Analisis Intervensi Data Deret Waktu den

(1)

Skripsi-R_LaTeX

Kurnia Wanto

July 12, 2016

1 Libraries and Import Data

# Libraries

---library(forecast) # --- BoxCox Arima auto.arima function is in forecast package

## Loading required package: zoo

##

## Attaching package: 'zoo'

## The following objects are masked from 'package:base': ##

## as.Date, as.Date.numeric

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## This is forecast 7.2

library(MASS) # --- boxcox function is in MASS package

library(FitAR) # --- LjungBoxTest function is in FitAR package

## Loading required package: lattice

## Loading required package: leaps

## Loading required package: ltsa

## Loading required package: bestglm

##

## Attaching package: 'FitAR'

## The following object is masked from 'package:forecast': ##

## BoxCox

library(tsoutliers) # --- tso function is in tsoutliers package library(lmtest) # --- coeftest function is in lmtest package library(stargazer) # --- stargazer function is in stargazer package

##

## Please cite as:

## Hlavac, Marek (2015). stargazer: Well-Formatted Regression and Summary Statistics Tables.

## R package version 5.2. http://CRAN.R-project.org/package=stargazer

library(TSA) # --- arimax function is in TSA package

## Loading required package: locfit

## locfit 1.5-9.1 2013-03-22

## Loading required package: mgcv

## Loading required package: nlme

##

## Attaching package: 'nlme'

## The following object is masked from 'package:forecast': ##

## getResponse

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## Loading required package: tseries

##

## Attaching package: 'TSA'

## The following objects are masked from 'package:forecast': ##

## fitted.Arima, plot.Arima

## The following objects are masked from 'package:timeDate': ##

## kurtosis, skewness

## The following objects are masked from 'package:stats': ##

## acf, arima

## The following object is masked from 'package:utils': ##

## tar

# 0_Import_Data.R

---options(width=80)

raw_data <- read.csv("/media/kurnia/Brain/Skripsi/DATA/World\ Bank/inflation.csv")

PDB_ID <- ts(raw_data$GDP, start=1970, end=2013)

#PDB_ID <- PDB_ID/1000000000 PDB_ID

## Time Series: ## Start = 1970 ## End = 2013 ## Frequency = 1

## [1] 9656740014 9849117953 11605084560 17171181163 27227710999

## [6] 32147953008 39328674730 48396143465 54298158340 55122620334

## [11] 78013206038 92473878832 94715163814 85369201879 87612439197

## [16] 87338874330 80060657612 75929617715 88787623310 101000000000

## [21] 114000000000 128000000000 139000000000 158000000000 177000000000

## [26] 202000000000 227000000000 216000000000 95445548017 140000000000

## [31] 165000000000 160000000000 196000000000 235000000000 257000000000 ## [36] 286000000000 365000000000 432000000000 510000000000 540000000000 ## [41] 709000000000 846000000000 877000000000 868000000000

# stat.desc(rwpdb)

plot(PDB_ID, xlab="Waktu", ylab="US Dollar", col="blue", type="l",

main="Plot PDB Indonesia 1970-2013") points(PDB_ID, cex = .5, col = "red") abline(v=1998, col=1, lty=2)

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Plot PDB Indonesia 1970−2013

Waktu

US Dollar

1970

1980

1990

2000

2010

0e+00

4e+11

8e+11

1998

(t=29)

2 Pemodelan ARIMA

2.1 Plot

N

_t

# 1-Plot_Nt.R

---# Creating Nt

Nt <- ts(PDB_ID[1:28], start=1970, end=1997)

# Creating Layout 1,1;2,3

m <- rbind(c(1, 1), c(2, 3))

layout(m)

par(mar = c(3, 3, 1, 1))

plot(Nt, xlab="Waktu", ylab="US Dollar", col="blue", type="p",

main="Plot PDB Indonesia 1970-1997 (Nt)")

points(Nt, cex = .5, col = "red")

# Trend Linear

t <- 1970:1997

trend_PDB <- glm(Nt~t)

abline(trend_PDB, col="dark blue", lwd=2)

# ACF

acf(Nt, 20, xlim=c(1,20)) text(10,0.8, "ACF")

# PACF

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Plot PDB Indonesia 1970−1997 (Nt)

Waktu

1970 1975 1980 1985 1990 1995

5.0e+10

# Show the ACF & PACF value acf(Nt, 27, plot=FALSE)

##

## Autocorrelations of series 'Nt', by lag ##

## 1 2 3 4 5 6 7 8 9 10 11

## 0.872 0.704 0.552 0.419 0.303 0.206 0.123 0.058 0.011 -0.021 -0.020

## 12 13 14 15 16 17 18 19 20 21 22

## -0.019 -0.031 -0.058 -0.095 -0.157 -0.233 -0.300 -0.336 -0.369 -0.390 -0.392

## 23 24 25 26 27

## -0.381 -0.350 -0.291 -0.206 -0.099

pacf(Nt, 27, plot=FALSE)

