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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?

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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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  • $\begingroup$ So in the OLS framework the regressors are not fixed? Just the sample size is fixed? So for example in my example it would mean the following: $n$ number of people run the following experiment: take $m$ different dates and try to predict the stock price of $S$ as these $m$ dates using the prices of the $S_i$ at these dates. Then the variance of the beta among these $n$ different person is going to be small and on average they will have the correct beta? $\endgroup$ Commented Feb 26 at 20:26
  • $\begingroup$ actually in most cases we are in the fixed design framework so the input data is the same for everyone. Just the output varies which doesn't make much sense in finance for example where everyone observes the ouput price of the stock. $\endgroup$ Commented Feb 26 at 20:32
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    $\begingroup$ @confucius_is_confused, OLS can be used with either fixed or random regressors. There is finite sample and asymptotic distribution theory for both cases. In finance, most of the time series (asset prices, exchange rates, accounting variables, ...) are best though of as random processes. We condition on the past values (a single realized trajectory) of the process, but the future values are random (many possible trajectories). Note that past data are always fixed but the question is whether we assume they come from a random process or from someone deliberately setting these values (nonrandom). $\endgroup$ Commented Feb 26 at 20:56
  • $\begingroup$ @confucius_is_confused, in your example, you could think of $n$ not as people that are running an experiment (as there is no experiment; we mainly work with observational data) but as possible past trajectories of the random process of interest. Alternative versions of the universe might be a way to think about it. So whichever version of the universe we happen to inhabit, if the bias of our estimator is zero and variance is small, we have a high probability of recovering something close to the true parameter values of the data generating process (a random one). $\endgroup$ Commented Feb 26 at 21:01
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    $\begingroup$ @confucius_is_confused, yes, though its formulation is slightly different. Under random design, we are conditioning things on $X$. (Under fixed design, such conditioning does not do anything, so it is skipped.) See e.g. p. 13 and then p. 27 in Hayashi "Econometrics" (a graduate level econometrics textbook). $\endgroup$ Commented Feb 27 at 7:11

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