Proceedings of 7th Global Business and Social Science Research Conference
13 - 14 June, 2013, Radisson Blu Hotel, Beijing, China, ISBN: 978-1-922069-26-9
Predicting Aggregate Retail Sales Using Hybrid ARIMA
Youqin Pan*
This paper demonstrates that aggregate retail sales forecasting
benefits from empirical mode decomposition (EMD). The hybrid
forecasting methods integrating EMD and ARIMA were employed to
predict retail sales during the periods which economic recession
occurred. Both in-sample forecasts and out-of-sample one-period
forecasts show that EMD-ARIMA outperforms the classical ARIMA
model in term of forecasting accuracy.
1. Introduction
Accurate forecasts of future retail sales can help improve effective operations in
retail business and retail supply chains (Chu & Zhang, 2003) since strategic planning
and operational decisions along the supply chain depend on the forecast of retail
sales. Various decisions have been affected by the forecasts of sales, at the
organizational level, retail sales forecasts are critical inputs to many business
decisions such as marketing, purchasing, production (Mentzer & Bienstock, 1996).
Most retailers are eager to reduce their costs and increase profits to stay completive,
accurate forecasts will definitely help retailers to achieve this goal. Due to the global
competition and quick market changes, sales forecasting comes to play an important
role in improving the quality of decision making and improving the performance of
retail companies. Moreover, unstable economic environment has motivated retail
practitioners to explore new forecasting methods to obtain accurate forecasts. For
instance, recent financial crisis in 2008 has brought significant nonlinearity and
irregularity to the economic system. These irregularity or nonlinearity makes it hard
for traditional forecasting models to capture and predict.
In literature, a variety of linear and nonlinear forecasting methods have been applied
to predict retail sales. Comparative studies between linear and nonlinear models in
forecasting retail sales have been extensively studied (Alon et al., 2001, Chu &
Zhang, 2003; Zhang & Qi, 2005). For example, Alon et al. (2001) found that linear
models such as Winters Exponential Smoothing and Autoregressive Integrated
Moving Average (ARIMA) perform well during stable economic conditions. However,
Artificial Neural Networks (ANNs) are superior to the traditional linear models in a
volatile economic environment in which recession or high inflation occurs. It is wellknown that traditional methods (Winters Exponential Smoothing and ARIMA) are
essentially linear. The easy implementation and interpretation make linear models
more popular in retail sales forecasting. If the linear models are able to capture the
underlying data generating process well, they should be the preferred models (Chu
& Zhang, 2003).
Among the linear models, ARIMA is more flexible and powerful in forecasting linear
time series. The ARIMA model is suitable to forecast future retail sales since US
aggregate retail sales usually exhibit strong trends and seasonal patterns. However,
*Dr.Youqin Pan, Department of Marketing and Decision Science, Salem State University, U.S.A.
Email : ypan@salemstate.edu
Pan
ARIMA model becomes inadequate to approximate complex nonlinear systems
(Zhang, 2003) especially when there is a great deal of nonlinearity and irregularity in
the time series. Thus, researchers attempt to use the ANNs to forecast nonlinear
time series. ANN forecasting models have the potential to provide more accurate
and robust solutions for problems where traditional methods cannot be applied
(Zhang & Qi, 2005). It is often used when the true distribution of the demand is
unknown, especially when the demand process exhibits nonlinear activities. Despite
the potentials of ANNs, research has shown that findings are mixed with regard to
whether the flexible nonlinear approach is better than the linear method in
forecasting (Adya & Collopy, 1998). ANNs also have their own limitations, for
example, a great deal of computational effort is required to minimize overfitting. In
addition, they are also sensitive to parameter selection (Xu, et al. 2010). The
prediction performance of ANNs can be significantly different with minor change in
parameter selection (Plummer, 2000; Pan, 2009). This may be one of the reasons
why mixed findings exist in literature.
