ISTITUTO DI STUDI E ANALISI ECONOMICA
VECTOR-AUTO-REGRESSION APPROACH TO
FORECAST ITALIAN IMPORTS
Carmine Pappalardo
ISAE - Piazza Indipendenza, No. 4, 00185 Rome (Italy).
E-mail: c.pappalardo@isae.it
Tel. +39 06 4448 2355
and
Gianfranco Piras
Tor Vergata University, Faculty of Economics
via Columbia, No. 2, 00133 Rome (Italy)
ISAE - Piazza dell’Indipendenza, No. 4, 00185 Rome (Italy).
E-mail: g.piras@isae.it
Tel. +39 06 4448 2347
Working paper No. 42
February 2004
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2
ABSTRACT
Imports represent a relevant component of total economic resources. For the
Italian case, they mainly consist of raw materials and intermediate goods.
In this paper, we evaluate several econometric models performing shorthorizon forecasts of Italian imports of goods. Year-to-year growth rate of the
monthly seasonally unadjusted series is the variable to predict. VAR
forecasting ability has been compared to that of a linear univariate benchmark
(ARIMA) model. Main forecast diagnostics have been presented. Finally, we
perform two types of forecast encompassing tests (Diebold-Mariano, 1995;
Fair-Shiller, 1990) for which we present main results.
JEL Classification: C53, C52, C32.
Keywords: Forecasting, VAR model, Import, Forecast evaluation.
We are grateful to an anonymous referee for the helpful comments and to all the
participants at the Meeting "Modelli Stocastici e Metodi di Simulazione per Analisi di Dati
Dipendenti", University of Molise, Faculty of Economics, 28-29 April 2003, and to all the
participants at the ISAE seminar. A particular thank to Claudio Lupi, Giancarlo Bruno,
Sergio de Nardis for the useful discussions and suggestions, and thanks to Gianna Foschi for
her efficient editorial assistance. The opinions expressed in this paper are solely the
responsibility of the authors and should not be interpreted as reflecting the views of
ISAE or its staff.
3
NON TECHNICAL SUMMARY
Imports represent a relevant component of total economic resources. For the
Italian case, they mainly consist of raw materials and intermediate goods. For
this reason, imports can be taken as a significant leading indicator of the
aggregate business cycle. This feature, though extremely useful in assessing
very short-run dynamics of Italian economy, cannot be properly exploited due
to the lack of the availability of the statistical information. Quantity indexes,
released from ISTAT, are made available about three months late with respect
to the reference period. Our aim is to provide very short run forecasts useful
to integrate the available information.
In this paper, we evaluate several econometric models performing shorthorizon forecasts of Italian imports of goods. Year-to-year growth rate of the
monthly seasonally unadjusted series is the variable to predict. VAR
forecasting ability has been compared to that of a linear univariate benchmark
(ARIMA) model. Main forecast diagnostics have been presented. Finally, we
perform two types of forecast encompassing tests (Diebold-Mariano, 1995;
Fair-Shiller, 1990) for which we present main results.
4
LA PREVISIONE DI BREVE TERMINE DELLE IMPORTAZIONI DI
BENI IN QUANTITA’
SINTESI
Le importazioni di beni costituiscono una componente particolarmente
rilevante nella composizione delle risorse complessive di un paese. In Italia,
l’acquisto di beni dall’estero è in prevalenza costituito da materie di base e
prodotti semilavorati, utilizzati nelle fasi iniziali e intermedie del processo
produttivo. Per questa caratteristica, le importazioni presentano un
comportamento anticipatore dell’economia nazionale.
L’esercizio di previsione consiste in un’applicazione della metodologia VAR
(Vector Autoregression) alla serie dei volumi mensili di beni importati e alle
variabili coincidenti e
anticipatrici di tale indicatore. L’analisi della
performance previsiva è condotta rispetto ad un insieme di previsione ottenute
da un modello univariato ARIMA, corretto per tenere conto di valori anomali
ed effetti di calendario. I modelli sono stati confrontati utilizzando due diverse
metodologie di test di forecast encompassing (Diebold-Mariano, 1995; FairShiller, 1990).
Classificazione JEL: C53, C52, C32.
Parole chiave: Previsione, Modelli VAR, Import, Diagnostiche di previsione.
5
CONTENTS
1. INTRODUCTION
2. PRELIMINARY ANALYSIS
2.1. Potentially leading indicators
2.2. Stationarity and seasonality
2.3. Cyclical components
2.4. Testing for nonlinearity
Pag.
“
“
“
“
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7
8
9
10
12
13
3.
THE FORECASTING MODEL
3.1. Reduction diagnostics
“
“
17
19
4.
FORECAST EVALUATION
“
22
5.
FORECAST ENCOMPASSING TESTS
“
23
“
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28
29
CONCLUDING REMARKS
REFERENCES
6
1. INTRODUCTION
In Italy, external balance represents a very crucial component in determining
the size and pattern of real product. As far as external demand is concerned,
exports may be considered as a significant share of total national production
of goods and services sold abroad. Particularly, during expansive cycles,
domestic production of goods of traditional specialization of Italian industry
(such as textiles and clotures, furniture, industrial machines) has been mostly
sustained by external demand. Also benefiting of significant exchange rate
devaluations, Italian exports have largely contributed to the growth of
national economy in the last decades.
Raw and intermediate materials, which domestic firms use as inputs in the
initial phases of productive process, are mostly bought on international
markets. As a consequence, increase of imports (mainly of goods) can also be
considered as a signal of a successive rise in output dynamics, driven by a
growth of internal and/or external demand components (investments in capital
goods, consumption, exports). Moreover, as directly related to the industrial
cycles (one of the main sources of aggregate fluctuations), imports represents
one of the most significant leading indicators of short-term patterns of Italian
economy.