##

## Partial autocorrelations of series 'Nt', by lag ##

## 1 2 3 4 5 6 7 8 9 10 11

## 0.872 -0.231 -0.012 -0.034 -0.035 -0.022 -0.036 -0.001 -0.011 0.006 0.082

## 12 13 14 15 16 17 18 19 20 21 22

## -0.051 -0.055 -0.068 -0.056 -0.151 -0.104 -0.043 0.019 -0.105 -0.019 -0.023

## 23 24 25 26 27

## -0.040 0.010 0.062 0.080 0.113

2.2 Box–Cox Transformation

# - 2-BoxCox_Trans.R

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t1 <- 1:length(PDB_ID)

# --- Search for optimal lambda par(mar = c(2, 4, 0, 1))

MASS::boxcox(lm(PDB_ID~t1), lambda= seq(-1,1,1/10),

ylab = "log-Likelihood")

−1.0

−0.5

0.0

0.5

1.0 −100

−80

−60

−40

log−Lik

elihood

95%

lambda.model <- forecast::BoxCox.lambda(PDB_ID)

# --- Box-Cox Transformation

Nt_box <- forecast::BoxCox(Nt, lambda = lambda.model)

# --- It is ambigous, either use lambda for Nt (0.5) or PDB_ID (0.2)

2.3 Plot

W

t

# --- 3-Plot_Wt.R

---# --- Generating Wt

Wt <- as.numeric(diff(Nt_box))

# --- Creating Layout 1,1;2,3

m <- rbind(c(1, 1), c(2, 3))

layout(m)

par(mar = c(3, 3, 1, 1))

plot(Wt, xlab="Waktu", ylab="US Dollar",

col="blue", type="o",

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)

points(Wt, cex = .5, col = "red")

# --- Trend Linear

t <- 1:27

trend_PDB <- glm(Wt~t)

abline(h=mean(Wt))

# --- ACF

0.4

Series Wt

ACF

2.4 Identifikasi Model ARIMA

# --- 4-Id_Model.R

---# Save ARIMA model

Nt_arima <- auto.arima(Nt, max.d=2, seasonal=FALSE, lambda=lambda.model,

stepwise=TRUE, trace=TRUE, max.p=5, max.q=5)

##

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## ARIMA(0,1,0) with drift : 236.1509

## ARIMA(1,1,0) with drift : 234.0414

## ARIMA(0,1,1) with drift : 233.3114

## ARIMA(0,1,0) : 250.4358

## ARIMA(1,1,1) with drift : 235.8706

## ARIMA(0,1,2) with drift : 235.7134

## ARIMA(1,1,2) with drift : Inf

## ARIMA(0,1,1) : 240.4512

##

## Best model: ARIMA(0,1,1) with drift

2.5 Estimasi Parameter ARIMA

# --- 5-Est_Par.R

---# --- Creating All Possible Models

model_010 <- Arima(Nt, order=c(0,1,0), lambda = lambda.model, include.drift=FALSE)

model_010d <- Arima(Nt, order=c(0,1,0), lambda = lambda.model, include.drift=TRUE)

# --- Function for rounded value # AIC

raic <- function (model) { round(model$aic,2) }

# AICc

raicc <- function (model) { round(model$aicc,2) }

# BIC

rbic <- function (model) { round(model$bic,2) }

# --- Calculate parameters P-Value PValue <- function (model) {

(1-pnorm(abs(model$coef)/sqrt(diag(model$var.coef))))*2

}

# --- Estimating parameters for all possible models: #

# Generating Function

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print(model_010$coef) cat("\nP-Values:\n") print(PValue(model_010))

# --- ARIMA(0,1,0) with drift EstimasiParamater(model_010d)

## Parameter:

## ar1 drift

## 0.4123134 15.2953779

##

## P-Values:

## ar1 drift

## 0.022776839 0.003271827

# --- ARIMA(0,1,1)

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## Parameter:

## ma1 drift

## 0.4436015 15.6496441

##

## P-Values:

## ma1 drift

## 0.002482773 0.000348842

# --- ARIMA(1,1,1)

EstimasiParamater(model_111)

## Parameter:

## ar1 ma1

## 0.6093421 0.1197083 ##

## P-Values:

## ar1 ma1

## 0.01929868 0.76162785

## Parameter:

## ar1 ma1 drift

## 0.1523234 0.3491976 15.4399096

##

## P-Values:

## ar1 ma1 drift

## 0.634934540 0.201132736 0.001321079

2.6 Diagnosis Model ARIMA

# --- 6-Diag_Mod.R

---# --- Ljung-Box Test for Nt

# --- OR can use stats::Box.test, lag = min(10,n/5) <-- Rob J. Hyndman Box.test(Nt_arima$residuals, lag = round(length(Nt)/5,0),

