can use the POLICY AND TRANSITION FUNCTIONS. Is that right??

And why are the values of the autocorrelations different to those of the policy function??

[jpfeifer]

You can do that, but it requires you to extract the required coefficients from the decision rule matrices stored in oo_.dr.

[Daniel Bendel]

just to get things right, since the confusing regarding how dynare works with predeterminded variabels.

Suppose I have the following policy function (b is predeterminded):

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                               b       
Constant                    0.000043
S(-1)                       0.069240
A(-1)                      -0.057798
b(-1)                       0.961578
Ptilde(-1)                 -0.324475
M(-1)                       0.249524
lambda(-1)                 -0.002588
chis(-1)                    0.069812

Does this mean, that:
b_t+1 = Constant + 0.961578*b_t + 0.069240*S_t-1 -0.057798*A_t-1 + ... ????

And if this is right, how would I get to the coefficients of the following equation (where _ss means corresponding steady state):
b_t+1 - b_ss = coff1*(b_t - b_ss) + coff2* (S_t-1 - S_ss) + ... ???

[jpfeifer]

The displayed equation uses Dynare's stock at the end of period notation. To bring this into your preferred stock at the beginning of period notation, you need to shift the predetermined variables by one period. Your first equation is correct if b is the only predetermined stock. Otherwise, some of the other variables need to be shifted as well.

Regarding the second point. Note that the constant in the policy rule is just the sum of all constant terms, i.e. it lumps together all coff*X_ss terms. This leaves the slope coefficients unaltered, i.e. the coefficients are the ones displayed.