xlvii reunión anual
TRANSCRIPT
XLVII Reunión AnualNoviembre de 2012
ISSN 1852-0022ISBN 978-987-28590-0-8
FORECASTING ARGENTINE CONSUMER EXPENDITURE DURING BREAKS: CAN COMMODITY PRICES HELP?
Ahumada HildegartGaregnani María Loren
ANALES | ASOCIACION ARGENTINA DE ECONOMIA POLITICA
1
FORECASTING ARGENTINE CONSUMER EXPENDITURE DURING BREAKS:
CAN COMMODITY PRICES HELP?
HILDEGART A. AHUMADA† AND MARIA LORENA GAREGNANI‡
† Di Tella University, Miñones 2159 (1428), Buenos Aires, Argentina E-mail: [email protected]
‡Central Bank of Argentina, Reconquista 266 (1003). Buenos Aires, Argentina*. UNLP-UdeSA. E-mail: [email protected]
August 2012
Abstract
We evaluate the forecasting performance of an Equilibrium-Correction Model (EqC) based on “the solved-out consumption function” approach. EqCs are usually estimated when applying this approach but EqCs are also prone to forecasting failures because of equilibrium-mean changes. We focus on examining the effects of breaks during the forecasting period, for which several methods are explored. We found that forecasts of consumption growth improve when we recursively update estimates of the EqCs. Moreover, including a broken trend and the soybean price, each as an “outside variable”, only for the forecasting period to recursively adjust forecasts, we achieved much better accuracy.
Keywords: Forecasting – Equilibrium-Correction – Robust devices – External break – Outside variable.
JEL: C53, E21
* The views expressed herein are solely our own and should not be interpreted as those of the Central Bank of Argentina.
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1. Introduction
A key issue in economic forecasts is to understand the sources and effects of breaks. To
this end, a vast literature has focussed on forecasting in the face of structural breaks.
Specifically, since location shifts have been found to be the main responsible for poor
economic forecasts, different ways of improving econometric models and new methods -
including naïve mechanisms - have been suggested to cope with them (Clements and
Hendry, 2006, 1999,1998).
Under macroeconomic instability even modelling aggregate consumer behaviour can be
difficult, given that parameter changes have usually been considered as a main obstacle
for “the solved-out consumption function” approach (Muellbauer and Lattimore, 1995).
Cointegration analysis and Error Correction models are often employed for consumer
expenditure, as they are currently applied by time series modellers.
However, equilibrium-mean changes are often detected in “Error Correction” models,
which have been consequently renamed “Equilibrium Correction” models (EqC) since such
an equilibrium may actually be wrong when the true one has shifted (Clements and
Hendry, 1999).
In a previous study (Ahumada and Garegnani, 2011) we examined the presence of breaks
in EqC models of Argentine consumption and found that the temporary breaks we
detected may be associated with liquidity constraints after the government defaulted and
abandoned a convertibility regime. Although the deterministic components of consumption
and income varied across the sub-samples, a long run relation of homogeneity lasts for the
whole sample, suggesting the co-breaking nature of the series.1 Apart from disposable
income, only the real exchange rate was found to have a significant effect in the short run.
Although the model obtained was exhaustively evaluated its forecasting performance was
not assessed.
It is worth noting, however, that “breaks… will generally be more comprehensible
retrospectively than at the time of occurrence. In practice there will always be events that
are essentially unknowable ex ante, which will then cause breaks in the parameters of a
forecasting model” (Clements and Hendry, 2011, p272).
1 The data used in this paper have a different base-year (1993 instead of 1986) and the sample is extended by one year.
3
For this reason an interesting strategy recently suggested by Castle, Fawcett and Hendry
(2011), is forecasting during a new break. They consider two approaches depending on
the type of break: internal or external. An “external” break occurs when the forecasting
model stays constant and an “internal” break occurs when the model itself shifts. In the
first case there are changes in collinearity between explanatory variables (included and
omitted) that increased forecast error uncertainty. During an internal break estimation
uncertainty can be large due to the shift the model experiences.
Including an outside variable (when enters lagged) only for the forecast period (Castle et
al., 2011) and updating a learning (e.g. exponential) function for the evolving breaks
(Castle et al., 2010) can help forecasting during external and internal breaks, respectively.
They can be useful to track breaks soon after their occurrence but they should be
compared to robust devices - such as differencing devices (DD) or intercept correction (IC)
- which, at least from what can be seen from analysis and simulation, can perform as well
as the estimated models.
Argentina’s consumer expenditure during the 2000s provides a rich experience for both
modelling in sample breaks and evaluating alternative approaches for forecasting during
breaks. This paper is focussed on these issues. The next section describes the
consumption-income relationship. Section 3 presents the econometric models. Section 4
evaluates forecasts during breaks. Section 5 concludes.
