Solutions for HW7
TA: Dai Runyu
11/29/2020
Q5
1)
The plot is shown below
coef <- c(0.119,0.097,0.040,0.038,0.081,0.107,0.095,0.104,0.103,0.159
, 0.110,0.103,0.016)
plot(coef, type = "l")
2)
We see that the t-statistics of only lag two, three, and twelve are less than 2.
3)
The sum of the lag coefficients from zero through twelve =1.172 which is different from 1 but not too far away. The deviation from 1 may come from sampling error. The value close to 1 means that the variation of price can be explained by wage and its the past values.
4)
4)
Note that \[LRP = \theta = \delta_0 + \delta_1 +...+\delta_12.\] So \[y = \alpha_0 + \delta_0 z_t + \delta_1 z_{t-1} + ...+ \delta_{12} z_{t-12} + e_t,\] \[ = \alpha_0 + (\theta - \delta_1 -...-\delta_{12})z_t + \delta_1 z_{t-1} + ...+ \delta_{12} z_{t-12} + e_t, \] \[ = \alpha_0 + \theta z_t + \delta_1(z_{t-1}-z_t) + \delta_2(z_{t-2}-z_{t}) + ...+ \delta_{12}(z_{t-12}-z_{t}) + e_t\] Then we can directly get the standard error of LRP through \(z_t\) in the above regression.
5)
We need to add more 6 lags of wage, that is 13 through 18 of \(gwage_t\), to the equation. Then, we treat the previous regression in 4) as the restricted regression. The regression including more 6 lags of wage is treated as the unrestricted regression. Note that $ df_{un} = 273-6 = 267.$ Then the F-test of these 6 lags are \(F = [( R^2_{un} − R^2_{r} )/(1 − R^2_{un} )]*(267-20)/6.\)
C11.10
1)
library(wooldridge)
lm2 <- lm(wooldridge::phillips$cinf ~ wooldridge::phillips$unem)
summary(lm2)##
## Call:
## lm(formula = wooldridge::phillips$cinf ~ wooldridge::phillips$unem)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.0741 -0.9241 0.0189 0.8606 5.4800
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.8282 1.2249 2.309 0.0249 *
## wooldridge::phillips$unem -0.5176 0.2090 -2.476 0.0165 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.307 on 53 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.1037, Adjusted R-squared: 0.08679
## F-statistic: 6.132 on 1 and 53 DF, p-value: 0.0165We see these estimate are not noticablely changed compared to previous one.
2)
Natural rate of unemployment -2.8282/-0.5176 = 5.464 which is little bit smaller than 5.58 obtained in example 11.5.
3)
lm3 <- lm(wooldridge::phillips$unem ~ wooldridge::phillips$unem_1)
tidy(lm3)We see that first order auto-correlation of \(unem_t\) is about 0.74 which is not very close to 1.
4)
Just run the usual regression.
lm4 <- lm(wooldridge::phillips$cinf ~ wooldridge::phillips$cunem)
summary(lm4)##
## Call:
## lm(formula = wooldridge::phillips$cinf ~ wooldridge::phillips$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
## wooldridge::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: 0.005831We find that including the recent 56 years data gives us \(R^2\) = 0.1348 which is larger than that in i) (0.1037).
C18.1
1)
lm5 <- lm(wageprc$gprice ~ wageprc$gwage + wageprc$gprice_1 )
summary(lm5)##
## Call:
## lm(formula = wageprc$gprice ~ wageprc$gwage + wageprc$gprice_1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0114300 -0.0013730 -0.0000838 0.0012337 0.0151880
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0012884 0.0002756 4.675 4.57e-06 ***
## wageprc$gwage 0.0814330 0.0310760 2.620 0.00926 **
## wageprc$gprice_1 0.6396425 0.0446325 14.331 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.002514 on 281 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.4539, Adjusted R-squared: 0.4501
## F-statistic: 116.8 on 2 and 281 DF, p-value: < 2.2e-16The estimated impact propensity (IP) is 0.0814 while the estimated LRP is 0.0814/(1 – 0.640) = 0.226. The estimated lag distribution is graphed below.
lag_dis <- numeric(13)
for (t in 2:13)
{
lag_dis[1] <- 0.0814330
lag_dis[t] <- lag_dis[1]*0.6396425^(t-1)
}
plot(lag_dis, type = "l")
2)
The IP for the FDL model estimated in Problem 11.5 was 0.119, which is much greater than the estimated IP for the GDL model (0.0814).
The estimated LRP from GDL model (0.226) is much < than that for the FDL model (1.172). Thus, we cannot believe that the GDL model as a good approximation to the FDL model. One reason these are so different can be seen by comparing the estimated lag distributions (see below for the GDL model). We see that the largest lag coefficient is at the ninth lag, which is impossible with the GDL model (where the largest impact is always at lag zero).
3)
lm6 <- lm(wageprc$gprice ~ wageprc$gwage + wageprc$gwage_1 + wageprc$gprice_1 )
summary(lm6)##
## Call:
## lm(formula = wageprc$gprice ~ wageprc$gwage + wageprc$gwage_1 +
## wageprc$gprice_1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0110135 -0.0013896 -0.0001353 0.0012590 0.0150225
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0010814 0.0002993 3.613 0.000359 ***
## wageprc$gwage 0.0898050 0.0313361 2.866 0.004474 **
## wageprc$gwage_1 0.0551139 0.0316775 1.740 0.082985 .
## wageprc$gprice_1 0.6185942 0.0460885 13.422 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.002505 on 280 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.4598, Adjusted R-squared: 0.454
## F-statistic: 79.44 on 3 and 280 DF, p-value: < 2.2e-16- The estimated IP is 0.0898
- The estimated LRP = (0.0898 + 0.0551)/(1-0.6186) = 0.380
The lag distribution is graphed below
lag_dis_1 <- numeric(12)
for (t in 3:12)
{
lag_dis_1[1] <- 0.0898
lag_dis_1[2] <- 0.0898*0.6186 + 0.0551
lag_dis_1[t] <- lag_dis_1[2]*0.6186^(t-2)
}
plot(lag_dis_1, type = "l")
We see that these two values in RDL are both higher than in GDL model. The RDL model is more flexible than the GDL model. But notice that here the maximum effect is at the first lag which is not consistent with the FDL estimates in 11.5 (at the 9th lag).