I discussed in a previous post about the shortcomings of the Big Mac Index and how these issues might lead to misuses of this index. As one would expect, there is a thin academic literature which discusses or uses the Big Mac Index, however the studies that exist make some interesting points which deserve to be mentioned.
In their article in The Economic Journal, Parsley and Wei study the real exchange rate based on the Big Mac prices. I would say surprisingly, they found some advantages in using Big Mac prices over the standard CPI. Moreover, what is rather considered a weak point of the Big Mac Index, the fact that the share of non-tradables is rather large (which they estimated it at about 60%), becomes an advantage when one considers that the structure of the Big Mac Index is known better than the one of the CPI. Some of the key findings of this study: the Big Mac Index and the CPI are correlated which, according to them, suggest the possibility of extending the results based on Big Mac to those based on CPI; using this exchange rate based on Big Mac, they were also able to reassess a few puzzles in exchange rate economics (persistence and the proposition that exchange rate movements are determined only by deviations in the law of one price gap in tradables).
Another interesting paper is due to Clements, Lan and Pei. The authors are pretty much aware about the limitations of the Big Mac Index. However, when its biases are eliminated or diminished, the Big Mac Index can be used to predict the real exchange rates. On top of that, its forecasting performance is even better than the one based on random walk, mostly in the medium to the long run.
In the end, these studies show that the Big Mac Index has some informative value; however its value becomes clearer when economists are aware about its limitations and biases and correct them in a meaningful way.
Macroeconomics Wikipedia: Real Effective Exchange Rate