Introduction

\[ \newcommand{\N}[1]{\mathsf{N}\left(#1\right)} \newcommand{\E}[1]{\mathsf{E}\left(#1\right)} \newcommand{\V}[1]{\mathsf{Var}\left(#1\right)} \newcommand{\ols}[1]{\widehat{#1}} \newcommand{\cond}[1]{\left|#1\right.} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\pred}[3]{\widehat{#1}_{#2\left|#3\right.}} \]

• Forecasting: Essential for life.
  • Forecast the work required to get slides ready…
• Reminder:
  • Sir Prof David F. Hendry: “Statistical Model Selection with Big Data”
  • 4pm, room G09, Henley Business School.
  • Cannot recommend more highly your attendance for some if not all!
• Thus far:
  • Judgemental forecasting.
  • Linear regression.
  • ARIMA.
  • Model selection and impulse saturation.
  • Time series decomposition.
• Today:
  • Exponential smoothing.
• Next week:
  • Tuesday: Recap and talking through the sample midterm paper.
  • Thursday: Midterm (sample paper: http://rpubs.com/jjreade/sample_midterm_i)

Another one that’s tricky to call…

Preliminaries

• In order that these slides can be replicated using R, full code posted.

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") #this is my authentication code - please sign up at www.quandl.com for your own!
#tourists_AU <- Quandl("EUROSTAT/TOUR_OCC_NIM_1")
tourists_US <- Quandl("BTS/AIRPASS")
tourists_US.t <- ts(tourists_US$Total[order(tourists_US$Date)],start=c(2000,1),freq=12)
vehicle_sales <- Quandl("FRED/TOTALNSA")
vehicle_sales.t <- ts(vehicle_sales$Value[order(vehicle_sales$Date)],start=c(1976,1),freq=12)
UKcc <- Quandl("UKONS/LMS_BCJB_M")
UKcc.eng.t <- ts(UKcc$Value[order(UKcc$Year)],start=c(1983,6),frequency=12)

7.1 Simple Exponential Smoothing

• Exponential smoothing popularised in 1950s.
• Produces effective forecasting models with minimal effort.
• Idea mixes two naive forecasting models:
  • Random walk forecast: \(\hat{y}_{T+h|T} = y_T\).
  • Average method: \(\hat{y}_{T+h|T} = \frac1T \sum_{t=1}^T y_t\)
• Instead we attach larger weight to more recent observations: \[ \hat{y}_{T+1|T} = \alpha y_T + \alpha(1-\alpha) y_{T-1} + \alpha(1-\alpha)^2 y_{T-2}+ \cdots.\label{eq-7-ses} \]

7.1 Simple Exponential Smoothing

7.1 Simple Exponential Smoothing

• Alternative expression in terms of weighted average.
• Express forecast as weighted average of most recent observation and forecast: \[ \hat{y}_{t+1|t} = \alpha y_t + (1-\alpha) \hat{y}_{t|t-1} \]
• Recursively back to the beginning of time: \[ \begin{align} \hat{y}_{2|1} &= \alpha y_1 + (1-\alpha) \ell_0,\\ \hat{y}_{3|2} &= \alpha y_2 + (1-\alpha) \hat{y}_{2|1},\\ \hat{y}_{4|3} &= \alpha y_3 + (1-\alpha) \hat{y}_{3|2},\\ &\vdots\\ \hat{y}_{T+1|T} &= \alpha y_T + (1-\alpha) \hat{y}_{T|T-1}. \end{align} \]

7.1 Simple Exponential Smoothing

• Substituting recursively: \[ \begin{align} \hat{y}_{3|2} &= \alpha y_2 + (1-\alpha) \left[\alpha y_1 + (1-\alpha) \ell_0\right]\\ &= \alpha y_2 + \alpha(1-\alpha) y_1 + (1-\alpha)^2 \ell_0\\ \hat{y}_{4|3} &= \alpha y_3 + (1-\alpha) [\alpha y_2 + \alpha(1-\alpha) y_1 + (1-\alpha)^2 \ell_0]\\ &= \alpha y_3 + \alpha(1-\alpha) y_2 + \alpha(1-\alpha)^2 y_1 + (1-\alpha)^3 \ell_0\\ &~~\vdots \\ \hat{y}_{T+1|T} &= \sum_{j=0}^{T-1} \alpha(1-\alpha)^j y_{T-j} + (1-\alpha)^T \ell_{0}. \end{align} \]