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##

## Box-Ljung test

##

## data: Nt_arima$residuals

## X-squared = 4.6519, df = 5, p-value = 0.4598

# --- Kolmogorov-Smirnov Test

ks.test(Nt_arima$residuals, "pnorm", mean(Nt_arima$residuals), sd(Nt_arima$residuals))

##

## One-sample Kolmogorov-Smirnov test

##

## data: Nt_arima$residuals

## D = 0.12779, p-value = 0.7032 ## alternative hypothesis: two-sided

3 Analisis Intervensi

3.1 Identifikasi Orde Intervensi

# --- 7-Id_Intv.R

---# --- Box-Cox Transformation on PDB_ID, not sure using lambda for Nt or PDB (?) PDB_box <- forecast::BoxCox(PDB_ID, lambda.model)

Nt_forecast <- forecast(Nt_arima, h = 16)

# --- h-step forecast for Nt Nt_forecast

## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95

## 1998 220497998535 197099056251 2.460650e+11 185541793986 2.605269e+11

## 1999 239159146510 196702923844 2.886377e+11 176804075377 3.179553e+11

## 2000 259061074266 201691531495 3.287799e+11 175728796103 3.713233e+11

## 2001 280264506146 209057531124 3.696427e+11 177709273908 4.254125e+11

## 2002 302832106925 218017345778 4.121581e+11 181528673216 4.816195e+11

## 2003 326828511902 228252002513 4.567977e+11 186676314234 5.406397e+11

## 2004 352320356974 239609173819 5.038712e+11 192893799595 6.029214e+11

## 2005 379376308699 252012697584 5.536162e+11 200037356226 6.688034e+11

## 2006 408067094348 265426772546 6.062350e+11 208023418758 7.385694e+11

## 2007 438465531944 279839323613 6.619105e+11 216803369306 8.124736e+11

## 2008 470646560285 295253378801 7.208153e+11 226350450744 8.907534e+11

## 2009 504687268962 311682226900 7.831165e+11 236652419316 9.736372e+11

## 2010 540666928360 329146527131 8.489786e+11 247707157733 1.061349e+12

## 2011 578667019651 347672499230 9.185654e+11 259519925416 1.154110e+12

## 2012 618771264767 367290745815 9.920414e+11 272101565937 1.252142e+12

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# --- Identification intervention order with plot of model residuals

error_idintv <- rep(0,44)

error_idintv[1:28] <- model_011d$residuals

error_idintv[29:44] <- PDB_ID[29:44] - Nt_forecast$mean

plot(error_idintv, type="h", xlab="Waktu (T)", ylab = "Residual", xaxt = "n")

abline(h=c(-3*sd(model_011d$residuals), 3*sd(model_011d$residuals)), col="blue", lty=2) abline(v = 29, col = "red", lty = 3, lwd = 1.5)

text(29, 200, "T=29",cex = .8, pos = 2)

axis(1, at = c(0,10,20,30,40), labels = c("T-29" ,"T-19", "T-9", "T+1", "T+11"))

−1e+11

1e+11

Waktu (T)

Residual

T=29

T−29

T−19

T−9

T+1

T+11

error_idintv

## [1] 4.858446e-01 -1.246534e+01 6.359059e+00 2.457999e+01 2.869940e+01

## [6] -7.179901e+00 1.422428e+01 6.659505e+00 -2.208022e+00 -1.249470e+01

## [11] 4.190256e+01 -7.418031e+00 -8.507764e+00 -2.844920e+01 1.075969e+00

## [16] -1.662309e+01 -2.195533e+01 -1.412280e+01 1.512103e+01 -1.582605e+00

## [21] 5.067219e+00 1.711697e+00 -2.171179e+00 7.913453e+00 1.363048e+00

## [26] 8.221202e+00 2.866832e+00 -2.641994e+01 -1.250525e+11 -9.915915e+10

## [31] -9.406107e+10 -1.202645e+11 -1.068321e+11 -9.182851e+10 -9.532036e+10

## [36] -9.337631e+10 -4.306709e+10 -6.465532e+09 3.935344e+10 3.531273e+10

## [41] 1.683331e+11 2.673330e+11 2.582287e+11 2.069343e+11

# --- Plot PDB_ID vs forecasting of Nt

plot(forecast(model_011d, h=16), main =NA, ylab="US Dollar" ) points(PDB_ID, cex=.5, col="dark red", pch=19)

lines(PDB_ID, col="red") abline(v=1998, lty=2)

text(1998, PDB_ID[41], "1998\n(t=29)", pos=2)

legend("topleft", legend = c("PDB_ID", "Peramalan Nt"), cex=0.75, lty=1,

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US Dollar

1970

1980

1990

2000

2010

0.0e+00

8.0e+11

1998

(t=29)

