Trend Regression Bookkeeping

Question

You're fitting a linear trend to 20 days of P&L by least squares, and the slope formula needs sum_t=1^20 t^2. Compute it exactly, state the general formula for sum_t=1^N t^2, and give a one-line sanity check on the order of magnitude.

Accepted Answer

Formula. sum_t=1^N t^2 = fracN(N+1)(2N+1)6. With N = 20: frac20 cdot 21 cdot 416 = frac172206 = mathbf2870. Mental route: frac20 cdot 216 = 70, then 70 cdot 41 = 2870. Where the formula comes from (telescoping again, the workhorse of closed forms): sum (k+1)^3 - k^3 = 3k^2 + 3k + 1 over k = 1..N. The left side collapses to (N+1)^3 - 1; the right side is 3sum k^2 + 3fracN(N+1)2 + N. Solve for sum k^2 and factor. Sanity check. A sum of squares is roughly the integral int_0^N x^2,dx = fracN^33, here approx 2667, same ballpark as 2870, and the formula is literally fracN^33 plus lower-order terms. Estimating sums by integrals is the fastest error-catcher in mental math. Why it earns rent on a desk. OLS against time needs sum t and sum t^2; the centered version sum (t - bar t)^2 = fracN(N^2-1)12 is the denominator of every trend slope and the variance (times N) of a discrete uniform, the same fracN^212 that prices "uniform on a tick grid" rounding error. The slope of a fitted trend has standard error proportional to 1/sqrtsum(t-bar t)^2 sim N^-3/2: doubling your window cuts slope noise by 2sqrt2, which is why trend estimates on short windows are mostly noise.

Trend Regression Bookkeeping

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