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 Yahoo! Finance data.

import datetime as dt
import pandas as pd
import pandas_datareader.data as web

start = dt.datetime(1951, 1, 1)
end = dt.datetime(2014, 1, 1)
sp500 = web.DataReader('^GSPC', 'yahoo', start=start, end=end)
low = sp500.index.min()
high = sp500.index.max()
monthly_dates = pd.date_range(low, high, freq='M')
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(0)
    sigma = np.sqrt(12 * r.var(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        8.148534
sigma    14.508540
SR        0.561637
dtype: float64

>>> param_cov
             mu     sigma        SR
mu     3.729435 -0.442891  0.273945
sigma -0.442891  0.495087 -0.049454
SR     0.273945 -0.049454  0.020830

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.