# UIP condition and initial response of nominal exchange rate

#1

Hello everyone,

I have a very general question, but I cant seem to find an answer that I understand.

In open economy models we typically use the UIP condition to determine future changes of the interest rate. In a very simple linearized form this is given by:

R_{t} - R_{t}^{f}*riskpremium = E_{t}*S_{t+1} - S_{t} = \Delta S_{t+1}

Thus, whenever the domestic interest rate R_{t} falls below the foreign interest rate R_{t}^{f}, the domestic currency has to appreciate (RHS is negative) in expectation to compensate investors for the lower interest rate. It can do so by a) an increase in S_{t}, i.e. a depreciation of the currency today or b) an appreciation of the currency tomorrow, i.e. a fall in E_{t}*S_{t+1}.

For a monetary policy shock we typically see, that a) is chosen and thus the currency depreciates on impact, if the interest rate falls and then appreciates from t+1 on.

My question here is: What determines the initial reaction of the nominal exchange rate in time t S_{t}? What decides the initial response of the nominal exchange rate and thereby the first deviation from steady state ( S_{t} - S_{t-1} = \Delta S_{t})

Any intuition or help would be very much appreciated.

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#2

Hi,
The determination of the nominal exchange rate depends on the sum of interest rate differentials and the terminal value of the nominal exchange rate, that is usually undefined for models with inflation targeting regimes. So, iterating forward the UIP condition you will have

S_{t} = - SUM(R_{t} - R_{t}^{f}*riskpremiun + S_{bar}

In order to compute the terminal value S_{bar} in dynare, you can use the cumsum function of matlab in the impulse responde function for Delta_{t}.

best

D

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#3

Hey,

thank you very much for the response. I understood your point, but I dont think that it answered my question. At least I didnt understand it then

My question was, what determines the first interest rate differential. I.e. S_{t} - S_{t-1} = \Delta S_{t}. The UIP condition only determines the interest rate differential one period ahead \Delta S_{t+1}.

I know that I could just accumulate the changes in the nominal exchange rate, to get the level of the real exchange rate. But I want to understand, how the initial response is determined.

For instance, after an expansionary monetary policy shock, we usually see a large depreciation on impact, followed by an appreciation of the currency. Surely this is consistent with the UIP condition, but as stated above, the UIP condition only determines the expected future change in the exchange rate. The UIP condition in this case states, that the nominal exchange rate is expected to appreciate in period t+1. But it does not state, that in order to do that, the nominal exchange rate has to depreciate on impact. The currency could have not moved on impact and appreciated in t+1 and the UIP condition would still be fulfilled. I want to know what decides the division of tasks in the UIP condition for the initial impact. As shown in the formular below, a fall in R_{t} can either be compensated by a fall in S_{t+1} or an increase in S_{t}. Sorry, if I wasnt clear.

R_{t} - R_{t}^{f}*riskpremium = E_{t}*S_{t+1} - S_{t} = \Delta S_{t+1}

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#4
1. I get the impression that you are thinking of S(t+1) and S(t) as somewhat independent, for ex as in two regressors in a regression. S(t) also influences S(t+1).
2. The direction and the quantum of the move of the exchange rate on impact are both determined by the interest rate differential. In the long run, S(t) will converge to a new level, i.e. a new steady-state, that is consistent with the money supplies at home and abroad.
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#5

I see, they depend on one another and thus its impossible to disentangle the two. Thanks for all your helpful answers. My questions comes about, because I am puzzled by a response to one of my shocks. I have a two country DSGE model, which I tested a lot and so far the results always seemed reasonable. I now used it for country specific uncertainty shocks and approximated it to a third order using dynare. I followed all of Prof. Pfeiffers advices, compute the IRFs from the Ergodic Mean in the Absence of shocks, checked for stationarity of the stochastic steady state of the model and so one. My problem is that, country x interest rate fall, while country y interest rate rises. But in contrast to my intuition, country x currency appreciates on impact. I ran a monetary policy level when using the third order approximation and everything seemed fine. I am struggling to understand what is going on and the result persists, even when I make prices and wages completely flexible and simplify the model drastically. If you have any intuition, please let me know. Thanks so much for your help.

