<p>It’s derived as follows. First start with original equation.</p>

<p><span class="math-container">$$(1+i_t)=(1+i^*_t)\frac{S_{t+1}}{S_t}$$</span></p>

<p>Take natural logs of both sides:</p>

<p><span class="math-container">$$\ln(1+i_t)=\ln(1+i^*_t)+\ln(S_{t+1}) -\ln(S_t)$$</span></p>

<p>Now you just use the following:</p>

<p><span class="math-container">$$s_t=\ln(S_t)$$</span></p>

<p>And use the well known fact that for small values of <span class="math-container">$i$</span> the following approximation holds:</p>

<p><span class="math-container">$$\ln(1+i)\approx i$$</span></p>

<p>And rearrange to get </p>

<p><span class="math-container">$$\hat{i}_t-\hat{i}^*_t=\hat{s}_{t+1}-\hat{s}_t$$</span></p>

<p>PS: Also actually the equation you derived is not log-linearized as it’s not linear in its log parameters </p>

CC BY-SA 4.0
2020-02-29T10:47:11.033Z
2020-02-29T10:53:22.843Z
2020-02-29T10:53:22.843Z
1muflon1
<p>I am trying to derive the log-linear version of the uncovered interest rate parity under complete asset markets.</p>

<p>I know that the UIP condition is given by</p>

<p><span class="math-container">$$(1+i_t)=(1+i^*_t)\frac{S_{t+1}}{S_t}$$</span></p>

<p>I have seen in papers that the log-linear version is simply given by</p>

<p><span class="math-container">$$\hat{i}_t-\hat{i}^*_t=\hat{S}_{t+1}-\hat{S}_t$$</span></p>

<p>but I cannot see how this is derived. According to me, the log-linear version should be</p>

<p><span class="math-container">$$\frac{i}{1+i}\hat{i}_t-\frac{i^*}{1+i^*}=\hat{S}_{t+1}-\hat{S}_t$$</span></p>

<p>Could someone please write down all steps to obtain the log-linear version of the UIP?</p>

<p>Thank you very much</p>

<macroeconomics><mathematical-economics><open-economy-macro><new-keynesian-economics><dsge>
Log-linear version of the uncovered interest rate parity
Alex Ruiz
# Log-linear version of the uncovered interest rate parity

(Asked by: username_0 on 2020-02-28)

I am trying to derive the log-linear version of the uncovered interest rate parity under complete asset markets.

I know that the UIP condition is given by

$$(1+i\_t)=(1+i^\*\_t)\frac{S\_{t+1}}{S\_t}$$

I have seen in papers that the log-linear version is simply given by

$$\hat{i}\_t-\hat{i}^\*\_t=\hat{S}\_{t+1}-\hat{S}\_t$$

but I cannot see how this is derived. According to me, the log-linear version should be

$$\frac{i}{1+i}\hat{i}\_t-\frac{i^\*}{1+i^\*}=\hat{S}\_{t+1}-\hat{S}\_t$$

Could someone please write down all steps to obtain the log-linear version of the UIP?

Thank you very much

**Tags:** macroeconomics, mathematical-economics, open-economy-macro, new-keynesian-economics, dsge

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## 1 answers

### Answer by username_1 on 2020-02-29. Score: 2

It’s derived as follows. First start with original equation.

$$(1+i\_t)=(1+i^\*\_t)\frac{S\_{t+1}}{S\_t}$$

Take natural logs of both sides:

$$\ln(1+i\_t)=\ln(1+i^\*\_t)+\ln(S\_{t+1}) -\ln(S\_t)$$

Now you just use the following:

$$s\_t=\ln(S\_t)$$

And use the well known fact that for small values of $i$ the following approximation holds:

$$\ln(1+i)\approx i$$

And rearrange to get

$$\hat{i}\_t-\hat{i}^\*\_t=\hat{s}\_{t+1}-\hat{s}\_t$$

PS: Also actually the equation you derived is not log-linearized as it’s not linear in its log parameters


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