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In general, what does the bias-variance tradeoff look like when estimating covariance matrices with varying return frequencies (i.e. daily, weekly, monthly returns)?

From my observations I've noticed that, on average, lower frequency (e.g. monthly) returns result in higher estimates of asset variance and higher estimates of asset correlations while also having higher variance of those estimations. This leads me to hypothesize that lower frequency returns result in higher estimate variance compared to higher frequency returns. I've been looking for papers or proofs that show this more robustly but I'm falling short. Furthermore, what can I say about the bias of these estimates?

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    $\begingroup$ In general if one is estimating the variance of a diffusion process, empirical research shows that higher frequency returns (up to about 5 minute returns) are better estimators than lower frequency ones. Roughly speaking - if one looks at the price of an asset - estimates of mean return don't improve (on a relative basis) because all that matters is the initial price, the last price and the number of periods (no path dependency). On the other hand, volatility is exactly this "path dependence" of your returns, so it would make sense the more you zoom the better your estimate(assuming no jumps). $\endgroup$ Commented Feb 8, 2023 at 21:50
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    $\begingroup$ "lower frequency (e.g. monthly) returns result in higher estimates of asset variance" this is not my experience. "...while also having higher variance of those estimations". Yes, rubikscube09 explained this one: the more often you sample the better (more precise) the estimate (Merton, 1980). $\endgroup$
    – nbbo2
    Commented Feb 9, 2023 at 6:31
  • $\begingroup$ @rubikscube09 (and nbbo2) - thank you, that's helpful for me to understand the impact of frequency on the variance of the estimates. I realize now that some of the assets I was sampling from exhibited some autocorrelation in their daily returns, which resulted in daily estimates of volatility being lower than weekly/monthly estimates, though the variance of those estimates was indeed still lower when using daily returns. Hence the tradeoff between bias and variance. $\endgroup$
    – Ringleader
    Commented Feb 9, 2023 at 14:38

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