Tough imports share relevant information for short-term investigations, such
features cannot be adequately used for business cycle purposes, given the
excessive delay with which official statistics are provided. Indicators in
volume, which we are much interested in for business cycle purposes, are
available about a quarter after the reference month. Hence, a three-step ahead
prediction is necessary to achieve a nowcast of the indicator itself. As far as
we are concerned, it has not been undertaken a systematic forecast of Italian
import of goods on a monthly basis. In this context, short-term forecasting
models play a significant role to face the gap between the availability of
official statistics and the needs of a timely short-term business cycle analysis.
In this paper, several short-term forecasting models of the Italian imports of
goods in volume are presented. In-sample-predictions from a simple
univariate model are used as benchmark to evaluate the forecasting
performances of alternative specifications of VAR (Vector Autoregression)
models. Forecast encompassing tests (Diebold and Mariano, 1995; Fair and
Shiller, 1990), based on comparison of the information from ARIMA and
VAR predictions, have been carried out as an additional evidence on
forecasting accuracy.
7
The paper is organized as follows: in the next section we present some
preliminary analysis of the time series concerned in the specification and
forecasting exercises. In particular, departures from the linearity assumption
have been tested. Section 3 develops VAR models identification, estimation
and forecasting procedures. Section 4 is dedicated to analyze forecast
evaluation. Prediction accuracy, evaluated through forecast encompassing
tests, has been developed in section 5. Section 6 concludes.
2. PRELIMINARY ANALYSIS
This section provides some descriptive but relevant preliminary information
concerning the series involved in the forecasting exercise. For the Italian case,
the building of a forecasting model of the annual dynamics of the total import
of goods in volume has not been undertaken on a systematic basis. The
excessive delay with which the information on the volumes imported is made
available, makes difficult the use of such series in common analysis.
Recently, a strong effort has been undertaken by the Italian National
Statistical Institute (ISTAT) to make quantity indicators on imports (and
exports) much more timely. As an additional difficulty, it is not actually
available a long time series to be used for prediction purposes. The reference
time series, actually updated by ISTAT on a monthly bases, starts from 1996;
such series is not immediately comparable with the previous ones due to a
deep revision both in statistical methodology both in classification standards
adopted.
A long time series has been obtained using all existing information. Actually,
1995 based series are directly provided by ISTAT. Starting from January
1996, data in volume are constructed as chain indexes with the reference basis
changing one time a year and covering a large set of goods traded. Also, they
take account of the dynamics of imports composition through time, which was
considered to be fixed by the old series. Data based on the previous
methodology, available over a long time period ranging from 1980 until 1998,
consist in a fixed based series (1985=100). The reconstruction of a long time
series was firstly based on the comparison of the two indicators (the old and
the new one) over a common time span covering the period 1996:01 1998:12.
Data before 1996 have been obtained from the old indicator scaled using the
parameters from a linear regression of the new and the old series, estimated
over the common sample of 36 observations. Such resulting series presents
the advantages to be available over a longer time span (starting from 1980)
8
and to be monthly updated on the basis of the indicators in volume currently
produced by ISTAT. Such indicators is that used for the identification and
estimation of very short-run forecasting econometrics models of the monthly
series of total imports in volume. Taking account of the delay with which the
official indicator is released, three steps ahead forecasts may be correctly
considered as a nowcast of the reference series. Really, true forecasts may be
obtained through predictions of the official figures over a longer forecasting
horizon.
Contrary to other researches (e.g., Bruno and Lupi, 2001) the set of indicators
for import is not as rich as those for other variables. At least at this initial
stage, roughly coincident variables have been considered too. The variables
selected as potential predictors of volumes imported are the following: the
general index of industrial production (IPI), the quantity of goods transported
by railways (TK), the business survey series of short-term production
prospects (TTP), the Italian exports of goods in volume (XWQ). 1
2.1. Potentially leading indicators
Industrial production is the key indicator for the monitoring of business cycles
dynamics. Since early stages of productive process is crucially dependent on
raw materials and intermediate goods, increases in production should be
relatively anticipated by a growth in imports. IPI has been included
considering the strict relationships between production and flows of goods
from abroad. Thus, it does not present any particular leading property, as the
descriptive analysis in the following sections will prove. The need to take
account of production dynamics in forecasting imports led to select two main
proxies of industrial production. First of all, following Bruno and Lupi
(2001), goods transported by railways proved to be a powerful predictor of
industrial production since they largely consist of intermediate and raw
materials. TTP is a variable taken from the business survey on manufacturing
sectors carried out by ISAE on a monthly basis. It represents industrial
operators’ opinions on production dynamics in the very short-term (three-four
months ahead).2 Cyclical analysis showed a long lead of production
expectations also over imports of goods. Finally, the monthly series of
1
The industrial production index is released monthly by ISTAT. The business survey series
of short-term production prospect is released by ISAE, the Institute for Studies and
Economic Analysis. The time series for tons of goods transported by railways are kindly
provided by Trenitalia, the Italian State railways company.
2
Operators are asked to express their opinion on production pattern, according to three
modalities: "up", "stable" and "down". The variable has been quantified through the
balance approach.
9
products exported has been considered. It shares the same features of the
imports and has been reconstructed in the same way. The assumption is that
products sold abroad can be viewed as a share of national production and, hence,
are able to activate imports of goods (especially with reference to given specific
sectors). In the following sections, univariate characteristics of the above set of
predictors will be explored. All series are considered as log transformed while
the expression -log((200/(TTP+100)-1) is applied to production expectations to
make the series unbounded.