7.1 Simple Exponential Smoothing

• Alternative representation: Component form.
  • Eventually level, trend and seasonal components.
• Currently just level, \(\ell_t\), component.
• Component form is: \[ \begin{align} \text{Forecast equation}&&y_{t+1|t} &= \ell_{t}\\ \text{Smoothing equation}&&\ell_{t} &= \alpha y_{t} + (1 - \alpha)\ell_{t-1}, \end{align} \]

7.1 Simple Exponential Smoothing

• Alternative representation: Error correction form.

• Rearrange level equation to get error correction form: \[ \begin{align} \ell_{t} &= \ell_{t-1}+\alpha( y_{t}-\ell_{t-1})\\ &= \ell_{t-1}+\alpha e_{t} \end{align} \]
• Here, \(e_{t}=y_{t}-\ell_{t-1}=y_{t}-y_{t+1|t}\).
  • \(e_t\) is one-step within-sample forecast error.

7.1 Longer Term Forecasts

• Simple exponential smoothing simply extrapolates level further ahead, hence: \[ y_{T+h|T}=y_{T+1|T}=\ell_T, \qquad h=2,3,\dots. \]

7.1 Initialisation

• What is initial value, i.e. \(\ell_0\)?
• Less important with larger \(T\) but important for small samples.
• Common approach: Set \(\ell_0=y_1\).
• Alternative is optimisation approach: Use \(\ell_0\) satisfying: \[ \text{SSE}=\sum_{t=1}^T(y_t -y_{t\cond{t-1}})^2=\sum_{t=1}^Te_t^2. \]
• Here though, solving for \(\ell_0\) and \(\alpha\) very non-linear.

7.1 Example

plot(UKcc.eng.t,main="Claimant Count in England",ylab="Claimant Count",xlab="Date")

fit1 <- ses(UKcc.eng.t, alpha=0.2, initial="simple", h=3)
fit2 <- ses(UKcc.eng.t, alpha=0.6, initial="simple", h=3)
fit3 <- ses(UKcc.eng.t, h=3)
plot(fit1, plot.conf=FALSE, ylab="Claimant Count",
  xlab="Date", main="", fcol="white", type="o",include=30)
lines(fitted(fit1), col="blue", type="o")
lines(fitted(fit2), col="red", type="o")
lines(fitted(fit3), col="green", type="o")
lines(fit1$mean, col="blue", type="o")
lines(fit2$mean, col="red", type="o")
lines(fit3$mean, col="green", type="o")
legend("bottomleft",lty=1, col=c(1,"blue","red","green"), 
  c("data", expression(alpha == 0.2), expression(alpha == 0.6),
  expression(alpha == 0.9999)),pch=1,bty="n")

plot(vehicle_sales.t,main="US Vehicle Sales",ylab="Sales",xlab="Date")

fit1 <- ses(vehicle_sales.t, alpha=0.2, initial="simple", h=3)
fit2 <- ses(vehicle_sales.t, alpha=0.6, initial="simple", h=3)
fit3 <- ses(vehicle_sales.t, h=3)
plot(fit1, plot.conf=FALSE, ylab="Sales",
  xlab="Date", main="", fcol="white", type="o",include=50)
lines(fitted(fit1), col="blue", type="o")
lines(fitted(fit2), col="red", type="o")
lines(fitted(fit3), col="green", type="o")
lines(fit1$mean, col="blue", type="o")
lines(fit2$mean, col="red", type="o")
lines(fit3$mean, col="green", type="o")
legend("bottomleft",lty=1, col=c(1,"blue","red","green"), 
  c("data", expression(alpha == 0.2), expression(alpha == 0.6),
  expression(alpha == 0.5003)),pch=1,bty="n")

