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Univariate and Multivariate Regression Analysis
1. Introduction
📌 Regression analysis is a statistical method used to examine relationships between independent (predictor) and dependent (outcome) variables.
📌 It helps in prediction, forecasting, and understanding causal relationships in economics, finance, and social sciences.
📌 There are two main types:
- Univariate Regression (Single predictor variable)
- Multivariate Regression (Multiple predictor variables)
✔ Example:
- Univariate Regression: Relationship between income (X) and consumption (Y).
- Multivariate Regression: Relationship between income (X₁), education level (X₂), and consumption (Y).
2. Univariate Regression Analysis
✔ Definition: Univariate regression examines the relationship between one independent variable (X) and one dependent variable (Y).
✔ It is also called simple regression.
🔹 (1) Linear Univariate Regression
✔ The most basic form is the Simple Linear Regression Model: Y=β0+β1X+ϵY = \beta_0 + \beta_1 X + \epsilon
where:
- YY = Dependent variable (outcome)
- XX = Independent variable (predictor)
- β0\beta_0 = Intercept (value of Y when X = 0)
- β1\beta_1 = Slope coefficient (change in Y for a one-unit change in X)
- ϵ\epsilon = Error term (unexplained variation)
✔ Example:
If we regress monthly consumption (Y) on income (X), the equation may look like: Consumption=200+0.5×Income\text{Consumption} = 200 + 0.5 \times \text{Income}
📌 Interpretation: A $1 increase in income leads to a $0.5 increase in consumption.
🔹 (2) Non-Linear Univariate Regression
✔ If the relationship between X and Y is not linear, we use non-linear functions: Y=β0+β1X2+ϵY = \beta_0 + \beta_1 X^2 + \epsilon
✔ Example: A quadratic function can model diminishing marginal returns (e.g., education and salary growth).
📌 When to Use Univariate Regression?
✅ When you need a simple model with one predictor.
✅ When relationships appear linear and straightforward.
✅ Useful for trend analysis and forecasting (e.g., inflation and GDP).
3. Multivariate Regression Analysis
✔ Definition: Multivariate regression examines the relationship between one dependent variable (Y) and multiple independent variables (X₁, X₂, X₃,…, Xₙ).
✔ It is used when multiple factors influence an outcome.
🔹 (1) Multiple Linear Regression
Y=β0+β1X1+β2X2+β3X3+…+βnXn+ϵY = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + … + \beta_n X_n + \epsilon
where:
- X1,X2,…,XnX_1, X_2, …, X_n = Multiple independent variables.
- β1,β2,…\beta_1, \beta_2, … = Coefficients (impact of each predictor on Y).
✔ Example: Predicting house prices (Y) based on:
- Square footage (X₁)
- Number of bedrooms (X₂)
- Distance from city center (X₃)
Price=50,000+300×Sq. Feet+10,000×Bedrooms−5,000×Distance\text{Price} = 50,000 + 300 \times \text{Sq. Feet} + 10,000 \times \text{Bedrooms} – 5,000 \times \text{Distance}
📌 Interpretation:
✅ A larger house increases price, but distance from the city reduces price.
🔹 (2) Interaction Terms in Multivariate Regression
✔ Sometimes, variables interact with each other: Y=β0+β1X1+β2X2+β3(X1×X2)+ϵY = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 (X_1 \times X_2) + \epsilon
✔ Example: Salary prediction may depend on both education (X₁) and experience (X₂), but their combined effect is stronger.
🔹 (3) Non-Linear Multivariate Regression
✔ When relationships are non-linear, we extend the model: Y=β0+β1X1+β2X12+β3X2+ϵY = \beta_0 + \beta_1 X_1 + \beta_2 X_1^2 + \beta_3 X_2 + \epsilon
✔ Example: Demand for a product may follow a logarithmic or quadratic trend.
4. Comparing Univariate vs. Multivariate Regression
| Feature | Univariate Regression | Multivariate Regression |
|---|---|---|
| Number of Predictors | One (XX) | Multiple (X1,X2,X3…X_1, X_2, X_3…) |
| Complexity | Simple | More complex |
| Use Cases | Trend analysis, forecasting | Explaining multiple factors |
| Accuracy | May miss key influences | More accurate but harder to interpret |
| Example | Income → Consumption | Income + Education → Consumption |
5. Assumptions in Regression Analysis
📌 Key assumptions for valid regression models:
✅ Linearity → The relationship between predictors and outcome is linear.
✅ No multicollinearity → Independent variables should not be highly correlated.
✅ Homoscedasticity → Variance of errors should be constant across observations.
✅ No autocorrelation → Errors should not be correlated in time-series data.
✅ Normality of residuals → Error terms should be normally distributed.
✔ Violation of assumptions can lead to biased or misleading results.
6. Applications of Regression Analysis in Economics
📌 Regression analysis is widely used in economics, finance, and business:
✅ Economic Growth: GDP growth (Y) based on investment (X₁), labor force (X₂), trade openness (X₃).
✅ Labor Market Analysis: Wage determination based on education, experience, gender, and industry.
✅ Inflation Forecasting: Inflation rate predicted by money supply, interest rates, and unemployment.
✅ Consumer Demand Analysis: Demand for a product based on price, income levels, and competitor prices.
✔ Example: A company might use multivariate regression to predict sales based on advertising spend, competitor pricing, and economic conditions.
7. Conclusion
✔ Univariate regression is useful for simple relationships, while multivariate regression is needed for complex interactions.
✔ Multivariate regression provides better predictions but requires more data and careful interpretation.
✔ Regression models help economists, policymakers, and businesses make informed decisions.