Figure 1 (first five panels) presents the plots of the set of series (to be predict/
predictors) we are dealing with. We provide a description of such series in
terms of their long run and medium term stochastic properties. Trend
components, which describe the long-run dynamics of the variables, have
been assumed to include frequencies exceeding 8 years. Cycle frequencies
range from 18 up to 96 months are extracted using Band-Pass filters (see
subsection 2.3). Seasonal fluctuations have been previously described
according to their evolving pattern. Further, for each variable, the null of
seasonal integration has also been tested (see subsection 2.2). Imports show
strong seasonal patterns changing over time. In particular, seasonal dynamics
present substantial changes at the beginning of 90’s, showing a more regular
behaviour. They are characterized by significantly increasing fluctuations in
the second part of the sample. Imports long run dynamic sharply increases
from the mid of 90’s, assuming rate of growth largely greater than those
observed in the past. The series of quantity exported shows long run dynamics
rising along the whole time period. Its seasonal pattern shares features
analogous to that of imports, changing over time with large oscillations in the
second half of the sample. Industrial production is characterized by a greater
regularity of its long term pattern. Oscillations at business cycle frequencies
appear to be prevailing respect to the trend, given its moderate rates of
growth. Seasonality presents strong and regular fluctuations (with large
though in August) so to assume a largely deterministic pattern. A similar
pattern shows the railways transport of goods: the series is mostly
characterized by medium term (business cycle) frequencies, and possible
outliers. A significant different behaviour concern the series of short-term
production expectations. The plot shows dominant cyclical patterns; over the
sample considered, it appears a lightly growing trend. Such features, will be
assessed in more detail in successive sub-sections.
2.2. Stationarity and seasonality
Some relevant stochastic properties of the time series we consider have been
evaluated through a unit root test. As we are dealing with raw data, the
10
presence of such roots has been detected both at the zero (regular) both at
seasonal frequencies. Empirically, the analysis has been carried out using the
test due to Beaulieu and Miron (1993). For each variable, test statistics are
obtained running an auxiliary regression, which deterministic part is specified
with a constant, a trend, eleven seasonal dummies and lags of the dependent
variable up to get white noise residuals. A synthetic evidence is reported in
Table 1. Such results reject the presence of a unit root at zero frequency for
imports, exports and industrial production variables. Considering TK series,
the null of stationarity is accepted both at the regular both at some seasonal
frequencies. Short-run production expectations does not show any unit root at
the seasonal frequencies. Testing also rejects the null at the zero frequency.
Such result does not seems to be coherent with recent prevailing evidences: it
could be due to the particular sample extension and to the presence of outliers,
which could affect testing. In all cases, the presence of a complete set of unit
roots is rejected with strong evidence. In spite of this, the assumption of nostationarity at frequencies other than the regular is confirmed for all the
variables considered in this paper. Dealing with raw data, the application of
the seasonal difference operator tends to remove more than due and could
induce overdifferencing. Nevertheless, can be assumed that HEGY
(Hylleberg, Hendry, Granger, Yoo, 1992) methodology leads to weak (but
relevant) indication on non-stationarity. For this reason, the application of the
seasonal filter could be a reasonable general practice in controlling for
seasonal fluctuations. It tends to assume a prevalent deterministic character
for industrial production and railways transported goods. Such evidence is
confirmed also for TTP, but less significantly. Exports and imports, on the
contrary, present seasonal fluctuations which strongly evolves over time (the
twelve monthly dummies have been estimated to be not significant along the
sample period). In that latter case, the application of annual differences could
not be sufficient and residual seasonality could produce bias in estimation
results. In these cases, such noise could be controlled augmenting the number
of lags for dependent variable and also controlling for the trading day effect.
11
FREQ.
0
π/ 6
π/ 3
π/ 2
2π/ 3
5π/ 6
π
−3.7
11.0 ∗∗
6 .1
8.4 ∗∗
3 .4
2 .9
−1.8 ∗
MQW(5)
−3.4
4 .8
2 .6
7 .0 ∗
8.6 ∗∗
3 .8 ∗
−1.7
IPI(3)
−2.4
3 .3 ∗
0 .6
0 .1
0 .8
0 .3
−0.5
XQW(3)
− 5 .3
18.7
19.2
15.6
16.3
11.2
− 2 .9
TTP(0)
∗∗
∗∗
∗∗
∗∗
∗∗
∗∗
∗∗
−4.3
2 .2
7 .5
12.0
8 .4
4 .7
−1.9
TK(4)
∗∗
∗∗
∗∗
∗
Table 1. Unit roots test. t-test for the 0 and π frequencies, F-test for the
others. ’*’ and ’**’ indicates significance at 5% and 1%, respectively.
Number of lags in parenthesis.
2.3. Cyclical components
The predictive ability of forecasting models for imports, deeply depends on
the leading properties of the corresponding indicators. In this section we pay
particular attention to the cyclical characteristics of the time series considered.
Our aim is to observe stylized facts which could help us in the selection of the
variables for model identification. As it is known, the series of import is
characterized by a large and deep cyclical component, with high variability.
Moreover, it shows a systematic lead on aggregate business cycle. This
feature makes more difficult the selection of variables which present a regular
lead on the import cycle. Cycle components of each series have been
extracted applying Band-Pass filter in the form developed by Baxter and King
(1999). Cyclical frequencies ranges from 18 to 96 months (up to 8 years).
