We have a random sample of workers from a large firm. In 2017, the firm ran a training program. Some workers did the training program, others did not. The firm now wants to assess the effect of the training on earnings.  We use the following model to estimate the effect of a training program on annual earnings in 2018: ln⁡(earn2018)=β0+β1train+β2ln⁡(earn2016)+β3educ+β4exper+u where earn2018 = individual total annual earnings in 2018 in dollars train = a dummy variable that takes the value 1 if the individual worker did the training in 2017 and 0 otherwise earn2016 = individual total annual earnings in 2016 in dollars educ = the individual's years of education    exper = the individual's years of experience     Now, I want to test whether the effect of an additional year of education increases earnings by twice as much as an additional year of experience. My null hypothesis is H0:β3=2β4. To get the standard error I need to conduct this hypothesis test, I rearrange or re-parameterise the model so when I estimate this new model I will have a coefficient estimate and its standard error that will allow me to test this hypothesis. Which of the following is the correct re-parameterised model?  a) ln⁡(earn2018) = β0 + β1train + β2ln⁡(earn2016) + (θ+2β4)educ + (θ-0.5β3)exper + u. where θ = β3 - 2β4 b) none of the answers are correct c) ln⁡(earn2018) = β0 + β1train + β2ln⁡(earn2016) + θ(educ + 2exper) + β4exper + u. where θ = β3 - 2β4 d) ln⁡(earn2018) = β0 + β1train + β2ln⁡(earn2016) + θeduc + β4(2educ + exper)+ u. where θ = β3 - 2β4 e) ln⁡(earn2018) = β0 + β1train + β2ln⁡(earn2016) + (θ - 2β4)educ + β4exper + u. where θ = β3 - 2β4

We have a random sample of workers from a large firm. In 2017, the firm ran a training program. Some workers did the training program, others did not. The firm now wants to assess the effect of the training on earnings. 

We use the following model to estimate the effect of a training program on annual earnings in 2018:

ln⁡(earn2018)=β₀+β₁train+β₂ln⁡(earn2016)+β₃educ+β₄exper+u

where

• earn2018 = individual total annual earnings in 2018 in dollars
• train = a dummy variable that takes the value 1 if the individual worker did the training in 2017 and 0 otherwise
• earn2016 = individual total annual earnings in 2016 in dollars
• educ = the individual's years of education   
• exper = the individual's years of experience  

Now, I want to test whether the effect of an additional year of education increases earnings by twice as much as an additional year of experience. My null hypothesis is H₀:β₃=2β₄.

To get the standard error I need to conduct this hypothesis test, I rearrange or re-parameterise the model so when I estimate this new model I will have a coefficient estimate and its standard error that will allow me to test this hypothesis.

Which of the following is the correct re-parameterised model? 

a) ln⁡(earn2018) = β₀ + β₁train + β₂ln⁡(earn2016) + (θ+2β₄)educ + (θ-0.5β₃)exper + u. where θ = β₃ - 2β₄

b) none of the answers are correct

c) ln⁡(earn2018) = β₀ + β₁train + β₂ln⁡(earn2016) + θ(educ + 2exper) + β₄exper + u. where θ = β₃ - 2β₄

d) ln⁡(earn2018) = β₀ + β₁train + β₂ln⁡(earn2016) + θeduc + β₄(2educ + exper)+ u. where θ = β₃ - 2β₄

e) ln⁡(earn2018) = β₀ + β₁train + β₂ln⁡(earn2016) + (θ - 2β₄)educ + β₄exper + u. where θ = β₃ - 2β₄