Introduction

\[ \newcommand{\N}[1]{\mathsf{N}\left(#1\right)} \newcommand{\E}[1]{\mathsf{E}\left(#1\right)} \newcommand{\V}[1]{\mathsf{Var}\left(#1\right)} \newcommand{\ols}[1]{\widehat{#1}} \newcommand{\cond}[1]{\left|#1\right.} \newcommand{\abs}[1]{\left|#1\right|} \]

• Forecasting is an essential part of life — business, social, private.
  • What clothes to wear tomorrow?
  • What clothes to buy tomorrow?
  • What clothes to produce tomorrow?
• This course equips you with means to forecast objects of interest.
  • Theory: What makes a good forecast model?
  • Practice: How to create that good forecast model?
  • Communicating: How to report that good forecast model?
• Online module description here

New Course Structure (with dates)

• As informed by email, significant change to structure of course.
• Midterm I: Feb 3 4pm (week 4), on theory of forecasting.
  • Mock by week 2 (please hold me to this!)
• Midterm II: March 4 11am (week 8), on practice of forecasting.
  • Mock by week 6 (please hold me to this!)
• Project: ``Forecast a variable of your choice.’’
  • Presentation: March 25 (week 11), on your project idea/progress.
    • Guidelines on good presentational skills,
    • Individual feedback will be provided.
  • Coursework: Due April 20, 4pm.

Weekly Forecasts

• Each week, submit your forecast online. For week 1, submit here.
  • Deadline each week: Thursday 6pm. Late forecasts will be penalised.
• Why so many?
  • Forecast different time horizons.
  • Check on progress through term.
• Additional series each week:
  • Week 1: Rightmove House Price Index (YoY) (Jan)
  • Week 2: GDP 2015 Q4 first estimate (released Jan 27 9:30am)
    • Data series from ONS is IHYQ.
  • Week 3: Mortgage approvals (released by BoE Jan 30)

UK GDP 2014Q4

library(zoo)

## 
## Attaching package: 'zoo'
## 
## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

library(tseries)
setwd("/home/readejj/Dropbox/Teaching/Reading/ec313/2015/Lecture-3/")
ukgdp <- read.csv("QNA_CSDB_DS.csdb.csv",stringsAsFactors=F)
ukgdp.z <- zooreg(ukgdp$IHYQ,start=c(1955,2),frequency=4)
plot(ukgdp.z,main="UK Quarterly GDP since 1955",ylab="% growth QoQ",xlab="Year")
abline(0,0)

Course Outline

• Week 1
  • Tuesday: Introduction to forecasting (and to R) (Ch. 1–2).
  • Wednesday: Judgemental (Ch. 3) and Regression-based Forecasts (Ch. 4–5).
• Week 2
  • Tuesday: ARIMA Models (Ch. 8), Non-stationarity and structural breaks.
  • Wednesday: Model selection and impulse saturation.
• Week 3
  • Tuesday: Structural Time Series Modelling (Ch. 6) and Exponential Smoothing Methods (Ch. 7).
  • Wednesday: Course recap.
• Week 4
  • Tuesday: Midterm I
  • Wednesday: Practical Introduction to R.
• Week 5
  • Tuesday: Introduction to R: Running regression models.
  • Wednesday: R Class I: BYOD (bring your own data).
• Week 6: Enhancement Week (forecast competition and Settlers)
• Week 7
  • Tuesday: Introduction to R: Forecasting.
  • Wednesday: R Class II: BYOD (bring your own data).
• Week 8
  • Tuesday: Introduction to R: Evaluating forecasts.
  • Wednesday: Midterm II
• Week 9
  • Tuesday: Project: How to conduct an empirical project.
  • Wednesday: Project Class I: Drop-in sessions.
• Week 10
  • Tuesday: Project: Example ideas I.
  • Wednesday: Project Class II: Drop-in sessions.
• Week 11
  • Tuesday: Project: Example ideas II and Presentation Prep.
  • Wednesday: Project Presentations

Last week: Mini-recap

• What is forecasting:
  • Statements about future realisations of random variables.
• Judgemental forecasting:
  • Forecasting without data.
  • With data: Principles to mitigate bias.
• Forecasting using linear regression:
  • Construct linear model as before:
    • Check \(e_t \sim iid\left(0,\sigma^2\right)\).
  • Need to forecast \(x_{1,t},\dots,x_{k,t}\) to make forecasts.

