Covariance Estimation¶

The bootstrap can be used to estimate parameter covariances in applications where analytical computation is challenging, or simply as an alternative to traditional estimators.

This example estimates the covariance of the mean, standard deviation and Sharpe ratio of the S&P 500 using monthly returns computed from the daily prices distributed with arch.

import numpy as np
import pandas as pd

import arch.data.sp500

sp500 = arch.data.sp500.load()
low = sp500.index.min()
high = sp500.index.max()
monthly_dates = pd.date_range(low, high, freq='ME')
monthly = sp500.reindex(monthly_dates, method='ffill')
returns = 100 * monthly['Adj Close'].pct_change().dropna()

The function that returns the parameters.

def sharpe_ratio(r):
    mu = 12 * r.mean(axis=0)
    sigma = np.sqrt(12 * r.var(axis=0))
    sr = mu / sigma
    return np.array([mu, sigma, sr])

Like all applications of the bootstrap, it is important to choose a bootstrap that captures the dependence in the data. This example uses the stationary bootstrap with an average block size of 12.

import pandas as pd
from arch.bootstrap import StationaryBootstrap

bs = StationaryBootstrap(12, returns)
param_cov = bs.cov(sharpe_ratio)
index = ['mu', 'sigma', 'SR']
params = sharpe_ratio(returns)
params = pd.Series(params, index=index)
param_cov = pd.DataFrame(param_cov, index=index, columns=index)

The output is

>>> params
mu        4.439391
sigma    14.468318
SR        0.306835
dtype: float64

>>> param_cov
              mu     sigma        SR
mu     12.521785 -3.272164  0.941381
sigma  -3.272164  2.137770 -0.280233
SR      0.941381 -0.280233  0.072791

Note

The covariance estimator is centered using the average of the bootstrapped estimators. The original sample estimator can be used to center using the keyword argument recenter=False.