Sure, let's break down this problem step by step.

1. First, let's understand the problem. We have a full rank linear model with two predictors x1 and x2, and we want to find a joint confidence region for the parameters β1 and β2. The errors are assumed to be independent and normally distributed with mean 0 and variance σ^2.

2. The joint confidence region for β1 and β2 is a region in the parameter space where, if the true values of the parameters were to fall in this region, the observed data would not be considered unusual at a significance level of α.

3. To derive the joint confidence region, we can use the fact that the marginal distribution over a subset of multivariate normal random variables is multivariate normal. This means that the joint distribution of β1 and β2 is also multivariate normal.

4. The mean vector and covariance matrix for this multivariate normal distribution can be obtained by dropping the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. In this case, the irrelevant variable is β0, so we drop it from the mean vector and the covariance matrix.

5. The mean vector for β1 and β2 is simply the vector of their expected values, which are the true values of β1 and β2. The covariance matrix is the matrix of their variances and covariance, which can be estimated from the data.

6. The joint 100(1 − α)% confidence region for β1 and β2 is then the region where the multivariate normal distribution of β1 and β2 exceeds the critical value for the χ^2 distribution with 2 degrees of freedom at the 1 − α level. This region is an ellipse in the β1-β2 plane centered at the estimated values of β1 and β2, with axes determined by the estimated variances and covariance of β1 and β2.

7. The exact expression for this ellipse depends on the estimated variances and covariance of β1 and β2, but in general it can be written as (β1 - β1_hat)^2 / var(β1_hat) + (β2 - β2_hat)^2 / var(β2_hat) - 2cov(β1_hat, β2_hat) / (var(β1_hat)var(β2_hat))

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Sure, let's break down this problem step by step.

1. First, let's understand the problem. We have a full rank linear model with two predictors x1 and x2, and we want to find a joint confidence region for the parameters β1 and β2. The errors are assumed to be independent and normally distributed with mean 0 and variance σ^2.

2. The joint confidence region for β1 and β2 is a region in the parameter space where, if the true values of the parameters were to fall in this region, the observed data would not be considered unusual at a significance level of α.

3. To derive the joint confidence region, we can use the fact that the marginal distribution over a subset of multivariate normal random variables is multivariate normal. This means that the joint distribution of β1 and β2 is also multivariate normal.

4. The mean vector and covariance matrix for this multivariate normal distribution can be obtained by dropping the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. In this case, the irrelevant variable is β0, so we drop it from the mean vector and the covariance matrix.

5. The mean vector for β1 and β2 is simply the vector of their expected values, which are the true values of β1 and β2. The covariance matrix is the matrix of their variances and covariance, which can be estimated from the data.

6. The joint 100(1 − α)% confidence region for β1 and β2 is then the region where the multivariate normal distribution of β1 and β2 exceeds the critical value for the χ^2 distribution with 2 degrees of freedom at the 1 − α level. This region is an ellipse in the β1-β2 plane centered at the estimated values of β1 and β2, with axes determined by the estimated variances and covariance of β1 and β2.

7. The exact expression for this ellipse depends on the estimated variances and covariance of β1 and β2, but in general it can be written as (β1 - β1_hat)^2 / var(β1_hat) + (β2 - β2_hat)^2 / var(β2_hat) - 2cov(β1_hat, β2_hat) / (var(β1_hat)var(β2_hat)) <= χ^2_(1-α, 2), where β1_hat and β2_hat are the estimated values of β1 and β2, var(β1_hat) and var(β2_hat) are their estimated variances, cov(β1_hat, β2_hat) is their estimated covariance, and χ^2_(1-α, 2) is the critical value of the χ^2 distribution with 2 degrees of freedom at the 1 − α level.

Knowee AI · Accepted Answer

Sure, let's break down this problem step by step.

1. First, let's understand the problem. We have a full rank linear model with two predictors x1 and x2, and we want to find a joint confidence region for the parameters β1 and β2. The errors are assumed to be independent and normally distributed with mean 0 and variance σ^2.

2. The joint confidence region for β1 and β2 is a region in the parameter space where, if the true values of the parameters were to fall in this region, the observed data would not be considered unusual at a significance level of α.

3. To derive the joint confidence region, we can use the fact that the marginal distribution over a subset of multivariate normal random variables is multivariate normal. This means that the joint distribution of β1 and β2 is also multivariate normal.

4. The mean vector and covariance matrix for this multivariate normal distribution can be obtained by dropping the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. In this case, the irrelevant variable is β0, so we drop it from the mean vector and the covariance matrix.

5. The mean vector for β1 and β2 is simply the vector of their expected values, which are the true values of β1 and β2. The covariance matrix is the matrix of their variances and covariance, which can be estimated from the data.

6. The joint 100(1 − α)% confidence region for β1 and β2 is then the region where the multivariate normal distribution of β1 and β2 exceeds the critical value for the χ^2 distribution with 2 degrees of freedom at the 1 − α level. This region is an ellipse in the β1-β2 plane centered at the estimated values of β1 and β2, with axes determined by the estimated variances and covariance of β1 and β2.

7. The exact expression for this ellipse depends on the estimated variances and covariance of β1 and β2, but in general it can be written as (β1 - β1_hat)^2 / var(β1_hat) + (β2 - β2_hat)^2 / var(β2_hat) - 2cov(β1_hat, β2_hat) / (var(β1_hat)var(β2_hat)) <= χ^2_(1-α, 2), where β1_hat and β2_hat are the estimated values of β1 and β2, var(β1_hat) and var(β2_hat) are their estimated variances, cov(β1_hat, β2_hat) is their estimated covariance, and χ^2_(1-α, 2) is the critical value of the χ^2 distribution with 2 degrees of freedom at the 1 − α level.

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