Today

1. Constructing forecasts submitted last week.
   • Plus some forecast errors.
2. ARIMA Models (Ch. 8).
3. Non-stationarity.
4. Structural breaks.

Constructing Dave Forecast

library(gets)
library(forecast)

## Loading required package: timeDate
## This is forecast 5.6

dave <- read.csv("Dave_140115.csv",stringsAsFactors=F)
dave$david.cameron[NROW(dave)] <- NA
dave$Week <- as.Date(substr(dave$Week,1,10))
plot(dave$Week,dave$david.cameron,
     main="Google Search Popularity of David Cameron",
     ylab="Search popularity",xlab="Date",type="l")

dave.isat <- isat(dave$david.cameron,ar=1:3,sis=T)

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## 
## GETS of union of retained IIS indicators... 
## 
## 
## SIS block 1 of 20...
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## 
## SIS block 2 of 20...
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## 
## SIS block 3 of 20...
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## 
## SIS block 4 of 20...
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## 
## SIS block 5 of 20...
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## 
## SIS block 6 of 20...
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## 
## SIS block 7 of 20...
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## 
## SIS block 8 of 20...
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## 
## SIS block 9 of 20...
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## 
## SIS block 10 of 20...
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## 
## SIS block 11 of 20...
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## 
## SIS block 12 of 20...
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## 
## SIS block 13 of 20...
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## 
## SIS block 14 of 20...
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## 
## SIS block 15 of 20...
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## 
## SIS block 16 of 20...
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## Searching path no. 1 out of 28
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## 
## SIS block 17 of 20...
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## Searching path no. 1 out of 28
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## 
## SIS block 18 of 20...
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## Searching path no. 1 out of 28
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## 
## SIS block 19 of 20...
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## 
## SIS block 20 of 20...
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## 
## GETS of union of retained SIS indicators... 
## 
## Searching path no. 1 out of 4
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## 
## GETS of union of ALL retained indicators...
## 
## regressor-matrix is column rank deficient, so dropping 3 regressors
## 
## Searching path no. 1 out of 4
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dave.0 <- cbind(dave[4:c(NROW(dave)-1),],dave.isat$aux$mX[,grep("is",dave.isat$aux$mXnames)])
dave.1 <- data.frame("Week"=seq(from=as.Date("2015-01-11"),as.Date("2015-03-22"),by="weeks"),
                     "david.cameron"=rep(NA,11),"iis331"=rep(0,11),"iis332"=rep(0,11),
                     "iis333"=rep(0,11),"sis300"=rep(1,11),"sis327"=rep(1,11),"sis331"=rep(1,11))
dave.mod <- Arima(dave.0$david.cameron, order = c(3,0,0), xreg = dave.0[,3:8])
dave.forc <- forecast(dave.mod,h=11,xreg = dave.1[,3:8])
print(round(dave.forc$mean))

## Time Series:
## Start = 573 
## End = 583 
## Frequency = 1 
##  [1] 5 5 5 5 5 5 5 5 5 5 5

plot(dave.forc,include=50)

Constructing FTSE Forecast

ftse <- read.csv("FTSE_140115.csv",stringsAsFactors=F)
ftse <- ftse[order(as.Date(ftse$Date)),]
plot(as.Date(ftse$Date),ftse$Close,main="FTSE since 1984",ylab="FTSE",xlab="Date",type="l")

ftse.ar <- auto.arima(ftse$Close)
ftse.forc <- forecast(ftse.ar,100)
plot(ftse.forc,include=200)

Constructing Rightmove Forecast

houses <- read.csv("rightmove_140115.csv",stringsAsFactors=F)
houses$DateTime <- as.Date(substr(houses$DateTime,1,8),"%Y%m%d")
houses <- houses[order(houses$DateTime),]
plot(houses$DateTime,houses$Actual,main="Rightmove HPI",ylab="Index",xlab="Date",type="l")
lines(houses$DateTime,houses$Previous,col=2,type="l")
legend("top",lty=1,col=c(1,2),legend=c("Actual","Previous"),bty="n",ncol=2)

houses.mod <- arima(houses$Actual,order=c(1,0,0))
houses.forc <- forecast(houses.mod,h=1)
plot(houses.forc,include=50)

Forecast Errors

dave19 <- read.csv("dave_190115.csv",stringsAsFactors=F)
ftse19 <- read.csv("FTSE_190115.csv",stringsAsFactors=F)
houses19 <- read.csv("rightmove_190115.csv",stringsAsFactors=F)

• Actual outcomes were:
  • FTSE: 2015-01-16, 6498.8, 6553.2, 6443.3, 6550.3, 0, 6550.3.
  • Dave: 2015-01-11 - 2015-01-17, 9.
  • Houses: 20150118 23:01:00, 8.2, NA, 7.

