Nonlinear Models
Suppose you believe there is a linear relationship between unit changes in an independent variable called \(X\) and percentage changes in a dependent variable called \(Y\). That is, you believe that when \(X\) goes up by one unit, \(Y\) goes up by a constant percentage change, regardless of the level of \(X\). Of the following equations, which one would you choose to model this relationship?
- \(Y_i = \beta_1X_i^{\beta_2}v_i\)
- \(Y_i = \beta e^{\beta_2X_i}v_i\)
- \(Y_i = \beta_1 + \beta_2X_i + u_i\)
- \(Y_i = \beta_1 + \beta_2 \ln(X_i) + u_i\)
Solution:
To model a relationship where unit changes in \(X\) correspond to constant percentage changes in \(Y\), we typically use a logarithmic transformation. This is because percentage changes in \(Y\) suggest a multiplicative relationship between \(X\) and \(Y\).
The correct model is: \(Y_i = \beta_1 + \beta_2 \ln(X_i) + u_i\)
Explanation:
Option 1: \(y_i = \beta_1X_i^{\beta_2}v_i\)
- This suggests a power relationship, which implies a constant elasticity, not a constant percentage change.
Option 2: \(y_i = \beta e^{\beta_2X_i}v_i\)
- This implies an exponential relationship, where the growth rate of \(Y\) changes with \(X\).
Option 3: \(y_i = \beta_1 + \beta_2X_i + u_i\)
- This implies a linear relationship, not a percentage change.
Option 4: \(y_i = \beta_1 + \beta_2 \ln(X_i) + u_i\)
- This is the logarithmic model, which means a unit change in \(X\) leads to a percentage change in \(Y\).
Therefore, the correct answer is Option 4.
Consider the following model attempting to explain a person’s earnings from work (\(EARNINGS\)) as a function of their years of education (\(S\)), years of work experience (\(WEXP\)), and gender (\(FEMALE\)):
\[
EARNINGS_i = \beta S_i^{\beta_1}e^{\beta_2WEXP_i}e^{\beta_3FEMALE_i}v_i
\]
For this regression, which of the following statements is closest to being correct?
- \(\beta_2\) gives the percentage change in earnings resulting from a one-year increase in years of schooling, holding constant the levels of work experience and gender.
- \((\beta_3 \times 100)\) gives the percentage change in earnings resulting from a one percent change in work experience, holding constant the levels of schooling and gender.
- \(\beta_2\) gives the percentage change in earnings resulting from a one percent change in years of schooling, holding constant the levels of work experience and gender.
- \(\beta_3\) gives the percentage change in earnings resulting from a one-year increase in work experience, holding constant the levels of schooling and gender.
Solution:
The model given is: \(EARNINGS_i = \beta S_i^{\beta_1}e^{\beta_2WEXP_i}e^{\beta_3FEMALE_i}v_i\)
We need to find the correct interpretation of the coefficients.
- \(\beta_2\) gives the percentage change in earnings resulting from a one-year increase in years of schooling, holding constant the levels of work experience and gender.
- \((\beta_3 \times 100)\) gives the percentage change in earnings resulting from a one percent change in work experience, holding constant the levels of schooling and gender.
- \(\beta_2\) gives the percentage change in earnings resulting from a one percent change in years of schooling, holding constant the levels of work experience and gender.
- \(\beta_3\) gives the percentage change in earnings resulting from a one-year increase in work experience, holding constant the levels of schooling and gender.
Explanation:
- Option 1: This is incorrect because \(\beta_2\) is associated with \(WEXP\), not schooling.
- Option 2: This is correct. In the exponential term \(e^{\beta_2 WEXP_i}\), \(\beta_2\) represents the continuous percentage change. When multiplied by 100, it gives the percentage change per unit of \(WEXP_i\), assuming the exponential model.
- Option 3: This is incorrect because \(\beta_2\) is associated with \(WEXP\) in an exponential form, not schooling.
- Option 4: This is incorrect because \(\beta_3\) is associated with the binary variable \(FEMALE\), not work experience.
Therefore, the correct answer is Option 2.