plot(tourists_US.t,main="Total Air Passengers in the US",ylab="Total",xlab="Date")

fit1 <- ses(tourists_US.t, alpha=0.2, initial="simple", h=3)
fit2 <- ses(tourists_US.t, alpha=0.6, initial="simple", h=3)
fit3 <- ses(tourists_US.t, h=3)
plot(fit1, plot.conf=FALSE, ylab="Total",
  xlab="Date", main="", fcol="white", type="o",include=50)
lines(fitted(fit1), col="blue", type="o")
lines(fitted(fit2), col="red", type="o")
lines(fitted(fit3), col="green", type="o")
lines(fit1$mean, col="blue", type="o")
lines(fit2$mean, col="red", type="o")
lines(fit3$mean, col="green", type="o")
legend("bottomleft",lty=1, col=c(1,"blue","red","green"), 
  c("data", expression(alpha == 0.2), expression(alpha == 0.6),
  expression(alpha == 0.6263)),pch=1,bty="n")

7.2 Holt’s Linear Trend Method

• Holt (1957) extended to allow forecasting with trending data. \[ \begin{align} \text{Forecast equation}&& \pred{y}{t+h}{t} &= \ell_{t} + hb_{t} \\ \text{Level equation}&& \ell_{t} &= \alpha y_{t} + (1 - \alpha)(\ell_{t-1} + b_{t-1})\\ \text{Trend equation}&& b_{t} &= \beta^*(\ell_{t} - \ell_{t-1}) + (1 -\beta^*)b_{t-1} \end{align} \]
• \(b_t\) is estimate of trend component, allowed to continuously update:
  • Weighted average of trend between \(t\) and \(t-1\) and all over all previous observations (\(b_{t-1}\)).

7.2 Example using UK GDP

wd <- "/home/readejj/Dropbox/Teaching/Reading/ec313/Resources/Datasets/"
gdp_file <- paste(wd,"gdp_1_2000.csv",sep="")
gdp_all <- read.csv(gdp_file)
uk.gdp.t <- ts(gdp_all$England.GB.UK[19:nrow(gdp_all)],start=1800,frequency=1)
uk.gdp.t0 <- window(uk.gdp.t,end=2008)
uk.gdp.t1 <- window(uk.gdp.t,start=2009)
uk.gdp.t.aan <- ets(uk.gdp.t0,model="AAN")
uk.gdp.t.aan.fit <- forecast(uk.gdp.t.aan,h=5)
plot(window(uk.gdp.t,start=1950),ylim=range(min(window(uk.gdp.t,start=1950)),max(uk.gdp.t.aan.fit$mean)))
lines(uk.gdp.t.aan$fitted,col=2)
lines(uk.gdp.t.aan.fit$mean,col=3)
legend("topleft",lty=1,col=c(1,2,3),legend=c("Actual","Fitted","Forecast"),bty="n")

7.3 Exponential Trend Method

• Multiplicative time trend can be specified: \[ \begin{align} y_{t+h\cond{t}} &= \ell_{t} b_{t}^h\\ \ell_{t} &= \alpha y_{t} + (1 - \alpha)(\ell_{t-1} b_{t-1})\\ b_{t} &= \beta^*\frac{\ell_{t}}{ \ell_{t-1}} + (1 -\beta^*)b_{t-1} \end{align} \]
• Now \(b_t\) is estimated growth rate, trend exponential not linear.
• Error correction form is: \[ \begin{align} \ell_{t} &= \ell_{t-1} b_{t-1}+\alpha e_{t}\\ b_{t} &= b_{t-1}+ \alpha \beta^*\frac{e_{t}}{\ell_{t-1}} \end{align} \]
• Here, \(e_t=y_{t}-(\ell_{t-1}b_{t-1})=y_{t}-\pred{y}{t}{t-1}\).