2. Describing the consumption-income relationship
This section describes the main relationship we studied using quarterly data over 1980Q1-
2011Q4 (see Appendix A1 for data definitions and sources). Private consumer expenditure
and disposable income (matched by mean and ranges) are plotted in Figure 1.
Argentine consumer expenditure has varied greatly since 1980. As we can often observe
in Figure 1, not only the magnitude of its rate of change but also its sign was uncertain. In
brief, the eighties were characterised by low activity and consumption levels along with
high-inflation and even outbreaks of hyper-inflation. The nineties, by contrast, were a
period of income and consumption expansion after price stability was obtained through a
convertibility regime (1991-2001), although unemployment and external indebtedness also
4
increased. The relative tranquillity of this latter period was temporarily interrupted in 1995
due to the regional consequences of the Mexican devaluation (known as “Tequila effect”).
Although the convertibility regime withstood this external shock, it was an initial sign of the
vulnerability of this monetary regime. It collapsed in early 2002. Private consumption
abruptly fell after the default and devaluation took place. Besides, an asymmetric
“pesification” of deposits and loans left banks insolvent and the economy without financing.
Recovery soon started and consumption and income reached never before known levels
after 2005. This was also driven since 2004 by the strong growth that the economy
experienced after the prolonged recession that it had suffered for several years. Demand
stimuli policy and unprecedented commodity prices have been indicated as the sources of
the economic growth.
In 2008 there was again economic uncertainty due to both domestic and international
factors. During the first half of the year this uncertainty was associated with a distribution
conflict derived from large increases in the price of exports. During the second part of the
year it was associated with the international crisis. But as no large volumes of new debt
had been acquired since the 2002 crisis, the international financial restrictions did not
affect Argentina. When the economy started to decelerate, the government used an
expansionary fiscal policy to maintain domestic expenditure and by the end of 2009 credit
expansions also stimulated the economic activity. During 2010 and 2011 again, output
grew at high rates as did foreign trade, with both exports and imports reaching record
levels.
For the sample period as a whole, we can observe how similar the pattern shown by
consumption and income is.2 Indeed cointegration analysis (see Ahumada and Garegnani,
2011) indicates that also co-breaking3 between consumption and income seems to exist
for many different regimes of the Argentine economy (see Clements and Hendry, 1999
and Hendry and Massmann, 2007). However results appeared to change from 2005
onwards, when we started our forecasting evaluation.
2Dickey Fuller unit root tests indicate that private consumption and disposable income series are I(1) (integrated of first order). Although the deterministic components (constant and trend) seems to be different for the sub-samples both series have the same for each sub-sample. Similar conclusions can be obtained from a recursive estimation of the t-statistics of the Augmented Dickey Fuller test (ADF), which allows the constant and the trend to change with each new observation. The hypothesis of unit roots cannot be rejected for either consumption or income series (according to critical values for maximum and minimum reported by Banerjee, et. al., 1992). 3Co-breaking consists in the removal of deterministic shifts through the linear combination of the variables.
5
Figure 1. Time series of consumption and income
1980 1985 1990 1995 2000 2005 2010
11.7
11.8
11.9
12
12.1
12.2
12.3
12.4
12.5
12.6 c y
3. Ex post EqC Estimation.
3.1. An initial EqC In this section we present the estimation of EqCs considering the whole sample 1980Q1-
2011Q4, that is, any break would be modelled in-sample. Looking for a forecasting model
of consumer expenditure we update one previously estimated although using a new base
year and an extended sample. Assuming that previous cointegration results hold and
selecting the dynamics4, the following EqC was estimated,
4 We used Autometrics, an automatic algorithm to select variables. Oxmetrics 6.3 (Doornik and Hendry, 2009) was used in the estimations of both, the systems and single equations.
6
Table 1. Estimated equation of the initial EqC
Δct= 0.1786 -0.2169 Δc
t-1 +0.294 Δc
t-4 +0.9484 Δy
t
(HCSE) (0.0769) (0.0567) (0.0869) (0.0331) +0.2351 Δy
t-1 -0.341 Δy
t-4 -0.2041 c
t-1
(0.0584) (0.0726) (0.0744) +0.1839 y
t-1 -0.07008 Cseas8592 -0.0183 CSeasonal_1
(0.0692) (0.0117) (0.0056) -0.0103 CSeasonal_2 -0.03527 I:1988(1) +0.0498 I:1989(1) (0.0046) (0.0068) (0.0074) -0.03543 I:1990(3) +0.03985 I:2011(1) (0.0049) (0.0042) R2=0.972 F(14,106)=261.3 [0.000]** σ=0.012
Residual tests and Regression specification test AR 1-5 test: F(5,101) = 1.6844 [0.1452] ARCH 1-4 test: F(4,113) = 2.0052 [0.0985] Normality test: Chi^2(2) = 4.0519 [0.1319] Hetero test: F(18,98) = 1.1259 [0.3398] Hetero-X test: F(62,54) = 1.0245 [0.4660] RESET23 test: F(2,104) = 1.3484 [0.2641]
The lower part of the Table 1 reports diagnostic statistics for testing residual
autocorrelation (AR), autoregressive conditional heteroscedasticity (ARCH), skewness and
excess kurtosis (Normality), heteroscedasticity (Hetero and Hetero-X, which uses squares
and squares and cross-products of the original regressors and RESET (RESET23 which
uses squares and cubes). See Doornik and Hendry (2009a) for details and references.
