11.6

i)

\(H_0:~\beta_1 = 1\), and the t-statistic = (1.104-1)/0.039 = 2.667. Then by checking the table for two sided tails, we can easily reject the null hypothisis at 1% significance level, impying that this coefficents practically different from 1.

ii)

\(H_0:~\beta_1 = 1\), and the t-statistic = (1.053-1)/0.039 = 1.359, so here \(\beta_1\) is not siginifcantly different from 1.

\(H_0:~\beta_2 = 0\), and the t-statistic = 0.480/0.109 = 4.404 which implies that the \(\beta_2\) is signifcantly different from 0. Since it is positive and siginicant different from 0, you should invest in the T-bills.

iii)

This high correlation implies that \(hy3_t\) is unit root which cannot give us a reasonable t-test.

iv)

To test the seasonality, we can add dummy variable for each quater as the season variable, say Q2, Q3, Q4. Then we need to do joint test of all the variables (include the season variable).

C5

i)

The regression results are shown below:

library(wooldridge)
lm1 <- lm(fertil3$cgfr ~  + fertil3$cpe +fertil3$cpe_1 +fertil3$cpe_2 +fertil3$t)
tidy(lm1)

We see that the time variable is not significant so that there is no need to include it.

ii)

We just need to run the restricted regression and unrestricted regression, and to get the joint test of \(pill\) and \(ww2\). We see that the F-statistic = 2.823072 and p-value = 0.0669, which implies that these two variables are not jointly significantly at 5% level.

lm2 <- lm(fertil3$cgfr ~  fertil3$cpe +
            fertil3$cpe_1 +fertil3$cpe_2 + fertil3$pill+fertil3$ww2)
tidy(lm2)
lm2_res <- lm(fertil3$cgfr ~  fertil3$cpe +
                fertil3$cpe_1 +fertil3$cpe_2)
tidy(anova(lm2,lm2_res))

iii)

The regression results are shown below:

lm3 <- lm(fertil3$cgfr ~ fertil3$cpe +
            fertil3$cpe_1 +fertil3$cpe_2 + fertil3$pill+ fertil3$ww2 + fertil3$t)
tidy(lm3)

We see that t becomes 0.094 which is much more significant than before (0.00787), and the coef of \(pill\) decrease from -1.67 to -4.89 and much more significant.

iv)

Note that LRP = \(\theta = \beta_0+ \beta_1+ \beta_2\), but the standard error of it is difficult to find. So we use the same algebra in past HWs to transform the original regression to \(\Delta gfr = \alpha_0 + \theta \Delta pe + \beta_1(\Delta pe_1-\Delta pe) + \beta_2(\Delta pe_2-\Delta pe_1) + \beta_3 pill_t+ \beta_4 ww2_t + \beta_5 t.\) Here the \(\theta\) is our target.

lm4 <- lm(fertil3$cgfr ~ fertil3$cpe +
            I(fertil3$cpe_1-fertil3$cpe) + I(fertil3$cpe_2-fertil3$cpe_1) + fertil3$pill+ fertil3$ww2 + fertil3$t)
tidy(lm4)

We see that the t-statistic for LRP is about -0.0944, implying that LRP is practically 0, which is different from that in 10.19.

C6

i)

To test \(E(gc_t|I_{t-1}) = E(gc_t)\). i.e \(E(gc_t)\) does not depend on the past information, is to test \(H_0: \beta_1 = 0\) against \(H_1: \beta_1 \neq 0\). The regression results are shown below:

lm5 <- lm(consump$gc ~ consump$gc_1)
tidy(lm5)

The results shows that the t-statistics for \(\beta_1\) is 2.859 suggesting that we should reject PIH. Also, there is evidence of auto-correlation in the consumption growth, \(gc_t\).

ii)

We have the following regression results:

consump$inf_1 <- c(NA, consump$inf[1:(length(consump$inf)-1)])
consump$i3_1 <- c(NA, consump$i3[1:(length(consump$i3)-1)])

lm6 <- lm(consump$gc ~ consump$gc_1 + consump$gy_1 + consump$i3_1 + consump$inf_1)
tidy(lm6)
tidy(anova(lm5, lm6))

We see that all the included regressors based on the p-values are not individually signifcant. The F-sta = 1.512245 with p-value = 0.23 for joint test also shows insgnificance at 5% level. Neither individually nor jointly significant for these variables.

iii)

The p-value increases to from 0.007 to 0.144 in ii) showing insignificance of $ gc_1$. This change may comes from that gc_1 are correlated with at least one of the extra 3 regressors in part ii), namely mulit-colinearity problem occurs. But we should not say that the PIH is now supported by the data since \(gc_1\) is just separately tested here and joint tests are required.

iv)

summary(lm6)
## 
## Call:
## lm(formula = consump$gc ~ consump$gc_1 + consump$gy_1 + consump$i3_1 + 
##     consump$inf_1)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.0249090 -0.0075867  0.0000855  0.0087231  0.0188620 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)   
## (Intercept)    0.0225944  0.0070892   3.187  0.00335 **
## consump$gc_1   0.4335777  0.2896546   1.497  0.14487   
## consump$gy_1  -0.1079113  0.1946394  -0.554  0.58340   
## consump$i3_1  -0.0007467  0.0011107  -0.672  0.50653   
## consump$inf_1 -0.0008281  0.0010041  -0.825  0.41606   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01134 on 30 degrees of freedom
##   (2 observations deleted due to missingness)
## Multiple R-squared:  0.3038, Adjusted R-squared:  0.211 
## F-statistic: 3.273 on 4 and 30 DF,  p-value: 0.02431

We get F = 3.27 with 4 and 30 df. And the p-value is about 0.02431, implying that we should reject the PIH at the 5% level. This results are in lined with in i) but with less rejection power.