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 predictions. The formula for $\hat{\sigma}^2$ 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 underlying assumptions of linear regression. The key assumption is that the residuals, $\mathbf{Y} - \mathbf{X} \hat{\beta}$, are normally distributed with mean 0 and variance $\sigma^2$. This assumption is crucial in deriving the distribution of $\hat{\sigma}^2$.

Derivation of the Distribution

Let's start by defining the residuals as $\mathbf{e} = \mathbf{Y} - \mathbf{X} \hat{\beta}$. Then, we can rewrite the formula for $\hat{\sigma}^2$ as:

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

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

Properties of the Normal Distribution

One of the key properties of the normal distribution is that the sum of normally distributed variables is also normally distributed. This property can be used to derive the distribution of $\hat{\sigma}^2$.

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

Using the properties of the normal distribution, we can show that $\hat{\sigma}^2$ follows a scaled chi-squared distribution with $n - p - 1$ degrees of freedom. The scaling factor is $\frac{1}{n - p - 1}$.

Proof

To prove that $\hat{\sigma}^2$ follows a scaled chi-squared distribution, we can use the following steps:

1. Show that the residuals, $\mathbf{e}$, are normally distributed with mean 0 and variance $\sigma^2$.
2. Show that the sum of the squared residuals, $\mathbf{e}^\top \mathbf{e}$, is a chi-squared random variable with $n - p - 1$ degrees of freedom.
3. Use the scaling factor, $\frac{1}{n - p - 1}$, to derive the distribution of $\hat{\sigma}^2$.

Step 1: Normality of Residuals

The residuals, $\mathbf{e}$, are normally distributed with mean 0 and variance $\sigma^2$ because the response variable, $\mathbf{Y}$, is normally distributed and the design matrix, $\mathbf{X}$, is fixed.

Step 2: Chi-Squared Distribution

The sum of the squared residuals, $\mathbf{e}^\top \mathbf{e}$, is a chi-squared random variable with $n - p - 1$ degrees of freedom because the residuals are normally distributed and the sum of the squared normal variables is a chi-squared variable.

Step 3: Scaling Factor

The scaling factor, $\frac{1}{n - p - 1}$, is used to derive the distribution of $\hat{\sigma}^2$. This scaling factor is necessary to ensure that the expected value of $\hat{\sigma}^2$ is equal to the true variance, $\sigma^2$.

Conclusion

In conclusion, the estimated variance, $\hat{\sigma}^2$, follows a scaled chi-squared distribution with $n - p - 1$ degrees of freedom. This result is crucial in assessing the goodness of fit of the linear regression model and making predictions.

Implications

The distribution of $\hat{\sigma}^2$ has several implications in linear regression:

• Confidence Intervals: The distribution of $\hat{\sigma}^2$ can be used to construct confidence intervals for the true variance, $\sigma^2$.
• Hypothesis Testing: The distribution of $\hat{\sigma}^2$ can be used to test hypotheses about the true variance, $\sigma^2$.
• Model Selection: The distribution of $\hat{\sigma}^2$ can be used to compare the fit of different linear regression models.

Real-World Applications

The distribution of $\hat{\sigma}^2$ has several real-world applications in linear regression:

• Predicting Stock Prices: The distribution of $\hat{\sigma}^2$ can be used to predict stock prices by estimating the variance of the residuals.
• Forecasting Demand: The distribution of $\hat{\sigma}^2$ can be used to forecast demand by estimating the variance of the residuals.
• Risk Analysis: The distribution of $\hat{\sigma}^2$ can be used to analyze risk by estimating the variance of the residuals.

Conclusion

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

A: The distribution of $\hat{\sigma}^2$ is a scaled chi-squared distribution with $n - p - 1$ degrees of freedom.

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

A: The distribution of $\hat{\sigma}^2$ is important in linear regression because it is used to assess the goodness of fit of the model and make predictions. It is also used to construct confidence intervals and test hypotheses about the true variance, $\sigma^2$.

Q: How is the distribution of $\hat{\sigma}^2$ related to the normal distribution?

A: The distribution of $\hat{\sigma}^2$ is related to the normal distribution because the residuals, $\mathbf{e}$, are normally distributed with mean 0 and variance $\sigma^2$. This normality assumption is crucial in deriving the distribution of $\hat{\sigma}^2$.

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

A: The scaling factor used in the distribution of $\hat{\sigma}^2$ is $\frac{1}{n - p - 1}$. This scaling factor is necessary to ensure that the expected value of $\hat{\sigma}^2$ is equal to the true variance, $\sigma^2$.

Q: How is the distribution of $\hat{\sigma}^2$ used in confidence intervals?

A: The distribution of $\hat{\sigma}^2$ is used in confidence intervals to estimate the true variance, $\sigma^2$. The confidence interval for $\sigma^2$ is given by:

$$\left( \frac{(n - p - 1) \hat{\sigma}^2}{\chi^2_{\alpha/2, n - p - 1}}, \frac{(n - p - 1) \hat{\sigma}^2}{\chi^2_{1 - \alpha/2, n - p - 1}} \right)$$

where $\chi^2_{\alpha/2, n - p - 1}$ and $\chi^2_{1 - \alpha/2, n - p - 1}$ are the critical values from the chi-squared distribution with $n - p - 1$ degrees of freedom.

Q: How is the distribution of $\hat{\sigma}^2$ used in hypothesis testing?

A: The distribution of $\hat{\sigma}^2$ is used in hypothesis testing to test hypotheses about the true variance, $\sigma^2$. The null and alternative hypotheses are:

$$H_0: \sigma^2 = \sigma_0^2$$

$$H_1: \sigma^2 \neq \sigma_0^2$$

The test statistic is given by:

$$\frac{(n - p - 1) \hat{\sigma}^2}{\sigma_0^2}$$

The p-value is calculated using the chi-squared distribution with $n - p - 1$ degrees of freedom.

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

A: Some real-world applications of the distribution of $\hat{\sigma}^2$ include:

• Predicting stock prices: The distribution of $\hat{\sigma}^2$ can be used to predict stock prices by estimating the variance of the residuals.
• Forecasting demand: The distribution of $\hat{\sigma}^2$ can be used to forecast demand by estimating the variance of the residuals.
• Risk analysis: The distribution of $\hat{\sigma}^2$ can be used to analyze risk by estimating the variance of the residuals.

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 normality assumption: The normality assumption is crucial in deriving the distribution of $\hat{\sigma}^2$. Ignoring this assumption can lead to incorrect conclusions.
• Using the wrong scaling factor: The scaling factor used in the distribution of $\hat{\sigma}^2$ is $\frac{1}{n - p - 1}$. Using the wrong scaling factor can lead to incorrect conclusions.
• Not accounting for the degrees of freedom: The distribution of $\hat{\sigma}^2$ has $n - p - 1$ degrees of freedom. Not accounting for the degrees of freedom can lead to incorrect conclusions.