Although these diagnostic tests do not reject the null, we found non-constant parameters,
apart from a change in the deterministic seasonal pattern,5 a change in the short-run effect
of income indicates a lower effect. Also, as long run homogeneity was rejected, this
restriction was not imposed in the equilibrium correction term and unrestricted levels of
lagged income and consumption were allowed for. Besides, the effect of changes in the
real exchange rate was no longer significant. An important drawback that can be observed
in the next figure was a tendency to over-predict consumption since 2004,
5 Both deterministic and stochastic seasonality was found.
7
Figure 2. Forecasts using an EqC (3.1)
Static Forecasts Δc
2004 2005 2006 2007 2008 2009 2010 2011 2012
-0.10
-0.05
0.00
0.05
0.10
0.15 Static Forecasts Δc
To understand this behaviour, a closer inspection of the series in Figure 1 shows that
consumer expenditure remained clearly below income after 2005 and until 2009. The
lower rate of growth of consumption relative to income can be modelled by including in the
cointegration space a broken trend for this period, which does not preclude finding
cointegration since it depends only on the stochastic components.6 A similar view can be
obtained from the ratio of consumer expenditure to disposable income along with its level
in Figure 37. After 2005 the ratio falls below previous values but starts growing again after
2008.
6 Two variables can have different deterministic trends but the same stochastic trend (see Juselius, 2006). In previous research, although the trend was initially included in the cointegration space to determine the rank, it turned out to be non-significant indicating that both determinist and stochastic trends are the same for consumption and income. 7 Unobserved components are estimated using STAMP in Oxmetrics 6.3, see Koopman et al. (2007).
8
Figure 3. The ratio of consumer expenditure to disposable income, trend and seasonal
components.
c-y Level
1985 1990 1995 2000 2005 2010
-0.40
-0.35
-0.30
c-y Level
LconsprivdispincN-Seasonal
1985 1990 1995 2000 2005 2010
-0.025
0.000
0.025
0.050LconsprivdispincN-Seasonal
Figure 3 also shows the seasonal component, where the stochastic seasonality can be
perceived as well as the large variation from 1985 to 1992 (just before a different base-
year of the national accounts was adopted).
3.2. An EqC with a broken trend
Because of the observed behaviour, we analysed cointegration including a broken trend
for 2005Q1-2009Q4 in the cointegration space, using the system-based procedure from
Johansen (1988) and Johansen and Juselius (1990). Only one long run relationship
(between consumption and income) was obtained for the whole sample. Critical values for
the trace statistic from the response surface of Johansen, et al. (2000) indicate one
cointegration vector at 1%.The hypotheses of homogeneity and a valid conditional model
9
of consumption expenditure (see Johansen, 1992; Urbain, 1992) are not rejected. The
main results are reported in Table 2.
Table 2. Cointegration Analysis with broken trend
Rank k k=0 k≤ 1log-lik 648.5 659.7λk 0.1692
k=0 k≤ 1λtrace 25.49 ** 3.06
λtraceadj 23.39 ** 2.81Prob 0.002 0.094
Variable SE
c 1 0y -0.95913 0.0189
trend0509 0.002336 0.0009constant u
Variable SE
c -0.54604 0.1835y -0.2787 0.1831
Other unrestricted variables included are: cseas8592, cseasonal and cseasonal_1 and dummies for: 2011Q2; 2009Q3; 1992Q1; 1990Q4; 1990Q3; 1989Q1; 1988Q3; 1984Q4; 1984Q3 and 1982Q2.The hypothesis of long run homogeneity of y is not rejected at 1%and 5% using LR statistic
Eigenvectors β
Adjustment Coeficients α
Null hypothesis
10
Given these cointegration results, a conditional EqC was estimated to model consumption
on income also including the log differences of the real exchange rate. The following
model was obtained,8
Table 3. Estimated equation of the EqC with broken trend
Δct = 0.04921 +0.3876 Δc
t-4 +0.9412 Δy
t -0.1279 Δy
t02on
(HCSE) (0.0093) (0.0825) (0.0329) (0.0349) -0.4426 Δy
t-4 -0.0176 Δrexch
t -0.3062 EqCt-1
(0.0810) (0.0059) (0.0592) -0.06439 Cseas8592 +0.0421 I:1984(3) -0.03617 I:1988(3) (0.0102) (0.0016) (0.0033) +0.05479 I:1989(1) -0.0453 I:1990(3) (0.0074) (0.0034) R2= 0.973 F(11,109)=359 [0.000]** σ=0.0114 Residual tests and Regression specification test AR 1-5 test: F(5,104) = 1.3693 [0.2419] ARCH 1-4 test: F(4,113) = 0.035761 [0.9975] Normality test: Chi^2(2) = 1.6370 [0.4411] Hetero test: F(14,102) = 0.76868 [0.7001] Hetero-X test: F(33,83) = 1.5121 [0.0675] RESET23 test: F(2,107) = 0.18746 [0.8293]
This EqC with the broken trend showed a different short run effect of national disposable
income on private consumption too9 but the rest of the variables showed parameter
constancy as can be observed in Figure 4 (apart from the different deterministic
seasonality, Cseas8592). Since 2002:1 the contemporaneous effect of national disposable
income growth has been lower than before. It changed from 0.94 to 0.81. There is also an
additional effect due to income acceleration taken into account in the negative effect of
0.44 from Δyt-4.
