Question

Readers may have encountered the multiple regression model in a previous course in statistics. We can briefly write it as follows. Suppose we have a vector of \(n\) observations \(\mathbf{Y}\) which has the distribution \(N_{n}\left(\mathbf{X} \boldsymbol{\beta}, \sigma^{2} \mathbf{I}\right)\), where \(\mathbf{X}\) is an \(n \times p\) matrix of known values, which has full column rank \(p\), and \(\beta\) is a \(p \times 1\) vector of unknown parameters. The least squares estimator of \(\boldsymbol{\beta}\) is $$ \widehat{\boldsymbol{\beta}}=\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1} \mathbf{X}^{\prime} \mathbf{Y} $$

Short answer

The least squares estimator of \(\boldsymbol{\beta}\) is given by \(\widehat{\boldsymbol{\beta}}=\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1} \mathbf{X}^{\prime} \mathbf{Y}\) which involves the steps of transposing the matrix \(\mathbf{X}\), multiplication of matrices and inversion of a matrix, followed by multiplication with the vector \(\mathbf{Y}\).

Step-by-step solution

Matrix Transposition

Transpose the matrix \(\mathbf{X}\) to get \(\mathbf{X}^{\prime}\). The transpose of a matrix is obtained by interchanging its rows and columns. This means, the \((i, j)\) entry of the original matrix becomes the \((j, i)\) entry of the transposed matrix.

Matrix Multiplication

Multiply \(\mathbf{X}^{\prime}\) by \(\mathbf{X}\), i.e., calculate \(\mathbf{X}^{\prime} \mathbf{X}\). A resultant square matrix is obtained. In this multiplication, each entry of the resultant matrix can be calculated as the dot product of the corresponding row of the first matrix and the column of the second matrix.

Matrix Inversion

Find the inverse of the resultant matrix, i.e., calculate \(\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1}\) after ensuring that the determinant of the matrix is non-zero. If the determinant is zero, the matrix does not have an inverse. Once the inverse is computed, each entry in the final inverted matrix can be calculated using a specific formula, which involves dividing the corresponding cofactor of that entry by the determinant of the original matrix.

Matrix Vector Multiplication

Multiply the obtained matrix with \(\mathbf{X}^{\prime}\) and \(\mathbf{Y}\) in sequence, i.e., calculate \(\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1} \mathbf{X}^{\prime} \mathbf{Y}\). This yields the least squares estimator for \(\boldsymbol{\beta}\).

Key concepts

Least Squares Estimator

In the world of statistics, especially when dealing with linear regression models, the least squares estimator becomes a crucial tool. It is essentially a method used to find the best-fitting line through a set of data points, which minimizes the sum of the squared differences between the observed and predicted values. This approach helps estimate the coefficients or parameters of a linear regression model.

For the multiple regression model, if we have a vector of observations \(\mathbf{Y}\) and a matrix of independent variables \(\mathbf{X}\), the least squares estimator for the parameter vector \(\boldsymbol{\beta}\) is provided by the formula:

• \(\widehat{\boldsymbol{\beta}} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{Y}\).

This formula is derived from setting the derivative of the sum of squared residuals to zero and solving for \(\boldsymbol{\beta}\). It involves several matrix operations, like transposition, multiplication, and inversion, to calculate the estimator efficiently.

Matrix Transposition

Matrix transposition is a fundamental operation in linear algebra, vital for the computation of the least squares estimator. When you transpose a matrix, you swap its rows with its columns. For a given matrix \(\mathbf{X}\), its transpose, denoted \(\mathbf{X}'\), has its elements rearranged such that the \((i, j)\)th entry of \(\mathbf{X}\) becomes the \((j, i)\)th entry of \(\mathbf{X}'\).

Transposing a matrix is crucial when dealing with regression models because the transposed matrix \(\mathbf{X}'\) is used in further calculation of the least squares estimator. For example, if \(\mathbf{X}\) is an \(n \times p\) matrix, then \(\mathbf{X}'\) becomes a \(p \times n\) matrix, ready for the next steps of matrix multiplication.

Matrix Multiplication

Matrix multiplication is another essential operation that helps link different matrices together in regression analysis. It allows us to perform operations on data transformations within these matrices effectively. After obtaining \(\mathbf{X}'\), the next step is to calculate \(\mathbf{X}'\mathbf{X}\).

This multiplication involves taking each row vector of \(\mathbf{X}'\) and computing the dot product with each column vector of \(\mathbf{X}\). It results in a \(p \times p\) square matrix because \(\mathbf{X}'\) is \(p \times n\) and \(\mathbf{X}\) is \(n \times p\). Each element of \(\mathbf{X}'\mathbf{X}\) is essentially a sum of products of corresponding entries of the row from \(\mathbf{X}'\) and the column from \(\mathbf{X}\).

• This step is pivotal as it leads to the generation of a matrix that, if invertible, can aid in finding the least squares estimator precisely.

Matrix Inversion

Matrix inversion is a key concept in deriving solutions for linear equations, crucial for obtaining the least squares estimator. When we talk about matrix inversion, it is about finding a matrix \(\mathbf{A}^{-1}\) such that when it is multiplied by \(\mathbf{A}\), yields the identity matrix \(\mathbf{I}\).

In the context of the least squares estimator, we seek to find \((\mathbf{X}'\mathbf{X})^{-1}\). This step can only be done if the determinant of \(\mathbf{X}'\mathbf{X}\) is non-zero, indicating that the matrix is invertible. The calculation involves complex operations, but it is essential as it paves the way for further multiplication needed to compute \(\widehat{\boldsymbol{\beta}}\).

• Finding the inverse can sometimes be computationally expensive, but it is crucial for the least squares method.
• Once inverted, \(\mathbf{X}'\mathbf{X}\) allows us to finalize the computation and provide the best estimates for our regression parameters.