Let's address each part step-by-step: --- # (a) Autoregressive Process AR(1) Given: \( x_t = .2 + .8x_{t-1} + \varepsilon_t \), where \( \varepsilon_t \sim IIDN(, \sigma_\varepsilon^2) \) ## (i) Find \( E(x_t | x_{t-1}) \) Recall, the conditional expectation is the mean given \( x_{t-1} \): \[ E(x_t | x_{t-1}) = E[.2 + .8x_{t-1} + \varepsilon_t | x_{t-1}] \] Since \( \varepsilon_t \) is mean and independent of \( x_{t-1} \): \[ E(x_t | x_{t-1}) = .2 + .8x_{t-1} \] --- ## (ii) Show that \( Var(x_t | x_{t-1}) = \sigma_\varepsilon^2 \) Recall, variance is only affected by the random component: \[ Var(x_t | x_{t-1}) = Var(.2 + .8x_{t-1} + \varepsilon_t | x_{t-1}) \] Since \( .2 + .8x_{t-1} \) is deterministic given \( x_{t-1} \), only \( \varepsilon_t \) contributes: \[ Var(x_t | x_{t-1}) = Var(\varepsilon_t) = \sigma_\varepsilon^2 \] --- # (b) Wage Equation Regression Results Regression equation: \[ \log(wage) = \beta_ + \beta_1 \text{exper} + \beta_2 \text{expersq} + \beta_3 \text{educ} + \beta_4 \text{age} + \beta_5 \text{kidslt6} + \beta_6 \text{kidsge6} + u \] ## Interpretation of Results ### Model Fit - **R-squared = .1582**: About 15.8% of the variation in log(wage) is explained by the model. - **F(6, 421) = 13.78, p < .01**: The model is jointly significant. ### Coefficients | Variable | Coefficient | Std. Err. | t | P>|t| | 95% Conf. Interval | Interpretation | |------------|-------------|-----------|-------|------|--------------------|---------------| | exper | .039819 | .015258 | 2.61 | .009| (.0098, .0698) | Each extra year of experience increases log(wage) by ~.04, holding other variables constant. Significant at 1% level. | | expersq | -.0007812 | .0004097 | -1.91 | .057| (-.00157, .000241)| Squared experience has a small, negative effect on log(wage), suggesting diminishing returns to experience. Marginally significant (p ~ .057). | | educ | .1078319 | .0136235 | 7.92 | .000| (.081, .1346) | Each additional year of education increases log(wage) by ~.108. Highly significant. | | age | -.0014653 | .0059351 | -.25 | .805| (-.0131, .0102) | Age has a negative but statistically insignificant effect on log(wage). | | kidslt6 | -.0607106 | .1061006 | -.57 | .567| (-.2696, .1487) | Having children under 6 has a negative but not significant effect on log(wage). | | kidsge6 | -.014591 | .0293505 | -.50 | .619| (-.0722, .0431) | Having children 6 or older has a negative but not significant effect on log(wage). | | _cons | -.4290078 | .3183346 | -1.32 | .187| (-1.0466, .2042) | Intercept; not significant. | ### Summary: - **Experience and education** are significant predictors of women's wages (education is highly significant and has a strong positive effect). - **Squared experience** is marginally significant and negative, indicating diminishing wage returns from more experience. - **Age, children under 6, and children 6 or older** are not statistically significant predictors of log(wage) in this sample. --- ## Final Answers ### Part (a) 1. \( E(x_t | x_{t-1}) = .2 + .8x_{t-1} \) 2. \( Var(x_t | x_{t-1}) = \sigma_\varepsilon^2 \) ### Part (b) - The regression shows that **education** and **experience** significantly increase married working women's wages. **Additional years of education** have a relatively large positive effect. **Experience** also increases wages, but at a diminishing rate. **Age** and **number of children (both below and above 6 years)** have no statistically significant effect on wages in this sample. The overall model explains about **16% of the wage variation**. --- Let me know if you need further clarification or more detailed interpretations!