In this case, the change in the real exchange rate has again a significant and negative
effect of approximately 2% on private consumption. It can be interpreted as being derived
from “wealth perception” as discussed in previous studies (see Ahumada and Garegnani,
2011 and Garegnani, 2008). 8 Autometrics helped us to select the variables of this model. The dummies are a subset of those chosen by Impulse saturation (see Doornik and Hendry, 2009). 9 We include Δyt02on = Δyt for t = 2002Q1 to 2011Q4.
11
Taking into account the breaks discussed above Recursive Chow statistics (“forecast
horizon” one-step, descendent and ascendant) are below the 5% critical value and the
recursive estimates of the main coefficients are within the previous ± 2 standard errors
intervals.
Figure 4. Recursive Graphics EqC with broken trend
Dc_4 × +/-2SE
1990 2000 20100.20.40.60.8
Dc_4 × +/-2SE Constant × +/-2SE
1990 2000 2010
0.0250.0500.075
Constant × +/-2SE Dy × +/-2SE
1990 2000 20100.9
1.0
1.1
Dy × +/-2SE
Dy_4 × +/-2SE
1990 2000 2010-0.75
-0.50
-0.25Dy_4 × +/-2SE Dy02on × +/-2SE
1990 2000 2010
-0.2
0.0Dy02on × +/-2SE Drexch × +/-2SE
1990 2000 2010
-0.04
-0.02
0.00 Drexch × +/-2SE
EqC_1 × +/-2SE
1990 2000 2010
-0.4
-0.2
0.0 EqC_1 × +/-2SE Cseas8592 × +/-2SE
1990 2000 2010
-0.075
-0.050
-0.025 Cseas8592 × +/-2SE 1up CHOWs 1%
1990 2000 2010
0.0
0.5
1.01up CHOWs 1%
Ndn CHOWs 1%
1990 2000 2010
0.5
1.0 Ndn CHOWs 1% Nup CHOWs 1%
1990 2000 2010
0.5
1.0 Nup CHOWs 1%
3.3. An EqC including commodity prices
The model of consumer expenditure presented in 3.2 ex-post incorporates breaks; a
change in the deterministic seasonality pattern and a different short run income elasticity
of disposable income after 2002. Moreover, the model includes a broken trend during
2005Q1-2009Q4. For this period, an omitted “outside” variable may explain why
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consumption grew less than disposable income but so far we identified that outside
variable with a trend. However, another explanatory variable may be the “economic cause”
of the fall observed in the consumption income ratio. From 2000 the prices of the
Argentina´s commodities exports show an upward trend but from 2004 they deviate
downwards from this trend. The following figure plots the consumption-income ratio
alongside the soybean price which has become (along with derived products) the main
component of Argentine exports.
Figure 5. The ratio of consumer expenditure to disposable income and the soybean price
c-y lsoybeanp
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7c-y lsoybeanp
Therefore, we developed another EqC incorporating the soybean price in the cointegration
space. Using the system-based procedure only one long run relationship between
consumption, income and soybean price was obtained for the whole sample. The
hypotheses of homogeneity of income and a valid conditional model of consumption
expenditure on income and soybean price are not rejected. These results are reported in
Table 4.