Extracted cycles have been plotted in Figure 1. Industrial production cycle
appears to be coincident with that of imports. Correlation measures, reported
in table 2, show a little lead of the latter series, according to consolidated
empirical evidence. With reference to the other indicators of industrial
activity, the series of railway transport of goods seems to be coincident. As it
results from the figure, it shows a more defined lead of imports only in the
second part of the considered time span. Production prospects represents the
series showing the greater lead (two months on average) and the larger
variability. The same indicator confirms previous findings about the leading
properties of TTP on industrial production (about 5 months). Finally, exports
series results to be coincident with imports, partly contrary to consolidated
evidence.
12
150
50
mqw
125
ttp
100
25
100
ipi
75
0
50
-25
1980
150
1990
2000
xqw
1980
2500
1990
2000
tk
1980
0.05
1990
mqw
2000
xqw
2000
100
0.00
1500
-0.05
50
1000
1980
0.05
1990
mqw
2000
ipi
1980
mqw
0.05
0.00
0.00
-0.05
-0.05
1990
2000
1995
tk
mqw
2000
ttp
0.2
0.0
-0.2
1995
2000
1995
2000
1995
2000
Figure 1. First five plots represent seasonally unadjusted series. In the remaining,
indicators’cyclical components are plotted against that of MQW.
SERIES
XQW
IPI
TK
TTP
TTPvsIPI
σ
0.034
0.022
0.041
0.140
ρ(0)
0.678
0.871
0.794
0.723
0.586
ρ(max) lead(+)/lag(-)
0.678
0.888
0.794
0.797
0.764
0
−1
0
+2
+4
Table 2. Cyclical analysis. ρ (0) is the correlation between
the series and MQW; ρ (max) indicates the maximum crosscorrelation; lead(+)/lag(-) is the interval in months at which
ρ (max) is observed.
2.4. Testing for nonlinearity
Over the last period, the interest which the literature has shown for non-linear
time series models has been steadily increasing. The main idea under this
approach is the fact that some economic time series show the characteristic to
have a non-linear mean over the period of observation. In this section, we test
13
the null hypothesis of linearity against a well specified non-linear alternative,
consisting in a particular class of regime-switching models, known as smooth
transition autoregressive (STAR, Granger and Terasvirta, 1993).
The hypothesis testing in the STAR framework involves tests of linearity
against the alternatives of LSTAR or ESTAR nonlinearity and
heteroskedasticity. First of all, the testing problem is complicated by the
existence of so-called nuisance parameters under the null hypothesis. The
presence of such parameters causes the lack of availability of the standard
statistical theory for test statistics. In particular, STAR models present
parameters which are not restricted under the null. A large literature has been
developed to face the problem of identifiability of nuisance parameters under
the null hypothesis. In this paper we adopt the approach developed by
Luukkonen, Saikkonen and Terasvirta (1988), according to which the
transition function can be approximated by a Taylor series. This fact is a
solution to the identification problem and linearity can be tested using usual
methods and distributions. Testing is performed taking into account two
different specifications of such function. The transition function is, firstly,
assumed to follow a (first-order) logistic function, leading to the logistic
STAR model (LSTAR); secondly, the exponential function, getting the
exponential STAR (ESTAR). In the case the alternative is assumed to be
LSTAR nonlinearity, the auxiliary regression becomes
y t = β 0 ′ xt + β 1 ′ xt st + ε t
(1)
′′
Testing for linearity is equivalent to test the null hypothesis H0 : β1 = 0 . This
test statistics can be constructed as an LM type test with a χ 2 distribution
with p + 1 degrees of freedom under the null hypotesis of linearity. It is
reported as LM1 statistics in the following tables. A variant of this statistics
has been developed by Luukkonen et al. (1988), since he noticed LM1 has no
power when only the intercept differs across regimes. A third order
approximation of the logistic transition function yields the auxiliary
regression
yt = β0′ xt + β1′ xt st + β2′ xt st2 + β3′ xt st3 + εt
(2)
In such equation, the test for null hypothesis, named as LM3, reduces to
H 0′′ : β1 = β2 = β3 = 0 , which again can be tested by a standard LM type test.
Testing against an ESTAR alternative, assuming the transition function as an
exponential, requires an auxiliary equation constructed on the basis of a
14
second-order Taylor approximation of the type
yt = β0′ xt + β1 ′ xt st + β2′ st2 + β3′ st3 + β4′ st4 + εt
(3)
The null hypothesis to be tested is H0′: β1 = β2 = β3 = β4 = 0. To evaluate
empirical findings we use the F-version of the LM test statistics, denoted
LM4 in the following tables. Especially in small samples, it is a good strategy
to use F-version of LM-type tests as it results much more robust than χ 2
variant. Both tests can be constructed making use of two auxiliary
regressions. For LM4, the F-version is
LM 4 = T (SSR0 − SSR1 )/SSR0
(4)
where SSR1 and SSR0 represent, respectively, the residual sum of squares of
the OLS estimates with and without interaction terms.
All the variables are taken in seasonal differences; the transition variables we
consider are lagged values of such series. The maximum value of the delay
parameter is equal to 6. Finally, deterministic trend has been considered as an
additional transition variable. Auxiliary equations have been specified with
constant, trend, eleven seasonal dummies and the level of the endogenous
variables lagged up to the 12th lag. First, concentrating on standard evidence
(LM1 test), it emerges significant nonlinearity for all variables considered,
with transition function lagged by 1 to 3. Import and production prospects
make exception. Under a more robust evidence (LM3 test), only nonlinearity
at the 10 % significance level is detected for import and industrial
production.3 Weak evidence of ESTAR nonlinearity has been found for
export and TK. No evidence appears in the case of linear and deterministic
transition function. Considering heteroskedasticity robust tests, all the
variables satisfy the linearity assumption, for whatever specification of the
transition function. As a preliminary conclusion, any non linearity present in
the series may be considered as moderate since the linearity tests does not
reject the null with strong evidence.
