EC313 Business and Financial Forecasting Lecture 7

Introduction

• Recap session for course thus far.
• Midterm exam tomorrow on theory of forecasting:
  • All material covered so far.
• Format of exam: 60 minutes.
  • Section A: Mathematical, choose 2 from 6.
  • Section B: Theoretical, choose 2 from 6.
  • Sample exam: http://rpubs.com/jjreade/sample_midterm_i.
• Tips:
  • Plan your time well. 15 minutes per question.
  • Answer the question set.
  • Mark clearly what question you’re answering.
  • Write clearly.

Free Coffee

Mortgage Approvals

library(forecast)
mg <- read.csv("/home/readejj/Dropbox/Teaching/Reading/ec313/2015/Lecture-7/mortgages_280115.csv",stringsAsFactors=F)
mg.t <- ts(mg$LPMVTVX[is.na(mg$LPMVTVX)==F],start=c(1993,4),freq=12)
mg.t.arima <- auto.arima(mg.t)
mg.t.ets <- ets(mg.t)
mg.t.arima.f <- forecast(mg.t.arima,h=1)
plot(mg.t.arima.f)

Dave last week

• His squirming over election debates gives traction…

dave27 <- read.csv("Dave_270115.csv",stringsAsFactors=F)
dave27$Week <- as.Date(substr(dave27$Week,1,10))
plot(dave27$Week,dave27$david.cameron,
     main="Google Search Popularity of David Cameron",
     ylab="Search popularity",xlab="Date",type="l",ylim=range(0,20))

dave27.ts <- ts(dave27$david.cameron,start=c(2004,1),freq=52)
tsdisplay(dave27.ts)

dave27.arima <- auto.arima(dave27.ts)
dave27.forc <- forecast(dave27.arima,h=9)
plot(dave27.forc,include=100)

dave27.forc$mean

## Time Series:
## Start = c(2015, 7) 
## End = c(2015, 15) 
## Frequency = 52 
## [1] 4.979808 4.971005 4.967168 4.965495 4.964765 4.964447 4.964309 4.964248
## [9] 4.964222

dave03 <- read.csv("Dave_030215.csv",stringsAsFactors=F)
tail(dave03,1)

##                        Week david.cameron
## 579 2015-02-01 - 2015-02-07             6

FTSE last week

ftse <- Quandl("YAHOO/INDEX_FTSE")
ftse <- ftse[order(ftse$Date),]
ftse.0 <- ftse[ftse$Date<as.Date("2015-01-30"),]
ftse.auto <- auto.arima(ftse$Close)
ftse.forc <- forecast(ftse.auto,h=50)
plot(ftse.forc,include=100)

ftse.forc$mean[seq(1,45,5)]

## [1] 6787.733 6790.772 6794.285 6797.858 6801.430 6805.003 6808.576 6812.148
## [9] 6815.721

ftse[ftse$Date==as.Date("2015-01-30"),]

##         Date   Open High    Low  Close    Volume Adjusted Close
## 2 2015-01-30 6810.6 6844 6749.4 6749.4 799357300         6749.4

League Table…

• Of sorts…
• Week 1: Max is currently doing best.
• Week 2: Mark Carney is currently in the lead.
• Week 3: Chris Tucker is ahead.

A Distraction?

• After tomorrow’s bloodshed, another forecast?
• Recall what’s at stake:
  • Forecasts to be judged on MAPE to 1DP.
  • Tie-breaker is number of forecasts submitted.
• Forecast Friday’s trade balance announcement:
  • http://www.fxstreet.com/economic-calendar/event.aspx?id=a5d475bb-27fa-4a82-90ba-9fc271c5b9bd

trade <- read.csv("trade_030215.csv",stringsAsFactors=F)
trade$date <- as.Date(substr(trade$DateTime,1,8),"%Y%m%d")
plot(trade$date,trade$Actual,type="o",ylab="Total Trade Balance (£)",xlab="Date")

• Submit forecasts at:
  • http://goo.gl/forms/4PNKOuhe21

Recap - Lecture 1

• What is a forecast
• Forecast errors
• Optimal forecasts
• Forecast failure
• Basic forecast models

Recap - Lecture 2

• Judgemental Forecasting (Ch. 3):
  • Limitations
  • Key principles
  • Delphi method
  • Scenario forecasting
• Regression-based forecasts (Chs. 4–5)
  • OLS
  • Regression and correlation
  • Evaluating the model/residual diagnostics
  • Useful predictors
  • Inference

Recap - Lecture 3

• 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: \[ \begin{array}{c c c c} (1-\phi_1B - \cdots - \phi_p B^p) & (1-B)^d y_{t} &= &c + (1 + \theta_1 B + \cdots + \theta_q B^q)e_t\\ \uparrow & \uparrow & &\uparrow\\ \text{AR($p$)} & \text{$d$ differences} & &\text{MA($q$)}\\ \end{array} \]

Recap - Lecture 4

• Model selection:
  • How good is our model?
  • Need to select a model, but how?
• Impulse saturation:
  • Need to account for outliers.
  • Need to account for structural breaks.

Recap - Lecture 5

• Decompose time series \(y_t\) formally into three components:
  • Additively: \(y_t = S_t + T_t + E_t\).
  • Multiplicatively: \(y_t = S_t \times T_t \times E_t\).
• \(S_t\) is seasonal component, \(T_t\) trend component, \(E_t\) irregular.
  • Additive more appropriate if seasonal/trend components consistent throughout.
  • Multiplicative if components tend to vary with size of data series.

1. If \(m\) even, compute trend-cycle component using 2x\(m\)-MA to obtain \(\ols{T}_t\).
   • If \(m\) odd, computer trend-cycle component using \(m\)-MA.
2. Calculate detrended data series \(y_t - \ols{T}_t\).
3. Estimate seasonal component by averaging detrended values for each period.
   • Monthly: Average for all January, February, , December observations in sample.
   • Adjust to ensure seasonal components sum to zero. Yields \(\ols{S}_t\).
4. Remainder \(\ols{E}_t\) calculated using \(\ols{S}_t\) and \(\ols{T}_t\): \[ \ols{E}_t = y_t - \ols{T}_t - \ols{S}_t. \]

Recap - Lecture 6

• 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, \(e_t=y_{t} - (\ell_{t-1} + b_{t-1}+s_{t-m})=y_{t} - \pred{y}{t}{t-1}\).

Sample Exam - Section A

1. Detail the steps involved in constructing a forecast using an ARIMA(1,1,1) model.

2. Compare forecast errors from using a first order autoregressive model, and a random walk model. Under which circumstances would you choose one over the other?

3. Show that forecast errors are non-decreasing in the forecast horizon.

4. Describe the Augmented Dickey-Fuller test. How important would you describe its role in the forecasting process?

5. Describe the Simple Exponential Smoothing (SES) method. What is the forecast equation?

6. Describe the Classical Time Series Decomposition approach. Is it recommended?

Sample Exam - Section B

1. List the sources of forecast uncertainty; which would you suggest is most important, and why?

2. Compare and contrast the stepwise and general-to-specific model selection strategies.

3. Why is it not necessarily the case that the best possible in-sample model provides the best out-of-sample forecasts?

4. `Judgemental forecasts are always and everywhere a bad thing’. Evaluate this statement.

5. `The best forecast model is the most parsimonious one’. Evaluate this statement.

6. Describe impulse saturation; what benefits and costs do you see from the procedure?