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I'm look for historical data on the 30-Day Federal Funds futures contract, in order to replicate the results of the following article:

https://www.sciencedirect.com/science/article/pii/S0304393208000494

I want the expiration to be rolling, such that I for each observation, get the price that market participants were willing to pay for a contract that expires in 30 days.

I have so far been looking at the contract called "ff1 comdty", which refers to something called a "Generic 1st Fed Funds Futures" contract. What does it mean that it is "Generic 1st"? And is it the correct contract type for my purpose?

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The real Fed Fund Futures are constructed by the exchange and set to run from start to end of a calendar month, so unfortunately there is no rolling future price. The "generic 1st" contract is just a placeholder which rolls from one contract to the next at the beginning of the month.

The closest you are likely to get is the 1m USD OIS price; this is a fixed rate in exchange for daily-compounded Fed Fund rates. Its length will vary with the calendar, but it should largely suffice.

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Eurodollar futures are a good approximation of fed funds expectation. You can get it on BBG via ED ticker. Often Eurodollar futures are used to determine probability of hikes/cuts. One thing to note, Eurodollar have some credit risk because it is related to LIBOR. To remove it you can strip it with the OIS 1m.

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I do not understand why you need rolling expiration. My understanding of that article of Piazzesi Swanson 2008 is that they use the daily last price of the "Generic 1st Fed Funds Futures".

Their goal is to extract daily risk-free changes in market expectations about monetary policy. Since the price of a Fed Funds Futures contract is for the monthly average of the effective Fed Funds rate you have to interpolate that value to get a daily change. In particular, one considers the daily change in the FF1 last price and multiply it by a factor based on the day of the month.Intuitively, price changes at the end of the month are more relevant that at the beginning. See KUTTNER Kenneth 2000 Monetary Policy Surprises and Interest Rates: Evidence from the Fed Funds Futures Market for a detailed discussion of how and why this factor is computed.

In general, the swap markets for interest rates is newer (less time series data), smaller and less liquid that CBOE Fed Funds Futures market.

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