3
We are considering transition variable lagged by 2 and by 3 and specified as logistic.
15
trend
d12y(-1)
d12y(-2)
d12y(-3)
LM1
0.966
0.706
0.805
0.457
Standard test
LM3
1.000
0.260
0.052
0.030
LM4
1.000
0.508
0.125
0.127
Heteroskedasticity robust test
LM1
LM3
LM4
0.848
1.000
0.998
0.937
0.663
0.859
0.935
0.260
0.869
0.685
0.530
0.807
Table 3. LM-type test for STAR non-linearity: IMPORT OF GOODS
Trend
d12y(-1)
d12y(-2)
d12y(-3)
LM1
0.642
0.038
0.046
0.107
Standard test
LM3
0.255
0.150
0.114
0.140
LM4
0.541
0.013
0.350
0.181
Heteroskedasticity robust test
LM1
LM3
LM4
0.783
0.965
0.984
0.296
0.644
0.879
0.432
0.845
0.968
0.629
0.959
0.993
Table 4. LM-type test for STAR non-linearity: EXPORT OF GOODS
Trend
d12y(-1)
d12y(-2)
d12y(-3)
LM1
0.582
0.011
0.032
0.053
Standard test
LM3
0.952
0.248
0.504
0.518
LM4
0.879
0.070
0.744
0.359
Heteroskedasticity robust test
LM1
LM3
LM4
0.725
0.962
0.897
0.238
0.796
0.823
0.390
0.949
0.994
0.436
0.917
0.994
Table 5. LM-type test for STAR non-linearity: INDUSTRIAL PRODUCTION
Trend
d12y(-1)
d12y(-2)
d12y(-3)
LM1
0.583
0.035
0.025
0.018
Standard test
LM3
0.998
0.126
0.130
0.064
LM4
0.974
0.225
0.419
0.224
Heteroskedasticity robust test
LM1
LM3
LM4
0.422
0.973
0.995
0.056
0.695
0.833
0.078
0.992
0.958
0.107
0.968
0.778
Table 6. LM-type test for STAR non-linearity: TONS/Km OF GOODS
TRANSPORTED BY RAIL
Trend
d12y(-1)
d12y(-2)
d12y(-3)
LM1
0.977
0.434
0.022
0.119
Standard test
LM3
1.000
0.655
0.315
0.117
LM4
1.000
0.732
0.058
0.091
Heteroskedasticity robust test
LM1
LM3
LM4
0.724
0.991
0.990
0.442
0.872
0.813
0.169
0.840
0.946
0.274
0.956
0.958
Table 7. LM-type test for STAR non-linearity: PRODUCTION PROSPECTS
16
3. THE FORECASTING MODEL
An explicit aim of this work is to find out a reliable and simple model to
forecast the Italian import of goods, in order to overcome the problems arising
from the delay with which official information is developed. Empirical
evidence on the non-linearity of the series suggests to consider linear models.
This methodology, with simple seasonal component, offers advantages over
more complicated ones in terms of their short-term forecasting accuracy.
Nevertheless, the single equation framework offers an oversimplified option
and does not allow for multi-step dynamic forecasts. For all these reasons we
decide to consider the well established VAR (Vector Autoregression)
framework.
As we shown in Section 2, the four time series that we consider have different
seasonal properties, but there was a strong evidence that the presence of a
complete set of unit roots has to be rejected. This implies that if we
parameterize the VAR in seasonal differences, we are likely to overdifference the series. Nevertheless, there is some evidence on the effect on
forecasting performance deriving from imposing all the seasonal roots at unity
when this is not the case in reality. There are indications that filtering out only
the correct unit roots, in general, does not produce superior forecasts. In
particular, Lyhagen and Löf (2001) suggest that when the model is not known
and the aim of the modelling exercise is forecasting, a VAR in annual
differences may be a better choice than a seasonal error correction model
based on seasonal unit roots pre-testing. Moreover, Osborn, Heravi and
Birchenhall (1999) find that, despite the series typically providing evidence
against seasonal integration, models based on seasonal differences produce
forecasts that are at least as accurate as those based on deterministic
seasonality. Clemens and Hendry (1997) conclude, from their analysis, that
imposing seasonal unit roots and using the model based on seasonal
differences may improve accuracy even if the imposition is not warranted
according to the outcomes of unit root tests. As it has been shown also in Paap
et al. (1997), models based on seasonal differences improve forecast even in
presence of structural breaks occurring during the forecast period. Therefore,
we parameterize our VAR in seasonal differences.
The model has been specified with reference to the non seasonal adjusted
series and, in its more general formulation, takes the form:
∆∆12 yt = β ∆12 yt −1 + ∑ γ j′ ∆∆12 yt − j + φ ′ dt + ε t
′
13
j =1
17
(5)
where ∆ = (1− L) ,
∆12 = (1 − L12 ) , L
is the usual lag operator such that
Lzpt = zt − p , yt = (IPIt ,TKt ,TTPt , XQWt ) , and dt are the deterministic components.
′
As discussed in the previous sections, we have considered four variables for
the forecasting exercise. These variables are those which best represent the
path of the series we want to forecast. Combining in different ways these
variables we have obtained four VAR models which we, now, describe in
detail.
The first one of them (VAR1) consists of three variables: imports, which is
the series that we are trying to forecast, exports and the series of goods
transported on railways. The deterministic part considers the correction for
working days and two seasonal dummies (January 1993 and December 1996).
