A dollar neutral strategy invests the same amount of money long and short without accounting for the volatility (risk) of either side. Depending on volatility you either end up positively or negatively correlated with the market.
A market-neutral strategy eliminates the correlation to the market by hedging the long side with an equally risky (=same volatility) short side. E.g. A hedge of a stock portfolio with a short position on the S&P500 Future. The size of the short position is chosen in a way that the resulting strategy doesn’t correlate with the S&P any more.
I have a dollar neutral strategy that is positively correlated with market but the value of net beta is not zero. What should be the appropriate benchmark for this strategy? Are there any methods to construct a portfolio that can be a benchmark for this strategy ( like regressing market returns with dollar neutral etc) ?