uk.gdp.t.mmn <- ets(uk.gdp.t0,model="MMN")
uk.gdp.t.mmn.fit <- forecast(uk.gdp.t.mmn,h=5)
plot(window(uk.gdp.t,start=1950),ylim=range(min(window(uk.gdp.t,start=1950)),
                                            max(uk.gdp.t.mmn.fit$mean)),
     xlim=range(1950,2015))
lines(uk.gdp.t.mmn$fitted,col=2)
lines(uk.gdp.t.mmn.fit$mean,col=2,lty=2)
lines(uk.gdp.t.aan.fit$mean,col=3)
legend("topleft",lty=c(1,1,2,1),col=c(1,2,2,3),
       legend=c("Actual","Fitted","Mult Forecast","Add Forecast"),bty="n")

7.4 Damped Trend Methods

• Experience suggests trend methods often over-predict growth.
• Led to development of damped trend methods: \[ \begin{align} \pred{y}{t+h}{t} &= \ell_{t} + (\phi+\phi^2 + \dots + \phi^{h}) b_{t} \\ \ell_{t} &= \alpha y_{t} + (1 - \alpha)(\ell_{t-1} + \phi b_{t-1})\\ b_{t} &= \beta^*(\ell_{t} - \ell_{t-1}) + (1 -\beta^*)\phi b_{t-1}. \end{align} \]
• C.f. earlier method had \(\pred{y}{t+h}{t} = \ell_{t} + h b_{t}\).
• Error correction form: \[ \begin{align} \ell_{t} &= \ell_{t-1} + \phi b_{t-1} + \alpha e_{t} \\ b_{t} &= \phi b_{t-1}+ \alpha \beta^*e_{t}. \end{align} \]

uk.gdp.t.aand <- ets(uk.gdp.t0,model="AAN",damped=TRUE)
uk.gdp.t.aand.fit <- forecast(uk.gdp.t.aand,h=5)
plot(window(uk.gdp.t,start=1950),ylim=range(min(window(uk.gdp.t,start=1950)),
                                            max(uk.gdp.t.aan.fit$mean)))
lines(uk.gdp.t.aand$fitted,col=2)
lines(uk.gdp.t.aand.fit$mean,col=2,lty=2)
lines(uk.gdp.t.mmn.fit$mean,col=3)
lines(uk.gdp.t.aan.fit$mean,col=4)
legend("topleft",lty=c(1,1,2,1,1),col=c(1,2,2,3,4),
       legend=c("Actual","Fitted","Additive Damped Forecast","Multiplicative Forecast","Additive Forecast"),bty="n")

7.4 Damped Trend Methods

• Multiplicative damped trend: \[ \begin{align} \pred{y}{t+h}{t} &= \ell_{t}b_{t}^{(\phi+\phi^2 + \dots + \phi^{h})} \\ \ell_{t} &= \alpha y_{t} + (1 - \alpha)\ell_{t-1} b^\phi_{t-1}\\ b_{t} &= \beta^*\frac{\ell_{t}}{ \ell_{t-1}} + (1 -\beta^*)b_{t-1}^{\phi}. \end{align} \]
• Suggestion this produces yet more conservative forecasts.
• Error correction form: \[ \begin{align} \ell_{t} &= \ell_{t-1} b^\phi_{t-1}+\alpha e_{t}\\ b_{t} &= b^\phi_{t-1}+ \alpha \beta^*\frac{e_{t}}{\ell_{t-1}}. \end{align} \]

uk.gdp.t.mmnd <- ets(uk.gdp.t0,model="MMN",damped=TRUE)
uk.gdp.t.mmnd.fit <- forecast(uk.gdp.t.mmnd,h=5)
plot(window(uk.gdp.t,start=1950),ylim=range(min(window(uk.gdp.t,start=1950)),
                                            max(uk.gdp.t.mmn.fit$mean)),
     xlim=range(1950,2015))
lines(uk.gdp.t.mmnd$fitted,col=2)
lines(uk.gdp.t.mmnd.fit$mean,col=2,lty=2)
lines(uk.gdp.t.aand.fit$mean,col=3)
lines(uk.gdp.t.mmn.fit$mean,col=4)
lines(uk.gdp.t.aan.fit$mean,col=5)
legend("topleft",lty=c(1,1,2,1,1,1),col=c(1,2,2,3,4,5),
       legend=c("Actual","Fitted","Multiplicative Damped Forecast",
                "Additive Damped Forecast","Multiplicative Forecast","Additive Forecast"),bty="n")

