HW 2

Question 1

What is the assumption of the Pooled OLS (POLS) model to estimate the effect of exec consistently? In which way does the Fixed Effects (FE) model relax this assumption?

Answer 1

• POLS assumes a composite error term c_it = μ_i + v_i
  
  • requires that the error term is uncorrelated with all regressors: Cov(c_it, exec_it) = 0
  • this includes Cov(μ_i, exec_it) = 0
• FE is a time-demeaned estimation
  
  • therefore state specific effect μ_i drops out (it is constant over time)
  • only requires Cov(v_it, exec_it) = 0

Question 2

Compare the results of two specifications for the POLS model: one without and one with the south dummy.

Discuss how the effect of exec changes between these specifications. What does this suggest concerning the validity of the assumption behind the POLS model?

(both models show positive effect of exec, w/o the south dummy, there is a larger coefficient for exec and higher significance)

Answer 2

we see from excluding south that:

• the effect of exec is biased upwards
• suggests that state-specific effects are correlated with exec
• exogeneity assumption Cov(exec_it,ϵ_it)=0 is likely not met -> there are probably even other state-specific variables that are not controlled for -> POLS is not consistent

Question 3

Interpret the results of the FE model:

• FE: β_exec: -0.15
• POLS: β_exec: 0.26***

Give an economic interpretation as to why the effect of exec differs so much compared to the POLS model.

Why does the variable south drop out in the FE model?

Answer 3

1. Interpretation: one additional execution is associated with .15 fewer murders (FE) and .26 more murders (POLS) holding all other constant
2. Economic reasoning: in POLS, state specific effects such as culture or poverty are not accounted for. If they are included (FE), executions do not have a significant effect any more.
3. South dummy drops out in FE, because it is state-specific and time-constant

Question 4

Show formally that for T = 2, the First Differences (FD) estimator is equal to the FE estimator.

Answer 4

Question 5

what are stochastic regressors?

Answer 5

The term stochastic regressor means that the regressors, i.e. the explanatory variables are random with the change of time

Question 6

Consider the one-way error component regression model with random effects below.

Which estimator for β would you use in case of known variance components σ_μ² and σ_v² ? Explain why.

Answer 6

GLSE, because it is feasible and BLUE (more efficient)

Question 7

Consider the one-way error component regression model with random effects below.

Give the pooled OLS estimator of θ in matrix notation and show that it is unbiased. Does this also hold in the fixed effects model? Explain briefly.

Answer 7

1. Find ^θ_POLS.

• ^θ_POLS = (Z’Z)^-1Z’Y = (Z’Z)^-1Z’(Zθ+ε) = (Z’Z)^-1Z’Z θ + (Z’Z)^-1Z’ε
• ^θ_POLS = θ + (Z’Z)^-1Z’ε

2. Find E [^θ_POLS].

• E[^θ_POLS] = E[θ + (Z’Z)^-1Z’ε] = θ + (Z’Z)^-1Z’E[ε]
• E[ε] = 0 by assumption
• E[^θ_POLS] = θ

3. Fixed Effects model?

• in FE, E[ε] = E[Gμ+v] = Gμ
• POLS is biased unless all individual specific effects μ are zero!

Question 8

Show that the within-estimator ^β_w=(X’QX)^-1X’QY is an unbiased estimator of β.

• Q = I_NT - P
• P = G(G’G)^-1G’

What are additional assumption(s) required for that property?

Answer 8

1. Use F-W to simplify estimator ^β_w=(X’QX)^-1X’QY:

• ^β_w = (X’QX)^-1 X’Q (α1_NT+ Xβ + Gμ + v)
  
  • Due to F-W: Q1_NT = 0, QG = 0
• ^β_w = β + (X’QX)^-1 X’Q v

2. Show unbiasedness

• E[^β_w] = β + (X’QX)^-1 X’Q E[v]
  
  • By assumption, E[v] = 0
• E[^β_w] = β

3. Additional assumptions:

• X’QX needs to be invertible
• rk (x, G) = k + N

Question 9

Consider the one-way error component regression model with random effects below. Suppose that the explanatory variables are stochastic and at least partly correlated with the individual-specific effects μ_i such that E[μ|X] ≠ 0, but E[v|X] = 0.

Would you still use β_POLS? Explain briefly your answer.

Answer 9

No, because E[μ|X] ≠ 0 menas E[ε|X] ≠ 0 (endogenous regressors)

GLSE is now biased!

Question 10

Consider the one-way error component regression model with random effects below. Suppose that the explanatory variables are stochastic and at least partly correlated with the individual-specific effects μ_i such that E[μ|X] ≠ 0, but E[v|X] = 0.

Name a test to check the null hypothesis E[μ|X] = 0.

Answer 10

Hausman test

Question 11

Consider the one-way error component regression model with random effects below. Suppose that the explanatory variables are stochastic and at least partly correlated with the individual-specific effects μ_i such that E[μ|X] ≠ 0, but E[v|X] = 0.

Verify that the within-estimator ^β_w=(X’QX)^_-1X’QY remains unbiased. In which sense are the assumptions required for the unbiasedness of the within-estimator weaker than those required for the unbiasedness of the GLS estimator?

Answer 11

• E[^β_w] = β + E[(X’QX)^_-1X’Qv]
• = β + E[E[(X’QX)^_-1X’Qv] ] -> Law of iterated expectations
• = β + E[(X’QX)^_-1X’QE[v|X] ]
• = β due to E[v|X] = 0

Weaker assumptions because within estimator does not require E[μ|X] = 0