# 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.