# Nominal interest rate as jump variable

I use Dynare to calculate a second-order approximation to a New Keynesian model already available as a linearized version in the literature. There the gross nominal interest rate R is assumed being a jump variable. To verify the correctness of my model I calculate a first-order approximation of my model to compare the IRF to those resulting from the linearized system available in the literature.

Assuming R as a jump variable in the linearized system is unproblematic while I have to assume R as predetermined in the first-order approximation of my model because otherwise Dynare has a problem with the Blanchard-Kahn conditions.

To get a better idea here are the equations from the optimization problem of the household causing the problems together with some explanations.

(1) c(t)^(-sigma)=lambda(t) + mu(t)
where lambda and mu are the Lagrange multiplier of the budget constraint and the CIA constraint, respectively

Euler equation
(2) lambda(t)=beta*E_t[lambda(t+1)*R(t+1)/Pi(t+1)]
where Pi is the inflation rate and R is the gross nominal interest rate

(3) lambda(t)=beta*E_t{[lambda(t+1)+mu(t+1)]/Pi(t+1)}

Because of certainty equivalence when linearizing the system one can transform (1) into

(1a) lambda(t)= c(t)^(-sigma)/R(t)

As a result there is only an R(t) left but no R(t+1) (R(t+1) in the Euler equation cancels out).

Nevertheless I have to pay attention to the expectation operators and can not do this simplification. But this leaves me with R(t+1) in the Euler equation which causes a Blanchard-Kahn condition problem.

I understand that it is per se problematic to have the interest rate determined in t+1. The question now is if Dynare is not able to ignore the expectational operators in a first-order approximation. Because if I use equation (1a) instead of (1) as part of my equation , that is do the simplification by hand, this gives me the same results as the linearized system.

Regards,
Bj

Dear Bjorn,

1. I’m a little suspicious of the way the problem is solved for linearization. I wonder whether the original dynamic assumption isn’t modified by the substitution.

2. Could you explain what is different in the determination of the interest rate and what makes it a jump variable?

Best

Michel

Dear Michel,

Equation (1a) is the result of

• Equating (2) and (3) ignoring the expectation operators E_t

–> (4) R(t+1)=1+(mu(t+1)/lambda(t+1))

-Changing 1 to
c(t)^(-sigma)=lambda(t)*(1+mu(t)/lambda(t))

Inserting (4) in (1) yields (1a) where again the expectation operators are ignored because pushing R(t+1) one period back to R(t) would actually depend on E_(t-1).
But I thought that in a first-order approximation ignoring the expectation operators would be okay. And the question is if Dynare does not ignore the expectation operators so that for Dynare the simplifications actually change the dynamics of the system.

In the linearized version using (1a) the interest rate R(t) is determined in period t. While without the simplification the interest rate is determined in period t+1 according to the Euler equation. Besides the Euler equation there is no equation that contains the nominal interest rate R.

The assumption of the interest being a jump variable is based on the following proposition of the paper where the linearized case was examined: (The head next to the variable denotes the percentage deviation from steady state: x^s=log(x^s/x*))

In the log-linear approximation to the model with exogenous money, there exists a unique rational expectations equilibrium converging to the steady state if prices are sufficiently rigid so that
θ < (1 + β)/(σ[c+g]/c - 1). The impact multipliers of a shock to government expenditures at time s then are ∂y

Could you send me an electronic version of the original paper?

Thanks

Michel

Hi Michel,

please find attached the electronic versions.

Thanks,
Bj
Article.pdf (416 KB)
Appendix.pdf (131 KB)

The difficulty that you are encountering may be coming from the monetary policy rule (equation 13). In the text it is

R(t+1) = r_x E(t)(pi_hat(t+1))+r_yE(t)(y_hat(t+1))

This is equivalent to say that this period interest rate is set as

R(t) = r_x E(t-1)(pi_hat(t))+r_yE(t-1)(y_hat(t))

The difficulty is then to modelize the lagged conditional expectation in Dynare.
You need to introduce two auxiliary variables, epi and ey, and the two following equations

epi = pi_hat(t+1)
ey = y_hat(t+1)
Those are the conditional expectation in period t. Then you can write the monetary policy rule in period t as

R = r_xepi_hat(-1)+r_yey(-1)

If you do that, you can keep R(+1) in the Euler equation.

Tell me if it helps

Kind regards

Michel

Hi Michel,

Unfortunately I do not use the equation for the interest rate rule because the monetary authority follows the constant money growth rule (eq. 12) so that the interest rate is determined endogenously.

Kind regards,
Bj

OK…
Why don’t you use directly equation (5):
R(t) = c(t)^-sigma/R(t)

Can you send me the *.mod file?

Thanks

Michel

Hi Michel,

thanks for spending your time on that.

I do not use equation (5) because I would like to directly compare the results resulting from the linearized model with those from the second-order approximation. But if I can not use the same timing assumption for R the comparison would be meaningless.

(1) linearized version
(2) the model using eq. (5) from the article
(3) the model without simplifications for second-order approximation

Regards,
Bj
Modified.mod (1.65 KB)
second_order.mod (1.64 KB)
linearized.mod (1.5 KB)

Thanks, I will get back to you in a couple of days

Best

Michel

It takes me longer than planed, but I haven’t forgotten…

Michel

Hi Michel,

no problem. I had some problems accessing this page in the last two weeks, that is why I come back to you so late.

In the meantime I found out that independent of the definition of the nominal interest rate as a jump variable or as being predetermined the dynamics of the system in response to a shock do not change except for the nominal interest rate. But this is inconsequential due to the fact that the real interest rate in the Euler equation changes as well if one assumes the nominal interest rate being a jump variable.

Regards,
Bj