Forecast Error Distributions

errors <- c(449.7,-100.3,-30.3,-170.3,-100.3,-8.3,-73.3,-199.3,-210.3,123.7,
            -5.3,-0.3,-70.3,-150.3,-161.84,-250.3,-350.3,-50.3,-120.3)
hist(errors,main="Forecast Errors for FTSE Forecasts")

• Yours truly forecast 5 for Dave, 6545.8277299 for the FTSE and 6.9255011162461 for Rightmove.

Today: ARIMA (Ch. 8)

• ARIMA (AutoRegressive Integrated Moving Average) models.
• Very common forecasting approach — very flexible.
• ARIMA models focus on autocorrelation in data.
• Components of ARIMA:
  • Integration
  • AutoRegression
  • Moving Average

8.1 Stationarity and Differencing

• Stationary data series: Properties do not depend on when observed.
  • Formally: If \(y_t\) stationary then \((y_t,\dots,y_{t+s})\) independent of \(t\).
• Trending or seasonal time series cannot be stationary.
• White noise is stationary.
• Cyclical (not seasonal) data are stationary.

Plots of Candidate Stationary Data Series

library(forecast)
library(Quandl)

## Loading required package: xts

library(quantmod)

## Loading required package: TTR
## Version 0.4-0 included new data defaults. See ?getSymbols.

Quandl.auth("y8y3ezt48WZeqUqq1yQd")
par(mfrow=c(3,3))
nikkei <- read.csv("^nkx_d.csv",stringsAsFactors=F)
nikkei.t <- zooreg(nikkei$Close,order.by=as.Date(nikkei$Date))
plot(nikkei.t,main="(a)",ylab="Daily Closing Price of Nikkei")
plot(diff(nikkei.t,differences=1),main="(b)",ylab="Daily Difference in Nikkei")
getSymbols('TRFVOLUSM227NFWA',src='FRED',return.class="zoo")

##     As of 0.4-0, 'getSymbols' uses env=parent.frame() and
##  auto.assign=TRUE by default.
## 
##  This  behavior  will be  phased out in 0.5-0  when the call  will
##  default to use auto.assign=FALSE. getOption("getSymbols.env") and 
##  getOptions("getSymbols.auto.assign") are now checked for alternate defaults
## 
##  This message is shown once per session and may be disabled by setting 
##  options("getSymbols.warning4.0"=FALSE). See ?getSymbol for more details

## [1] "TRFVOLUSM227NFWA"

plot(TRFVOLUSM227NFWA,main="(c)",ylab="Vehicla Miles Travelled")
football <- read.csv("football.csv",stringsAsFactor=F)
football.z <- zooreg(football$x,order.by=football$Group.1)
plot(football.z,ylab="Ratio of home to away wins",main="(d)")
getSymbols('UNRATE',src='FRED',return.class="zoo")

## [1] "UNRATE"

plot(UNRATE,ylab="Unemployment Rate (in US)",main="(e)")
getSymbols('PCECC96',src='FRED')

## [1] "PCECC96"

pce.t <- ts(PCECC96,start=c(1947,1),frequency=4)
plot(window(pce.t,start=c(1960,1)),main="(f)",ylab="Personal Consumption Expenditures")
plot(diff(window(log(pce.t),start=c(1960,1)),differences=1),main="(g)",ylab="% change in Personal Consumption Expenditures")
rigs <- Quandl("BKRHUGHES/RIGS_BY_PROV_ALBERTA_LAND")
plot(rigs,type="l")
weather.data.2012 <- read.csv(url("http://www.wunderground.com/history/airport/EGLF/2012/1/1/CustomHistory.html?dayend=31&monthend=12&yearend=2012&req_city=NA&req_state=NA&req_statename=NA&format=1"),stringsAsFactors=F)
weather.data.2013 <- read.csv(url("http://www.wunderground.com/history/airport/EGLF/2013/1/1/CustomHistory.html?dayend=31&monthend=12&yearend=2013&req_city=NA&req_state=NA&req_statename=NA&format=1"),stringsAsFactors=F)
weather.data.2014 <- read.csv(url("http://www.wunderground.com/history/airport/EGLF/2014/1/1/CustomHistory.html?dayend=31&monthend=12&yearend=2014&req_city=NA&req_state=NA&req_statename=NA&format=1"),stringsAsFactors=F)
weather.data.2015 <- read.csv(url("http://www.wunderground.com/history/airport/EGLF/2015/1/1/CustomHistory.html?dayend=31&monthend=12&yearend=2015&req_city=NA&req_state=NA&req_statename=NA&format=1"),stringsAsFactors=F)
weather.data <- rbind(weather.data.2012,weather.data.2013,weather.data.2014,weather.data.2015)
weather.data$date <- as.Date(weather.data$GMT,format="%Y-%m-%d")
plot(weather.data$date,weather.data$Mean.TemperatureC,type="l",ylab="Mean Daily Temperature in Reading",xlab="Date")

