Question

Often in regression the mean of the random variable \(Y\) is a linear function of \(p\) -values \(x_{1}, x_{2}, \ldots, x_{p}\), say \(\beta_{1} x_{1}+\beta_{2} x_{2}+\cdots+\beta_{p} x_{p}\), where \(\boldsymbol{\beta}^{\prime}=\left(\beta_{1}, \beta_{2}, \ldots, \beta_{p}\right)\) are the regression coefficients. Suppose that \(n\) values, \(\boldsymbol{Y}^{\prime}=\left(Y_{1}, Y_{2}, \ldots, Y_{n}\right)\) are observed for the \(x\) -values in \(\boldsymbol{X}=\left[x_{i j}\right]\), where \(\boldsymbol{X}\) is an \(n \times p\) design matrix and its ith row is associated with \(Y_{i}, i=1,2, \ldots, n .\) Assume that \(Y\) is multivariate normal with mean \(\boldsymbol{X} \boldsymbol{\beta}\) and variance-covariance matrix \(\sigma^{2} \boldsymbol{I}\), where \(\boldsymbol{I}\) is the \(n \times n\) identity matrix. (a) Note that \(Y_{1}, Y_{2}, \ldots, Y_{n}\) are independent. Why? (b) Since \(\boldsymbol{Y}\) should approximately equal its mean \(\boldsymbol{X} \boldsymbol{\beta}\), we estimate \(\boldsymbol{\beta}\) by solving the normal equations \(\boldsymbol{X}^{\prime} \boldsymbol{Y}=\boldsymbol{X}^{\prime} \boldsymbol{X} \boldsymbol{\beta}\) for \(\boldsymbol{\beta}\). Assuming that \(\boldsymbol{X}^{\prime} \boldsymbol{X}\) is non- singular, solve the equations to get \(\hat{\boldsymbol{\beta}}=\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{Y}\). Show that \(\hat{\boldsymbol{\beta}}\) has a multivariate normal distribution with mean \(\boldsymbol{\beta}\) and variance-covariance matrix $$ \sigma^{2}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} $$ (c) Show that $$ (\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})^{\prime}(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})=(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta})^{\prime}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta})+(\boldsymbol{Y}-\boldsymbol{X} \hat{\boldsymbol{\beta}})^{\prime}(\boldsymbol{Y}-\boldsymbol{X} \hat{\boldsymbol{\beta}}) $$ say \(Q=Q_{1}+Q_{2}\) for convenience. (d) Show that \(Q_{1} / \sigma^{2}\) is \(\chi^{2}(p)\). (e) Show that \(Q_{1}\) and \(Q_{2}\) are independent. (f) Argue that \(Q_{2} / \sigma^{2}\) is \(\chi^{2}(n-p)\). (g) Find \(c\) so that \(c Q_{1} / Q_{2}\) has an \(F\) -distribution. (h) The fact that a value \(d\) can be found so that \(P\left(c Q_{1} / Q_{2} \leq d\right)=1-\alpha\) could be used to find a \(100(1-\alpha)\) percent confidence ellipsoid for \(\beta\). Explain.

Short answer

This exercise involves mathematical proofs related to several quantities derived from observations and predictors that are organized into a matrix in multiple linear regression. With normal distribution, identity covariance matrix and linear relationship between predictors and observations, we derived several important properties and relationships like normal distribution of estimates, independence of residuals, chi-square and F-distributions for squared quantities and statistical inference foundations for confidence intervals.

Step-by-step solution

Part (a)

Given that the variance-covariance matrix is \(\sigma^{2} \boldsymbol{I}\), where \(\boldsymbol{I}\) is the identity matrix, it implies all the off-diagonal elements representing the covariance of any two different \(Y\) values are zero. Thus, \(Y_{1}, Y_{2}, \ldots, Y_{n}\) are uncorrelated and since they are normal, they are also independent.

Part (b)

Given the normal equation \(\boldsymbol{X}^{\prime} \boldsymbol{Y}=\boldsymbol{X}^{\prime} \boldsymbol{X} \boldsymbol{\beta}\), we can solve for \(\boldsymbol{\beta}\) to get the least squares estimates \(\hat{\boldsymbol{\beta}}=\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{Y}\). Under normal distribution assumption, the vector \(\boldsymbol{\beta}\) also has a multivariate normal distribution with mean \(\boldsymbol{\beta}\) and variance-covariance matrix \(\sigma^{2}\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1}\) from elements variance for multivariate normal distribution.

Part (c)

To prove the equation, expand both sides, as \((a-b)'(a-b) = a'a - 2 a'b + b'b\). After simplification, both sides will be equal.

Part (d)

The quantity \(Q_{1} / \sigma^{2} =(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta})^{\prime}\left(\boldsymbol{X}^{\prime}\boldsymbol{X}\right)(\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}) / \sigma^{2}\) follows a \(\chi^2\) distribution with \(p\) degrees of freedom by the definition of chi-square distribution and proved under (b)

Part (e)

The quantities \(Q_1\) and \(Q_2\) can be shown to be independent because they are functions of uncorrelated random variables.

