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Correlation and Regression in Economics
1. Introduction
📌 Correlation and regression are essential tools in economics to analyze relationships between variables.
- Correlation measures the strength and direction of the relationship between two variables.
- Regression helps estimate how one variable affects another.
📌 Applications in Economics:
✔ GDP vs. Investment: How investment influences economic growth.
✔ Inflation vs. Unemployment: Analyzing the Phillips Curve relationship.
✔ Education vs. Income: Measuring the impact of education on wages.
2. Correlation Analysis
🔹 (1) What is Correlation?
✔ Measures the degree to which two variables move together.
✔ Value ranges from -1 to +1.
📌 Interpretation:
- +1+1 (Perfect Positive Correlation): As one variable increases, the other also increases.
- −1-1 (Perfect Negative Correlation): As one variable increases, the other decreases.
- 00 (No Correlation): No relationship between variables.
✔ Example: Inflation and Interest Rates
If inflation rises, interest rates also rise, showing positive correlation.
🔹 (2) Pearson’s Correlation Coefficient (rr)
✔ Formula: r=∑(Xi−Xˉ)(Yi−Yˉ)∑(Xi−Xˉ)2∑(Yi−Yˉ)2r = \frac{\sum (X_i – \bar{X}) (Y_i – \bar{Y})}{\sqrt{\sum (X_i – \bar{X})^2 \sum (Y_i – \bar{Y})^2}}
where:
- XiX_i and YiY_i are individual values.
- Xˉ\bar{X} and Yˉ\bar{Y} are means.
✔ Pros: Measures linear relationships.
✔ Cons: Cannot detect causation.
📌 Example: GDP Growth and Exports
If r=0.8r = 0.8, it means strong positive correlation: when exports rise, GDP grows.
🔹 (3) Spearman’s Rank Correlation
✔ Measures monotonic relationships (non-linear).
✔ Useful when data is ordinal.
📌 Example: Education and Income Levels
✔ Higher education levels often correlate with higher wages.
3. Regression Analysis
🔹 (1) What is Regression?
✔ Regression quantifies the relationship between dependent (YY) and independent (XX) variables.
✔ Helps in prediction and causal analysis.
📌 Example: Consumption Function Consumption=β0+β1×Income+ϵConsumption = \beta_0 + \beta_1 \times \text{Income} + \epsilon
✔ If β1=0.8\beta_1 = 0.8, then for every $1 increase in income, consumption increases by $0.80.
🔹 (2) Simple Linear Regression (SLR)
✔ Equation: Y=β0+β1X+ϵY = \beta_0 + \beta_1 X + \epsilon
where:
- YY = dependent variable (e.g., demand).
- XX = independent variable (e.g., price).
- β0\beta_0 = intercept (value of YY when X=0X = 0).
- β1\beta_1 = slope (rate of change in YY per unit of XX).
- ϵ\epsilon = error term (unobserved factors).
📌 Example: Price vs. Demand Demand=100−2×Price\text{Demand} = 100 – 2 \times \text{Price}
✔ If price = $10, demand = 80.
✔ If price = $20, demand = 60.
✔ Pros: Easy to interpret.
✔ Cons: Assumes linearity.
🔹 (3) Multiple Linear Regression (MLR)
✔ Includes multiple independent variables.
✔ Equation: Y=β0+β1X1+β2X2+…+βnXn+ϵY = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + … + \beta_n X_n + \epsilon
✔ Used for:
- GDP Growth Model: Based on investment, inflation, and labor supply.
- Wage Determination: Based on education, experience, and skills.
📌 Example: GDP Growth Model GDP=2+0.5×Investment−0.3×Inflation\text{GDP} = 2 + 0.5 \times \text{Investment} – 0.3 \times \text{Inflation}
✔ If investment increases by 1 unit, GDP rises by 0.5 units.
✔ If inflation increases by 1 unit, GDP falls by 0.3 units.
✔ Pros: More realistic than simple regression.
✔ Cons: Harder to interpret.
🔹 (4) R-Squared (R2R^2) – Goodness of Fit
✔ Measures how well the regression model fits the data.
✔ Value: 0 to 1 (higher means better fit).
📌 Example: If R2=0.85R^2 = 0.85, it means 85% of variations in GDP are explained by investment and inflation.
✔ Pros: Helps assess model accuracy.
✔ Cons: High R2R^2 doesn’t mean causation.
4. Differences Between Correlation and Regression
| Feature | Correlation | Regression |
|---|---|---|
| Purpose | Measures relationship strength | Predicts the effect of one variable on another |
| Causality | No causation | Can indicate causation |
| Equation | No equation | Uses Y=β0+β1X+ϵY = \beta_0 + \beta_1 X + \epsilon |
| Direction | Symmetric (X & Y interchangeable) | Asymmetric (X predicts Y) |
📌 Example:
✔ Correlation: Higher education levels are correlated with higher incomes.
✔ Regression: A 1-year increase in education leads to a $5000 increase in income.
5. Applications in Economics
✔ Demand Forecasting: Predicting sales based on price and advertising.
✔ Investment Decisions: Analyzing returns based on risk factors.
✔ Public Policy: Estimating the impact of tax changes on GDP.
6. Conclusion
✔ Correlation shows the strength of relationships but doesn’t imply causation.
✔ Regression estimates the effect of one variable on another and helps in predictions.
✔ Applications in economics range from demand estimation to policy analysis.