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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
70
80
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
-
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
-
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
-
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
-
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 2 by 4 Moving Average
MAt = (2*xt + 2*xt-1 + xt-2)/5
A Store Retail Series
-
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
-
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 2005Q1 2005Q2 2005Q3 2005Q4
Raw Data 2 by 4 Moving Average
Revisions
Retail - Predominantly Non-food Store
-
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 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
0
50
100
150
200
250
300
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)
0
50
100
150
200
250
300
350
400
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
0
50
100
150
200
250
300
350
400
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?
0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
160.0
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
Original Series
Outlier
Outlier
Original Series = Trend-cycle + Seasonal Component + Irregular Component
0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
160.0
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
Original Series Prior-Adjusted Original Series
Automatically identified as an ‘unusual’ value and effect scaled
0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
160.0
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
Prior-Adjusted Original Series 2x4 term moving average (Initial Estimate of Trend)
Prior Adjusted Series – Initial Estimate of trend = Seasonal + Irregular Component
-50
-40
-30
-20
-10
0
10
20
30
40
50
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
0
5
10
15
20
25
30
35
40
45
-25
-20
-15
-10
-5
0
-40
-35
-30
-25
-20
-15
-10
-5
0
0
5
10
15
20
25
Decomposing Seasonal-Irregular Components into individual quarters…
-50
-40
-30
-20
-10
0
10
20
30
40
50
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
0
5
10
15
20
25
30
35
40
45
-25
-20
-15
-10
-5
0
-40
-35
-30
-25
-20
-15
-10
-5
0
0
5
10
15
20
25
-50
-40
-30
-20
-10
0
10
20
30
40
50
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)
0
5
10
15
20
25
30
35
40
45
-25
-20
-15
-10
-5
0
-40
-35
-30
-25
-20
-15
-10
-5
0
0
5
10
15
20
25
Combining Seasonal Components for the individual quarters…
0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
160.0
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
Original Series
-40
-30
-20
-10
0
10
20
30
40
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)
0.0
20.0
40.0
60.0
80.0
100.0
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
0
20
40
60
80
100
120
140
160
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.
0
20
40
60
80
100
120
140
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
0
20
40
60
80
100
120
140
160
180
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?
0
20
40
60
80
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?
0
20
40
60
80
100
120
140
160
180
200
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