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# Time Series #
###############
library(wooldridge)
# ls("package:wooldridge")
data(fertil3, package='wooldridge')
fertil3[1:5,c("gfr","pe","year","pill")]

# time series plot
par(mfrow = c(1, 2))
plot(gfr~year,data=fertil3,type="l")
plot(pe~year,data=fertil3,type="l")
 
# static model
summary(lm(gfr~pe+ww2+pill,data=fertil3))$coef

# DL model and multicollinearity
n = nrow(fertil3)
fertil3$pelag1 = c(NA,fertil3$pe[1:(n-1)])
fertil3$pelag2 = c(NA,NA,fertil3$pe[1:(n-2)])
summary(lm(gfr~pe+pelag1+pelag2+ww2+pill,data=fertil3))$coef
m = lm(gfr~pe+pelag1+pelag2+ww2+pill,data=fertil3)
library("car")
linearHypothesis(m, c("pe=0", "pelag1=0","pelag2=0"))
summary(lm(pe~pelag1+pelag2,data=fertil3))$r.squared

# restricted DL model
linearHypothesis(m, c("pe=pelag1", "pe=pelag2"))
fertil3$z=fertil3$pe+fertil3$pelag1+fertil3$pelag2
summary(lm(gfr~z+ww2+pill,data=fertil3))$coef

# LRP and transformed regression
fertil3$dpe1 = fertil3$pe-fertil3$pelag1
fertil3$dpe2 = fertil3$pe-fertil3$pelag2
summary(lm(gfr~pe+dpe1+dpe2+ww2+pill,data=fertil3))$coef

# ADL model
fertil3$gfrlag1 = c(NA,fertil3$gfr[1:(n-1)])
summary(lm(gfr~gfrlag1+pe+ww2+pill,data=fertil3))$coef

# AR(1) model
summary(lm(gfr~gfrlag1,data=fertil3))$coef
co = coef(lm(gfr~gfrlag1,data=fertil3))
cat("one-step ahead forecast is", co[2]*fertil3$gfr[72],"\n")
cat("two-step ahead forecast is", co[2]*co[2]*fertil3$gfr[72],"\n")