**Econometric Sense**, and kindly contributed to R-bloggers)

In a previous post, I worked through the theory behind intervention analysis. In his time series course, University of Georgia political science professor Jamie Monogan demonstrates how to implement intervention analysis in R. The following example is from this course. It investigates the impact of the terrorist attacks of 911 on president Bush’s approval ratings. An excerpt from the data set follows:** **

year month approve t s11

1 2001 1 45.41947 1 0

2 2001 2 55.83721 2 0

3 2001 3 55.91828 3 0

4 2001 4 58.12725 4 0

5 2001 5 55.79231 5 0

.

.

.

etc

Note in the plot of the series above, observation 10 represents October 2001, post 9/11/2011. It appears that there is a drastic shift in the series, that slowly decays and eventually returns to previous levels. This may indicate an intervention model with a pulse function. Following the Box-Jenkins methodology, an ARIMA(1,0,1) model with the intervention can be specified in R as follows:

model <- arimax(bush$approve, order=c(1,0,0), xtransf=bush$s11, transfer=list(c(1,0)))

where:

bush$approve = Yt , Bush’s approval ratings

order =c(1,0,0) indicates and AR(1) process

xtransf=bush#s11 is the intervention series, flagging Pt = 1 to indicate September 11, 2001

transfer = list(c(1,0)) indicates the functional form of the transfer function. The first parameter indicates the denominator, while the second indicates a numerator factor. As specified in the code above the transfer function is specified as follows:

[w0/ 1- δB ]

Based on SAS documentation (http://support.sas.com/documentation/cdl/en/etsug/60372/HTML/default/viewer.htm#etsug_arima_sect014.htm ) this could also be specified in SAS via PROC ARIMA as follows:

proc arima data=bush;

identify var=approve crosscorr=s11;

estimate p =1 input=( / (1) s11 );

run;

Using R, the estimated model coefficients are:

Coefficients:

ar1 intercept T1-AR1 T1-MA0

0.8198 58.2875 0.8915 23.6921

s.e. 0.1184 4.6496 0.0166 3.9415

Based on the output above, w0 = 23.6921 and δ = .8915. The plot below shoes the fitted values which look very similar the to graphs for this specification shown in the previous post.

The code below is was used to conduct this analysis, and is excerpted from Jamie Monogan’s original R program.

# ------------------------------------------------------------------

# | PROGRAM NAME: R_ARIMAX

# | DATE:1/21/12

# | CREATED BY: Matt Bogard

# | PROJECT FILE: /Users/wkuuser/Desktop/Briefcase/R Programs

# |----------------------------------------------------------------

# | PURPOSE: illustration of ARIMA model with transfer function/intervention series

# |

# |

# |

# |------------------------------------------------------------------

# | COMMENTS: R code and data originally sourced from : http://monogan.myweb.uga.edu/teaching/ts/index.html

# |------------------------------------------------------------------

#libraries

rm(list=ls())

library(foreign)

library(TSA)

setwd('/Users/wkuuser/Desktop/briefcase/R Data Sets') # mac

#load data & view series

bush <- read.dta("BUSHJOB.DTA")

names(bush)

print(bush)

plot(y=bush$approve, x=bush$t, type='l')

#identify arima process

acf(bush$approve)

pacf(bush$approve)

#estimate arima model

mod.1 <- arima(bush$approve, order=c(0,1,0))

mod.1

#diagnose arima model

acf(mod.1$residuals)

pacf(mod.1$residuals)

Box.test(mod.1$residuals)

#Looks like I(1). Note: There is a theoretical case against public opinion data being integrated.

#estimate intervention analysis for september 11 (remember to start with a pulse

mod.2 <- arimax(bush$approve, order=c(0,1,0), xtransf=bush$s11, transfer=list(c(1,0))); mod.2

summary(mod.2)

#Notes: the second parameter directs the numerator. If 0 only, then concurrent effect only. The first parameter affects the denominator. c(0,0) replicates the effect of just doing "xreg"

#Our parameter estimates look good, no need to drop delta or switch to a step function.

#Graph the intervention model

y.diff <- diff(bush$approve)

t.diff <- bush$t[-1]

y.pred <- 24.3741*bush$s11 + 24.3741*(.9639^(bush$t-9))*as.numeric(bush$t>9)

y.pred <- y.pred[-1]

plot(y=y.diff, x=t.diff, type='l')

lines(y=y.pred, x=t.diff, lty=2)

#suppose an AR(1) process

mod.2b <- arimax(bush$approve, order=c(1,0,0), xtransf=bush$s11, transfer=list(c(1,0))); mod.2b

y.pred <- 58.2875 + 23.6921*bush$s11 + 23.2921*(.8915^(bush$t-9))*as.numeric(bush$t>9)

plot(y=bush$approve, x=bush$t, type='l')

lines(y=y.pred, x=bush$t, lty=2)

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