The second one (VAR2) is made up of imports, the series of tons of goods
transported on railways, and the series of future production prospect released
monthly by ISAE. While the first two equations of the VAR model include
the trading days, the third one contains only a seasonal dummy (November
1992).
The third model (VAR3)is formed by the series of imports, exports and the
industrial production index released monthly by ISTAT. Trading days are
contained in the deterministic part. It has not been necessary to use any
dummy variable.
The last one (VAR4) is different from the others. It is made up of four
variables: imports, exports, tons of goods transported on railways and future
production prospect. It contains, in its deterministic part, the working days’
correction.4
The large use of dummy we have made has its practical justification in the
presence of many outliers in the data. Moreover the use of dummies to correct
the anomalous data is a very spread practice in this kind of literature. The
deterministic part include also, in some cases, the variables ∆12log (TDt ) and
∆12log(TDt −1 ) , with TDt the number of trading days in month t . The number of
trading days significantly influences manufacturing activity. The use of the
lagged value is not very common in practice, but, in presence of particularly
4
We have used the expressions working days’ and trading days’ corrections with the same
significance only for simplicity.
18
unfavorable (favorable) trading days configurations, it is legitimate to expect
that firms tend to compensate lower (higher) realized production in the
following month. In order to include the working days’ correction in the
deterministic part of the models, we have tested the significance of such
regressors in each equation of the VAR models. The estimated coefficients of
the two variables, are, generally, both highly significant, and seem to confirm
this point of view.
As it is known, models like these are subject to the course of dimensionality:
the number of parameters grows as the square of the number of variables
times the maximum lag contained in the more general specification of the
model. For this reason the VAR models have been sequentially simplified to
obtain a more parsimonious parameterization using a "general-to-specific"
reduction (Krolzig, 2000; Krolzig and Hendry, 2001). Nevertheless, by
reducing the complexity of the VAR, it is necessary to ensure,
simultaneously, that the parsimonious subset VAR will contain all the
information embodied in the unrestricted one. To assure this, in each step of
the reduction, a statistical test is made. The reduction procedure stop when it is
not possible to eliminate another variable without loosing information from
the general model. Table 8 reports the lags’ structure after the reduction.
VAR1
VAR2
Lags’ structure 1, 3, 4, 6,12,13 1, 2, 3, 5, 8,12
VAR3
1, 2, 5, 9
VAR4
1, 2, 3, 5,12
Table 8. Lags’ structure after the reduction
Only one model represents a significative parsimonious version of the more
general one. In many cases, even if the restricted VAR is more parsimonious
that the starting one, it is still rather highly parameterized including a large
number of lags. It is interesting to notice that in all, or nearly, the restricted
specifications are present the lags from 1 to 3, which capture the
autoregressive components of the model, and the lag 12, which is
characterized from the seasonal components.
3.1. Reduction diagnostics
The main statistics and diagnostics of the VAR models estimated over the
period 1990.01-1999.05 are reported in the following tables. The tables report
the standard error of each equation in the VAR (σ ) , the correlation of actual
and fitted values ( ρ ) , the p-value of the LM test for residuals autocorrelation
up to the twelfth order (AR 1-12), and the p-value of the test for residual
19
normality (Normality). The lower part of the tables reports the p-values, in
their F-form, of the parameter constancy forecast tests. The first one of them
does not consider parameter uncertainty.
σ
ρ
Normality
0.040
0.867
0.163
dd12 lmqw
0.253
0.048
0.816
dd 12 lxqw
0.101
0.718
0.428
dd12 tk
0.355
VAR
Parameter stability test on the forecasting interval 1997.06 − 1999.05
0.287
FΩ
FV( e )
FV( E )
AR 1─12
0.737
0.167
0.776
0.639
0.766
Table 9. Main VAR diagnostic: estimation period 1990.01 − 1999.05
σ
ρ
Normality
0.042
0.860
0.666
dd 12 lmqw
0.048
0.849
0.105
dd12 lxqw
0.281
0.098
0.760
dd 12 tk
0.314
VAR
Parameter stability test on the forecasting interval 1997.06 − 1999.05
0.225
FΩ
0.621
FV( e )
FV( E )
AR 1─12
0.864
0.545
0.568
0.658
Table 10. Main VAR diagnostic: estimation period 1990.01 − 1999.05
All the VAR’s reduced specifications show good diagnostics over the
considered estimation period. In particular, the correlation of actual and fitted
values seems to be a strong result. In general, this diagnostic satisfies our
expectation about the good specification of the models. The tests for
parameter consistency, calculated over the forecast evaluation sample, do not
reject structural stability. They seem to be very robust except for the first one
( FΩ , which does not consider parameter uncertainty).
20
σ
ρ
Normality
0.056
0.723
0.145
dd12 lmqw
0.173
0.051
0.759
dd 12 lxqw
0.035
0.827
0.758
dd12 tk
0.383
VAR
Parameter stability test on the forecasting interval 1997.06 − 1999.05
0.992
FΩ
FV( e )
FV( E )
AR 1─12
0.284
0.159
0.186
0.998
0.999
Table 11. Main VAR diagnostic: estimation period 1990.01 − 1999.05
σ
ρ
Normality
0.045
0.852
0.736
dd 12 lmqw
0.043
0.856
0.414
dd12 lxqw
0.176
0.100
0.779
dd 12 tk
0.044
0.888
0.665
dd12ttp
0.675
VAR
Parameter stability test on the forecasting interval 1997.06 − 1999.05
0.183
FΩ
FV( e )
FV( E )
AR 1─12
0.240
0.125
0.895
0.826
0.722
0.798
Table 12. Main VAR diagnostic: estimation period 1990.01 − 1999.05
21
0.2
VAR4 one step ahead
import
ARIMA one step ahead
0.2
0.1
0.1
0.0
0.0
-0.1
-0.1
2000
0.2
2001
VAR4 three steps ahead
import
2002
ARIMA three steps ahead
2000
0.2
0.1
0.1
0.0
0.0
-0.1
-0.1
2000
2001
VAR4 two steps ahead
import
2002
VAR4 four steps ahead
import
2000
ARIMA two steps ahead
2001
2002
ARIMA four steps ahead
2001
2002
Figure 2. Forecast: graphical analysis.
