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# Mathematical Review

This is the basic mathematical definitions and calculation methods for Econometircs.

### Best Predictor

Given a random vector $$X$$, we want to forecast $$Y$$, Let $$g(X)$$ be a predictor of $$Y$$. For Prediction Error, it is defined as $$Y - g(X)$$, and this prediction error can be treated as a random variable, and  it can take positive and negative values. To minimize this prediction error, we define the the mean squared error (MSE) of predictor $$g(X)$$ as $$\mathbb{E}\[(Y-g(X))^2]$$. We can have that the CEF $$m(x) = \mathbb{E}(Y|X=x)$$ is the best predictor, which has the smallest mean squared prediction error. Which means if we have $$\mathbb{E}(Y^2) < \infin$$, then for any predictor $$g(X)$$, we have:

&#x20;                                                   $$\mathbb{E}\[(Y-g(X))^2] \geq \mathbb{E}\[(Y-m(X))^2]$$

&#x20;**Proof**:

$$\mathbb{E}\[u^2] = \mathbb{E}\[(Y-g(X))^2]=\mathbb{E}\[(Y-m(X) + m(X) -g(X))^2]$$

&#x20;          $$=\mathbb{E}\[(Y-m(X))^2] + \mathbb{E}\[(m(X)-g(X))^2]+2\mathbb{E}\[(Y-m(X))(m(X) -g(X))]$$

&#x20;          $$\geq \mathbb{E}\[(Y-m(X))^2]$$

since: $$\mathbb{E}\[(Y-m(X))(m(X) -g(X))] = \mathbb{E}\[\mathbb{E}\[(Y-m(X))(m(X) -g(X))|X]]$$

as under condition $$X$$, $$m(X) -g(X)$$ is no longer a random variable, by the definition of $$m(X)$$

$$\mathbb{E}\[\mathbb{E}\[(Y-m(X))(m(X) -g(X))|X]] = \mathbb{E}\[(m(X) -g(X))\mathbb{E}\[(Y-m(X))|X]] = 0$$

So above inequality becomes equality when $$m(X) = g(X)$$, therefore, $$m(X)$$ is the smallest.