This Great Graphic show the dollar against the yen (white line) and the Nikkei (yellow line), since the eve of the election that swept the LDP back into power in Japan, and heralded Abenomics.  

Click on picture to enlarge

It shows the two track each other, and sometimes, link now, very tightly. On a day-to-day basis, it may not always be clear which is leading, but the co-movement is nearly intuitive. A bullish dollar yen view is also a bullish Nikkei view and vice versa.  

This is confirmed in the correlation between the level of dollar-yen and the level of the Nikkei.   The two time series move in the same direction about 86% of time on a 60-day rolling basis. This is down slightly from the year's high in early October near 90%.  

However, it is also important for investors to consider the correlation of the returns of the two instruments. Among other things, this is critical for hedging decisions and other money management issues.   And here the correlation is considerably lower, but still not insignificant.  The correlation of the percentage change of the dollar-yen exchange rate and the percentage change in the Nikkei is around 32.5%, on a 60-day rolling basis.  In August it peaked above 48%.  In June 2013 it was above 58%.  

The correlation of returns is also not particularly stable. Consider this 60-day rolling correlation had been inverse for a few weeks this past July.  In addition, the correlation might not be causal. The combination of the BOJ's Halloween surprise, the same day the Fed formally ended QE3+, and the pension fund diversification helped spur both the Nikkei and the dollar-yen. This has been followed by two important things. First, there does not seem to be much of an official push back against the yen's relatively sharp drop. We think there is a reasonable chance of this as the JPY120 area is approached.  Second, the decision to postpone the sales tax may have also favored shares and the dollar over the yen.   

Nevertheless, some analysts insist on projecting the target of one--say dollar-yen--to the price of the other. For example, if one expects the dollar to rise another 10% against the yen to say JPY130, then some analysts will translate that to say a 13% move in the Nikkei or roughly 19500.  What this means is that dollar-based investors will give up the bulk of their possible gains in a basket of Japanese stocks through the potential exchange rate developments. There are various ways institutional investors can hedge the currency risk.  

Retail investors have a few choices.  One of the popular ETFs in the space is DXJ, which is a Wisdom-Tree ETF that invests in a basket of dividend paying Japanese stocks and hedges out the yen exposure). The small cap variant is DXJS. This is not a recommendation but a description of ways that investors can and do separate the equity investment from the currency risk.