What are the various assumptions on which the multivariate regression analysis rests?

Multivariate regression analysis, a statistical technique used to understand the relationship between one dependent variable and two or more independent variables, is based on several key assumptions. Ensuring these assumptions are met is crucial for the validity of the regression model. The primary assumptions are:

1. Linearity

• Assumption: The relationship between the dependent and independent variables is linear.
• Implication: Non-linear relationships are not accurately captured by a linear model.
• Verification: Scatter plots of residuals vs. predicted values or each independent variable vs. the dependent variable can help check for linearity.

2. Independence

• Assumption: Observations are independent of each other.
• Implication: The presence of correlation between observations (autocorrelation) can lead to unreliable and unstable regression coefficients.
• Verification: Durbin-Watson test can be used to detect autocorrelation.

3. Multivariate Normality

• Assumption: The residuals (errors) are normally distributed.
• Implication: Non-normal residuals can lead to biases in the estimation process.
• Verification: Normal probability plots (Q-Q plots) or statistical tests like the Shapiro-Wilk test can be used to assess normality.

4. No or Little Multicollinearity

• Assumption: Independent variables are not highly correlated with each other.
• Implication: High multicollinearity can make it difficult to determine the individual effect of independent variables.
• Verification: Variance Inflation Factor (VIF) and Tolerance or correlation matrices can be used to detect multicollinearity.

5. Homoscedasticity

• Assumption: The variance of error terms (residuals) is constant across all levels of the independent variables.
• Implication: Heteroscedasticity (non-constant variance of residuals) can lead to inefficient estimates.
• Verification: Scatter plots of residuals vs. predicted values or independent variables can be used to check for homoscedasticity.

6. No Endogeneity of Regressors

• Assumption: The independent variables are not correlated with the error term.
• Implication: If this assumption is violated, it can result in biased and inconsistent estimates.
• Verification: Often difficult to test, but instrumental variable methods can be used if endogeneity is suspected.

7. Adequate Sample Size

• Assumption: The sample size should be sufficiently large relative to the number of independent variables.
• Implication: Small sample sizes can lead to overfitting and unreliable estimates.
• Verification: Generally, having at least 10-15 observations per independent variable is recommended.

8. Model Specification

• Assumption: The model is correctly specified, including all relevant variables and excluding irrelevant ones.
• Implication: Omitted variable bias or inclusion of irrelevant variables can distort the true relationship.
• Verification: Domain knowledge and stepwise regression techniques can help in model specification.

Conclusion

Meeting these assumptions is crucial for the reliability and interpretability of a multivariate regression analysis. Violations of these assumptions can lead to biased, inconsistent, or inefficient estimates, affecting the conclusions drawn from the model. It's important to conduct diagnostic tests and consider remedial measures if any of these assumptions are violated.