Q1

We have \(Var(y_t) = Var(e_t + e_{t-1}+ ...+ e_1 + y_0) = t\sigma_e^2,\) and \(Var(y_{t+h}) = Var(e_{t+h}+ e_{t+h-1} +...+ e_t + e_{t-1}+ ...+ e_1 + y_0) = (t+h)\sigma_e^2.\)

\[ \begin{aligned} Cov(y_t,y_{t+h}) &= E(y_ty_{t+h})-E(y_t)E(y_{t+h}) \\ &= E(y_ty_{t+h}) \\ &= E[(e_t + e_{t-1}+ ...+ e_1 + y_0)(e_{t+h}+ e_{t+h-1} +...+ e_t + e_{t-1}+ ...+ e_1 + y_0)]\\ &= t\sigma_e^2. \end{aligned} \]

Therefore, we have \[Corr(y_t,y_{t+h}) = Cov(y_t,y_{t+h})/{\sqrt{(Var(y_t)Var(y_{t+h})}} = {t\sigma_e^2}/{\sqrt{(t+h)\sigma_e^2t\sigma_e^2}} = \sqrt{t/(t+h)}.\]

C2

i)

library(dplyr)
library(tidyverse)
library(broom)
library(wooldridge)
lm1 <- lm(earns$ghrwage ~ earns$goutphr + earns$goutph_1 )
tidy(lm1)

Yes, it is statistically significant.

ii)

lm2 <- lm(earns$ghrwage ~ earns$goutphr + I(earns$goutphr - earns$goutph_1))
tidy(lm2)

So the t-test for the hypothesis \(\theta = \beta_1+\beta_2 = 1\) is (1.185999-1)/0.203142 = 0.9156 which is not significant and we fail to reject this hypothesis.

iii)

lm3 <- lm(earns$ghrwage ~  earns$goutphr + earns$goutph_1 + earns$goutph_2)
tidy(lm3)

The t-statistic for \(goutph_{t-2}\) is 0.4091787, which is not siginicant. So there is no need for the model to include \(goutph_{t-2}\).

C3

i)

lm4 <- lm(phillips$cinf ~ phillips$cunem)
summary(lm4)
## 
## Call:
## lm(formula = phillips$cinf ~ phillips$cunem)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -7.4790 -0.9441  0.1384  1.0889  5.4551 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)   
## (Intercept)    -0.07214    0.30584  -0.236  0.81443   
## phillips$cunem -0.83281    0.28984  -2.873  0.00583 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.267 on 53 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.1348, Adjusted R-squared:  0.1185 
## F-statistic: 8.256 on 1 and 53 DF,  p-value: 0.005831

The t statistic on \(\Delta unem\) is about -2.87, which is very significant. Also, we see that the estimated coefficient is not statistically different from –1 because (-0.833 + 1)/0.290 ≈ 0.578. This implies that there is a 1-1 trade off between \(\Delta inf\) and \(\Delta unem\).

ii)

We prefer the model in (i) as its \(R^2\) = 0.1348 which is higher than that in 11.19 (\(R^2\) = 0.108).