8.1 Differencing

• Nikkei was non-stationary in (a), stationary in (b).
• Taking differences (differencing) of a time series can make it stationary.
• Differencing eliminates levels information in data series:
  • Hence removes trend and seasonal components from data.
• Autocorrelation Function (ACF) plot useful for identifying stationarity (or otherwise):
  • ACT plots autocorrelation \(r_k\) for \(k=1,2,\dots\).
  • ACF (very) gently slopes down with non-stationary series.
• Measure autocorrelation using: \[ r_{k} = \frac{\sum\limits_{t=k+1}^T(y_{t}-\bar{y})(y_{t-k}-\bar{y})}{\sum\limits_{t=1}^T(y_{t}-\bar{y})^2}, \]
• Partial autocorrelation (PACF) function:
  • Control for intermediate lags.
  • Principle of domino rally. \[ r^*_{k} = \frac{\sum\limits_{t=k+1}^T(y^*_{t}-\bar{y}^*)(y^*_{t-k}-\bar{y}^*)}{\sum\limits_{t=1}^T (y^*_{t}-\bar{y}^*)^2}, \]
• where \(y^*_t\) is \(y_t\) controlling for \(y_{t-1},\dots,y_{t-k+1}\).

ACF of Nikkei

plot(nikkei.t,main="Nikkei")

acf(coredata(nikkei.t),main="ACF of Nikkei")

plot(diff(coredata(nikkei.t),differences=1),main="Daily Returns, Nikkei",type="l")

acf(diff(coredata(nikkei.t),differences=1),main="ACF of Daily Returns, Nikkei")

ACF of unemployment

plot(UNRATE,ylab="Unemployment Rate",xlab="Date")

acf(coredata(UNRATE))

plot(diff(UNRATE,differences=1,lag=12),main="Change in unemployment",type="l")

acf(diff(coredata(UNRATE),differences=1,lag=12),main="ACF of Daily Returns, Nikkei")

8.1 Random Walk Model

• Can write differences series as: \[ y'_t = y_t - y_{t-1}. \]
  • Differences series has \(T-1\) observations relative to \(T\) of \(y_t\).
• If differenced series is white noise, we write: \[ y_t - y_{t-1} = e_t \quad\text{or}\quad {y_t = y_{t-1} + e_t}. \]
  • Very common model in economics and finance. Basis for naive forecast model.
• We may allow differences to have non-zero mean: \[ y_t - y_{t-1} = c + e_t\quad\text{or}\quad {y_t = c + y_{t-1} + e_t}. \]
  • This is random walk with drift model: \(c>0\) implies positive drift.

8.1 Second-Order Differencing

• Occasionally we need to difference twice to achieve stationarity: \[ y''_{t} = y'_{t} - y'_{t - 1} \\ = (y_t - y_{t-1}) - (y_{t-1}-y_{t-2})\\ = y_t - 2y_{t-1} +y_{t-2}. \]
• \(y''_t\) models ‘change in changes’ or ‘rate of change’.
  • Can sometimes be effective forecast device, especially during structural breaks.

8.1 Seasonal Differencing

• Often helpful to take seasonal differences: \[ y'_t = y_t - y_{t-m} \qquad\text{where $m=$ number of seasons.} \]
  • Seasonally-naive model: Seasonally differenced data is white noise. \[ y_t = y_{t-m}+e_t. \]

Seasonal Differencing on Vehicle Miles

plot(TRFVOLUSM227NFWA,main="Raw data")

plot(log(TRFVOLUSM227NFWA),main="Log transformation")

plot(diff(log(TRFVOLUSM227NFWA),differences=1,lag=12),main="Seasonally Differenced")

8.1 Unit Root Tests

• Determining how many times to difference is quite subjective.
• Attempt to be more objective is to use unit root test.
  • For example, augmented Dickey-Fuller (ADF) test. Rearrange AR(\(k\)): \[ y'_t = \phi y_{t-1} + \beta_1 y'_{t-1} + \beta_2 y'_{t-2} + \cdots + \beta_k y'_{t-k}. \]
  • Test is of null hypothesis that \(\phi=0\) (non-stationarity).
• KPSS test: Null is stationarity.

