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.

Key concepts

Regression Coefficients

In regression analysis, the regression coefficients are the heart of our model. They are crucial because they quantify the relationship between each predictor and the outcome variable. In our exercise, we work with a matrix representing these coefficients, typically denoted as \( \boldsymbol{\beta} \). This matrix has a dimension of \(3 \times 1\), matching the number of predictors in the matrix \( \boldsymbol{X} \). The regression coefficients help us understand how changes in predictor variables affect the outcome variable. Each coefficient tells us the expected change in the outcome variable associated with a one-unit change in the respective predictor, assuming all other predictors are held constant. Understanding these coefficients requires us to perform calculations using matrix algebra. Specifically, in the context of a multivariate normal distribution, the formula for these coefficients is \( \hat{\boldsymbol{\beta}} = (\boldsymbol{X}^{\prime} \boldsymbol{X})^{-1} \boldsymbol{X}^{\prime} \boldsymbol{Y} \). This tells us that from the observed data, the estimates of our coefficients can be derived by these matrix operations. When exercising this theory with real data, such as the observed \( \boldsymbol{Y}' = (6,1,11,3) \) in our problem, we perform these computations to find \( \hat{\boldsymbol{\beta}} \). This will elucidate which predictors have the most substantial impact on the outcome, represented by \( \boldsymbol{Y} \).

Covariance Matrix

The covariance matrix is a fundamental tool in multivariate statistics, providing insight into the relationships between different random variables. In our scenario with the multivariate normal distribution, the covariance matrix for the estimated coefficients \( \hat{\boldsymbol{\beta}} \) is expressed as \( \sigma^2 (\boldsymbol{X}' \boldsymbol{X})^{-1} \). This matrix offers a map of the variance of each regression coefficient and their covariances with one another.Covariances are particularly informative because they indicate how much two variables change together. A positive covariance implies a tendency for variables to increase and decrease together, whereas a negative covariance suggests an inverse relationship. Understanding these nuances aids in interpreting the complexity of multivariate data.Another key point is how minimizing the covariance can lead to more efficient estimates of \( \hat{\boldsymbol{\beta}} \). In practice, smaller covariances between predictor variables are preferable as they often lead to more reliable regression coefficients. This covariance matrix is dependent upon both \( \boldsymbol{X} \) and the variance \( \sigma^2 \) of the error terms. It exemplifies the connection between variation in predictors and how it translates into approximation error in estimating the coefficients.

Matrix Algebra

Matrix algebra plays a central role in multivariate statistics, especially in solving systems of equations and performing transformations. The application of matrix algebra is essential when working with linear regression models in multivariate settings, as number-crunching often involves matrices and their manipulation.In the given problem, matrix algebra allows us to compute \( \hat{\boldsymbol{\beta}} \) by utilizing operations such as transposition, multiplication, and inversion. Specifically, to find \( (\boldsymbol{X}' \boldsymbol{X})^{-1} \), we first transpose the matrix \( \boldsymbol{X} \), multiply it by \( \boldsymbol{X} \), and then calculate the inverse. Each step requires comprehending these matrix operations:

• **Transposition**: Changing the orientation of the matrix from rows to columns or vice versa.
• **Multiplication**: Combining matrices to synthesize new information, such as \( \boldsymbol{X}' \boldsymbol{Y} \).
• **Inversion**: Finding the inverse matrix, akin to dividing in arithmetic, crucial for deriving coefficients.

By mastering these operations, students can solve matrix equations that symbolize the underlying linear relationships in multivariate models. This makes matrix algebra an indispensable part of statistical analysis and estimation in multivariate contexts.