# Exogenous variables in model estimation

I am trying to estimate a small open economy model. The model is rather large: to simplify, let’s say I have four equations (IS-PC-MPR-exchange rate) with four endogenous variables (interest rate, output gap, inflation, exchange rate delta), three shocks (one per equation) and one exogenous variable, Fed Funds rate.

So the model looks something like this:

``````i = c+ alpha1*i(-1) + alpha2*(pi - pi_target) + v
pi = g + beta*outputgap + beta1*pi(-1) + beta2*fx(-1) + u
outputgap = k + gamma*(i(-1) - pi(-1)) + e
fx = i - i_fed
``````

This is a simplification, so please do let me know if I’m unclear. I am trying to replicate a paper by the BCB which I can provide if needed.

I would be very grateful if you could please help me with the treatment of the fed funds rate. I’m very much a beginner, so please do redirect me elsewhere if I have missed a previous explanation. From what I understand, there are three ways to incorporate the path we know of the fed funds rate when estimating the parameters of the model:

1. treating it as a varobvs, which however in this case presents both a practical issue (not enough equations per endogenous variables) and a theoretical issue (the fed doesn’t care about this particular economy, so it’s not endogenous, and also it cannot be predicted from within the model)
2. treating it as varexo, in which case we have two choices: either we say we’re in a stochastic world, and then we can only set the variance of shocks, not their path - which loses the very convenient information that we gain from knowing the path of the fed funds rate at the same time we were observing the rest of the variables; or we say it’s deterministic and we set a path, which however means that the agents will incorporate the information about it from the very start, which however isn’t the case, since the fed isn’t observed until time t in this model, and there’s not other way to predict what they will do.

I guess my question is then: how do we set the path of exogenous variables for the purpose of estimation, if the path of these variables is known? if useful: I have also been reading the documentation on histval, which however as far as I understand 1. doesn’t take a vector as argument, so I would have to input the values manually 2. doesn’t make the dates of the variable correspond to the dates of the endogenous variables. Similarly initval and endval don’t allow for the path but only for some equivalent of a steady state value (which makes sense to me on the basis of the documentation).

I’m sure I have missed something enormous but I am unable to find what it is! Please do redirect me to some documentation/example files if you have some - all the ones I find have varexos being strictly shocks, so more the equivalent of e, v and u as above.

Thanks so much in advance, and apologies if I’m missing some important chunks of Dynare.

Let’s step back for a moment. From the perspective of the small open economy, the foreign variables are exogenous, but they still follow some endogenous dynamics. The typical way to proceed then is to specify the foreign block as a VAR (e.g. Bayesian estimation of an open economy DSGE model with incomplete pass-through - ScienceDirect). In that case, there is some predictable dynamics but you maintain an equality between the number of equations of variables.

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That makes sense, thank you so much for your answer. I’ll probably proceed as you’re recommending; however I am still curious about “true” exogenous variables.

An example could be earthquakes for a fiscal spending model or the weather for a food prices model - for these we do have historical data but they’re virtually unpredictable, so that intuitively I would think we could use the information in our estimation by incorporating their paths, but they wouldn’t really be possible to forecast endogenously. Could you please help me understand how these would be incorporated in Dynare? Just by specifying their variance in the shocks ; block, or otherwise?

In that case, you would simply add an equation telling you that an exogenous shock is observed when it appears.

How would we then chose the stderr of the observed shock? Would it make sense to calibrate the value of stderr close to zero?

That depends on what you are trying to do. At first order, the standard deviation does not matter for other properties of the model due do certainty equivalence.

I am interested in the estimation of the model. As of now, I set the stderr equal to the value calculated from the exogenous series. I would be interested in understanding better how the stderr affects the estimation of all the other parameters in the model, when using the kalman filter.

If this is an exogenous series and observed, it should not have interaction effects with the rest of the model.