Interpreting Regression Coefficients

This worksheet will have practice exercises about reading and interpreting regression output. We'll work through a single dataset, building from simple to complex: basic interpretation, non-unit changes, omitted variable bias, quadratics and cubics, log outcomes, and interaction terms.

Dataset: We use sysuse nlsw88 (National Longitudinal Survey of Women, 1988), which is built into Stata. Key variables:

📊 Part 1: Basic Coefficient Interpretation

Regression 1a

Question 1: Interpret the coefficient on ttl_exp.

Question 2: What does the constant (_cons) mean?

Regression 1b: Adding a binary variable

Question 3: Interpret the coefficient on union.

Question 4: Why did the coefficient on ttl_exp change from 0.331 to 0.323 when we added union?

Regression 1c: Adding another binary variable

Question 5: Interpret the coefficient on collgrad.

Question 6: What is the predicted wage for a non-union, non-college graduate with 10 years of experience?

Question 7: What is the predicted wage for a union member, college graduate with 10 years of experience?

📏 Part 2: Non-Unit Changes

Regression 2a

Question 8: The coefficient on age is -0.068. Does this mean being one year older is associated with earning 7 cents less per hour? Is that a meaningful quantity?

Question 9: What is the association between a 10-year increase in age and wages?

Regression 2b

Question 10: Working one more hour per week is associated with earning $0.09 more per hour. What is the association with working one standard deviation more hours per week? (The SD of hours is about 10.5)

Regression 2c

Question 11: The coefficient on tenure is 0.042 and on experience is 0.264. A colleague says “tenure barely matters.” Is that a fair statement? (The SD of tenure is 5.5 and the SD of experience is 4.6.)

⚠️ Part 3: Omitted Variable Bias (Sign of Bias)

The omitted variable bias formula tells us how a coefficient changes when we add a variable to the regression:

OVB Formula: $\hat{\beta}_{short} = \hat{\beta}_{long} + \hat{\delta}_1 \cdot \hat{\gamma}$

Where: - $\hat{\beta}_{short}$ = coefficient on X in the regression without the omitted variable - $\hat{\beta}_{long}$ = coefficient on X in the regression with the omitted variable - $\hat{\delta}_1$ = coefficient of the omitted variable on Y (from the long regression) - $\hat{\gamma}$ = coefficient of X on the omitted variable (from an auxiliary regression)

The sign of the bias = sign($\hat{\delta}_1$) x sign($\hat{\gamma}$)

Example 1: Omitting union status

Short regression (without union):

reg wage ttl_exp

     ttl_exp |   .3314291

Long regression (with union):

reg wage ttl_exp union

     ttl_exp |   .3234277
       union |   1.282932

Auxiliary regression (union on experience):

reg union ttl_exp

     ttl_exp |   .0050413

Question 12: Using the OVB formula, what is the sign of the bias from omitting union from regression reg wage ttl_exp?

Example 2: Omitting experience

Short regression (without experience):

reg wage collgrad

    collgrad |   3.615502

Long regression (with experience):

reg wage collgrad ttl_exp

    collgrad |   3.260086
     ttl_exp |   .2983893

Auxiliary regression (experience on college):

reg ttl_exp collgrad

    collgrad |   1.191116

Question 13: What is the sign of the bias from omitting ttl_exp in the wage-on-college regression?

Example 3: Omitting college

Short regression (without college):

reg wage tenure

      tenure |   .1858747

Long regression (with college):

reg wage tenure collgrad

      tenure |   .1629178
    collgrad |    3.43025

Auxiliary regression (college on tenure):

reg collgrad tenure

      tenure |   .0066925

Question 14: What is the sign of the bias from omitting collgrad?

Question 15 (Thought Experiment): Suppose you run reg wage ttl_exp and you suspect that ability (unobserved) is an omitted variable. You believe more able people earn more wages and more able people also have more work experience. What is the sign of the bias?

📈 Part 4: Quadratic and Cubic Terms (Non-Linear Marginal Effects)

Quadratic: tenure and tenure-squared

The equation is: Wage = 6.327 + 0.344(Tenure) - 0.009(Tenure^2)

Question 16: Can you interpret 0.344 as “one more year of tenure is associated with $0.34 more in wages”?

Question 17: What is the formula for the marginal effect of tenure?

Question 18: What is the marginal effect of one more year of tenure for someone with 1, 5, 10, 15, and 20 years of tenure?

Question 19: At what level of tenure is the marginal effect zero (the “turning point”)?

Cubic

Question 20: What is the marginal effect formula for the cubic? Evaluate it at of 1,5,10,15, and 20

📝 Part 5: Log Dependent Variable – ln(y)

Regression 5a

Question 21: Interpret the coefficient on ttl_exp.

Question 22: What about a 5-year increase in experience?

Regression 5b: ln(wage) with a binary variable

Question 23: Interpret the coefficient on union.

Regression 5c: ln(wage) with multiple controls

Question 24: Interpret the coefficient on collgrad.

ln(y) with a quadratic

Question 25: What is the marginal effect of tenure at tenure = 5?

🔗 Part 6: Interaction Terms

Interaction terms let the effect of one variable depend on the level of another variable.

Continuous x Binary interaction

The equation is: Wage = 2.98 + 0.333(Exp) + 1.819(Union) - 0.041(Exp x Union)

Question 26: What is the effect of one more year of experience for non-union workers?

Question 27: What is the effect of one more year of experience for union workers?

Question 28: What does the coefficient on union (1.819) mean in the interaction model?

Binary x Binary interaction

Question 29: Interpret each coefficient.

Question 30: What is the predicted wage for each of the four groups?

Question 31: What is the “effect of college” for each group?

Question 32: What is the “effect of being in a union” for each group?