4. FORECAST EVALUATION
In this section we evaluate the forecasting ability of the four VAR models as
opposed to an ARIMA model. We are interested to investigate not only if all
the VAR models offers a best prediction with respect to the benchmark
produced by the ARIMA, but also which of the VAR models produce a best
prediction of the Italian import of goods. To make the evaluation more
interesting, the ARIMA model has been enriched with a deterministic part
that includes trading days and Easter effects. This model has been estimated
recursively by maximum likelihood and the forecasts have been produced
using TRAMO:5 a model like this, represent a good benchmark very difficult
to exceed. The comparison has been made over a fairly long period (1999:72002:5).
5
For more details see Gómez and Maravall (1998), Maravall (1995).
22
STEPS AHEAD
VAR1
VAR2
VAR3
VAR4
ARIMA
1
4.33
3.51
3.83
2.82
4.03
3.14
3.75
2.87
5.57
4.14
2
4.71
3.79
3.95
3.15
4.37
3.28
3.68
2.99
5.85
4.47
3
5.26
4.40
4.37
3.65
4.80
3.86
4.13
3.49
6.00
4.74
4
6.05
5.29
4.92
4.08
5.31
4.32
4.83
4.10
7.01
5.52
Table 13. Forecast evaluation diagnostics. For each
model the first row reports the root mean square
error (RMSE), the second one indicates the mean
absolute error (MAE)
In Table 13, the root mean square error and the mean absolute error in each
steps ahead of the forecast are reported for each VAR model and for the
benchmark produced with the ARIMA model. It is easy to see that each VAR
model presents best results in term of prediction evaluation if compared with
the ARIMA. In some cases, the difference began very large (in the order of
two points percent) in each step. The best performance seems to be that of the
VAR model made up of four variable (VAR4). At the first step, the root
mean square error of this VAR model (3.75 percent) is about two point per
cent above of the ARIMA’s one. This difference remain in all the other steps.
Similar consideration could be made about the mean absolute error.
5. FORECAST ENCOMPASSING TESTS
In order to compare the predictive accuracy of our models, we have
performed some econometric tests. The forecast encompassing tests are a way
to compare the quantity of information contained in two models. In other
terms, if a model encompasses another, we can not conclude that it has a great
quantity of information, but only that it contains a part of information that is
not in the other model. Nevertheless, it could be true that the second model
(the one which is encompassed), contains itself a quantity of information
which is not contained in the first (the one which encompasses). For this
reason, it is necessary to compare the models one with another and viceversa.
In this paper, we have used two different ways to perform forecast
encompassing tests: one was firstly proposed by Diebold and Mariano (1995),
the second, is due to Fair and Shiller (1990). In the following, we shortly
explain the way in which the two different tests perform.
23
Diebold and Mariano
n
Suppose one has two series of n forecasts each to be compared. Let {eit }t =1 be
dt = eit (eit − ejt )
an arbitrary function. The null hypothesis of equality of expected forecast
n
−1
performance is E (dt ) = 0 . It is natural to consider d = n ∑ t =1 dt , so that
h − step ahead forecast error deriving from model i . Denote by
n(d − µd )→d N(0, 2π fd (0)) , where µ d is the population mean of d t , and
f d (0) is the spectral density of d t at frequency zero. Diebold and Mariano
(1995) propose basing the test of equal forecasting accuracy on
(6)
which, under the null, tends to a standardized normal distribution when
is a consistent estimate of f d (0) . In order to correct for the size
distortions noticed in the test based on DM , Harvey et al. (1989,1997,1998)
propose modifying the test in this way:
n + 1 − 2h + n−1h(h −1) 2
∗
DM =
DM .
n
1
(7)
Using this statistics comport that, under the null, forecast i encompasses
forecast j and E(dt ) = 0 . On the other hand, under the alternatives, forecast i
could be improved by incorporating some of the features present in forecast
j . In this paper, we have used the DM ∗ version of the test. In order to obtain
a consistent estimate of fd (0) , we use an unweighted sum of the sample
autocovariances up to h − 1 , of the form:
with γ k the lag −k sample autocovariance.
Fair and Shiller
To better compare the performance of the models, we also propose another
exercise of forecast encompassing. This second approach we refer to, was
firstly proposed by Fair and Shiller (1990). The main idea is to compare the
24
information contained in the forecast produced by the different models taken
two at a time. In other words, the approach is based on the comparison of the
series forecasted across different models only through a simple regression of
the form:
∆12 yt =α + βM1( yˆ M1,t − yt−12 ) + βM2 ( yˆ M2,t − yt−12 ) + et ,
(8)
where yˆ M 1,t and yˆ M 2,t are the forecast obtained using model M 1 and M 2
respectively. If only one of the two models contains relevant information, the
corresponding estimated coefficient will be significant.