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#6

@BSchumann You need to keep in mind that your UIP condition above is based on a linearization, i.e. only holds at first order. But you are considering a higher order approximation.

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#7

Hello Prof. Pfeifer,

thats of course true and in the code I also use the non-linear one. I just linearized for the purpose of an easier understanding. I also though that my results may be due to nonlinearities in the function and therefore I linearized it once in the code in order to compare the differences. The result still persists. After an uncertainty shock the policy rate increases (interest rate differential are positive), but the currency is depreciating.

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#8

But a linearized equation cannot give you the intuition for a shock that works via Jensen’s Inequality.

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#9

I understand. As I said, I only linearized the equation once, to see if my results come from the non-linearities in the UIP condition. But you are absolutely right, that in order to get an intuition, I need the nonlinear equations. What i am trying to do is a bit similar to what you did in your paper on terms of trade uncertainty. The major difference being, that I use a two country model instead of a small open economy one.

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#10

I think you need to argue about covariances. See the original 2011 AER “Risk Matters” paper.

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#11

Dear Prof. Pfeifer,

thanks a lot for the hint. It really helped a lot and I am now much closer. I am just still a bit puzzled how the initial response of the nominal exchange rate is determined in open economy models. Even If we abstract from higher order approximations, I still cant seem to find an answer. The UIP condition in time t determines the change in the exchange rate in time t+1, that I get. If we go one step back in time, the UIP condition in t-1 determines the change in the exchange rate in t? That cant be, as the exchange rate in t, would thus allways be the steady state exchange rate I am not mistaken. So I am wondering, how the change in the exchange rate in t is determined.

In Period t:

In Period t-1

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#12

does not necessarily have to hold as shocks would have realised. UIP only has to hold in expectation.
R_ss/R_f_ss= (1+DeltaNEER_ss);

That is true.

Reuben

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#13

I agree, but in a linearized model or a first order approximation, we have certainty equivalence, so the change in the exchange rate should allways be determined by the previous period interest rate differential. As I said, my puzzle is just that the UIP condition just determines the change in the exchange rate in t+1. I am wondering where to look, when I want to understand the change in the exchange in t. I know that those two are not independend and in a DSGE modell, everything depends on each other, but I am looking for an equation to explain the intuition for the change of the exchange rate in t. The UIP is just not suited for that I guess.

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#14

I am not sure I understand your confusion. Take an Euler equation. It describes the relation between consumption today and consumption tomorrow given interest rates. Put differently, it determines consumption growth. What determines consumption today then? Iterating the Euler equation forward, you get the complete future path of consumption growth rates. What determines the initial value for consumption from which you start growing? It’s the transversality condition. For any other path than the saddle path, either debt would explode or consumption along the full path would be inefficiently low. A similar logic applies the the UIP.

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#15

Thanks for your answer Prof. Pfeiffer. My UIP conditions (one for each country) are given by:

Home country:

`R_{t}/R_{t,f}*riskpremium =`

`E_{t}[UC_{t+1} * S_{t+1}/{S_t} * P_{t}/P_{t+1}] / Et[UC_{t+1}*P_{t}/P_{t+1}]`

Foreign country:

`R_{t,f}/R_{t}*riskpremium =`

`E_{t} [UC_{f,t+1} * S_{t}/S_{t+t} * P_{f,t}/P_{f,t+1}] / Et [UC_{f,t+1}*P_{f,t}/P_{f,t+1}]`

With UC as the utility of consumption and P as the CPI of the country.

To follow your logic I rewrote the UIP condition and cancelled the lagrange multipliers and the price terms (which I can only do for a first order approximation).

