Some examples of the comparison Direct vs. Indirect adjustment for the Mexican
economic indicators
Blanca Rosa Sainz López
National Institute of Statistics and Geography (INEGI)
Introduction
The users´ questioning on why seasonally adjusted series calculated by the direct method
and the indirect method can show contrary signs about the magnitude of growth in certain
periods, led to the analysis of the results of both methods to find the reason for these
differences.
a) Total Investment: Discrepancy between the aggregate (direct adjustment) and the
sum of its components (indirect adjustment).
Users´ questioning: in recent quarters it has been observed that the behavior of the
directly seasonally adjusted total investment has differed its implicit evolution from the
addition of the seasonally adjusted components. In the fourth quarter of 2010 and the first
of 2011 the total investment seasonally adjusted directly showed a slowdown, while the
total investment obtained by the sum of its components has shown an increase in its
quarterly growth rate (see figure 1).
Figure 1
Total Investment: Seasonally adjusted series
a) Annual percentage variation
b) Quarterly percentage variation
15
8
6
10
4
5
2
0
0
-2
-5
Direct
-10
Indirect
Direct
-4
Indirect
-6
-8
-15
I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV I
2004
2005
2006
2007
2008
2009
2010
-10
I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV I
2003
2011
2004
2005
2006
2007
2008
2009
2010 2011
Response: when you use the indirect method to the aggregate you are adding the
calendar and seasonal effects of the components, and it is not necessarily such way. It is
convenient to analyze each case; there are cases where for one or more of the
components the calendar effects are not significant and for other components they are
significant. When you add the components you assume that the adjustment in the total is
in the same way as in its components, but it can be that in the aggregate the calendar
effect is more significant than it is in the components. That’s why we consider it is
important to analyze the aggregate independently of the effects of its components and
define its own model. It may also be happen at the time of adding several components that
the effects are cancelled and are not significant in the aggregate. For example, there is an
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outlier in one of the components; if we calculate the adjustment of the aggregate as a sum
of the components adjustments, the outlier of the component will appear in the aggregate
decomposition. Probably if we adjust directly the aggregate, maybe that outlier is not
significant or less significant or can happen there are outliers in the aggregate that are not
in the components.
If we analyze the case of Mexican Total Investment which has two components, the Public
and Private Investment, in the Public Investment the Easter effect is not significant and in
the Private Investment it is significant with a parameter value of 3 days. For 2010 when
Easter Sunday was on April 4th, for the Private Investment the adjustment for the Easter
effect was equal to the one in 2009 and 2011. As a consequence the Easter adjustment
for the aggregate obtained by the sum of its components in 2010 was equal, to the one
2009. However, if we calculate the adjustment directly for the aggregate, we have that the
Easter effect is significant with a value of 8 days, greater than the value of its components,
which could indicate that if the effect is not significant for a component that doesn’t mean
there isn’t any effect in the component. With the value of the Easter effect for the
aggregate the adjustment for 2010 and 2011 is in the following way:
Total Investment
(Annual percentage variation)
Period
2010
2011
Original Serie
I
II
III
IV
I
(-)2.73
1.85
3.81
6.40
7.74
Original Serie adjusted for
Easter effect, direct method
(-)1.12
0.20
3.81
6.40
5.99
Original Serie adjusted for
Easter effect, indirect method
(-)2.73
1.85
3.81
6.40
7.69
As we can appreciate the adjustment for the Easter effect is very different with the direct
method than with the indirect one, and even more in the first and second quarter of 2010
and the first quarter of 2011. So if we use the indirect adjustments this would led to
different conclusions than if we use the direct results.
Below are the figures 2 and 3 of the Original series adjusted by the Easter effect with the
direct and indirect method and the corresponding factors.
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Figure 2
Total Investment: Original series adjusted by Easter effect
Annual percentage variation
15
10
5
0
Direct
Indirect
Differences
-5
-10
I
II
III
IV
I
2004
II
III
IV
I
II
2005
III
IV
I
2006
II
III
IV
I
2007
II
III
IV
I
2008
II
III
IV
I
II
2009
III
IV
2010
I
-15
2011
Figure 3
Total Investment: Easter factors
103
102
101
100
99
Direct
Indirect
98
I
II
III IV
2003
I
II
III IV
2004
I
II
III IV
2005
I
II
III IV
2006
I
II
III IV
2007
-3-
I
II
III IV
2008
I
II
III IV
2009
I
II
III IV
2010
I
2011
97
Additionally, there aren’t significant outliers in the components in the study period, but in
the total there is an outlier in LS2009.1 (change of level in the first quarter of 2009).
It is the figure 4 of the seasonal factors with both methods, where you can see that the
factor is substantially changed over time with the indirect method and in recent years
seasonality becomes more marked, while the factors calculated with the direct method
have remained approximately with the same pattern only slight changes. Moreover, it is
also reflected in an irregular factor with more volatility obtained with the indirect method
(see figure 5).