7.5 Holt-Winters Seasonal Method

• Holt (1957) and Winters (1960) extended trend model to include seasonal component.
  • Three smoothing equations: Level, trend and seasonal components.
  • Additive/multiplicative variants based on nature of seasonality.

\[ \begin{align} y_{t+h\cond{t}} &= \ell_{t} + hb_{t} + s_{t-m+h_m^+} \\ \ell_{t} &= \alpha(y_{t} - s_{t-m}) + (1 - \alpha)(\ell_{t-1} + b_{t-1})\\ b_{t} &= \beta^*(\ell_{t} - \ell_{t-1}) + (1 - \beta^*)b_{t-1}\\ s_{t} &= \gamma (y_{t}-\ell_{t-1}-b_{t-1}) + (1-\gamma)s_{t-m}, \end{align} \] - Here, \(h_m^+=\lfloor(h-1)\mod~m\rfloor+1\). + Notation \(\lfloor u \rfloor\) means largest integer not greater than \(u\).} + Ensures seasonal component for forecast comes only from final year of sample. - Error correction form: \[ \begin{align} \ell_{t} &= \ell_{t-1} + b_{t-1}+\alpha e_{t}\\ b_{t} &= b_{t-1} + \alpha\beta^*e_{t}\\ s_{t} &= s_{t-m}+\gamma e_t, \end{align} \] - Here, \(e_t=y_{t} - (\ell_{t-1} + b_{t-1}+s_{t-m})=y_{t} - \pred{y}{t}{t-1}\).

7.5 Holt-Winters Seasonal Method

tourists_US.t0 <- window(tourists_US.t,end=c(2011,12))
tourists_US.t1 <- window(tourists_US.t,start=c(2012,1))
tourists_US.t.aaa <- ets(tourists_US.t0,model="AAA")
tourists_US.t.aaa.fit <- forecast(tourists_US.t.aaa,h=NROW(tourists_US.t1))
plot(tourists_US.t,pch=2,type="o",main="US Air Passengers")
lines(tourists_US.t.aaa$fitted,col=2,type="o")
lines(tourists_US.t.aaa.fit$mean,col="darkred",lty=2,lwd=2,type="o")
legend("topleft",lty=c(1,1,2),col=c(1,2,"darkred"),legend=c("Actual","ETS(A,A,A)","Forecast"),pch=c(2,1,1),bty="n")

7.5 Holt-Winters Seasonal Method

• Multiplicative method: \[ \begin{align} \pred{y}{t+h}{t} &= (\ell_{t} + hb_{t})s_{t-m+h_{m}^{+}}. \\ \ell_{t} &= \alpha \frac{y_{t}}{s_{t-m}} + (1 - \alpha)(\ell_{t-1} + b_{t-1})\\ b_{t} &= \beta^*(\ell_{t}-\ell_{t-1}) + (1 - \beta^*)b_{t-1} \\ s_{t} &= \gamma \frac{y_{t}}{(\ell_{t-1} + b_{t-1})} + (1 - \gamma)s_{t-m} . \end{align} \]
• Error correction form: \[ \begin{align} \ell_{t} &= \ell_{t-1} + b_{t-1}+\alpha\frac{e_{t}}{s_{t-m}}\\ b_{t} &= b_{t-1}+\alpha\beta^*\frac{e_{t}}{s_{t-m}}\\ s_{t} &= s_{t}+\gamma \frac{e_{t}}{(\ell_{t-1} + b_{t-1})} \end{align} \]

