Let's break down and solve the problem step by step: --- # **(a) Autoregressive Process AR(1)** Given: \[ x_t = .2 + .8x_{t-1} + \epsilon_t, \quad \epsilon_t \sim IIDN(, \sigma^2) \] --- ## **(i) Find \( E(x_t | x_{t-1}) \):** The conditional expectation is: \[ E(x_t | x_{t-1}) = E[.2 + .8x_{t-1} + \epsilon_t \mid x_{t-1}] \] Since \( \epsilon_t \) is independent of \( x_{t-1} \) and has mean zero: \[ E(x_t | x_{t-1}) = .2 + .8x_{t-1} + E(\epsilon_t) = .2 + .8x_{t-1} \] --- ## **(ii) Show that \( Var(x_t | x_{t-1}) = \sigma^2 \):** Conditional variance: \[ Var(x_t | x_{t-1}) = Var(.2 + .8x_{t-1} + \epsilon_t \mid x_{t-1}) \] \(.2 + .8x_{t-1}\) is just a constant given \(x_{t-1}\), so the variance is just the variance of \( \epsilon_t \): \[ Var(x_t | x_{t-1}) = Var(\epsilon_t) = \sigma^2 \] --- # **(b) Wage Equation for Married, Working Women** Model: \[ \log(wage) = \beta_ + \beta_1 \text{exper} + \beta_2 \text{exper}^2 + \beta_3 \text{educ} + \beta_4 \text{age} + \beta_5 \text{kidslt6} + \beta_6 \text{kidsge6} + u \] Regression summary (key coefficients): | Variable | Coef. | Std. Err. | t | P>|t| | 95% Conf. Interval | |-----------|------------|-----------|-------|------|-----------------------------| | exper | .039819 | .015258 | 2.61 | .009| [.009281, .06981] | | expersq | -.0007812 | .0004097 | -1.91 | .057| [-.001586, .000241] | | educ | .1078319 | .0136235 | 7.92 | .000| [.081053, .1346016] | | age | -.001463 | .0059351 | -.25 | .805| [-.013313, .010208] | | kidslt6 | -.0607106 | .1061006 | -.57 | .567| [-.269263, .147824] | | kidsge6 | -.014591 | .0293505 | -.50 | .619| [-.072289, .043109] | | _cons | -.4290078 | .3183346 | -1.32 | .187| [-1.046631, .2041854] | --- ## **(c) Interpret the Results** ### **Statistical Significance (P-values):** - **Experience (\(exper\))**: Significant positive effect (\(p = .009 < .05\)). Each year of experience increases log(wage) by about .04, holding other variables constant. - **Experience squared (\(expersq\))**: Marginally significant (\(p = .057\)), negative coefficient suggests diminishing returns to experience. - **Education (\(educ\))**: Highly significant (\(p = .000\)), positive effect. Each year of education increases log(wage) by about .108. - **Age**: Not significant (\(p = .805\)), effect is close to zero. - **Children under 6 (\(kidslt6\))**: Not significant (\(p = .567\)), small negative effect. - **Children 6 or older (\(kidsge6\))**: Not significant (\(p = .619\)), very small negative effect. - **Constant**: Not significant (\(p = .187\)). ### **Model Fit:** - \(R^2 = .1582\): The model explains about 15.8% of the variation in log(wage). - \(F(6, 421) = 13.78\), \(p < .001\): Model is jointly significant. ### **Economic Interpretation:** - **Experience and Education**: Both are important for higher wages (statistically significant), though the effect of experience diminishes at higher levels. - **Age and Children**: No significant effect on wages found for age or for having children, whether under 6 or older. --- ## **Summary Table** | Variable | Effect on log(wage) | Significance | Interpretation | |------------|--------------------|--------------|-----------------------------------------------------| | exper | Positive | Significant | Experience increases wages; effect diminishes at higher experience (see expersq). | | educ | Positive | Significant | More education increases wages. | | age | Near zero | Not significant | No strong evidence age matters after controlling for other factors. | | kidslt6 | Slight negative | Not significant | Having young children does not significantly affect wages. | | kidsge6 | Slight negative | Not significant | Having older children does not significantly affect wages. | --- ## **Final Answers** ### **(a)** 1. \( E(x_t | x_{t-1}) = .2 + .8x_{t-1} \) 2. \( Var(x_t | x_{t-1}) = \sigma^2 \) ### **(b)** - Experience and education significantly increase wages for married, working women. - Age and having children (either under or over 6) do not significantly affect wages. - The model explains a moderate share (15.8%) of the variation in wages. --- **If you need further breakdown of the math or more detail in interpretation, just ask!**