An approximation of quarterly growth from monthly growth

An approximation from Justin Wolfers: “Quarterly growth ˜ 0.33 * this month’s growth + 0.67 * (t-1) + 1.0 * (t-2) + 0.67 * (t-3) + 0.33*(t-4)”.

I stared at this one for a while. But it’s actually pretty easy to prove, assuming that we’re expressing growth as a difference and not a quotient. Say it’s month t now. The quarterly growth of some quantity f which varies with time, for the most recent quarter over the quarter before that, is f(t) + f(t-1) + f(t-2) - f(t-3) - f(t-4) - f(t-5).

The weighted sum of monthly growths there is

{1 \over 3} (f(t) - f(t-1)) + {2 \over 3} (f(t-1) - f(t-2)) + {3 \over 3} (f(t-2) - f(t-3)) + {2 \over 3} (f(t-3) - f(t-4)) + {1 \over 3} (f(t-4) - f(t-5))

and most terms here cancel, leaving

{1 \over 3} (f(t) + f(t-1) + f(t-2) - f(t-3) - f(t-4) - f(t-5)).

This is one-third the quarterly growth in Wolfers’ tweet – but if both figures are annualized, as is conventional in economic data, that takes care of the factor of 3.

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