Temperature correction of
energy consumption time
series
Sumit Rahman, Methodology Advisory
Service, Office for National Statistics
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Gigawatt hours
Final consumption of energy – natural
gas
• Energy consumption depends strongly on air
temperature – so it is seasonal
Gas consumption
120000
100000
80000
60000
40000
20000
0
Average monthly temperatures
• But temperatures do not exhibit perfect
seasonality
deviations in temperature from long-term monthly averages
+4.0
+2.0
+1.0
+0.0
-1.0
-2.0
-3.0
Jan-10
Jan-09
Jan-08
Jan-07
Jan-06
Jan-05
Jan-04
Jan-03
Jan-02
Jan-01
Jan-00
Jan-99
Jan-98
Jan-97
Jan-96
Jan-95
Jan-94
Jan-93
Jan-92
-4.0
Jan-91
deviation (degrees Celsius)
+3.0
Seasonal adjustment in X12-ARIMA
•
•
•
•
Y=C+S+I
Series = trend + seasonal + irregular
Use moving averages to estimate trend
Then use moving averages on the S + I for
each month separately to estimate S for each
month
• Repeat two more times to settle on estimates
for C and S; I is what remains
Seasonal adjustment in X12-ARIMA
• Y=C×S×I
• Common for economic series to be modelled
using the multiplicative decomposition, so
seasonal effects are factors (e.g. “in January
the seasonal effect is to add 15% to the trend
value, rather than to add £3.2 million”)
• logY = logC + logS + logI
Temperature correction – coal
• In April 2009 the temperature deviation was
1.8°(celsius)
• The coal correction factor is 2.1% per degree
• So we correct the April 2009 consumption
figure by 1.8 × 2.1 = 3.7%
• That is, we increase the consumption by
3.7%, because consumption was understated
during a warmer than average April
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unadjusted
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thousands of tonnes
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Current method – its effect
Coal consumption
seasonally adjusted
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7000
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3000
2000
1000
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thousands of tonnes
Coal consumption
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Current method – its effect
seasonally adjusted
8000
temperature corrected and seasonally
adjusted
7000
6000
5000
4000
3000
2000
1000
0
Regression in X12-ARIMA
• Use xit as explanatory variables (temperature
deviation in month t, which is an i-month)
• 12 variables required
• In any given month, 11 will be zero and the
twelfth equal to the temperature deviation
Regression in X12-ARIMA
• Why won’t the following work?
12
log Yt    i xit  log Ct St I t
i 1
Regression in X12-ARIMA
• So we use this:
12
log Yt    i xit  ARIMA
i 1
Regression in X12-ARIMA
• More formally, in a common notation for
ARIMA time series work:
12
 ( B)( B )(1  B) (1  B ) (log Yt    i xit ) 
12
d
12 D
i 1
 ( B)( B ) t
12
• εt is ‘white noise’: uncorrelated errors with zero
mean and identical variances
Regression in X12-ARIMA
• An iterative generalised least squares
algorithm fits the model using exact maximum
likelihood
• By fitting an ARIMA model the software can
fore- and backcast, and we can fit our linear
regression and produce (asymptotic)
standard errors
Coal – estimated coefficients
coefficient (percentage)
20
15
10
5
0
-5
-10
-15
Jan Feb Mar Apr May Jun
Jul Aug Sep Oct Nov Dec
Interpreting the coefficients
• For January the coefficient is -0.044
• The corrected value for X12 is log Yt 
• The temperature correction is
e
12
 x
i 1
  i xit
• If the temperature deviation in a January is
0.5°, the correction is e
 ( 0.0440.5)
 1.022
• We adjust the raw temperature up by 2.2%
• Note the signs!
i it
Interpreting the coefficients
• If  i xit is small then
e
  i xit
 1   i xit
• So a negative coefficient is interpretable as a
temperature correction factor as currently
used by DECC
• Remember: a positive deviation leads to an
upwards adjustment
Coal – estimated coefficients
coefficient (percentage)
20
15
10
5
0
-5
-10
-15
Jan Feb Mar Apr May Jun
Jul Aug Sep Oct Nov Dec
Gas – estimated coefficients
coefficient (percentage)
12
10
8
6
4
2
0
Jan Feb Mar
Apr
May Jun
Jul
Aug Sep Oct Nov Dec
Smoothing the coefficients for coal
Coefficients for coal
20
Coefficients for gas
15
12
10
coefficient (%)
coefficient (%)
10
5
0
8
6
4
2
-5
0
Jan
-10
-15
Jan Feb Mar Apr May Jun Jul
Aug Sep Oct Nov Dec
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
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thousands of tonnes
Comparing seasonal adjustments
Coal consumption, seasonally adjusted
8000
proposed new factors
7500
7000
current method of temperature correction
6500
6000
5500
5000
4500
4000
3500
3000
Heating degree days
• The difference between the maximum
temperature in a day and some target
temperature
• If the temperature in one day is above the
target then the degree day measure is zero
for that day
• The choice of target temperature is important
Smoothing the coefficients, heating
degree days model (Eurostat measure)
Coefficients for gas
0.20
0.15
correction factor, per unit
deviation from the average
degree day
correction factor, per unit deviation from the
average degree day
Coefficients for coal
0.10
0.05
0.00
-0.05
-0.10
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
Jan Feb Mar Apr May Jun
-0.15
-0.20
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Jul Aug Sep Oct Nov Dec
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8000
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thousands of tonnes
Effect on coal seasonal adjustment
Coal consumption, seasonally adjusted
using Eurostat degree days
7500
7000
current method of temperature correction
6500
6000
5500
5000
4500
4000
3500
3000
The difference temperature correction
can make!
Million tonnes of oil equivalent
Primary
energy
Temperature
consumption Unadjusted
2009
211.1
adjusted
212.6
2010
217.3
211.3
Annual change +2.9%
-0.6%