8.1 Unit Root Testing

adf.test(nikkei.t,alternative="stationary")

## 
##  Augmented Dickey-Fuller Test
## 
## data:  nikkei.t
## Dickey-Fuller = -1.8428, Lag order = 25, p-value = 0.6451
## alternative hypothesis: stationary

kpss.test(nikkei.t)

## Warning in kpss.test(nikkei.t): p-value smaller than printed p-value

## 
##  KPSS Test for Level Stationarity
## 
## data:  nikkei.t
## KPSS Level = 34.5268, Truncation lag parameter = 30, p-value =
## 0.01

adf.test(TRFVOLUSM227NFWA,alternative="stationary")

## 
##  Augmented Dickey-Fuller Test
## 
## data:  TRFVOLUSM227NFWA
## Dickey-Fuller = -2.0629, Lag order = 8, p-value = 0.5517
## alternative hypothesis: stationary

kpss.test(TRFVOLUSM227NFWA)

## Warning in kpss.test(TRFVOLUSM227NFWA): p-value smaller than printed
## p-value

## 
##  KPSS Test for Level Stationarity
## 
## data:  TRFVOLUSM227NFWA
## KPSS Level = 8.9275, Truncation lag parameter = 5, p-value = 0.01

8.2 Backshift Notation (The Lag Operator)

• For future models, it will be helpful to use : \[ By_t = y_{t-1}. \]
  • More conventionally denoted \(L\) and referred to as .
  • Operator that shifts data series back one period.
• Applied twice, series goes back two periods: \[ B(By_{t}) = B^{2}y_{t} = y_{t-2}. \]
• Hence seasonal differencing for monthly data is \(B^{12}y_{t}\).

8.2 Backshift Notation (The Lag Operator)

• \(B\) helps us describe differencing succintly: \[ y'_{t} = y_{t} - y_{t-1} = y_t - By_{t} = (1 - B)y_{t}. \]
• Second differencing: \[ y''_{t} = y_{t} - 2y_{t - 1} + y_{t - 2} = (1-2B+B^2)y_t = (1 - B)^{2} y_{t}. \]

8.2 Backshift Notation (The Lag Operator)

• General \(d\)th order differencing: \[ (1 - B)^{d} y_{t}. \]
• Can apply usual algebra rules to operator:

8.3 Autoregressive Models

• Autoregressive model of order \(p\), AR(\(p\)) model, can be written as: \[ y_{t} = c + \phi_{1}y_{t-1} + \phi_{2}y_{t-2} + \dots + \phi_{p}y_{t-p} + e_{t}. \]
• Categorising the AR(1) model:
  • \(\phi_1=0\): \(y_t\) is white noise.
  • \(\phi_1=1\) and \(c=0\): \(y_t\) is random walk.
  • \(\phi_1=1\) and \(c\neq0\): \(y_t\) is random walk with drift.
  • \(\phi_1<0\): \(y_t\) oscillates between positive and negative values.
• For stationarity in AR(1), require \(-1 < \phi_1 < 1\).
• For AR(2), require \(\phi_1+\phi_2 < 1\), \(\phi_1+\phi_2 < 1\) and \(\phi_2-\phi_1 < 1\).

8.4 Moving Average Models

• Instead of using past values of \(y_t\), moving average uses past forecast errors: \[ y_{t} = c + e_t + \theta_{1}e_{t-1} + \theta_{2}e_{t-2} + \dots + \theta_{q}e_{t-q}. \]

  • This is MA(\(q\)).
  • Not to be confused with MA smoothing. MA(\(q\)) generates forecasts.
• Can write any AR(\(p\)) process as MA(\(\infty\)) model:

8.4 Moving Average Models

• If \(-1 < \phi_1 < 1\) then \(\phi_1^k\) gets smaller with \(k\) so: \[ y_t = e_t + \phi_1 e_{t-1} + \phi_1^2 e_{t-2} + \phi_1^3 e_{t-3} + \dots \]

• Reverse true with constraints on MA parameters:

  • For MA(1): \(-1<\theta_1<1\).
  • For MA(2): \(-1<\theta_1<1\), \(\theta_1+\theta+2>-1\), \(\theta_1-\theta_2<1\).

