Theoretical Derivation
The formal model underlying the hypothesis is the uncovered Interest Rate Parity condition which states that in absence of a risk premium, arbitrage will ensure that the depreciation or appreciation of a country's currency vis a vis another will be equal to the nominal interest rate differential between them. Since under a peg, the exchange rate cannot change, short of devaluation or abandonment of the peg, this means that the two countries' nominal interest rates have to be equalized.
This in turn implies that the pegging country has no ability to set its nominal interest rate, and hence no independent monetary policy. The only way then that the country could have both a fixed exchange rate and an independent monetary policy is if it can prevent arbitrage in the foreign exchange rate market from taking place - institutes capital controls on international transactions.
Read more about this topic: Impossible Trinity
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