Understanding Economic Indicators

Scottish GDP as a case study in Indexation

and Time Series Methods

What is GDP

• “Size” of economic output

• Overall Value (Annual)– Blue book, IO tables

• Short Term Trend Indicators– More frequent (quarterly) – (ONS do three estimates that successively

incorporate three types of data.)

GVA concept

• Turning grapes into wine generates GVA

• Opening the bottle for you in a nice environment generates GVA

• Burning coal and transmitting power along lines generates GVA

• It’s a measure of “economic activity”

• GDP is the sum of all the GVA in the economy

Main Techniques 1

• Sample Surveys– Mainly collected in cash values at current

prices– Aggregated using standard techniques

• Ratio estimation

• Deflation– To convert current price to volume (constant

price)

Main Techniques 2

• Index numbers – To generate series that are comparable

between different industries – there are no “units”

– To weight together disparate measures to provide a whole economy picture

• Time Series methods– To allow publication of comparable quarterly

figures for industries that are not comparable quarter by quarter

Simple Volume Indexation

• Imagine the price of your favourite commodity.

100.00=100x(£2.41/£2.41)£2.412000

134.4=100x(£3.24/£2.41)£3.242009

129.9=100x(£3.13/£2.41)£3.132008

124.5=100x(£3.00/£2.41)£3.002007

119.5=100x(£2.88/£2.41)£2.882006

115.8=100x(£2.79/£2.41)£2.792005

112.0=100x(£2.70/£2.41)£2.702004

108.7=100x(£2.62/£2.41)£2.622003

105.8=100x(£2.55/£2.41)£2.552002

102.9=100x(£2.48/£2.41)£2.482001

IndexFormulaPriceYear

Man cannot live on beer alone

Obvious Strategy• Is to track the rate of change of a weighted

sum of the quantities of interest.

• E.g. price of an evenings entertainment:

2 x + 1 x + 2/77 x

But what about appropriate weights?

General price indices use a “basket” of goods

“Currently, around 120,000 separate price quotations areused every month in compiling the indices, covering some 650 representativeconsumer goods and services”

ONS CPI Note

http://www.statistics.gov.uk/articles/nojournal/CPI-Basket-of-Goods-2009.pdf

Price vs Volume

• A volume index:– Aims to track change in quantities– Market price is an often used weight

• A price index:– Aims to track price

• i.e. inflation

– Typically based on a basket of “output”

Base Weighted Volume IndexIndex of weighted volume

Weights come from base year

Also known as Laspeyres

Current Weighted Volume Index

Index of volume

Weights come from current year

Also known as Paasche

Examples of Volume Index Calculations

Year

price (£)

Number purchased per annum

Amount spend on

CDs MP3s CDs MP3s CDs MP3s

2004 12 8 9 3 108 24

2005 13 6 6 9 78 54

2006 14 5 4 14 56 70

Exercise: Calculate Base and Current Weighted Volume Indices for these data.

Comparison

Number purchased per annum

Laspeyres volume index

Paasche volume index

CDs MP3s

9 3 100.0 100.0

6 9 109.1 97.8

4 14 121.2 89.4

Economics

• People buy more things that get cheaper– And less things that get more expensive

• Known as the “Substitution effect”• Laysperes index ignores this

– Artificially high weight to fast growing/falling price commodities

• Paasche over weights its influence– Artificially low weight to fast growing/falling

priced commodities

More Economics

• Laysperes generally considered an upper bound for growth

• Paasche generally considered a lower bound for growth

• “True Growth” is somewhere in between

Geometric Mean

=

Fisher “ideal” index

Comparison

Number purchased per annum

LaspeyresVolumeindex

PaascheVolumeindex

FisherVolume

Index

CDs MP3s

9 3 100.0 100.0 100.0

6 9 109.1 97.8 103.3

4 14 121.2 89.4 104.1

Chainlinking

• Fisher is indeed an “ideal” measure

• But to compute it, you need price and volume data with the same resolution you want to publish

• In practice we use “chainlinking” on Laspeyres type indices

Chainlinking isBeyond the scope of this seminar

60

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90

100

110

120

130

140

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000

Old weights

New weights

Chained series

1995 = link year

But it looks a bit like this.

