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I am new to method of moments and want to figure out how to derive the method of moment estimator for $\beta$ in the linear equation with a constant term and three corresponding moments, namely, I have

$$y_i = \alpha+ x_i \beta + u_i$$

And I have the following moments: $E[x_i - \mu] = 0$, $E[y_i - \alpha+ x_i \beta_1] = 0$ and $E[x_i(y_i - \alpha+ x_i \beta)] = 0$

Thanks in advance

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Using the methods of moment, you need to compare the sample statistics and population statistics.

For a classical linear regression, your residue $u_i \sim N(0, \sigma^2)$ and there should no correlation between residue $u_i$ and your feature vector $x_i$ as you mentioned in the problem statement(strict exogeneity). Then MoM gives the following equations

$$E(u) = 0 = \frac{1}{N}\sum_{i=1}^n (y_i-\beta x_i), (1)$$

$$E(x(y-\beta x)) = \mathbf{0} = \frac{1}{N}\sum_{i=1}^n x_i(y_i-\beta x_i), (2)$$

Keep in mind the second formula is p-dim.

You do not have to look at higher-order moments ($E[(y_i-\beta x_i)^2] = \sigma^2$) as they are not useful for deriving the closed form in linear regression.

The most straight forward way is to convert the formula (2) into a matrix product form $$ X^T(Y-X\beta) = \mathbf{0}$$ Then you have the standard OLS estimator $$\hat{\beta} = (X^TX)^{-1}X^Ty $$ when $rank(X) = n$.

P.S, with MoM, it is easier to treat data with heteroscedascity, or X-column correlation because you can do the second-order estimation.

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