Introduction
Regression models are a core component of econometrics and statistics, used to explain the relationship between a dependent variable and one or more independent variables. There are several functional forms of regression models, each suited for different types of data behavior and analysis objectives. This response also solves the numerical portion based on the regression results provided for 46 districts in Uttar Pradesh (UP) for the year 2020.
Various Functional Forms of Regression Models
1. Linear Regression Model
The linear regression model is of the form:
Y = α + βX + ε
Here, a unit change in X leads to a constant change in Y. This is the most commonly used form and assumes a linear relationship between variables.
2. Log-Linear Model
log Y = α + βX + ε
Used when the dependent variable grows at an exponential rate relative to X. It implies that a one-unit change in X leads to a percentage change in Y.
3. Linear-Log Model
Y = α + β log X + ε
Used when a percentage change in X leads to an absolute change in Y.
4. Log-Log Model
log Y = α + β log X + ε
This is commonly used for elasticity analysis. The coefficient β gives the elasticity directly. This model is also referred to as a constant elasticity model.
5. Polynomial Regression
Y = α + β1X + β2X² + ... + ε
Used when the relationship between X and Y is non-linear, but can be expressed as a polynomial.
6. Exponential Model
Y = αe^(βX) + ε
This form is used when the rate of change in Y increases or decreases exponentially with X.
Given Regression Equation (Log-Log Model)
From the problem:
log C = 4.30 - 1.34 log P + 0.17 log Y
Standard errors: (0.91), (0.32), (0.20)
R² = 0.27
Answers to Sub-Questions
i. What is the elasticity of demand for cigarettes with respect to price?
Since this is a log-log model, the coefficient of log P directly gives the price elasticity of demand:
Price Elasticity of Demand = -1.34
This indicates that a 1% increase in price leads to a 1.34% decrease in cigarette consumption, implying that demand is elastic with respect to price.
ii. What is the income elasticity of demand for cigarettes? Is it statistically significant?
Income elasticity is given by the coefficient of log Y:
Income Elasticity = 0.17
To determine statistical significance, we calculate the t-statistic:
t = coefficient / standard error = 0.17 / 0.20 = 0.85
This t-value is quite low and generally not significant at 5% significance level (critical value ~1.96 for a two-tailed test). Hence, the coefficient is not statistically significant. This means we cannot confidently say income affects cigarette consumption in this data set.
iii. How would you retrieve R² from the adjusted R² given above?
Here, R² is already provided as 0.27. But if we were given Adjusted R² instead, we could retrieve R² using the formula:
Adjusted R² = 1 - [(1 - R²) * (n - 1)/(n - k - 1)]
Where:
- n = number of observations (46)
- k = number of independent variables (2: log P and log Y)
If Adjusted R² was given, you could rearrange this equation to solve for R².
Conclusion
This problem provides a good example of a log-log regression model, commonly used to measure elasticity. We interpreted the elasticity of price and income, and evaluated statistical significance using t-tests. Understanding different functional forms helps in selecting the right model depending on the economic behavior being studied.