Traditionally, when we think about currency pegging, we think that if two countries are exposed to highly correlated shocks, then pegging is not a bad idea. E.g. if Austria and Germany have the same business cycle, it is not a bad idea for Austria to adopt German monetary policy. By this logic, it is a bad idea for Qatar to adopt US monetary policy.
Now suppose there are two countries which have common shocks but dissimilar per capita income and thus different CPI consumption baskets. This leads to poor correlation between the two CPI measures. Example: one is a poor country and has a high weight for food and fuel while the other is a rich country where this is not the case.
Now suppose the poor country runs a pegged exchange rate to the rich country. Even though the two countries have common shocks, the inflation process in the poor country will be different when compared with the rich country. A monetary policy that delivers low and stable inflation in the rich country will fail to do this for the poor country even if the two countries have common macroeconomic shocks. The rich country will achieve the frontier in terms of low output gaps with a low and stable inflation rate, but the same would not be the case with the poor country.
By this logic, if Canada and Mexico have similar correlations with the US business cycle, it's less sensible for Mexico to adopt US monetary policy. By this logic, there are two strikes against India adopting US monetary policy by running a pegged rupee-dollar rate: Neither are the shocks common nor are the CPI baskets alike.
Differences in CPI baskets have another interesting implication. If one measures competitiveness by using the REER, but with CPIs that have very different weights, one could come to potentially odd conclusions. Even if PPP holds perfectly, CPIs with different weights can diverge. One country will show a loss of competitiveness, even though by definition competitiveness hasn't changed at all.