How To Derive The Distribution Of $ \hat{\sigma}^2$?

Introduction

In linear regression, the estimated variance, denoted as $\hat{\sigma}^2$, is a crucial component in assessing the goodness of fit of the model and making inferences about the population. The formula for the estimated variance is given by:

$$\hat{\sigma}^2 = \frac{1}{n - p - 1} ( \mathbf{Y} - \mathbf{X} \hat{\beta} )^\top ( \mathbf{Y} - \mathbf{X} \hat{\beta} )$$

where $\mathbf{Y}$ is the vector of response variables, $\mathbf{X}$ is the design matrix, $\hat{\beta}$ is the vector of estimated coefficients, $n$ is the number of observations, and $p$ is the number of predictors.

Understanding the Formula

To derive the distribution of $\hat{\sigma}^2$, we need to understand the components of the formula. The term $(\mathbf{Y} - \mathbf{X} \hat{\beta})$ represents the vector of residuals, which is the difference between the observed response variables and the predicted values based on the estimated coefficients.

Assumptions of Linear Regression

Before we proceed with deriving the distribution of $\hat{\sigma}^2$, it is essential to recall the assumptions of linear regression. The assumptions are:

• Linearity: The relationship between the response variable and the predictors is linear.
• Independence: Each observation is independent of the others.
• Homoscedasticity: The variance of the residuals is constant across all levels of the predictors.
• Normality: The residuals are normally distributed.
• No multicollinearity: The predictors are not highly correlated with each other.

Deriving the Distribution of $\hat{\sigma}^2$

To derive the distribution of $\hat{\sigma}^2$, we can use the following steps:

1. Express $\hat{\sigma}^2$ in terms of the residuals: We can rewrite the formula for $\hat{\sigma}^2$ as:

$$\hat{\sigma}^2 = \frac{1}{n - p - 1} \mathbf{e}^\top \mathbf{e}$$

where $\mathbf{e}$ is the vector of residuals.

2. Use the properties of the normal distribution: Since the residuals are normally distributed, we can use the properties of the normal distribution to derive the distribution of $\hat{\sigma}^2$.

3. Apply the chi-squared distribution: The sum of the squares of the residuals, $\mathbf{e}^\top \mathbf{e}$, follows a chi-squared distribution with $n - p - 1$ degrees of freedom.

4. Derive the distribution of $\hat{\sigma}^2$: Since $\hat{\sigma}^2$ is a function of the sum of the squares of the residuals, it also follows a chi-squared distribution with $n - p - 1$ degrees of freedom.

Properties of the Chi-Squared Distribution

The chi-squared distribution has the following properties:

• Mean: The mean of the chi-squared distribution is equal to the number of degrees of freedom.
• Variance: The variance of the chi-squared distribution is equal to twice the number of degrees of freedom.
• Shape: The chi-squared distribution is skewed to the right, with a longer tail on the right side.

Interpretation of the Distribution of $\hat{\sigma}^2$

The distribution of $\hat{\sigma}^2$ provides valuable information about the variability of the residuals. A small value of $\hat{\sigma}^2$ indicates that the residuals are small, suggesting that the model is a good fit to the data. On the other hand, a large value of $\hat{\sigma}^2$ indicates that the residuals are large, suggesting that the model is not a good fit to the data.

Conclusion

In conclusion, the distribution of $\hat{\sigma}^2$ is a crucial component in assessing the goodness of fit of a linear regression model. By understanding the properties of the chi-squared distribution, we can derive the distribution of $\hat{\sigma}^2$ and interpret its meaning in the context of the model.

References

• Hosmer, D. W., & Lemeshow, S. (2000). Applied logistic regression. Wiley-Interscience.
• Kutner, M. H., Nachtsheim, C. J., & Neter, J. (2005). Applied linear regression models. McGraw-Hill Irwin.
• Weisberg, S. (2005). _Applied linear regression**. Wiley-Interscience.**
  Frequently Asked Questions about the Distribution of $\hat{\sigma}^2$ ====================================================================

Q: What is the distribution of $\hat{\sigma}^2$?

A: The distribution of $\hat{\sigma}^2$ is a chi-squared distribution with $n - p - 1$ degrees of freedom, where $n$ is the number of observations and $p$ is the number of predictors.

Q: Why is the distribution of $\hat{\sigma}^2$ important?

A: The distribution of $\hat{\sigma}^2$ is important because it provides valuable information about the variability of the residuals. A small value of $\hat{\sigma}^2$ indicates that the residuals are small, suggesting that the model is a good fit to the data. On the other hand, a large value of $\hat{\sigma}^2$ indicates that the residuals are large, suggesting that the model is not a good fit to the data.

Q: What are the properties of the chi-squared distribution?

A: The chi-squared distribution has the following properties:

• Mean: The mean of the chi-squared distribution is equal to the number of degrees of freedom.
• Variance: The variance of the chi-squared distribution is equal to twice the number of degrees of freedom.
• Shape: The chi-squared distribution is skewed to the right, with a longer tail on the right side.

Q: How can I use the distribution of $\hat{\sigma}^2$ to make inferences about the population?

A: You can use the distribution of $\hat{\sigma}^2$ to make inferences about the population by:

• Calculating the standard error of $\hat{\sigma}^2$: The standard error of $\hat{\sigma}^2$ can be calculated using the formula:

$$\text{SE}(\hat{\sigma}^2) = \frac{\hat{\sigma}^2}{\sqrt{2(n - p - 1)}}$$

• Constructing a confidence interval for $\sigma^2$: A confidence interval for $\sigma^2$ can be constructed using the formula:

$$\hat{\sigma}^2 \pm t_{\alpha/2, n - p - 1} \text{SE}(\hat{\sigma}^2)$$

where $t_{\alpha/2, n - p - 1}$ is the critical value from the t-distribution with $n - p - 1$ degrees of freedom.

Q: What are some common mistakes to avoid when working with the distribution of $\hat{\sigma}^2$?

A: Some common mistakes to avoid when working with the distribution of $\hat{\sigma}^2$ include:

• Ignoring the assumptions of linear regression: The assumptions of linear regression, such as linearity, independence, homoscedasticity, normality, and no multicollinearity, must be met in order to use the distribution of $\hat{\sigma}^2$.
• Failing to check for outliers: Outliers can significantly affect the distribution of $\hat{\sigma}^2$, so it is essential to check for outliers before using the distribution.
• Using the distribution of $\hat{\sigma}^2$ without considering the sample size: The sample size must be sufficient to ensure that the distribution of $\hat{\sigma}^2$ is reliable.

Q: What are some real-world applications of the distribution of $\hat{\sigma}^2$?

A: The distribution of $\hat{\sigma}^2$ has many real-world applications, including:

• Predicting stock prices: The distribution of $\hat{\sigma}^2$ can be used to predict stock prices by modeling the volatility of the stock.
• Analyzing the effectiveness of a treatment: The distribution of $\hat{\sigma}^2$ can be used to analyze the effectiveness of a treatment by modeling the variability of the treatment outcomes.
• Forecasting energy demand: The distribution of $\hat{\sigma}^2$ can be used to forecast energy demand by modeling the variability of the energy demand.

Conclusion

In conclusion, the distribution of $\hat{\sigma}^2$ is a crucial component in assessing the goodness of fit of a linear regression model. By understanding the properties of the chi-squared distribution and avoiding common mistakes, you can use the distribution of $\hat{\sigma}^2$ to make inferences about the population and make predictions in real-world applications.