7.5 Holt-Winters Seasonal Method

tourists_US.t.mam <- ets(tourists_US.t0,model="MAM")
tourists_US.t.mam.fit <- forecast(tourists_US.t.mam,h=3)
plot(tourists_US.t,type="b",main="Total Passengers in the US")
lines(tourists_US.t.mam$fitted,type="b",col=2,pch=2)
lines(tourists_US.t.mam.fit$mean,type="b",col=2,lty=2,pch=3)
lines(tourists_US.t.aaa$fitted,type="b",col=3,pch=2)
lines(tourists_US.t.aaa.fit$mean,type="b",col=3,lty=2,pch=3)
legend("topleft",lty=c(1,2,2,3,3),col=c(1,2,2,3,3),pch=c(1,2,3,2,3),
       legend=c("Actual","ETS(M,A,M)","ETS(M,A,M) Forecast",
                "ETS(A,A,A)","ETS(A,A,A) Forecast"))

7.6 Taxonomy of Exponential Smoothing Methods

• All combinations of level, trend and seasonal components possible:
  • Multiplicative or additive components, damped trends.
• Fifteen combinations in all; various options for initialisation.

7.7 Innovations State Space Models for Exponential Smoothing

• Notation: Vast array of potential models.
• Building on error correction form, we create state-space models.
  • Measurement (state) equation: Observed data.
  • Transition (space) equation: Unobserved data.
• ETS function in R: , where:
  1. Error: Additive (A) or Multiplicative (M)?
  2. Trend: None (N), Additive (A or A\(_\textsf{d}\)) or Multiplicative (M or M\(_\textsf{d}\)).
  3. Seasonal: None (N), Additive (A) or Multiplicative (M).

ETS(A,N,N): Back to 7.1

• Error correction form from earlier: \[ \ell_t=\ell_{t-1}+\alpha e_t. \] where \(e_t = y_t - \ell_{t-1}\) and \(\pred{y}{t}{t-1} = \ell_{t-1}\).
• Hence \(e_t = y_t - \pred{y}{t}{t-1}\) is one-step forecast error and \(y_t = \ell_{t-1} + e_t\).
• Need to assume \(e_t =\varepsilon_t\sim\text{NID}(0,\sigma^2)\) to write: \[ \begin{align} \textbf{Measurement equation}&:\,\,\,y_t = \ell_{t-1} + \varepsilon_t, \\ \textbf{State equation}&:\,\,\,\ell_t = \ell_{t-1}+\alpha \varepsilon_t. \end{align} \]
• Measurement equation is evolution of observed terms
• State equation is evolution of unobserved components.
  • Unobserved component modelling is common name for these methods.
  • We never actually observe the seasonal component, or trend.

ETS(M,N,N): Multiplicative errors

• Multiplicative errors: \[ \varepsilon_t=\frac{y_t-\pred{y}{t}{t-1}}{\pred{y}{t}{t-1}} \]
  • where \(\varepsilon_t \sim \text{NID}(0,\sigma^2)\).
• Substituting \(\pred{y}{t}{t-1}=\ell_{t-1}\) gives \(y_t = \ell_{t-1}+\ell_{t-1}\varepsilon_t\) and \(e_t = y_t - \pred{y}{t}{t-1} = \ell_{t-1}\varepsilon_t\).
  • We can then write state-space form as: \[ \begin{align} y_t&=\ell_{t-1}(1+\varepsilon_t)\\ \ell_t&=\ell_{t-1}(1+\alpha \varepsilon_t). \end{align} \]

Forecasting Japanese exports With ETS(M,N,N)

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

##     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] "XTEXVA01JPM664N"

exp.t.ets.mnn.fit <- forecast(ets(XTEXVA01JPM664N,model="MNN"),100)
plot(exp.t.ets.mnn.fit,include=500)

ETS(A,A,N): Holt inear Model with Additive Errors

• Assume \(\varepsilon_t=y_t-\ell_{t-1}-b_{t-1} \sim \text{NID}(0,\sigma^2)\) such that: \[ \begin{align} y_t&=\ell_{t-1}+b_{t-1}+\varepsilon_t\\ \ell_t&=\ell_{t-1}+b_{t-1}+\alpha \varepsilon_t\\ b_t&=b_{t-1}+\beta \varepsilon_t, \end{align} \]
• For simplicity, \(\beta=\alpha \beta^*\).