Home country:

`R_{t}/R_{f,t}*riskpremium = E_{t}[S_{t+1}/S_{t}] -->`

`S_{t} = R_{f,t}*riskpremium/R_{t} *E{t}[S_{t+1}]`

Foreign country:

`R_{f,t}/R_{t}*riskpremium = E_{t}[S_{t}/S_{t+1}] -->`

`S_{t} = R_{f,t}/R_{t}*riskpremium/*E_{t}[S_{t+1}]`

I see your point now, that current the current level of the exchange rate is a function of current interest rates and the future expected exchange rate. I also get your point, that the covariance terms will matter for a third order approximation given the formula above. Given that I have three forward looking variables in the expectation operator in the true equation (the first one above), it becomes much more complicated to track covariances and their impact.

Also the model gives two UIP equations, with the only real difference being the risk premium. As the risk premium is a function of bond holdings, the difference bond holdings determine the exchange rate and vice versa, which is something that I also never thought about.

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#16

Hi what exactly do you mean by this statement? With 2 interest rates and 2 currencies, one should only get one UIP condition, right? One can derive UIP for both countries, but they are essentially the same equation, right?
Only the net bond position, is identified in a two-country model right? The risk premium is a function of the net bond position.

Reuben

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#17

Hi Reuben,

I willl try to explain you, why I think its important two have both UIP conditions instead of only 1 when we go for higher order approximations.

The resource constraint is given by the following equation

( S_{t}*B_{f,t}^{H} is the nominal value of Bonds issued by foreign households and held by domestic households converted to domestic currency, ,CA is the current account):

S_{t}*(B_{f,t}^{H}/ R_{f,t} - B_{f,t-1}^{H}) = B_{h,t}^{F} /R_{t} - B_{ht-1}^{F} + CA

–> The difference between the nominal Flows of Bonds is the current account.

The optimal allocation of foreign bonds held by domestic households is given by (UC is the lagrangian)

UC_{t} = R_{f,t}*exp(-riskpremium(B_{f,t}^{H}) * E_{t} [UC_{t+1} * S_{t+1}/S_{t}]

Here we added the risk premium to pin down the level of foreign households held by domestic agents in steady state (0).

The optimal allocation of home bonds held by foreign households is given by:

UC_{f,t} = R_{t}*exp(-riskpremium(B_{f,t}^{H}) * E_{t} [UC_{f,t+1} * S_{t}/S_{t+1}]

Again, we added the risk premium to pin down the level of bonds held by foreign households.

We could combine the two equations with the FOC for domestic bonds to get the two UIPs, but I think its easier to make the point here.

What you suggest is to set one of the level of bond holdings to 0 (its determinstic steady state) or simply not deriving the FOC for the optimal level of bonds held by foreigners, such that the resource constraint becomes.

S_{t}*(B_{f,t}^{H}/R_{f,t} - B_{f,t-1}^{H}) = 0 - 0 + CA

Given the steady of B_{f,t}^{H} of 0, this unique identifies the level of bond holdings. Up to a first order approximation, this is fine. But for higher order approximation you will get a covariance term in the FOCs (thanks to you I got that).

UC_{f,t} = R_{t}*exp(-riskpremium(B_{f,t}^{H}) * E_{t} [UC_{f,t+1} * S_{t}/S_{t+1}]

–>

UC_{f,t} = R_{t}*exp(-riskpremium(B_{f,t}^{H}) * E_{t} [UC_{f,t+1}] * E_{t}[S_{t}/S_{t+1}]] + Cov(UC_{f,t+1}, S_{t}/S_{t+1}])

As you can see, if Cov(UC_{f,t+1}, S_{t}/S_{t+1}]) is unequal to 0, the stochastic steady state of bond holdings is no longer 0. In fact in my models, Bond holdings in the stochastic steady state are positiv, as the expected depreciation of the exchange rate is positvely correlated with a fall in consumption (an increase utility).

By effectively setting the level of foreign bond holdings to 0 (which is equivalent to not deriving the equation), you neglect the effect of uncertainty on bond holdings for this country. In a symetric model, you will have asymetric stochastic steady states for instance.

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