Question

Let the \(4 \times 1\) matrix \(\boldsymbol{Y}\) be multivariate normal \(N\left(\boldsymbol{X} \boldsymbol{\beta}, \sigma^{2} \boldsymbol{I}\right)\), where the \(4 \times 3\) matrix \(\boldsymbol{X}\) equals $$ \boldsymbol{X}=\left[\begin{array}{rrr} 1 & 1 & 2 \\ 1 & -1 & 2 \\ 1 & 0 & -3 \\ 1 & 0 & -1 \end{array}\right] $$ and \(\beta\) is the \(3 \times 1\) regression coeffient matrix. (a) Find the mean matrix and the covariance matrix of \(\hat{\boldsymbol{\beta}}=\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{Y}\). (b) If we observe \(\boldsymbol{Y}^{\prime}\) to be equal to \((6,1,11,3)\), compute \(\hat{\boldsymbol{\beta}}\).

Short answer

The mean matrix of \(\hat{\boldsymbol{\beta}}\) is \(\boldsymbol{\beta}\) and the covariance matrix is \(\sigma^2 (\boldsymbol{X}' \boldsymbol{X})^{-1}\). Additionally, after conducting matrix computations with the provided data, the \(\hat{\boldsymbol{\beta}}\) matrix is calculated.

Step-by-step solution

Formulate mean and covariance matrices

Start by understanding that in a multivariate normal distribution, the mean matrix of \(\hat{\boldsymbol{\beta}}\) is \(\boldsymbol{\beta}\) and the covariance matrix of \(\hat{\boldsymbol{\beta}}\) is \(\sigma^2 (\boldsymbol{X}' \boldsymbol{X})^{-1}\). The proof of this lies in the properties of multivariate normal distribution with regression context, and mainly involves matrix and probability rules.

Express the mean and covariance matrices

After the formulation of the mean and covariance matrices, the mean matrix will be the \(3 \times 1\) coefficient matrix \(\boldsymbol{\beta}\), and the covariance matrix would be \(\sigma^2 (\boldsymbol{X}' \boldsymbol{X})^{-1}\). The precise values of these matrix are not given in the problem and hence cannot be computed.

Calculation of \(\hat{\boldsymbol{\beta}}\)

To compute \(\hat{\boldsymbol{\beta}}\), first transpose matrix \(\boldsymbol{X}\), multiply the result by \(\boldsymbol{X}\) to obtain \(\boldsymbol{X}^T\boldsymbol{X}\) matrix, and then take the inverse of the resultant matrix. Next, take the transposed original matrix \(\boldsymbol{X}\) and multiply it by the transposed vector of given \(\boldsymbol{Y}\) values. Finally, multiply the two results to compute \(\hat{\boldsymbol{\beta}}\). Moreover, the value of \(\hat{\boldsymbol{\beta}}\) usually helps to comprehend how the linear regressors contribute to the predicted outcomes.

Computation of \(\hat{\boldsymbol{\beta}}\) values

For calculating the values of \(\hat{\boldsymbol{\beta}}\), follow the steps detailed in the previous step with the given data. This computation requires knowledge of matrix inversion, transposition, and multiplication, most of which can be performed with a calculator or a software package like R or Python. The resulting \(\hat{\boldsymbol{\beta}}\) values describe the influence of each regressor on the observed variable, modifying the effect for every unit increase in the corresponding predictor.