ETS(M,A,N): Holt linear with Multiplicative Errors

• Define errors as relative errors: \[ \varepsilon_t=\frac{y_t-(\ell_{t-1}+b_{t-1})}{(\ell_{t-1}+b_{t-1})} \]
• Linear method with multiplicative errors: \[ \begin{align} y_t&=(\ell_{t-1}+b_{t-1})(1+\varepsilon_t)\\ \ell_t&=(\ell_{t-1}+b_{t-1})(1+\alpha \varepsilon_t)\\ b_t&=b_{t-1}+\beta(\ell_{t-1}+b_{t-1}) \varepsilon_t. \end{align} \]
• For further models, see Table 7.10 in textbook.

Modelling with various combinations

plot(UKcc.eng.t,main="Claimant Count in England",ylab="Claimant Count",xlab="Date")

fit.aaa <- ets(UKcc.eng.t,model="AAA")
plot(fit.aaa)

fit.mmm <- ets(UKcc.eng.t,model="MMM")
plot(fit.mmm)

plot(window(UKcc.eng.t,start=c(2005,1)), ylab="Claimant Count",xlab="Date", main="")
lines(fit.aaa$fitted, col="blue", type="l")
lines(fit.mmm$fitted, col="red", type="l")
legend("bottomleft",lty=1, col=c(1,"blue","red","green"), 
  c("data", expression(alpha == 0.2), expression(alpha == 0.6),
  expression(alpha == 0.9999)),pch=1,bty="n")

plot(vehicle_sales.t,main="US Vehicle Sales",ylab="Sales",xlab="Date")

fit1 <- ses(vehicle_sales.t, alpha=0.2, initial="simple", h=3)
fit2 <- ses(vehicle_sales.t, alpha=0.6, initial="simple", h=3)
fit3 <- ses(vehicle_sales.t, h=3)
plot(fit1, plot.conf=FALSE, ylab="Sales",
  xlab="Date", main="", fcol="white", type="o",include=50)
lines(fitted(fit1), col="blue", type="o")
lines(fitted(fit2), col="red", type="o")
lines(fitted(fit3), col="green", type="o")
lines(fit1$mean, col="blue", type="o")
lines(fit2$mean, col="red", type="o")
lines(fit3$mean, col="green", type="o")
legend("bottomleft",lty=1, col=c(1,"blue","red","green"), 
  c("data", expression(alpha == 0.2), expression(alpha == 0.6),
  expression(alpha == 0.5003)),pch=1,bty="n")

plot(tourists_US.t,main="Total Air Passengers in the US",ylab="Total",xlab="Date")

fit1 <- ses(tourists_US.t, alpha=0.2, initial="simple", h=3)
fit2 <- ses(tourists_US.t, alpha=0.6, initial="simple", h=3)
fit3 <- ses(tourists_US.t, h=3)
plot(fit1, plot.conf=FALSE, ylab="Total",
  xlab="Date", main="", fcol="white", type="o",include=50)
lines(fitted(fit1), col="blue", type="o")
lines(fitted(fit2), col="red", type="o")
lines(fitted(fit3), col="green", type="o")
lines(fit1$mean, col="blue", type="o")
lines(fit2$mean, col="red", type="o")
lines(fit3$mean, col="green", type="o")
legend("bottomleft",lty=1, col=c(1,"blue","red","green"), 
  c("data", expression(alpha == 0.2), expression(alpha == 0.6),
  expression(alpha == 0.6263)),pch=1,bty="n")