8.4 MA Models

e = rnorm(100)
plot(20 + e[2:100] + 0.8*e[1:99],type="l",main="MA(1)")

plot(e[3:100] - e[2:99] + 0.005*e[1:98],type="l",main="MA(2)")

8.5 Non-Seasonal ARIMA Models

• Combine AR, MA and differencing to stationarity into ARIMA(\(p,d,q\)) model: \[\label{eq-8-arima} y'_{t} = c + \phi_{1}y'_{t-1} + \cdots + \phi_{p}y'_{t-p} + \theta_{1}e_{t-1} + \cdots + \theta_{q}e_{t-q} + e_{t}. \]
• \(y'_t\) has been differenced \(d\) times in ().
• In backshift, or lag operator form: \[ \begin{array}{c c c c} (1-\phi_1B - \cdots - \phi_p B^p) & (1-B)^d y_{t} &= &c + (1 + \theta_1 B + \cdots + \theta_q B^q)e_t\\ \uparrow & \uparrow & &\uparrow\\ \text{AR($p$)} & \text{$d$ differences} & &\text{MA($q$)}\\ \end{array} \]

8.5 Non-Seasonal ARIMA Models

• All models used so far can be expressed as ARIMA models:
  • White Noise ARIMA(0,0,0)
  • Random walk ARIMA(0,1,0) with no constant
  • Random walk with drift ARIMA(0,1,0) with constant
  • Autoregression ARIMA(\(p\),0,0)
  • Moving average ARIMA(0,0,\(q\))

8.5 ARIMA

football.auto <- auto.arima(football.z)
plot(forecast(football.auto,h=10),include=80)

8.5 Understanding ARIMA Models

• auto.arima() function powerful (more later), but risky: What will it do?
• Important to think about impact of \(c\) and \(d\) parameters:
  • \(c=0,d=0\): Long-term forecasts tend to zero.
  • \(c=0,d=1\): Long-term forecasts tend to non-zero constant.
  • \(c=0,d=2\): Long-term forecasts follow time trend.
  • \(c\neq0,d=0\): Long-term forecasts go to mean of data.
  • \(c\neq0,d=1\): Long-term forecasts follow time trend.
  • \(c\neq0,d=2\): Long-term forecasts follow quadratic trend.

8.5 ACFs and PACFs

• Usually difficult to tell \(p\) and \(q\) from eyeballing data.
• Particular patterns in ACF and PACF may enable identification of \(p\) and \(q\):
  • ARIMA(\(p\),\(d\),0): If ACF gently declining (or sinusoidal), PACF has significant lags to \(p\), none after.
  • ARIMA(0,\(d\),\(q\)): If PACF gently declining (or sinusoidal), ACF has significant lags to \(q\), none after.
• Ignore isolated significant ACF/PACF lags — likely random chance.

ACF and PACF: Brent Crude Oil

boil <- Quandl("DOE/RBRTE")
plot(boil$Date,boil$Value,main="Brent Oil Price",type="l",ylab="Price ($/barrel)",xlab="Date")

acf(boil$Value,main="ACF of Brent Oil Price")

pacf(boil$Value,main="PACF of Brent Oil Price")

ACF and PACF: Copper

copper <- Quandl("OFDP/COPPER_6")
plot(copper$Date,copper$Mid,main="Copper Price",type="l",ylab="Price ($/tonne)",xlab="Date")
lines(copper$Date,copper$Bid,type="l",col=2)
lines(copper$Date,copper$Ask,type="l",col=3)

acf(copper$Mid,main="ACF of Copper Price")

pacf(copper$Mid,main="PACF of Copper Price")

ACF and PACF: Unemployment

plot(UNRATE,main="Unemployment")

acf(coredata(UNRATE))

pacf(coredata(UNRATE))

ACF and PACF: UK GDP

plot(ukgdp.z,main="UK GDP")

acf(coredata(ukgdp.z),main="ACF of UK GDP")

pacf(coredata(ukgdp.z),main="PACF of UK GDP")

8.6 Estimation and Order Selection

• ARIMA models can be estimated by least squares or Maximum Likelihood (MLE).
• Least squares find parameters \(c,\phi_1,\dots,\phi_p,\theta_1,\dots,\theta_q\) satisfying: \[ \min_{c,\phi_1,\dots,\phi_p,\theta_1,\dots,\theta_q} \sum_{t=1}^T e^2_t. \]
• MLE finds most likely \(c,\phi_1,\dots,\phi_p,\theta_1,\dots,\theta_q\) given data \(y_t\).

