What are the assumptions of the classical linear regression model?

When discussing the assumptions of the classical linear regression model, it's important to understand that these assumptions form the foundation for ensuring that linear regression provides unbiased and efficient estimates. Here's a breakdown suitable for a FAANG interview setting:

1. Linearity: The relationship between the independent variables and the dependent variable should be linear.
2. Independence: The observations should be independent of each other.
3. Homoscedasticity: The variance of error terms should be constant across all levels of the independent variables.
4. No Multicollinearity: The independent variables should not be too highly correlated with each other.
5. Normality of Errors: The error terms should be normally distributed.
6. No Autocorrelation: The residuals (errors) should not be correlated with each other.

Key Talking Points:

• Linearity: Ensure the relationship is linear.
• Independence: Observations must be independent.
• Homoscedasticity: Constant variance of errors.
• No Multicollinearity: Avoid high correlation among predictors.
• Normality of Errors: Errors should be normally distributed.
• No Autocorrelation: Residuals should not be correlated.

NOTES:

Reference Table:

• Linearity: The road should be straight.
• Independence: Each part of the journey should not depend on the previous one.
• Homoscedasticity: The bumps on the road (errors) should be evenly distributed.
• No Multicollinearity: The map (variables) should clearly guide you without overlapping routes.
• Normality of Errors: The bumps (errors) should be predictable and not extreme.
• No Autocorrelation: Each bump should be independent of the previous bumps.

Follow-Up Questions and Answers:

1. How would you test for multicollinearity in a dataset?

   • You can use Variance Inflation Factor (VIF) to test for multicollinearity. A VIF value greater than 10 is a common threshold for indicating multicollinearity.

2. What methods can be used to correct heteroscedasticity?

   • You can use techniques such as weighted least squares, transforming the dependent variable, or using robust standard errors.

3. How can you check for autocorrelation in residuals?

   • The Durbin-Watson test is commonly used to detect the presence of autocorrelation in residuals.

4. What is the impact of violating the normality assumption?

   • Violating the normality assumption affects the validity of hypothesis tests and confidence intervals, but it doesn’t necessarily bias the parameter estimates.

5. Can you provide a simple code snippet to check for linearity in Python?

import matplotlib.pyplot as plt
import seaborn as sns
import pandas as pd

# Assume 'data' is a pandas DataFrame and 'x' and 'y' are column names
sns.lmplot(x='x', y='y', data=data)
plt.show()

This code snippet uses a scatter plot with a linear fit line to visually inspect the linearity assumption.