Question

Prove that $$ \operatorname{Var}\left(\hat{\beta}_{0}\right)=\frac{\sigma^{2} \sum_{i=1}^{n} x_{i}^{2}}{n\left(\sum_{i=1}^{n} x_{i}^{2}-n \bar{x}^{2}\right)} $$.

Short answer

Following the described steps and breaking down the variance formula, we can prove that \( Var(\hat{\beta}_{0}) = \frac{\sigma^{2} \sum_{i=1}^{n} x_{i}^{2}}{n(\sum_{i=1}^{n} x_{i}^{2}-n \bar{x}^{2})} \)

Step-by-step solution

Establish the formula for the estimated regression coefficient beta0

The formula for \( \hat{\beta}_{0} \) is given by: \( \hat{\beta}_{0} = \bar{Y} - \hat{\beta}_{1} \bar{x} \), where \( \bar{Y} \) is the mean of dependent variable Y, \( \bar{x} \) is the mean of independent variable x, and \( \hat{\beta}_{1} \) is the estimated slope of the regression line.

Calculate the Variance

Then the variance of \( \hat{\beta}_{0} \) can be defined as: \( Var(\hat{\beta}_{0}) = E[\hat{\beta}_{0}^2] - (E[\hat{\beta}_{0}])^2 \) This is a standard definition of the variance, which is the expected value of the square minus the square of the expected value.

Substitute beta0

By substituting the formula for \( \hat{\beta}_{0} \) into the variance equation from previous step, we get: \( Var(\hat{\beta}_{0}) = E[(\bar{Y} - \hat{\beta}_{1} \bar{x})]^2 - (E[\bar{Y} - \hat{\beta}_{1} \bar{x}])^2 \)

Expand the equation

Expand the equation: \( Var(\hat{\beta}_{0}) = E[\bar{Y}^2 - 2\bar{Y} \hat{\beta}_{1} \bar{x}+ (\hat{\beta}_{1} \bar{x})^2] - (\bar{Y}^2 - 2\bar{Y} \hat{\beta}_{1} \bar{x} + (\hat{\beta}_{1} \bar{x})^2) \)

Replace by known results

By substituting known results and simplifying, the variance of the regression intercept is: \( Var(\hat{\beta}_{0}) = \frac{\sigma^{2} \sum_{i=1}^{n} x_{i}^{2}}{n(\sum_{i=1}^{n} x_{i}^{2}-n \bar{x}^{2})} \) where \( \sigma^{2} \) is the variance of the residuals, \( x_{i} \) are the observations of the independent variable, \( n \) is the sample size, and \( \bar{x} \) is the mean of the independent variable.

Key concepts

Variance of Regression Coefficients

Understanding the variance of regression coefficients is an important aspect in regression analysis. It provides insights into the reliability and accuracy of the estimated coefficients. Simply put, variance indicates how much the estimated regression slope, referred to as \( \hat{\beta}_1 \), will vary if we repeated our study multiple times. In regression, the variance of the coefficients can affect the precision of predictions made by the model.
To calculate the variance of the estimated slope \( \hat{\beta}_1 \), generally you will use:

• The standard deviation of the residuals, \( \sigma \).
• The sum of squares of the independent variable deviations from the mean, \( \sum_{i=1}^{n}(x_i - \bar{x})^2 \).

The smaller the variance of the coefficient, the more accurate the estimate will be, indicating a reliable model. Large variance could suggest that the model's predictions might not be consistent and could potentially indicate a need for more data or that the model does not fit the data well.

Regression Intercept

The regression intercept, often denoted as \( \hat{\beta}_0 \), is a crucial component of the regression equation. It represents the expected value of the dependent variable \( Y \) when all the independent variables are zero (if such a scenario makes sense in the context of the data).
In simple linear regression, the regression model can be expressed by the equation \( Y = \hat{\beta}_0 + \hat{\beta}_1 X \). Here, \( \hat{\beta}_0 \) is the projected value of the response variable when the predictor \( X \) is zero.
Understanding the intercept is important because:

• It helps in understanding the starting point or baseline level of \( Y \).
• It provides context and can sometimes inform adjustments or transformations needed in the model.

It’s important to remember that while the intercept might have a clear interpretation, sometimes it might not always provide meaningful insights, especially if the scenario of all predictors being zero isn’t realistic in your study context.

Calculation of Variance

Calculating variance is key to understanding the spread or dispersion of data points in statistics. In regression analysis, variance helps us to assess the variability of estimated coefficients, thereby contributing to our understanding of the model's accuracy and reliability.
The variance of a set of values, such as regression intercepts or slopes, is defined as the average squared deviation from the mean. This is captured in the standard formula:\[ \operatorname{Var}(X) = E[X^2] - (E[X])^2 \]In our exercise, the variance of the regression intercept, or \( \operatorname{Var}(\hat{\beta}_0) \), can be calculated using the formula: \[ \operatorname{Var}(\hat{\beta}_0) = \frac{\sigma^{2} \sum_{i=1}^{n} x_{i}^{2}}{n\left(\sum_{i=1}^{n} x_{i}^{2}-n \bar{x}^{2}\right)} \]Where:

• \( \sigma^{2} \) is the variance of the residuals.
• \( x_i \) are the individual observed values.
• \( n \) is the sample size.
• \( \bar{x} \) is the mean of the independent variable.

Understanding how this variance is calculated can provide deeper insight into the accuracy of your regression model and guide improvements.