Forecasting Japanese exports with various combinations

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

## [1] "XTEXVA01JPM664N"

jap.exp.t <- ts(XTEXVA01JPM664N,start=c(1955,1),freq=12)
jap.exp.t0 <- window(jap.exp.t,end=c(2007,12))
jap.exp.t1 <- window(jap.exp.t,start=c(2008,1))
exp.t.ets.mnn.fit <- forecast(ets(jap.exp.t0,model="MNN"),100)
exp.t.ets.ann.fit <- forecast(ets(jap.exp.t0,model="ANN"),100)
exp.t.ets.aan.fit <- forecast(ets(jap.exp.t0,model="AAN"),100)
exp.t.ets.man.fit <- forecast(ets(jap.exp.t0,model="MAN"),100)
#exp.t.ets.amn.fit <- forecast(ets(jap.exp.t0,model="AMN"),100)
exp.t.ets.mmn.fit <- forecast(ets(jap.exp.t0,model="MMN"),100)
plot(window(jap.exp.t,start=c(2000,1)),main="Japanese Exports, non-seasonal forecasts",ylab="Exports",xlab="Date")
lines(exp.t.ets.mnn.fit$mean,col=2,type="l")
lines(exp.t.ets.ann.fit$mean,col=3,type="l")
lines(exp.t.ets.aan.fit$mean,col=4,type="l")
lines(exp.t.ets.man.fit$mean,col=5,type="l")
lines(exp.t.ets.mmn.fit$mean,col=6,type="l")
legend("topleft",lty=1,bty="n",col=c(2:6),legend=c("MNN","ANN","AAN","MAN","MMN"))

exp.t.ets.mmm.fit <- forecast(ets(jap.exp.t0,model="MMM"),100)
#exp.t.ets.amm.fit <- forecast(ets(jap.exp.t0,model="AMM"),100)
#exp.t.ets.aam.fit <- forecast(ets(jap.exp.t0,model="AAM"),100)
#exp.t.ets.mma.fit <- forecast(ets(jap.exp.t0,model="MMA"),100)
#exp.t.ets.ama.fit <- forecast(ets(jap.exp.t0,model="AMA"),100)
exp.t.ets.maa.fit <- forecast(ets(jap.exp.t0,model="MAA"),100)
exp.t.ets.aaa.fit <- forecast(ets(jap.exp.t0,model="AAA"),100)
plot(window(jap.exp.t,start=c(2000,1)),main="Japanese Exports, seasonal forecasts",ylab="Exports",xlab="Date",
     ylim=range(c(window(jap.exp.t,start=c(2000,1)),
                  exp.t.ets.mmm.fit$mean,
                  exp.t.ets.maa.fit$mean,
                  exp.t.ets.aaa.fit$mean)))
lines(exp.t.ets.mmm.fit$mean,col=2,type="l")
lines(exp.t.ets.maa.fit$mean,col=3,type="l")
lines(exp.t.ets.aaa.fit$mean,col=4,type="l")
legend("topleft",lty=1,col=c(2,3,4),legend=c("MMM","MAA","AAA"),bty="n")

Maximum Likelihood Estimation

• Alternative to least squares estimation.
  • Likelihood function equal to joint density of all observations.
  • Intuitively: The likelihood of having observed our sample.
• Set up one density function with all parameters:
  • Smoothing parameters: \(\alpha\), \(\beta\), \(\gamma\) and \(\phi\).
  • Initial states: \(s_0,s_{-1},\dots,s_{-m+1}\).
• Must impose restrictions on smoothing parameters: \(0< \alpha,\beta^*,\gamma^*,\phi<1\).
  • Further restrictions implied by state-space for each model type.

Concluding

• Exponential smoothing methods:
  • More pretty pictures, informative decompositions.
  • Very easy to construct models, again reliant on extrapolation for forecasting.
• This afternoon:
  • 4pm: Sir Prof. David F. Hendry, G09 in HBS.
• Next week:
  • Tuesday: Revision session.
    • I will try and review everything covered.
    • Questions/Requests?
  • Thursday: 60 minute exam during the 11am-1pm window.