Proof that the Minimum Mean Squared Error (MMSE) Forecast is the Conditional Expectation

Students often struggle with the distinction between unconditional expectation \

E[Y] \

and conditional expectation \

E[Y | \\mathcal{F}] \

, failing to leverage the available information \

\\mathcal{F} \

. The proof's reliance on the property \

E[E[Y|X]|X] = E[Y|X] \

can also be a source of conceptual difficulty.

Institutional Reference: Time Series Analysis