Lecture 6.5 TimeSeries Application

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• Estimation sample : 1986-2005 (n=240)

• Hold-out forecast sample : 2006-2007 (n=24)

• Let \(y_t\) denote log(IP) or LOG(CLI) trend
  ADF:\(\Delta\)\(y_t\)=\(\alpha\)+\(\beta\)\(t\)+\(\rho\)\(y_{t-1}\)+\(\sum_{j=1}^3\)\(\gamma_j\)\(\Delta\)\(y_{t-j}\)+\(\epsilon_t\)

• -1.6 for LOGIP, -1.8 for LOGCLI, Not stationary

• Engel grandeur test on cointegration. Slise 4of 15
  • t-value 구하는 여러 방법은 아래와 같음

 y5.adf.nc.2

## 
## ############################################################### 
## # Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -0.5779 1.0282 1.4516

 summary(y5.adf.nc.2)@testreg$coef 

##                  Estimate   Std. Error    t value  Pr(>|t|)
## (Intercept)  8.919711e-04 7.820973e-04  1.1404861 0.2552498
## z.lag.1     -5.520094e-03 9.552198e-03 -0.5778873 0.5638959
## tt          -8.762567e-06 5.662198e-06 -1.5475556 0.1230807
## z.diff.lag   3.582966e-02 6.555305e-02  0.5465750 0.5851916

 y5.adf.nc.2@testreg$coefficients[2,3]

## [1] -0.5778873

• \(t-value\),\(\epsilon_{t-1}\) is -1.26>-3.8, Not cointegrated

Forecast GRIP with information {\(GRIP_{t-j}\),\(GRCLI_{t-j}\), \(j\)=3,4,5…}

• \(R_{retricted}^2\)=0.048
  • \(R_{unres}^2\)=0.1045
  • number of regressors (including constant) under unrestricted form =11
  • observations - #parameters = 240-11=229
  • number of restriction imposed by the null=9
• Mosel GRIP=\(GRIP_t\)=\(\alpha\)+\(\sum_{j=3}^L\)\(\beta_j\)\(GRIP_{t-j}\)+\(\epsilon_t\)

• \(H_0\) ;\(\beta_4\)=\(\beta_5\)=…\(\beta_{12}\)=0

• \(F\)=\(\frac{(0.0988-0.0519)/9}{(1-0.0988)/229}\)=1.3 -> 교과서
• \(F\)=\(\frac{(0.1045-0.048)/9}{(1-0.1045)/229}\)=1.6 -> 내가 한거

• 1.6<1.9209212, -> \(H_0\)를 기각할수 없음

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    + Item 3a
    + Item 3b

* Item 1
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    + Item 2a
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library(lmtest)
bgtest(GRIP~lag(GRIP,3),order=3)

## 
##  Breusch-Godfrey test for serial correlation of order up to 3
## 
## data:  GRIP ~ lag(GRIP, 3)
## LM test = 1.9032, df = 3, p-value = 0.5927

bgtest(GRIP~lag(GRIP,3)+lag(GRIP,4)+lag(GRIP,5)+lag(GRIP,6)
                           +lag(GRIP,7)+lag(GRIP,8)+lag(GRIP,9)+lag(GRIP,10)
                           +lag(GRIP,11)+lag(GRIP,12),order=3)

## 
##  Breusch-Godfrey test for serial correlation of order up to 3
## 
## data:  GRIP ~ lag(GRIP, 3) + lag(GRIP, 4) + lag(GRIP, 5) + lag(GRIP,     6) + lag(GRIP, 7) + lag(GRIP, 8) + lag(GRIP, 9) + lag(GRIP,     10) + lag(GRIP, 11) + lag(GRIP, 12)
## LM test = 1.9032, df = 3, p-value = 0.5927

plot(res03[-(229:237)],res12,xlab="residual model lag3",
       ylab="resid model lag 3-12",main="AR model for GRIP",col=c("red","blue")) #lec 6.5,slide 8/15

par(mfrow=c(1,2))
plot(GRIP[-(229:240)],res12,xlab="GRIP", ylab="RES LAG3-12",main="GRIP vs RES LAG 3-12")
plot(GRIP[-(238:240)],res03,xlab="GRIP", ylab="RES LAG3",main="GRIP vs RES LAG 3")

\(GRIP_t\)=0.0017+0.247\(GRIP_{t-3}\)+\(\epsilon_t\) (\(t_b\)=3.9,\(R^2\)=0.061)

ADL model for GRIP

GRC<-estSample$GRCLI
GRCLI<-ts(GRC)
regADL<-lm(formula=dyn(GRIP~lag(GRIP,3)+lag(GRCLI,6)))