90

95

100

105

110

115

120

1998 1999 2000 2001 2002 2003 2004 2005

1998 weights

1999 weights

2000 weights

2001 weights

2002 weights

2003 weights

Chainlinked

1999 volumes expressed in 1998 weights

The link factor for 1999 is equal to the ratio of the 2 estimates, in this case, approximately 2.1%. This is applied to all subsequent years also. The process is then repeated for the next year and so on.

1999 volumes expressed in 1999 weights

Price Index Calculations

• Handout.

Year BPI CTPI

2000 100.0 100.0

2001 102.9 103.2

2002 105.8 104.7

2003 108.7 98.9

2004 112.0 92.6

2005 115.0 90.8

2006 119.5 90.8

2007 124.5 89.3

2008 139.9 90.4

2009 134.4 89.9

Answers

Beer 2000 – 2004: 12.0%

Cheese Toasty 2000-2004: -7.4%

Beer 2004-2009:

Cheese Toasty 04-09:

Average Rate: Well,

i.e. 3.9%

Time Series Analysis

Typical input series

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100

300

500

700

1 2 3 4 5 6 7 8 910 11 12 13

1 2 3 4 5 6 7 8 910 11 12 13

1 2 3 4 5 6 7 8 910 11 12 13

1 2 3 4 5 6 7 8 910 11 12 13

1 2 3 4 5 6 7 8 910 11 12 13

1 2 3 4 5 6 7 8 910 11 12 13

1 2 3 4 5 6 7 8 910 11 12 13

1 2 3 4 5 6 7 8 910 11 12 13

1 2 3 4 5 6 7 8 910 11 12 13

2000 2001 2002 2003 2004 2005 2006 2007 2008

Smoothing and Moving Averages

• Some data sources are highly volatile and/or seasonal;

• We may not be interested in these short-term fluctuations;

• Smoothing reduces these fluctuations and makes it easier to identify long-term trends;

A Store Retail Series

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500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

2002

Q1

2002

Q2

2002

Q3

2002

Q4

2003

Q1

2003

Q2

2003

Q3

2003

Q4

2004

Q1

2004

Q2

2004

Q3

2004

Q4

MAt = average(xt-0.5,xt-1.5,xt+0.5,xt+1.5)

A Store Retail Series

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500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

2002

Q1

2002

Q2

2002

Q3

2002

Q4

2003

Q1

2003

Q2

2003

Q3

2003

Q4

2004

Q1

2004

Q2

2004

Q3

2004

Q4

Raw Data 4-Point Moving Average

MAt = (xt-2 + 2*(xt-1 + xt + xt+1) + xt+2)/8

A Store Retail Series

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500

1,000

1,500

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2,500

3,000

3,500

4,000

2002

Q1

2002

Q2

2002

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2002

Q4

2003

Q1

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Q2

2003

Q3

2003

Q4

2004

Q1

2004

Q2

2004

Q3

2004

Q4

Raw Data 4-Point Moving Average 2 by 4 Moving Average

MAt = (2*xt + 2*xt-1 + xt-2)/5

A Store Retail Series

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500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

2002Q1 2002Q2 2002Q3 2002Q4 2003Q1 2003Q2 2003Q3 2003Q4 2004Q1 2004Q2 2004Q3 2004Q4

Raw Data 2 by 4 Moving Average

A Store Retail Series

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500

1,000

1,500

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2,500

3,000

3,500

4,000

2002Q1 2002Q2 2002Q3 2002Q4 2003Q1 2003Q2 2003Q3 2003Q4 2004Q1 2004Q2 2004Q3 2004Q4 2005Q1 2005Q2 2005Q3 2005Q4

Raw Data 2 by 4 Moving Average

Revisions

Retail - Predominantly Non-food Store

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2,500

3,000

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4,000

2002Q1 2002Q2 2002Q3 2002Q4 2003Q1 2003Q2 2003Q3 2003Q4 2004Q1 2004Q2 2004Q3 2004Q4 2005Q1 2005Q2 2005Q3 2005Q4

Raw Data Previous 2 by 4 MA Revised 2 by 4 MA

Exponential Smoothing

• Applies exponentially decreasing weights to observations as they get older;

• Alpha is essentially the proportion of the most recent data point that is allowed through;

• Fresh data doesn’t cause revisions;• Movements are lagged compared with moving

averages.