8.6 Estimation and Order Selection

• Information criteria could be helpful in selection of \(p\) ahd \(q\).
• auto.arima uses: \[ \text{AIC} = -2 \log(L) + 2(p+q+k+1),\\ \text{AIC}_{\text{c}} = \text{AIC} + \frac{2(p+q+k+1)(p+q+k+2)}{T-p-q-k-2},\\ \text{BIC} = \text{AIC} + \log(T)(p+q+k-1). \]

8.7 ARIMA Modelling in R

• auto.arima() is algorithm for choosing \(p\), \(d\) and \(q\).
  • 1. Determine \(d\) via repeated KPSS tests.
  • 2. \(p\) and \(q\) then chosen via AICc using a stepwise method.
• Can choose own ARIMA model in R:

football.arima <- Arima(football.z,order=c(0,0,3))

• There exists alternative arima() function in R but:
  • Does not allow \(c\neq0\) if \(d>0\), does not allow subsequent forecasting.

Modelling Procedure

Example: UK QoQ GDP Growth

ukgdp.t <- ts(ukgdp$IHYQ,start=c(1955,2),frequency=4)
tsdisplay(ukgdp.t)

• Data series appears to have different properties around 1982.

ukgdp.t0 <- window(ukgdp.t,end=c(1982,2))
tsdisplay(ukgdp.t0)

ukgdp.t1 <- window(ukgdp.t,start=c(1982,3))
tsdisplay(ukgdp.t1)

• Focus on second half of data:
  • ACF and PACF suggest ARIMA(1,0,0).

ukgdp.a1 <- Arima(ukgdp.t1,order=c(1,0,0))
tsdisplay(ukgdp.a1$residuals)

• Check other ‘nearby’ models:

summary(ukgdp.a1)

## Series: ukgdp.t1 
## ARIMA(1,0,0) with non-zero mean 
## 
## Coefficients:
##          ar1  intercept
##       0.5799     0.5786
## s.e.  0.0710     0.1112
## 
## sigma^2 estimated as 0.2875:  log likelihood=-102.84
## AIC=211.69   AICc=211.88   BIC=220.26
## 
## Training set error measures:
##                       ME      RMSE       MAE  MPE MAPE      MASE
## Training set 0.002250977 0.5361739 0.3949923 -Inf  Inf 0.8583874
##                     ACF1
## Training set -0.07141299

ukgdp.a2 <- Arima(ukgdp.t1,order=c(2,0,0))
summary(ukgdp.a2)

## Series: ukgdp.t1 
## ARIMA(2,0,0) with non-zero mean 
## 
## Coefficients:
##          ar1     ar2  intercept
##       0.5056  0.1263     0.5789
## s.e.  0.0870  0.0869     0.1252
## 
## sigma^2 estimated as 0.2828:  log likelihood=-101.8
## AIC=211.59   AICc=211.92   BIC=223.03
## 
## Training set error measures:
##                       ME      RMSE       MAE  MPE MAPE      MASE
## Training set 0.002491449 0.5317836 0.3948206 -Inf  Inf 0.8580142
##                    ACF1
## Training set 0.01698083

ukgdp.a3 <- Arima(ukgdp.t1,order=c(1,0,1))
summary(ukgdp.a3)

## Series: ukgdp.t1 
## ARIMA(1,0,1) with non-zero mean 
## 
## Coefficients:
##          ar1      ma1  intercept
##       0.6835  -0.1565     0.5791
## s.e.  0.0980   0.1271     0.1232
## 
## sigma^2 estimated as 0.2841:  log likelihood=-102.09
## AIC=212.18   AICc=212.51   BIC=223.62
## 
## Training set error measures:
##                       ME     RMSE       MAE  MPE MAPE      MASE
## Training set 0.002325168 0.533027 0.3962993 -Inf  Inf 0.8612276
##                      ACF1
## Training set -0.009506348

• Finally, check against the computer:

ukgdp.auto <- auto.arima(ukgdp.t1)
summary(ukgdp.auto)

