Pairs spread & cointegration

hardQuantacceptance —~45 min

Implement pairs_spread(x, y) — the first step of a pairs trade: regress y on x with an intercept by ordinary least squares to get the hedge ratio beta, then measure the residual spread y - a - beta·x. Return [beta, spread_std] where spread_std is the population standard deviation of the residuals.

The hedge ratio sizes the two legs; the residual's spread is what you trade around. Return both rounded to 4 decimal places.

example 1

input

x=[-2.0, -1.0, 0.0, 1.0, 2.0], y=[1.0, 2.5, 4.0, 5.5, 7.5]

output

[1.6, 0.1414]

example 2

input

x=[10.0, 11.0, 12.0, 13.0, 14.0], y=[21.0, 22.5, 24.0, 25.5, 27.5]

output

[1.6, 0.1414]

constraints

3 ≤ len(x) == len(y) ≤ 5_000
x is not constant

teacher · 2 hints · the first is free

↗ read the theory

the chapter this problem is cut from

▶ open in the workbench

run the same math on a real tape

●solution.py· utf-8 · spaces:4

from statistics import pstdev


def pairs_spread(x: list, y: list) -> list:
    """[hedge ratio, residual spread std], both rounded to 4 dp."""
    # TODO: fit an intercept — this regression runs through the origin
    beta = sum(xi * yi for xi, yi in zip(x, y)) / sum(xi * xi for xi in x)
    resid = [yi - beta * xi for xi, yi in zip(x, y)]
    return [round(beta, 4), round(pstdev(resid), 4)]

ln 1, col 1tab indent · esc leave editor

python 3.13 · pyodide

+ 6 hidden on submit

input · keyword arguments to pairs_spread()

x=[-2.0, -1.0, 0.0, 1.0, 2.0], y=[1.0, 2.5, 4.0, 5.5, 7.5]

expected

[1.6, 0.1414]

the judge calls pairs_spread() with exactly these inputs · outputs are compared to 6 decimal places

● standing bypython 3.13 in your browser · pyodide 0.28.3 · judged on the server to 6 dp⌘↵ run · ⌘⇧↵ submit

paper only · educational · not advice