Calibrate to empirical steady state moments

Estimation SMM/GMM/IRF-Matching
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I want to calibrate some parameters to steady state moments of some macroeconomic ratio. I have some questions on the exact way in which I have to obtain those steady state values from real data. Particularly, I have these doubts, assuming I want to target steady state of x_t^{obs}/y_t^{obs}: 1) If the ratio is non-stationary (following unit root testing on it) would it still be valid to take the mean of the ratio as steady state target? 2) If it is stationary, should I take simply the mean of the ratio, or the mean of a best estimated ARMA model on the ratio? 3) When calculating the ratio it should be in levels not logs, if my theoretical model (dynare file) is in levels, right?

Thanks!

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  1. There is no general answer here. But usually your model will imply cointegration. If it does not hold, then you are in trouble when using perturbation solutions. So your unit root test is meaningless if theory tells you there is cointegration.
    2.Whether to use the level or the log depends on whether cointegration holds for levels or logs. Usually, it’s the ratio of the levels that is stationary.
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By

usually your model will imply cointegration.

you mean cointegration between variables in the theoretical dynare model?

And

So your unit root test is meaningless if theory tells you there is cointegration.

means that even if my real data ratio series is or not stationary don’t matter, in regard for the theoretical model calibration?

In summary, what I understand is that since the theoretical model implies cointegration (given that there exists a solution), then I should not care about the stationarity of the real data ratio, is that right?

In that sense, I should just take the average that the series gives me, and should not estimate the average of any ARMA on it?

Thanks!!

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What I am saying is: model and data should be consistent. If you think that your test not indicating cointegration is just due to low power of the test, then it’s fine. But if you really think that cointegrations is not present, then that is a problem for your model if it assumes cointegration to hold.

Of course, you can take the approach of assuming that the ARMA process models the failure of cointegration due to features outside of the model and what is left is what you put into the model.

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Hi. I just came up with an idea that seems to work econometrically. In order to obtain the empirical ss ratio, if I want to target for example investment to GDP ratio, I estimate the cointegration equation: I_t=\beta Y_t+\varepsilon_t (with no constant) by OLS, and then if \hat \varepsilon_t is stationary, my steady state target would be simply \hat \beta. In other words, if I find cointegration in the form of I_t=\beta Y_t+\varepsilon_t, then empirical steady state ratio is \hat \beta\equiv I_t/Y_t. Is something wrong with that reasoning? (apologies if maybe this was what you meant since the beginning, but I’ve been having a little trouble understanding which empirical ratios may be valid to target, since I want to be as rigorous as possible. Worth mentioning that at the end, using various forms of getting such a ratio, the results are quite similar between them).

Also, would it be too bad just to take the average of the ratio (without considering cointegration)? Since for example, It would help a lot the calibration process to have a real money balances to gdp ratio, but in the country I’m taking the data from, this ratio exhibits a clear non-stationary behavior, but maybe trying the previously explained method would work out well.

Thanks a lot!!

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I fail to grasp why this approach would help. If cointegration fails to hold in the data, then there exists no steady state ratio you should use. But your model seems to imply that such a ratio exists. Only one of the two can true!

If you believe that such a fixed stationary ratio exists, then usually the sample mean is a good estimator. A different estimator would be the \beta. Both should have similar values and usually you can’t go wrong with either for them.

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Thanks!!