Question

Statistics

Posted 6 months ago

`t the multivariate OLS formula, $\hat{\beta}^{O L S}=\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-\mathbf{1}} \mathbf{X}^{\prime} \mathbf{y}$ prod first row is $\hat{\beta}_{0}^{O L S}$ and whose second row is $\hat{\beta}_{1}^{O L S}$ from part`

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

To understand the multivariate OLS formula, we recognize that it is used to estimate the coefficients $\hat{\beta}^{OLS}$ in a linear regression model. The formula is given by $\hat{\beta}^{OLS} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y}$, where $\mathbf{X}$ is the matrix of explanatory variables, $\mathbf{y}$ is the vector of the dependent variable, and $\mathbf{X}'$ is the transpose of $\mathbf{X}$

step 2

To find $\hat{\beta}_{0}^{OLS}$ and $\hat{\beta}_{1}^{OLS}$, we need to compute the product $(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y}$ and extract the first and second rows of the resulting vector

step 3

Assuming the matrix multiplication and inversion have been correctly computed, $\hat{\beta}_{0}^{OLS}$ is the first element of the vector $\hat{\beta}^{OLS}$, and $\hat{\beta}_{1}^{OLS}$ is the second element of this vector

Answer

The first row of the vector $\hat{\beta}^{OLS}$ is $\hat{\beta}_{0}^{OLS}$ and the second row is $\hat{\beta}_{1}^{OLS}$.

Key Concept

Estimation of regression coefficients in OLS

Explanation

The OLS formula is used to estimate the coefficients of a linear regression model, where $\hat{\beta}_{0}^{OLS}$ represents the intercept and $\hat{\beta}_{1}^{OLS}$ represents the slope coefficient associated with the first explanatory variable.

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