Part (f)

The quantity \(Q_{2} / \sigma^{2} = (\boldsymbol{Y}-\boldsymbol{X}\hat{\boldsymbol{\beta}})^{\prime}(\boldsymbol{Y}-\boldsymbol{X}\hat{\boldsymbol{\beta}}) / \sigma^{2}\) follows a \(\chi^2\) distribution with \((n-p)\) degrees of freedom, by property of chi-square distribution where each of \(n-p\) residuals \((\boldsymbol{Y}-\boldsymbol{X} \hat{\boldsymbol{\beta}})\) squares contributes 1 degree.

Part (g)

To give \(c Q_{1} / Q_{2}\) an \(F\) -distribution, \(c\) must be the ratio of the degrees of freedom of \(Q_{1}\) and \(Q_{2}\), so \(c= p / (n-p)\) as they are the degrees of freedom from (d) and (f). An F-distribution describes ratio of two chi-square divided by their degrees of freedom.

Part (h)

Using \(Q_1\) and \(Q_2\) we can get an F distribution. We can also calculate a cut-off score \(d\) by defining our acceptable error to be \(\alpha\). Using the F-distribution table or function, we can find a value \(d\) that \[P\left(c Q_{1} / Q_{2} \leq d\right)=1-\alpha\]. This is the basis of statistical testing and establishing confidence intervals. We can obtain a \(100(1-\alpha)\%\) confidence ellipsoid for \(\beta\) by creating bounds using the relationship of \(\beta\) to the F-distribution.

Key concepts

Multivariate Normal Distribution

In regression analysis, understanding the multivariate normal distribution is crucial. It is a generalization of the univariate normal distribution to more than one variable. This concept comes into play when we examine the regression model \[ Y = X\beta + \epsilon \] where

• \(Y\) represents the vector of observed values,
• \(X\) is the design matrix containing rows that are associated with these observations,
• \(\beta\) is a vector of regression coefficients, and
• \(\epsilon\) is the error term.

When the random variable \(Y\) follows a multivariate normal distribution with mean \(X\boldsymbol{\beta}\) and a variance-covariance matrix \(\sigma^2 I\), it means that each element of \(Y\) is a normally distributed random variable, and they are independent of each other. This independence is due to the identity matrix \(I\), which indicates zero covariance between different variables of \(Y\). This feature of multivariate normal distribution helps simplify the process of estimating the regression parameters \(\beta\).

Least Squares Estimation

Least squares estimation is a method used to estimate unknown parameters in a regression model. In simple terms, it minimizes the sum of squared differences between the observed and predicted values. For regression models, the normal equation used is \[ X'Y = X'X\beta \] By solving this equation, we find the estimated coefficients, denoted as \(\hat{\beta}\), given by \[ \hat{\beta} = (X'X)^{-1}X'Y \] This method is chosen because it provides the best linear unbiased estimator (BLUE) under the assumption that errors are normally distributed with constant variance. Furthermore, because the errors are assumed normally distributed, \(\hat{\beta}\) itself follows a multivariate normal distribution with mean \(\beta\) and variance \(\sigma^2 (X'X)^{-1}\). This property is particularly useful because it allows us to make statistical inferences about the regression coefficients.

Chi-Square Distribution

The chi-square distribution is essential in regression analysis for hypothesis testing and constructing confidence intervals. It is used as a measure of the distribution of variance. In the context of regression, we consider quantities such as \[ Q_1 / \sigma^2 \] where \[ Q_1 = (\hat{\beta} - \beta)'(X'X)(\hat{\beta} - \beta) \] This expression yields a \(\chi^2\) distribution with \(p\) degrees of freedom, where \(p\) is the number of predictors in the regression model. This is because the term \(\hat{\beta} - \beta\) is normally distributed, and its quadratic form with respect to the matrix \(X'X\) produces a \(\chi^2\) distribution. Another form related to the error or residual component, \[ Q_2 / \sigma^2 \] where \[ Q_2 = (Y-X\hat{\beta})'(Y-X\hat{\beta}) \] also follows a chi-square distribution but with \(n-p\) degrees of freedom. Here, \(n\) stands for the number of observations. Understanding these distributions helps in evaluating the goodness-of-fit and conducting statistical tests in regression analysis.

Confidence Ellipsoid

A confidence ellipsoid is an extension of the confidence interval to multiple dimensions. In the regression context, it helps to give a bounded region where the true parameter \(\beta\) lies with a certain probability. By combining the chi-square distribution findings with the \(F\)-distribution, we can design a confidence ellipsoid for \(\beta\). Using statistics like \[ c Q_1 / Q_2 \] which follows an \(F\)-distribution, a critical value \(d\) can be determined such that \[ P(c Q_1 / Q_2 \leq d) = 1-\alpha \] where \(\alpha\) is the significance level.The ellipsoid is defined for a \(100(1-\alpha)\%\) confidence level. Geometrically, this region is shaped like an ellipsoid, illustrating the joint confidence in estimates of the multiple regression coefficients. This is crucial in assessing how far the estimated \(\hat{\beta}\) might stray from the actual \(\beta\) due to sampling variability, thus providing a multidimensional insight into the precision of our regression estimates.