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We want to predict Y based on some function of X, i.e., Y_hat = f(X). How can we show that the conditional expectation f*(X) = E(Y|X) is the mean-square optimal predictor, i.e., the function f* solves the minimization problem below?

min E{[Y - f(X)]^2}

Here the unrestricted mean-square optimal predictor is the conditional mean.

user321152
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  • It does not matter whether your conditional distribution is conditional on time series data, or on other predictors, or on nothing at all. The (conditional) expectation minimizes the expected square error for any (conditional) distribution, and the textbook proof works straightforwardly. – Stephan Kolassa May 09 '21 at 16:52
  • @StephanKolassa Could you please refer me to some textbook where I could have a look at the proof? Would really appreciate. – user321152 May 09 '21 at 17:00
  • [This thread](https://stats.stackexchange.com/q/192968/1352) gives one possible proof, you just need to translate the symbols to your case. – Stephan Kolassa May 09 '21 at 17:05
  • Actually, [this thread](https://stats.stackexchange.com/q/520286/1352) is much more informative. – Stephan Kolassa May 10 '21 at 19:20

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