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Table 4. Cointegration Analysis with soybean price
Rank k k=0 k≤ 1 k≤ 2log-lik 787.1 798.8 805.2λk 0.1746 0.1014
k=0 k≤ 1 k≤ 2λtrace 39.95 ** 14.72 1.77
λtraceadj 34.18 * 13.26 1.59Prob 0.014 0.106 0.207
Variable SE
c 1 0y -0.9338 0.0221
lsoybeanp -0.0497 0.0196constant u
Variable SE
c -0.3348 0.1183y -0.0889 0.1154
lsoybeanp -0.2084 0.3127Other unrestricted variables included are: cseas8592, cseas8592_2; cseasonal and cseasonal_1 and dummies for: 2008Q4; 2003Q4; 1989Q2; 1988Q1; 1987Q4 and 1983Q3.The hypothesis of long run homogeneity of y is not rejected at 1% and 5%using LR statistic
Eigenvectors β
Adjustment Coeficients α
Null hypothesis
It is worth noting that the effect of the soybean price on consumer expenditure could not
have been detected as significant until mid-2005, as can be observed in Figure 6 where
the recursive estimates of the long run coefficients are shown.
14
Figure 6. Recursive estimates of the long run coefficients of income and soybean price
beta1 y × +/-2SE
1990 1995 2000 2005 2010
-1.50
-1.25
-1.00
-0.75
-0.50
-0.25beta1 y × +/-2SE
beta2 lsoybeanp × +/-2SE
1990 1995 2000 2005 2010-0.50
-0.25
0.00
0.25beta2 lsoybeanp × +/-2SE
Given the cointegration results, a conditional EqC was estimated to model consumption on
income and soybean price, also including the log differences of the real exchange rate.
The following model was obtained,
15
Table 5. Estimated equation of the EqC with soybean price
Δct = 0.08072 +0.4601 Δc
t-4 +0.9126 Δy
t -0.1005 Δy
t02on
(HCSE) (0.0164) (0.0854) (0.0326) (0.0364) -0.4783 Δy
t-4 -0.0185 Δrexch
t -0.1957 EqCt-1
(0.0817) (0.0069) (0.040) 0.0271 Δlsoy_4-Δlsoy_5 -0.0580 Cseas8592 +0.0421 I:1984(3) (0.0079) (0.0109) (0.0013) 0.0359 I:1985(1) -0.0394 I:1988(3) +0.0524 I:1989(1) (0.0066) (0.0035) (0.0077) -0.0457 I:1990(3) -0.0322 I:2011(2) (0.0030) (0.0032) R2= 0.975 F(14,106)=297.3 [0.000]** σ=0.0112 Residual tests and Regression specification test AR 1-5 test: F(5,101) = 1.1738 [0.3273] ARCH 1-4 test: F(4,113) = 1.0573 [0.3811] Normality test: Chi^2(2) = 2.2587 [0.3232] Hetero test: F(16,98) = 0.91713 [0.5522] Hetero-X test: F(40,74) = 1.5090 [0.0631] RESET23 test: F(2,104) = 0.24003 [0.7870]
In Table 5, the EqC term (EqCt-1) indicates that about 0.20 of the disequilibria is corrected
in the first quarter in order to reach the long run relationship between consumption, income
and soybean price. The short run effects are similar to the model with the broken trend.
This model showed a different short run effect of national disposable income on private
consumption - since the default and end of the convertibility regime - too and therefore,
Δyt02on had to be included to obtain parameter constancy. However, the rest of the
variables showed parameter constancy. Since 2002Q1 the contemporaneous effect of
national disposable income growth changed from 0.91 to 0.81. There is also an additional
effect due to income acceleration, taking into account the negative effect of 0.48 from Δyt-4.
The change in the real exchange rate has a significant and negative effect of
approximately 2% on private consumption too. Besides, the soybean price has a short run
effect. The difference between the rate of growth of soybean price lagged by four and five
periods has a positive effect of approximately 3%. The rate of growth consumption lagged
16
by four periods has a positive effect on the contemporaneous rate of growth of
consumption indicating stochastic seasonality. A different deterministic seasonality was
detected for the first quarter of the period 1985-1992 (Cseas8592). Other additive dummy
variables were needed for 1984Q3, 1985Q1, 1988Q3, 1989Q1, 1990Q3 and 2011Q2.
Parameter constancy for the model of Table 5 was not rejected by their recursive
estimation, as can be observed in the following graphs. Recursive Chow statistics
(“forecast horizon” one-step, descendent and ascendant) are below the 5% critical value
and the recursive estimates of the main coefficients are within the previous ± 2 standard
errors intervals.
