I want to find out how black money in the economy affects the GDP of the country and the currency valuations. What are the variables I should investigate?
I trust someone will correct my mistakes in attempting to answer the question. I think the idea of "black money" refers to the income received in a black market exchange. I think most of us expect this income is intentionally hidden to escape taxation, hide illegal activity, or both. This is a rather narrow definition of "black money" and I will pause to elaborate.
Strictly speaking, GDP changes little if at all as a direct result of the illegal activity, because illegal goods and services are specifically excluded from GDP. The valuation of the currency relative to others is a potentially different answer because this is theoretically based on the relative demand and supplies of different currencies in the exchange market(s). Increased demand for goods in the international markets, legal or otherwise, generally increases the demand for the currency to purchase these goods. This should put upward pressure on the exchange rate (i.e., the price of the currency) of the country exporting these goods (holding everything else constant of course). This may be oversimplified for brevity.
The more interesting part of this question is about variables that bring out the relation between black money and the domestic economy. In addition, although GDP excludes illegal activity and sub rosa legal activity, it is still economic activity even it it is not measured. The so-called Grey-Market is related to the question and would add to the complexity.
Some of the variables include things like an estimate of money laundering, which may not be available. Some have noted the use of $100 bills in US currency might be a surrogate variable for some illegal activity and although the distribution from different Federal Reserve banks is known, it is a stretch to quantify the relationship if it exists. Stretching even further, one might detect changes in the the velocity of money while mathematically holding GDP and the money supply constant; obviously, things like this are questionable at best because the purchase of used goods is legal and would have a similar effect. In summary, I failed to offer a solid variable that accurately measures what you seek. Hopefully, it will stimulate ideas or other responses that will help.