Interpreting OLS Unbiasedness and Variance from One Sample
Summary
This exchange clarifies why properties such as unbiasedness and minimum variance matter even when a researcher observes only one historical sample. The accepted answer explains that estimator bias and variance are defined across repeated samples from a population or data-generating process, not by comparing different people who fit a regression to the same fixed dataset. In practice, the observed sample is treated as one possible realization among many that could have occurred.
If an estimator has zero bias, its average across those possible samples equals the true parameter; low variance means estimates tend to cluster around that average. Together, these properties provide a reason to expect an estimate from one sample to be near the true value. The exchange does not connect the theorem to a specific hedge-fund use case, and the answer explicitly leaves that application unanswered. It also offers no empirical test or discussion of whether regression assumptions hold for financial data.
Key ideas
- OLS bias and variance describe behavior across hypothetical repeated samples.
- A single observed dataset is treated as one realization from a data-generating process.
- Zero bias means the estimator’s repeated-sample average equals the true parameter.
- Low variance means estimates tend to stay near their repeated-sample average.
- The exchange does not identify a specific hedge-fund application of the theorem.
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Full text
# In which context do hedge funds use the Gauss Markov Theorem? # In which context do hedge funds use the Gauss Markov Theorem? Hedge Funds really like asking questions about linear regression during interviews. Especially about the properties of the OLS. But I don't understand in which context this is used. For example the fact that the OLS estimator is unbiased and has the minimum variance. In which scenario will they want that? For example if I am trying to predict the price of a stock $S$ with the price of other stocks: $S_i$. I calculate my OLS and find something but then meaning that the estimator is unbiased just means that if multiple people run the same linear regression they will on average find the true value, and each one will have close values for beta (estimator with low variance). Yet in this experiment nothing is random since the stock prices everyone have the same when training the model, so I am really confused in which kind of experiment we want the properties of the OLS? ## Answer by Richard Hardy (score 1, accepted) https://quant.stackexchange.com/a/78482 > Yet in this experiment nothing is random since the stock prices everyone have the same when training the model, so I am really confused in which kind of experiment we want the properties of the OLS? Indeed, we usually have just a single sample from a population or a data generating process, while estimators' properties such as bias and variance refer to repeated samples. The idea is, if an estimator has small variance, then it does not matter much which of the many possible samples we got, as with a high probability the estimate will be close to the mean of the estimates across all samples. And if an estimator has zero bias, that mean equals the true value. So with zero bias and small variance, with high probability our estimate will be close to the true value. > In which scenario will they want that? I don't know.
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