Figure 7. Recursive Graphics EqC with soybean price
Dc_4 × +/-2SE
2000 20100.25
0.50
0.75Dc_4 × +/-2SE Constant × +/-2SE
2000 2010
0.050.100.15
Constant × +/-2SE Dy × +/-2SE
2000 2010
0.91.01.1
Dy × +/-2SE
Dy_4 × +/-2SE
2000 2010-0.75
-0.50
-0.25Dy_4 × +/-2SE
Dy02on × +/-2SE
2000 2010
-0.3-0.2-0.10.0 Dy02on × +/-2SE Drexch × +/-2SE
2000 2010
-0.03
-0.01Drexch × +/-2SE
Cseas8592 × +/-2SE
2000 2010-0.075
-0.050
-0.025 Cseas8592 × +/-2SE EqC_1 × +/-2SE
2000 2010
-0.3
-0.1EqC_1 × +/-2SE 1up CHOWs 1%
2000 2010
0.0
0.5
1.0 1up CHOWs 1%
Ndn CHOWs 1%
2000 2010
0.5
1.0 Ndn CHOWs 1% Nup CHOWs 1%
2000 2010
0.250.500.751.00 Nup CHOWs 1%
17
Therefore, although there are no great differences regarding goodness of fitness among the
three models, they have diverse long run specifications which are discussed from a
forecasting perspective in the next section.
4. Forecasting consumer expenditure growth
When in-sample breaks were modelled, specifically by including a broken trend and a
commodity price in the long run, a congruent representation of Argentine consumer
expenditure over the period 1981-2011 was obtained and, in particular the analysis of
recursive estimates does not reject ex-post constancy of the conditional models. However,
the actual forecasting performance of an econometric model is a different issue. In this
section we study the forecasting performance of the initial conditional EqC model10, since
the effects of neither the broken trend nor the soybean price would have been detected had
they been included in the EqCs when estimated at the origin of the forecasting period. The
EqCs forecasts are compared to other approaches which are explained in 4.1. One step (a
quarter) ahead forecasts are calculated over 2006Q1-2011Q4.
A principal aspect of the next evaluation is that we will assume that the forecaster was
making the forecasts since 2005Q4 and therefore, he or she had observed by this time the
declining level of the consumption-income ratio at least for a year, as shown in Figure 3.
After observing a reverse trend for more than a year the forecaster assumed that the break
ends in 2009Q4.
Because of this assumption, forecasts are supposed to be made during an external break
due to an omitted variable; the broken trend or the soybean price. It is important to
emphasise that, although the EqC neither with the broken trend (Section 3.2) nor the
soybean price in the Cointegration space (Section 3.3) had been discovered at the origin of
the forecasts, their forecasts are also reported for comparisons.
10 We should bear in mind that they are conditional models which represent contingent plans of consumers who react to actual income and exchange rate values. Ahumada and Garegnani (2012) address the issue of having to forecast the exogenous variables in the case of a money demand. In this case we found that no great differences can be empirically appreciated using outside forecasted rather than actual values of the exogenous variables.
18
Therefore, the EqCs are recursively estimated since 2005Q1 until 2009Q4 but only the
initial one (Section 3.1) can be assumed to be the forecasting model. In addition, the other
approaches depicted in the next section are calculated for the forecast period. In particular
the use of an outside variable to adjust forecasts during the forecasting period is analysed.
4.1 Different approaches to forecasts during breaks
While the traditional theory of forecasting has assumed that the structure of the economy
will remain relatively unchanged empirical evidence has shown how inadequate this
assumption can be. Clements and Hendry (1998, 1999, 2005) have found that determinist
components are the main responsible for forecast failure. Since the major changes in
deterministic components are location shifts, long run means Vector equilibrium-correction
models (VEqCM) - and conditional EqC models as a special case - are particularly
affected. Therefore, they suggest different approaches to avoid such changes and so allow
forecasts to rapidly adapt to a new regime.
For example a VAR in differences (ΔVAR) is an alternative model to employ for forecasting
under breaks since by construction it does not include the equilibrium correction terms.
Being a closed system (a reduced form) this kind of model allows us also to compare their
forecasting performance to conditional EqC models. We recursively estimated a VAR(4)
model of Δc, Δy and Δrexch. Centred seasonals and dummies selected by impulse
saturation (at 1%) were also included in the model.
An alternative approach is intercept corrections (IC), or residual adjustments. In this case
the EqC model of xt can be adjusted after the break occurs. This can be done by putting
the forecast “back on track” when forecast errors are correlated as follows,
( )
(1) ˆ - ˆˆ111 −−− ΔΔ+Δ=Δ ttt
ICt xxxx
We calculate the adjustment when we detected systematic (but small) forecasts biases
since 2005.
19
Also, a simple model that removes deterministic components is the second difference
model. Since many economic variables do not continuously accelerate,
[ ] (2) 02 =Δ txE
a forecasting rule is,
(3) ˆ11 −+−++ Δ=Δ jTjTjT xx
It reduces the impact of the breaks by offsetting breaks in intercepts and trends as well.
For the difference model of consumer expenditure, Δct was forecasted using Δct-4. It is
reported as Δ1 Δ 4 given its seasonal behaviour. Therefore, the corresponding double
differences are supposed to have an expected value equal to zero [ ] 041 =ΔΔ txE
These forecasting approaches applied during breaks have been discussed in Castle et al.
