I know that empirically the uncovered interest rate parity fails. But let's set that aside for a moment.
The uncovered interest rate parity says:
$$(1+r_{USD}) E_{EUR,USD} = (1+r_{EUR}) $$
where $r_{USD}$ is the interest rate on the dollar deposit, $E_{EUR,USD}$ is the exchange rate (how many euros for 1 dollar) and $r_{EUR}$ is the interest rate on a euro deposit.
So if the interest rate on dollar increases we should expect the dollar to depreciate? Is my intuition correct?
Your equation is missing an important part. There is the current spot rate, as well as the future expected spot rate in the UIP equation.
$$(1+i_{\\\$})={\frac {E_{t}(S_{{t+k}})}{S_{t}}}(1+i_{c})$$
or rearranged:
$${{S_{t}}}\frac {(1+i_{\\\$})}{(1+i_{c})} = E_{t}(S_{{t+k}})$$
If you think of EURUSD now (how many USD per EUR, say 1.2, if US interest rate is 10% and EUR 5% you get (for a year), the value of
$${1.2}*\frac {(1+0.1)}{(1+0.05)} = 1.25714286$$
In other words, you need more USD per EUR - the USD depreciated, EUR appreciated.
Insofar, you are right that any higher interest in one country will be offset by an expected depreciation in that countries currency so that an investor will be equally well off. In other words it doesn't matter where you invest, as the future expected exchange rate offsets the interest rate differential.
This may sound counter-intuitive and leads to confusion because it makes sense to think people might be inclined to invest in the higher interest paying currency, thus leading to an appreciation of that currency. However, FX is super liquid. Therefore, spot will react asap (appreciate), so that later it can depreciate to restore equilibrium (parity).
There exists a widely used strategy called the "carry trade". For the carry trade to work, this cannot be the case (higher interest currencies do not depreciate as much).
Empirically, FX is more volatile than this relatonship suggests, which is why "overshooting models" were developed. These are part of the stock approach to FX modelling and consist of flexible and sticky price monetary models which combine capital markets, goods markets and money markets. Sticky price monetary models are also known as overshooting models, initially designed by Dornbusch (1976).
The essence of these models is that since FX reacts asap but goods prices are delayed, the spot rate must overshoot its value in the short run, to compensate for an even further depreciation ahead in time. This is for example shown in this PPT, which shows the mechanism of overshooting (in simply economics diagrams on slide 11/17). The picture is from FIGURE 4-12 of International Finance Theory and Policy 11th ed. by Krugman, Obstfeld and Melitz.
Actually, quite the contrary!
Indeed everything else been equal, dollar tends to appreciate. If you think the reason is quite simple: if ECB rate is 0% and FED moves from 0% to 0.25%, where do you want to have your money on? Dollar of course.
Everyone will have an incentive to buy Dollars (again, if interest rate was the only driver for FX) and its price would increase!
