ETC2410 Assignment 1 Q1

Author

Hector Wild

ETC2410 Assignment 1

Hector Wild (33843570)

Question 1: Regression Analysis

Question 1(a): Regression Estimation

We estimate the following regression equation using OLS:

\[ \hat{\text{wage}} = \beta_0 + \beta_1 \cdot \text{IQ} \]

The estimated coefficients and their standard errors are:

library(wooldridge)

data("wage2")

model <- lm(wage ~ IQ, data = wage2)

b0 <- round(coef(model)[1], 3)  # Intercept (beta_0)
b1 <- round(coef(model)[2], 3)  # Coefficient for IQ (beta_1)
se_b0 <- round(summary(model)$coefficients[1,2], 3)  # Standard error of beta_0
se_b1 <- round(summary(model)$coefficients[2,2], 3)  # Standard error of beta_1

b0
(Intercept) 
    116.992 
b1
   IQ 
8.303 
se_b0
[1] 85.642
se_b1
[1] 0.836

The estimated coefficients and their standard errors are:

Intercept (\(\beta_0\) = 116.992) with Standard Error: 85.642

IQ Coefficient (\(\beta_1\) = 8.303) with Standard Error: 0.836

Question 1(b): Interpretation of Estimated Coefficients

From part (a), we obtained the following regression model:

\[ \hat{\text{wage}} = 116.992 + 8.303 \cdot \text{IQ} \] The estimated coefficients and their interpretations are:

  • Intercept (β₀ = 116.992): This represents the expected wage for an individual with an IQ of 0. Although this value may not have a meaningful interpretation (since an IQ of 0 is unrealistic), it helps define the baseline level of wage in the model.

  • IQ coefficient (β₁ = 8.303): This represents the expected increase in wage for each one-point increase in IQ. For instance, if someone’s IQ increases by 1 point, their expected wage would increase by 8.303 units.

This interpretation supports the gut feeling that people with higher IQs typically make more money because they are more educated, have more talents, and perform better on the job. It’s crucial to remember that the particular context or data set being utilized may have an impact on how much of a change occurs. Despite being technically valid, the intercept could not make sense in practice since a person’s IQ of 0 is unrealistic.The interpretation assumes all other factors are held constant.

Question 1(c): Predicted Change in Wage for an Increase in IQ by Two Standard Deviations

We use the estimated regression equation:

\[ \hat{\text{wage}} = 116.992 + 8.303 \cdot \text{IQ} \]

sd_IQ <- sd(wage2$IQ)  # Standard deviation of IQ in the wage2 dataset


increase_IQ <- 2 * sd_IQ  # Increase in IQ by two standard deviations


wage_change <- b1 * increase_IQ  # Wage change for a two-standard-deviation increase in IQ


predicted_wage_90 <- b0 + b1 * 90  # Predicted wage when IQ = 90
predicted_wage_110 <- b0 + b1 * 110  # Predicted wage when IQ = 110


predicted_change_90 <- predicted_wage_90 + wage_change  # Predicted wage change for IQ = 90
predicted_change_110 <- predicted_wage_110 + wage_change  # Predicted wage change for IQ = 110

## Results: Output relevant variables for interpretation
sd_IQ  # Standard deviation of IQ
[1] 15.05264
increase_IQ  # Two standard deviations increase in IQ
[1] 30.10527
wage_change  # Predicted change in wage for an increase of two standard deviations
      IQ 
249.9641 
predicted_wage_90  # Predicted wage for IQ = 90
(Intercept) 
    864.262 
predicted_wage_110  # Predicted wage for IQ = 110
(Intercept) 
   1030.322 
predicted_change_90  # Predicted wage change for IQ = 90
(Intercept) 
   1114.226 
predicted_change_110  # Predicted wage change for IQ = 110
(Intercept) 
   1280.286

Results Summary: - Standard deviation of IQ: 15.05 - Wage change for 2 SD increase in IQ: $249.96 - Wage for IQ = 90 increases from $864.26 → $1114.23 - Wage for IQ = 110 increases from $1030.32 → $1280.29

Interpretation

A two-standard-deviation increase in IQ predicts an increase in wage of approximately $250.

  • For an individual with IQ = 90, wage increases from $864.26 to $1114.23.
  • For an individual with IQ = 110, wage increases from $1030.32 to $1280.29.

This demonstrates the linear effect of IQ in the model — the wage increase is the same regardless of starting IQ.

Question 1(d): Does IQ Explain Variation in Wage?

We now assess how well IQ explains the variation in wages using the R-squared and F-statistic from our regression model.

# Extract R-squared
r_squared <- summary(model)$r.squared

# Extract F-statistic and degrees of freedom
f_stat <- summary(model)$fstatistic[1]
f_df1 <- summary(model)$fstatistic[2]
f_df2 <- summary(model)$fstatistic[3]

# Display values for internal reference
r_squared
[1] 0.09553528
f_stat
   value 
98.54936 
f_df1
numdf 
    1 
f_df2
dendf 
  933

Summary and Interpretation:

  • The R-squared value is 0.095, meaning that approximately 9.5% of the variation in wage is explained by IQ in this model.
  • The F-statistic is 98.55 with 1 and 934 degrees of freedom, which is highly statistically significant, indicating that the relationship between IQ and wage is not due to random chance.

Conclusion: IQ does explain some of the variation in wage (9.5%), but it does not explain most of it. Therefore, while IQ is statistically significant, the overall explanatory power of the model remains low—suggesting other factors are more influential in determining wage outcomes.

Question 1(e): Can We Conclude Causality from IQ to Wage?

Analysis and Conclusion:

We cannot assume causality from this result alone. While the regression indicates a positive and statistically significant association between IQ and wage, this reflects a correlation rather than a causal relationship.

Possible omitted variables (e.g., education, motivation, family background) may explain both IQ and wage levels, leading to potential omitted variable bias.

Conclusion: We cannot conclude that higher IQs lead directly to higher wages. The observed association is likely influenced by unobserved factors. Without a controlled design or experimental data, causality cannot be established.

End of Assignment 1 Question 1