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)