00 xs 11 ttt sxs

Comparison of MA with Exponential Smoothing for Volatile Soure Data

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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Source Data 2*4 MA Exponentially Smoothed

Choice of Alpha

• Alpha can be between 0 and 1;

• Generally this is a judgement call;

• but if it looks like we need a small alpha (below 0.7) then…

• Optimal value is one that minimises the Mean Squared Error:– i.e. the sum of 21 tt xs

Summary

• Moving Average– Approximates the trend line;– Can remove seasonality;– Has difficulty at end points;– Prone to revisions.

• Exponential Smoothing– Lags movements in the data;– No Revisions.

Decomposing a time series

• A time series can be decomposed into:– The trend cycle component (medium and long term

growth and cycles in the series)– The seasonal component (effects that are largely

stable in timing, size and direction from year to year)– The irregular component (made up of anything

remaining e.g. short term fluctuations, sampling and non-sampling errors, unpredictable effects due to one-off events such as strikes or disasters

Additive and Multiplicative series

• Additive series – seasonal effects are constant

• Multiplicative series – seasonal effects grow as series grows (and vice versa)

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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

2000 2001 2002 2003 2004 2005 2006 2007 2008

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450

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

2000 2001 2002 2003 2004 2005 2006 2007 2008

Time Series Models

• The additive model is:Time Series = Trend Cycle + Seasonal Component + Irregular Component

Y = C + S + I

• The multiplicative model is:Time Series = Trend Cycle x Seasonal Component x Irregular Component

Y = C x S x I

X-12-ARIMA

• Developed by the US Census Bureau.

• Estimating and removing regular seasonal patterns from time series data.

• This leaves the long term trend and short term irregular movements

• Worked example – Mains Gas supply (a component series of GDP) which is an additive series.

Question

• What was the quarterly change in Mains Gas Supply in the second quarter of 2009?

• In 2009Q1 the index was 121 and in 2009Q2 it was 79 giving a 35 per cent decrease.

• Is this a sensible answer?

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20.0

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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Original Series

Outlier

Outlier

Original Series = Trend-cycle + Seasonal Component + Irregular Component

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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Original Series Prior-Adjusted Original Series

Automatically identified as an ‘unusual’ value and effect scaled

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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Prior-Adjusted Original Series 2x4 term moving average (Initial Estimate of Trend)

Prior Adjusted Series – Initial Estimate of trend = Seasonal + Irregular Component

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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Q1 Q2 Q3 Q4 Seasonal-Irregular Component

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Decomposing Seasonal-Irregular Components into individual quarters…

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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Q1 Q2 Q3 Q4 Seasonal-Irregular Component

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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Q1 Q2 Q3 Q4 Seasonal Component (Initial Estimate)

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Combining Seasonal Components for the individual quarters…

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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Original Series

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0

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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Seasonal Component (Initial Estimate)

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120.0

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Seasonally Adjusted Series (Initial Estimate)

‘Outliers’ put back in

89 23- = 66

89

23

66

X-12-ARIMA actual process

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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Seasonally Adjusted Series (X-12-ARIMA) Trend Original Series

Question

• What was the quarterly change in Energy Use in the second quarter of 2009?

• In 2009Q1 the index was 121 and in 2009Q2 it was 79 giving a 35 per cent decrease.

• In 2009Q1 the seasonally adjusted index was 89 and in 2009Q2 it was 95 giving a 7 per cent increase.

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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

2003 2004 2005 2006 2007 2008

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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

2003 2004 2005 2006 2007 2008

Level Shift

•A step change

•In GDP could be caused by companies opening/closing

Seasonal Break

•A change in the seasonal pattern

•In GDP could be caused by administrative changes

Exercise

• Discuss the charts on the handouts indentifying outliers, level shifts and seasonal breaks.

Index of sales of motor vehicles, motorcycles and parts

Index of sales biscuits, preserved pastry & cakes

1. Index of sales of motor vehicles, motorcycles and parts

Seasonal break

1999Q1 Additive Outlier

2001 Q4

Level Shift 2008 Q3?

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60

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100

120

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

2. Index of sales biscuits, preserved pastry & cakes

Seasonal break

1998Q3

Seasonal break

2002Q3

Additive Outlier

1995Q2

Additive Outlier

2009Q1?

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1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Revisions

• New data always gives you new information

• Which will tell you more about your modelling assumptions

• Revisions are good