In summary, among various forecasting techniques, few methods are as popular as
Autoregressive Integrated Moving Average (ARIMA). Liu (2006) argued that ARIMA
models can handle a wide variety of time series patterns and perform well under
many situations. However, when dealing with a great deal of nonlinearity and
irregularity in the time series, the forecasting performance of the ARIMA model
needs to be improved. To overcome the limitations of ARIMA and remain its merits,
this study intends to forecast retail sales immediately after financial crisis using a
hybrid methodology which integrates ARIMA and the empirical mode decomposition
(EMD) technique proposed by Huang et al. (1998). As a nonlinear, non-stationary
data process method, EMD has been widely used in the area of engineering.
Recently, EMD integrating with ARIMA and Neural Networks models has attracted
more attention in forecasting (Yu, Wang, & Lai, 2008; Xu, Qi, & Hua, 2010). In this
study, we will apply EMD to monthly retail sales data to forecast future retail sales in
2009. A sudden drop of the customer demand generates a great deal of nonlinearity
and irregularity in the time series due to the financial crisis in 2008, thus, EMD is a
suitable tool to capture the nonlinear activities. In order to test the robustness of the
EMD-ARIMA method, we use data from January 1977 and December1986 to
forecast retail sales for 1987 since this period was characterized with supply push
inflation, high interest rates, high unemployment and two recessions (Branson,
1989).
The purpose of this study is to demonstrate how robust the EMD-ARIMA model is in
forecasting aggregate retail sales during recession. The remainder of this paper is
organized as follows. Section 2 provides a brief literature review. Section 3
discusses the experimental plan. Section 4 reports the results of this study. Finally,
the conclusions and implications are given in section 5.
2. Literature Review
Alon (1997) found that the Winters models’ forecasts for aggregate retail sales
were more accurate than simple exponential and Holt's models. The results
indicated that Winters’ model is a robust model that can accurately forecast
aggregate retail sales. Alon, Qi & Sadowski (2001) further compared the artificial
neural networks (ANN) and traditional methods (such as ARIMA, Winters
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exponential smoothing and multiple regression) in forecasting aggregate retail sales
under two time periods. One period is from January 1978 to December 1985, the
other one is from January 1987 to April 1995. The results show that Winters
exponential smoothing and ARIMA may perform well during relatively stable
economic conditions while the ANNs is superior to the traditional models in a volatile
economic environment in which recession or high inflation occurs. It confirms that
ARIMA is more powerful and good at modeling trend pattern when economic
condition is stable but it is inadequate to approximate nonlinear-systems during
recession.
Chu & Zhang (2003) also conducted a comparative study of linear and nonlinear
models for aggregate retail sales forecasting. The authors demonstrated that
nonlinear model (neural network model) is superior to regressions models with
seasonal dummy variables and trigonometric variables. They also indicated that the
ARIMA model did not provide good forecasts in 1999 due to under-forecasting, and
the overall best model for retail sales forecasting is the neural network model with
deseasonalized time series data. In addition, they also reported that “direct Neural
Network model performs even worse than the ARIMA model as almost all forecasts
are relatively far below the actual values (Chu& Zhang, 2003, p. 225).” Therefore,
we cannot conclude that advanced nonlinear models such as NN models are always
superior to ARIMA models.
Zhang & Qi (2005) further investigated the application of neural networks in
forecasting time series with strong trend and seasonality because previous studies
on seasonal time series forecasting with neural networks yield mixed results. The
authors found that neural networks fail to capture seasonal or trend patterns
effectively if the data is not deseasonalized or detrended. However, the forecasting
errors will be reduced significantly if the data is preprocessed (such as detrending
and deseasonalization). In the same vein, Kuvulmaz et al. (2005) reached the
similar conclusion that the overall forecasting performance of ANNs is not better
than that of the classical linear methods in predicting retail sales without preprocessing the seasonal data.
Wong & Guo (2010) propose a hybrid intelligent model using extreme learning
machine and harmony search algorithm to forecast medium-term sales in fashion
retail supply chains. They demonstrate that the proposed model outperforms
traditional ARIMA models and two recently developed neural network models.
However, they also observe that the performance of the proposed model
deteriorated when the time series is irregular and random and they argue that “no
univariate time series forecasting model can forecast these abnormal sudden
changes (Wong & Guo, 2010, p. 620).” Thus, it is not clear whether the proposed
model works well with high irregularity and nonlinearity.
