I'm still confused.
If I have the model:
$y = β_1x_1 + β_2x_2 + u$.
I have to show that the OLS estimator is:
I think that I have to use $\beta =\begin{pmatrix} \beta_1 \\ \beta_2 \end{pmatrix}=(X^tX)^{-1}X^tY$. But I'm a bit confused. Now I think that: $$ X = \left[ \begin{array}{ccc} x_{11} & x_{12} \\ \vdots & \vdots \\ x_{n1} & x_{n2} \end{array} \right]. \qquad\qquad Y = \left[ \begin{array}{c} y_1 \\ \vdots \\ y_n \end{array} \right]. $$. And if I find $(X^tX):$ $$ X^tX = \left[ \begin{array}{ccc} \sum_{i} X_{i1}X_{i1} & \sum_{i} X_{i1}X_{i2} \\ \sum_{i} X_{i1}X_{i2} & \sum_{i} X_{i2}X_{i2} \end{array} \right]. \qquad\qquad $$. But I can't find $(X^tX)^{-1}$ because $\frac{1}{ad-bc}$ and the the denominator in the fraction will give $0$? What can I do in stead???
If $$ X^tX = \left[ \begin{array}{ccc} \sum_{i} X_{i1}X_{i1} & \sum_{i} X_{i1}X_{i2} \\ \sum_{i} X_{i1}X_{i2} & \sum_{i} X_{i2}X_{i2} \end{array} \right] = \left[ \begin{array}{ccc} \sum_{i} X^2_{i1} & \sum_{i} X_{i1}X_{i2} \\ \sum_{i} X_{i1}X_{i2} & \sum_{i} X^2_{i2}\end{array} \right]. \qquad\qquad $$
Then $$det(X^tX) = \left( \sum_{i} X^2_{i1}\right)\left(\sum_{i} X^2_{i2}\right) - \left(\sum_{i} X_{i1}X_{i2}\right)^2$$ which I don't see why should be zero. Then using: $$ (X^tX)^{-1} = \frac{1}{det(X^tX)} \left[ \begin{array}{ccc} \sum_{i} X^2_{i2} & - \sum_{i} X_{i1}X_{i2} \\ -\sum_{i} X_{i1}X_{i2} & \sum_{i} X^2_{i1} \end{array} \right]. \qquad\qquad $$ leads to the expected result for $\beta_1$.