## Series: ukgdp.t1 
## ARIMA(2,0,1)(2,0,0)[4] with non-zero mean 
## 
## Coefficients:
##          ar1     ar2     ma1     sar1     sar2  intercept
##       0.0360  0.4256  0.4910  -0.0314  -0.0995     0.5774
## s.e.  0.2671  0.1647  0.2772   0.0972   0.0953     0.1125
## 
## sigma^2 estimated as 0.2769:  log likelihood=-100.49
## AIC=214.98   AICc=215.91   BIC=235
## 
## Training set error measures:
##                       ME      RMSE       MAE  MPE MAPE      MASE
## Training set 0.002985056 0.5262004 0.3860422 -Inf  Inf 0.8389371
##                    ACF1
## Training set 0.01640524

tsdisplay(ukgdp.auto$residuals)

8.8 Forecasting

• Thus far have glossed over details of constructing forecasts. Method:
  • Expand ARIMA equation so only \(y_t\) on left hand side.
  • Replace \(t\) by \(T+h\) throughout.
  • On RHS replace future values by forecasts.
• Example: ARIMA(3,1,1) model: \[ (1-\phi_1B -\phi_2B^2-\phi_3B^3)(1-B) y_t = (1+\theta_1B)e_{t}. \]
• Expand LHS to get: \[ \left[1-(1+\phi_1)B +(\phi_1-\phi_2)B^2 + (\phi_2-\phi_3)B^3 +\phi_3B^4\right] y_t = (1+\theta_1B)e_{t}. \]

8.8 Forecasting

• Applying backshift/lag operator: \[ y_t - (1+\phi_1)y_{t-1} +(\phi_1-\phi_2)y_{t-2} + (\phi_2-\phi_3)y_{t-3} +\phi_3y_{t-4} = e_t+\theta_1e_{t-1}. \]
• Move all terms other than \(y_t\) to RHS: \[ y_t = (1+\phi_1)y_{t-1} -(\phi_1-\phi_2)y_{t-2} - (\phi_2-\phi_3)y_{t-3} -\phi_3y_{t-4} + e_t + \theta_1e_{t-1}. \]

8.8 Forecasting

• Second step is to move forward to forecast \(T+1\): \[ y_{T+1} = (1+\phi_1)y_{T} -(\phi_1-\phi_2)y_{T-1} - (\phi_2-\phi_3)y_{T-2} -\phi_3y_{T-3} + e_{T+1}+\theta_1e_{T}. \]
  • where we set \(e_{T+1}=0\), and \(e_T=\ols{e}_T\): \[ \hat{y}_{T+1|T} = (1+\phi_1)y_{T} -(\phi_1-\phi_2)y_{T-1} - (\phi_2-\phi_3)y_{T-2} -\phi_3y_{T-3} + \theta_1\hat{e}_{T}. \]
  • We then move forward to forecast \(T+2\) (and on and on): \[ \hat{y}_{T+2|T} = (1+\phi_1)\hat{y}_{T+1|T} -(\phi_1-\phi_2)y_{T} - (\phi_2-\phi_3)y_{T-1} -\phi_3y_{T-2}. \]

8.9 Seasonal ARIMA Modelling

• We can bolt on seasonal component for seasonal data:

• For example, an ARIMA(1,1,1)(1,1,1)\(_4\) is:

• Order of seasonal components could be chosen also via ACF/PACF.
• auto.arima detects seasonal patterns also.

Example: Japanese Exports

getSymbols('XTEXVA01JPM664N',src='FRED',return.class="zoo")

## [1] "XTEXVA01JPM664N"

exp.t <- ts(log(XTEXVA01JPM664N),start=c(1955,1),freq=12)
tsdisplay(exp.t)

exp.t.s <- diff(exp.t,differences=1,lag=12)
tsdisplay(exp.t.s)

#exp.t.s.1 <- diff(exp.t.s,differences=1)
#tsdisplay(exp.t.s.1)
exp.t.s.a1 <- Arima(exp.t,order=c(5,0,0),seasonal=c(0,1,0))
tsdisplay(exp.t.s.a1$residuals)

• Then check with the computer…

exp.t.auto <- auto.arima(exp.t)

## Warning in auto.arima(exp.t): Unable to fit final model using maximum
## likelihood. AIC value approximated

Export Forecasts

exp.forc <- forecast(exp.t.s.a1,h=20)
plot(exp.forc,include=100)

Concluding and Feedback

• ARIMA modelling:
  • Capture time series persistence of data.
  • Attempt to forecast using historical patterns.
• Don’t forget to forecast: http://goo.gl/forms/ZJ5iBbHx4A
• Don’t forget to feedback: http://goo.gl/forms/XdWvfgs0Le