M i /M j
ARIMA VAR1 VAR2
ARIMA
VAR1
VAR2
VAR3
VAR4
2.527
(0.016)
0.121
(0.904)
-0.596
(0.554)
0.220
(0.826)
2.551 3.244
(0.015) (0.002)
3.047
(0.004)
3.756
(0.000)
0.559 1.208
(0.579) (0.234)
0.471 0.095
(0.640) (0.924)
VAR3
VAR4
3.406
3.544
(0.001) (0.001)
2.910
3.364
(0.006) (0.001)
3.190
0.435
(0.002) (0.665)
1.473
(0.149)
0.956
(0.345)
Table 14. Modified Diebold-Mariano test. DM ∗
statistics evaluated for the one-step ahead forecast and
their p-value under the null are reported. The null
hypothesis is that the forecasts produced by model M i
encompass those produced by model M j .
Main results of forecast encompassing tests
In this section we look at the main results of the forecast encompassing tests
(Tables 14 - 17).
The test of forecast encompassing based on Diebold and Mariano statistics,
made up only for one and two steps ahead, show very strong results. The null
hypothesis is that the forecasts produced by model M i encompass those
produced by model M j . From our viewpoint it seems relevant to note that in
no cases the ARIMA projections seem to embody some piece of information
that would be useful for improving both the one-step and the two-step ahead
VAR forecast. Conversely, except for the VAR1, our models contains
information not contained in the ARIMA forecast. In some cases, as for the
25
VAR4, this evidence is particulary strong, but, in general, seems to be
strongly confirmed by the p-values. Between our models, the VAR4, is still
that which contains the greater part of information; in fact, it encompasses all
the other models, while is encompassed only by the VAR2. Thus, the latter
result has to be considered very anomalous as the VAR2 encompasses only
the VAR4 and not encompasses the other VAR models at one-step ahead
forecast. At the shorter horizon (one step ahead), the VAR2 seems to be the
once which incorporate less information with respect to the others. At the
longer horizon (two steps), it is more difficult to do similar consideration.
M i /M j
ARIMA VAR1 VAR2
VAR3
VAR4
3.214 4.051 4.108 4.806
(0.002) (0.000) (0.000) (0.000)
0.496
3.649 3.090 5.683
VAR1
(0.622)
(0.000) (0.003) (0.000)
0.718 1.630
2.339 0.947
VAR2
(0.477) (0.111)
(0.025) (0.349)
-0.611 0.366 1.804
1.985
VAR3
(0.545) (0.716) (0.079)
(0.054)
0.225 0.468 -0.453 0.623
VAR4
(0.822) (0.642) (0.652) (0.537)
Table 15. Modified Diebold-Mariano test. DM ∗
statistics evaluated for the two-steps ahead forecast and
their p-value under the null are reported. The null
hypothesis is that the forecast produced by model M i
ARIMA
encompass those produced by model M j .
M 1 /M 2
VAR3
VAR2
VAR1
VAR4
α
βM 1
βM 2
α
βM 1
βM 2
α
βM 1
βM 2
α
βM 1
βM 2
ARIMA VAR3 VAR2 VAR1
0.10
2.42
−1.85
−0.22
2.40
0.27
−0.08
0.20
2.48
0.33
1.12
−1.89
−1.67
0.10
1.47
−0.98
0.47
0.45
1.29
−0.61
−0.75
1.37
1.34
0.20
−0.93
−0.12
−1.01
0.28
−1.57
2.12
Table 16. Predictive accuracy tests (Fair-Shiller):
one step ahead forecast
26
M 1 /M 2
VAR3
VAR2
VAR1
VAR4
α
βM 1
βM 2
α
βM 1
βM 2
α
βM 1
βM 2
α
βM 1
βM 2
ARIMA VAR3 VAR2
0.35
2.38
−1.51
0.04
2.05
0.51
−0.44
0.53
2.66
0.62
1.38
−1.33
1.01
0.74
1.81
−1.42
−0.66
−0.95
0.15
3.05
3.06
0.21
0.48
0.21
−2.41
VAR1
−0.56
−2.62
0.50
−2.75
3.63
Table 17. Predictive accuracy tests (Fair-Shiller):
two step ahead forecast
The results for the two-step ahead are very similar and could be commented
in the same way. This evidence is only in part confirmed by the approach due
to Fair and Shiller. In fact, using this methodology, the VAR4 does not
encompass the ARIMA benchmark, which is, otherwise, encompassed by all
the other VAR models.
27
CONCLUDING REMARKS
In this paper, we evaluate several econometric models performing shorthorizon forecasts of Italian imports of goods. Year-to-year growth rate of the
monthly seasonally unadjusted series is the variable to predict. For the Italian
case, imports mainly consist of raw materials and intermediate goods. For this
reason, it can be taken as a significant leading indicator of the aggregate
business cycle. This feature, though extremely useful in assessing very shortrun dynamics of Italian economy, cannot be properly exploited due to the lack
of the availability of the statistical information.
Preliminary analysis on the series used in our forecast exercise have been
carried out. Among them, a set of nonlinearity tests have been applied to the
series of Italian total imports taken in seasonal differences. The null, that the
process is a linear autoregression, has been tested against the alternative of a
nonlinear self-exciting threshold autoregressive model (SETAR). A strong
evidence of linearity of the series of import has appeared, so stylized VAR
(VectorAutoregression) models have been specified along a restricted set of
variables (industrial production, exports among others). VAR forecasting
ability has been evaluated as opposed to that of a linear univariate benchmark
(ARIMA) model. Main forecast diagnostics and two types of forecast
encompassing tests have been presented. The principal result obtained
concerns the superiority of the VAR models which systematically outperform
the ARIMA benchmark model.
28
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