(2011). They also propose (and evaluate in their Monte Carlo study) a more novel strategy
to deal with external breaks,11 which are evolving while forecasting is going on. If an
outside variable (Zt) is known that enters lagged and it is omitted in the estimated model
(until T) then a new model of the forecast error,
( ) (4) ˆ - 1 tttt vZxx +=ΔΔ −λ
can be estimated only for the forecast period (t=T+1, T+2, …) and thus updating can
improve forecasting performance. It can be noted that although only a few observations
are used to estimate Equation (4) (starting with just one or two), the estimator of λ is
unbiased. We applied this approach and the results obtained are denoted by OVFP
(outside variable forecast period). In the consumer model Zt is i) the trend (plus a constant)
and ii) the one-quarter lagged soybean price (in logs).
11 If, instead, a location shift in the long run mean of the EqC were modelled as an internal break using an exponential function, nonlinear optimisation would be needed, see Castle et al. (2011).
20
A key issue to mimic a real case of forecasting is whether the forecaster knew the “outside
variable” and when its effect started. In our case the forecast of the (unobserved) level of
c-y indicated that for 2006Q1 until 2009Q4 the recursive estimation of the trend can be
used to adjust forecasts (see Figure 8).
Figure 8. Estimation of c-y level
c-y c-y Level
2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
-0.45
-0.40
-0.35
-.397
-0.413
c-y c-y Level
c-y c-y Level
2004 2005 2006 2007 2008 2009 2010 2011 2012
-0.45
-0.40
-0.35
-0.385-0.389
c-y c-y Level
In the case of the soybean price Figure 6 suggests that its effect can be present since the
forecast origin at 2006Q1. By this time this price had become a key variable, whose
behaviour macro analysts followed along with other indicators such as interest rates and
exchange rates.
21
4.2 Forecasts results
The next table reports the corresponding root mean squared error (RMSE) and the mean
absolute percentage error (MAPE) for each forecasting approach.
Table 6. One-step forecasts of Δc
Table 6.1. Forecasts using EqCs, robust devices and OVFP
Period EqC ΔVAR Δ1Δ4 IC OVFP OVFPno trend trend soyp
2006 RMSE 0.01358 0.05444 0.00531 0.00776 0.00317 0.00111MAPE 1.36788 1.39920 0.36538 0.33182 0.16998 0.11124
2007 RMSE 0.01672 0.03293 0.00589 0.00833 0.01411 0.00850MAPE 2.36077 1.57463 0.33003 1.36078 1.40448 1.35522
2008 RMSE 0.01458 0.10159 0.03678 0.00693 0.01391 0.00634MAPE 0.29286 0.95886 0.88780 0.29286 0.27087 0.10891
2009 RMSE 0.00507 0.20572 0.04093 0.00803 0.00398 0.00397MAPE 0.20551 0.79193 0.90019 0.20551 0.12631 0.12821
2010 RMSE 0.00907 0.00724 0.06138 0.00873 0.00785MAPE 0.23174 0.22984 2.84280 0.23174 0.36691
2011 RMSE 0.02255 0.02356 0.03208 0.03310 0.01814MAPE 2.45355 2.25446 2.55264 2.45355 1.42752
2006/11 RMSE 0.01468 0.06107 0.03626 0.01215 0.01038MAPE 1.15205 1.40149 1.31314 0.77846 0.74672
2006/09 RMSE 0.02815 0.09867 0.02780 0.00971 0.01059 0.00801MAPE 1.05676 1.18116 0.62085 0.72334 0.62039 0.60149
22
Table 6.2. Forecasts using OVFP and EqCs with long run modelled ex-post
Period OVFP EqC OVFP EqC trend with trend soyp soyp
2006 RMSE 0.00317 0.01406 0.00111 0.01273MAPE 0.16998 1.25877 0.11124 0.13715
2007 RMSE 0.01411 0.01648 0.00850 0.00787MAPE 1.40448 1.41711 1.35522 1.27482
2008 RMSE 0.01391 0.00830 0.00634 0.01003MAPE 0.27087 0.29586 0.10891 0.19835
2009 RMSE 0.00398 0.00417 0.00397 0.00115MAPE 0.12631 0.15418 0.12821 0.14825
2010 RMSE 0.00817 0.00785 0.00491MAPE 0.32154 0.36691 0.13215
2011 RMSE 0.01894 0.01814 0.01085MAPE 1.77668 1.42752 1.35594
2006/11 RMSE 0.01216 0.01038 0.00965MAPE 0.87069 0.74672 0.75198
2006/09 RMSE 0.01059 0.02491 0.00801 0.00795MAPE 0.62039 0.78148 0.60149 0.60097
We can observe that taking into account the broken trend to adjust forecasts
(OVFP_trend) improves the forecasts of the recursively estimated EqC model, shown in
the first column. If the forecaster included the soybean price both RMSE and MAPE are
still lower than the forecasts obtained by not including the trend in the EqC (except for one
case) even when it is also estimated recursively. This accuracy improvement was obtained
even when the first OLS (recursive) estimation of (4) has only one degree of freedom.