Figure 4
Total Investment: Seasonal factors
108
106
104
102
100
98
96
Direct
Indirect
I
II
III IV
2003
I
II
III IV
2004
I
II
III IV
2005
I
II
III IV
2006
I
II
III IV
2007
-4-
I
94
II
III IV
2008
I
II
III IV
2009
I
II
III IV
2010
I
2011
92
Figure 5
Total Investment: Irregular Factor
102
101
100
99
98
Direct
Indirect
I
II
III IV
2003
I
II
III IV
2004
I
II
III IV
2005
I
II
III IV
2006
I
II
III IV
2007
I
II
97
III IV
2008
I
II
III IV
2009
I
II
III IV
2010
I
96
2011
We also compared the diagnostics of the direct and indirect adjustments of the spectral
figures to see if there was residual seasonality or trading day in the results. In both cases
there isn’t any residual seasonality nor trading day. We couldn’t calculate the sliding
spans and the revision history diagnostics because the series is not long enough.
b) Demand for goods and services: Discrepancy between the aggregate (direct
adjustment) and the sum of its components (indirect adjustment).
Users´ questioning: In the last three quarters (3rd and 4th quarter 2010 and 1st quarter
2011) the direct seasonally adjusted aggregate demand for goods and services and the
one obtained by adding their components adjusted have showed contrary signs about the
magnitude of the growth (see figure 6).
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Figure 6
Demand for goods and services: Seasonally adjusted series
a) Annual percentage variation
b) Quarterly percentage variation
15
8
10
6
4
5
2
0
0
-2
-5
Direct
Indirect
Direct
Indirect
-10
-4
-6
-15
-8
-20
I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV I
2004
2005
2006
2007
2008
2009
2010
-10
I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV I
2011
2003
2004
2005
2006
2007
2008
2009
2010 2011
Concerning the case of aggregate demand, if you analyze the components we have that
only in the Gross Fixed Capital the Easter effect is significant, with a value of 8 days,
which added to obtain the indirect method would indicate, since this component represents
only 14.9%, that the Easter effect in the demand is less or not significant. But the Easter
effect is more significant when analyzing the total directly, with a value of 10 days. This
could indicate that if the effect is not significant for the other components that does not
mean there isn´t any effect in those components. With the value of that parameter for the
Easter effect the adjustment in 2010 and 2011 is as follows:
Demand for goods and services
(Annual percentage variation)
Period
2010
2011
Original Serie
I
II
III
IV
I
8.23
13.17
9.42
7.36
6.16
Original Serie adjusted for
Easter effect, direct method
8.84
12.54
9.42
7.36
5.57
Original Serie adjusted for
Easter effect, indirect method
8.53
12.87
9.42
7.36
5.88
As you can appreciate in the table, there are differences in the adjustment of the Easter
effect between the direct and indirect method, up to 0.3 percentage points in the first and
second quarter of 2010 and in the first quarter of 2011, causing differences in the
seasonally adjusted series also.
Below are figures 7 and 8 of the original series adjusted for the Easter effect with the direct
and indirect method and the corresponding factors.
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Figure 7
Demand for goods and services: Original series adjusted by Easter effect
Annual percentage variation
15
10
5
0
-5
Direct
Indirect
Differences
I
II
III
IV
I
2004
II
III
IV
I
II
2005
III
IV
I
2006
II
III
IV
I
2007
II
-10
III
IV
I
II
2008
III
IV
I
II
2009
III
IV
-15
I
2010
2011
Figure 8
Demand for goods and services: Easter factor
100.6
100.4
100.2
100
99.8
Direct
99.6
Indirect
I
II
III IV
2003
I
II
III IV
2004
I
II
III IV
2005
I
II
III IV
2006
I
II
III IV
2007
I
II
III IV
2008
I
II
III IV
2009
I
II
III IV
2010
I
99.4
2011
Additionally Total Consumption has two significant outliers LS2009.1 (change of level in
the first quarter of 2009) and AO2009.2 (additive in the second quarter of 2009), Exports
and Gross Fixed Capital Formation have an outlier LS2009.1 and Change in inventories
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doesn´t have significant outliers, while for the aggregate in the addition of the outliers
mentioned for the components also has a LS2008.4 (change of level in the fourth quarter
of 2008).
Seasonal factors are shown in figure 9 using both methods. You can see that factors
calculated with the indirect method differ in the first and second quarters of each year from
the factor obtained by the direct method and this is due to the difference in the Easter
effect adjustment. This is also reflected in an irregular factor more volatile with the indirect
method.
Figure 9
Demand for goods and services: Seasonal factor
104
102
100
98
96
Direct
Indirect
I
II
III IV
2003
I
II
III IV
2004
I
II
III IV
2005
I
II
III IV
2006
I
II
III IV
2007
-8-
I
II
III IV
2008
I
II
III IV
2009
I
II
III IV
2010
I
2011
94
Figure 10
Demand for goods and services: Irregular Factor
102
101
100
99
98
Direct
Indirect
97
I
II
III IV
2003
I
II
III IV
2004
I
II
III IV
2005
I
II
III IV
2006
I
II
III IV
2007
I
II
III IV
2008
I
II
III IV
2009
I
II
III IV
2010
I
96
2011
Finally, we have to take into account that if we want to preserve the additivity between the
aggregate and the seasonally adjusted components, we may not have consistent results
and we may even get opposite conclusions to the ones we obtain analyzing the aggregate
directly.
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