PDB_ID

Peramalan Nt

# --- Detecting Outliers with tsoutliers::tso

PDB_outlier <- tsoutliers::tso(PDB_box, types = c("AO","LS","TC"),

maxit.iloop=10, tsmethod = "auto.arima")

plot(PDB_outlier)

Original and adjusted series

600 1000

Outlier effects

−150

−50

0 1970

1980

1990

2000

2010

3.2 Estimasi Parameter Intervensi

# --- 8-Est_Intv.R

---# --- Pulse: Abrupt Temporary

cobalah2 <- TSA::arimax(PDB_box, order = c(0,1,1), xtransf = data.frame(

T29 = 1*(seq(PDB_ID)==29)), transfer = list(c(1,0)))

cobalah2

## ## Call:

## TSA::arimax(x = PDB_box, order = c(0, 1, 1), xtransf = data.frame(T29 = 1 *

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##

## Coefficients:

## ma1 T29-AR1 T29-MA0

## 0.4394 0.2055 -112.4912

## s.e. 0.1182 0.0804 14.5695

##

## sigma^2 estimated as 496.7: log likelihood = -194.6, aic = 395.19

# --- Test significance of model coeficients coeftest(cobalah2)

##

## z test of coefficients: ##

## Estimate Std. Error z value Pr(>|z|)

## ma1 0.439418 0.118198 3.7176 0.0002011 ***

## T29-AR1 0.205494 0.080446 2.5544 0.0106362 *

## T29-MA0 -112.491185 14.569525 -7.7210 1.154e-14 ***

##

---## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# stargazer(coeftest(cobalah2))

3.3 Diagnosis Model Intervensi

# 9-Diag_Intv.R

---# --- Generating f(It), based on cobalah2

pulse29 <- filter(1*(seq(PDB_ID)==29), filter = 0.2055,

method = "rec", sides = 1) * -112.4912

# --- Compute Model for PDB_ID

PDB_arima <- Arima(PDB_ID, lambda = lambda.model, order = c(0,1,1),

include.constant = TRUE, xreg = pulse29)

# --- Ljung-Box Test for Independence of residuals # , lag = min(10,n/5) <-- Rob J. Hyndman

Box.test(PDB_arima$residuals,

lag=round(length(PDB_ID)/5,0), type = "Ljung-Box", fitdf = 1)

##

## Box-Ljung test

##

## data: PDB_arima$residuals

## X-squared = 14.784, df = 8, p-value = 0.06348

# --- Kolmogorov-Smirnov Test for Normality of residuals ks.test(PDB_arima$residuals, "pnorm",

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##

## One-sample Kolmogorov-Smirnov test

##

## data: PDB_arima$residuals

## D = 0.14849, p-value = 0.2597 ## alternative hypothesis: two-sided

3.4 Peramalan Model Intervensi

# --- Generating f(It)

pulse29 <- filter(1*(seq(PDB_ID)==29),

filter = 0.2055, method = "rec",

sides = 1)*-112.4912

# --- Compute Model for PDB_ID

PDB_arima <- Arima(PDB_ID, lambda = lambda.model,

order = c(0,1,1),

include.constant = TRUE, xreg = pulse29)

# --- Future Value of xreg

xreg.rob = forecast(auto.arima(pulse29), h=5)$mean

# --- Forecasting

forecast(PDB_arima, xreg = xreg.rob)

## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95

## 2014 9.045913e+11 818883650574 9.973193e+11 776215514467 1.049392e+12

## 2015 9.669871e+11 813820112964 1.142359e+12 740978902953 1.244971e+12

## 2016 1.032775e+12 827954842699 1.276198e+12 733510015370 1.422443e+12

## 2017 1.102090e+12 850383812003 1.410060e+12 737126698072 1.598875e+12

## 2018 1.175072e+12 878177165612 1.547203e+12 747311925422 1.779178e+12

plot(forecast(PDB_arima, xreg = xreg.rob), main=NA)

Analisis Intervensi Data Deret Waktu den

Skripsi-R_LaTeX

Kurnia Wanto

July 12, 2016

Contents

1

Libraries and Import Data

Plot PDB Indonesia 1970−2013

Waktu

US Dollar

1970

1980

1990

2000

2010

0e+00

4e+11

8e+11

1998

(t=29)

2

Pemodelan ARIMA

2.1

Plot

N

Plot PDB Indonesia 1970−1997 (Nt)

2.2

Box–Cox Transformation

−1.0

−0.5

0.0

0.5

1.0

−100

−80

−60

−40

log−Lik

elihood

95%

2.3

Plot

W

Series Wt

2.4

Identifikasi Model ARIMA

2.5

Estimasi Parameter ARIMA

2.6

Diagnosis Model ARIMA

3