Regression: Estimated Partial Effect of Regressor? (Need help understanding answer)

Question

1. Estimate a linear regression model for log wages on education, experience, and experience squared. Report Results.
2. Predict the effect on average log wage of increasing everybody's education level by one year.

Hint: If the regression model is $$ \log (\text { wage })_{i}=\beta_{0}+\beta_{1} \times \text { educ }_{i}+\beta_{2} \times \operatorname{exper}_{i}+\beta_{3} \times \operatorname{exper}_{i}^{2}+\varepsilon_{i} $$ then the effect of increasing education level of individual $i$ by one year is $$ \theta_{i}=\beta_{1}-\beta_{2}-\beta_{3} \cdot\left(2 \cdot \operatorname{exper}_{i}-1\right) $$ because one year additional education implies one year less work experience.

The effect we are interested in is: $$ \theta=\beta_{1}-\beta_{2}-\beta_{3} \cdot(2 \cdot \overline{\operatorname{exper}}-1) $$

After you defined this, compare the task to that in lecture 5 where we consider the partial effect in a quadratic model. In lecture, we saw how this partial effect can be specified as a linear combination of the estimators. I.e. if $c$ is a K x 1 vector of constants then the partial effect can be expressed as a linear combination: $ c' \hat\beta$. In this regard, please see included answer below.

I do not understand how he came about the vector of constants or the standard error. More specifically, how do you get standard error of .0062 as shown at the very bottom of the answer below?

Any help is much appreciated.

Answer 1

Original regression points are

$$\log (W_i)=\beta_0+\beta_1\text{educ}_i+\beta_2\text{exper}_i+\beta_3(\text{exper}_i)^2+\epsilon_i$$

After giving everyone one more year of education and decreasing experience by one year, we get

$$\log (W_i)=\beta_0+\beta_1(1+\text{educ}_i)+\beta_2(-1+\text{exper}_i)+\beta_3(-1+\text{exper}_i)^2+\epsilon_i$$

The difference between these two is

$$\theta=\beta_1-\beta_2-\beta_3(2\text{exper}_i-1)$$

For all people, we have

$$\frac{\sum_{i=1}^n(\beta_1-\beta_2-\beta_3(2 experi_i-1)}{n}=\beta_1-\beta_2-\beta_3(2\overline{experi}_i-1)$$

Thus the coefficient of $\beta_0$ is 0, beta1 is 1, beta2 is -1, and beta3 is -(2$\overline{experi}$ -1). We get the vector

$$c=(0, 1, -1, -(2\overline{\text{exper}}_i-1))$$

Apparently the vector defined in the lecture has its fourth term to be the negative of the actual coefficient, or is a typo.

To get the estimated effect, plug in the coefficients as well as the average of the experi's for all the n subjects, to get the estimated effect: .0923-.0791-(-.0020)*(2$\overline{exper}_i$-1).

The standard error can be found by the square root of the following variance:

This is equivalent to $$\begin{split}c'Var(\hat\beta)c&=(0, 1, -1, -2\overline{exper}_i+1)'\begin{pmatrix}.2223^2&\sigma_{12}&\sigma_{13}&\sigma_{14}\\\sigma_{21}&.0076^2&\sigma_{23}&\sigma_{24}\\\sigma_{31}&\sigma_{32}&.0249^2&\sigma_{34}\\\sigma_{41}&\sigma_{42}&\sigma_{43}&.0009^2\end{pmatrix}\begin{pmatrix}0\\1\\-1\\-2\overline{exper}_i+1\end{pmatrix}\end{split}$$

I have never used matlab, but the LinearModel object should have a property called CoefficientCovariance that has what we need.