One interesting result (see Table 6.2) is that forecasts obtained using the OVFP
adjustment for the periods closer to the origin of the break (until 2009) are similar to, or
even better than those obtained when the corresponding models are estimated assuming
that the LR relationship (estimated for the whole sample is known), that is with EqC with
trend and EqC soyp (Equilibrium Correction model with soybean price).
23
The OVFP approach using the soybean price also has a better forecasting performance
(see Table 6.1) than using other mechanisms except in 2007 and 2010 when simple
device Δ1 Δ 4 and Δ VAR (particularly using MAPE) are more accurate.
Regarding significance, we calculate the difference of the forecast errors from both OVFP
with respect to Δ1 Δ 4 (taken as the benchmark) using a quadratic loss function (see
(Diebold and Mariano, 1995) and Morgan-Granger-Newbold statistic. Table 7 shows that
both OVFP are significantly different from the robust device.12
Table 7. Tests of comparative forecasting accuracy
MNGt-stat p-value
Difference OVFP with trend vs Δ1Δ4 (2006-2009) -2.96 0.0081 -2.95Difference OVFP soybeanp vs Δ1Δ4 (2006-2011) -3.39 0.0025 -4.89
DM
Since relative forecasting performance can change over time (see Giacomini, 2011) the
next figure allow us to observe the difference in the forecasting performance of both OVFP
with respect to Δ1 Δ 4 in each quarter. Larger differences are shown since 2008.
12 Using Heteroscedasticity and Autocorrelation Robust SE does not change this result.
24
Figure 9. Differences in squared forecast errors of the OVFP with respect to Δ1 Δ 4
Difference OVFP soybeanp vs D1D4 Difference OVFP trend vs D1D4
2006 2007 2008 2009 2010 2011 2012
-0.005
-0.004
-0.003
-0.002
-0.001
0.000
Difference OVFP soybeanp vs D1D4 Difference OVFP trend vs D1D4
This finding reveals that EqCs can still be useful for forecasting when their forecasts are
properly adjusted, in particular when an outside variable can be used.
5. Conclusions
This paper has focussed on the forecasting performance of EqCs of Argentine consumer
expenditure that have been used for forecasting the consumption growth during breaks.
Both in sample and out of sample breaks have been considered.
From an ex-post analysis, a long run relationship of consumption and income was found
allowing for a broken trend for the 2005-2009 period. When replacing this deterministic
variable by a commodity price, key for the Argentine economy, this new variable enters the
cointegration space and another EqC was estimated. In the short run of both models,
disposable income was also found to have a different effect after 2002. A changing pattern
of seasonality for the first quarters of 1985-1992 was detected. Taking into account these
25
changes ex post, the estimated models are congruent representations over 1981-2011.
These EqCs may be useful as forecasting models in the near future.
However, to mimic a real case of forecasting, forecasts for the last five years of the sample
were based on the recursive estimates of an EqC, which included neither the broken trend
nor the commodity prices, since they could not have been detected as significant at the
origin of the forecasting period. Forecasts from this model, recursively estimated were
compared to other forecasting approaches: a VAR in differences, intercept correction and
differences. In particular, we focussed on a recently suggested approach for external
breaks, which we applied to forecast consumption growth. Forecasts were adjusted for the
effect of an outside variable during the forecast period. This variable was either a trend or
the soybean price in the case we studied. When an outside variable was used to
recursively adjust forecasts from the EqC we achieved much better forecasting accuracy.
26
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Appendix 1: Data Definitions and Sources
Private Consumption: Aggregate expenditure in goods and services of private residents
and non-profit institutions (thousands of pesos at 1993 prices). Economic Commission
for Latin America (ECLAC), Buenos Aires and Dirección Nacional de Cuentas
Nacionales (INDEC).
Disposable Income: Gross national income plus current net transfers (thousands of pesos
at 1993 prices). Economic Commission for Latin America (ECLAC), Buenos Aires and
Dirección Nacional de Cuentas Nacionales (INDEC).
Real Exchange Rate: Real Exchange Rate peso/dollar and Ratio of wholesale to
consumer prices during the Convertibility regime. Dirección Nacional de Cuentas
Nacionales (INDEC) and BCRA.
Soybean price: Monthly averages of Chicago daily soybeans cash prices (N 1 Yellow)
obtained from Thomson Reuters Datastream. They are expressed in US dollar cents per
bushel, which are converted to US dollars per metric ton and deflated by the US CPI.