In summary, previous studies concentrate on the comparison of linear and nonlinear
models in forecasting retail sales during different time periods. None of these studies
investigates the effect of recessions on the forecasting performance of these
models. Moreover, there are some limitations of applying ARIMA and ANNs in
forecasting retail sales as mentioned above. Thus, it is necessary to investigate the
potential of any new forecasting model in predicting retail sales. Recently, more and
more researchers use the EMD technique to improve classical modeling and
forecasting method. The examples are the EMD-SVM (support vector machine)
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modeling which is proposed by Xu, Tian, & Qian (2007) and EMD-based neural
network ensemble methodology which is developed by Yu, Wang, & Lai (2008).
Due to the capability of the EMD to capture nonlinear activities and the power of
ARIMA model to model linear relationship, we integrate EMD and ARIMA to predict
the retail sales since this hybrid method should effectively capture both linear and
nonlinear relationships in a time series.
3. The Methodology and Model
3.1 Empirical Mode Decomposition
Empirical mode decomposition (EMD) proposed by Huang et al. (1998) is a powerful
tool for analysis of non-stationary and non-linear signals. It has many applications in
the area of engineering such as image processing and fault diagnosis. Recently, it
has gained popularity in forecasting because it is adaptive and applicable to nonlinear and non-stationary data. It assumes that different coexisting modes of
oscillations in a time series may occur at the same time, thus a complicated time
series can be decomposed into a finite and often small number of intrinsic mode
functions (IMFs). Each IMF must satisfy the following two conditions (Huang et al.,
1998):
(1) In the whole data series, the number of extrema (either maxima or minima)
and the number of zero crossing is the same, or differ at most by one.
(2) The mean value of the envelopes defined by local maxima and the envelopes
defined by the local minima must be zero at all points.
Readers who are interested in the detailed sifting procedures of EMD can consult
Huang et al. (1998).
3.2 ARIMA
ARIMA represents an autoregressive integrated moving average and was developed
by Box and Jenkins (Box & Jenkins, 1976). ARIMA model has become one of the
most powerful methods for time series forecasting since its introduction. However,
ARIMA model was designed to handle stationary time series. Thus, non-stationary
time series need to be differenced in order to use ARIMA model. Moreover, ARIMA
models are used as the baseline for forecasting comparison. When forecasts are
generated under a more complicated model such as neural network, they are often
compared with those obtained by an ARIMA model. If the forecasts obtained under
an ARIMA model are still more accurate than the forecasts obtained under a more
complicated model, it often indicates misspecification in the more complicated model
or the existence of outliers in the series (Liu, 2004). In this study, we compare the
forecasting performances of single ARIMA and EMD-ARIMA to see how much
improvement of forecasting accuracy can be achieved by applying the EMD to the
time series of the retail sales during recessions.
3.3 EMD-ARIMA
The EMD-ARIMA basically combines the EMD and ARIMA model to forecast a time
series. First, EMD applies to the original time series to extract IMF components and
the final residue. Second, ARIMA applies to each of the IMFs and the residue to get
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corresponding forecasts for each component extracted by EMD. Finally, the additive
property of the components of EMD decomposition allows us to add all the forecasts
for each component to get the forecast for EMD-ARIMA.
3.4 Data
We use monthly retail sales compiled by the US Bureau of the Census to conduct
this study. Data from two time periods were used. One of the sampling periods
examined is from January 2001 to December 2008 as shown in Figure 1, the
aggregate retail sales exhibit strong trend and seasonal patterns. Research
indicates that seasonal fluctuations which may dominate the remaining variations in
a time series will result in difficulty in effectively dealing with other time-series
components (Zhang & Kline, 2007). Due to this concern, we use adjusted retail
sales data instead of unadjusted retail sales to estimate the parameters of the
forecasting models for the two time periods in which we are interested. Figure 2
clearly shows that US adjusted retail sales from January 2000 to December 2008
contains a trend pattern which ended in the last quarter of 2008 when the financial
crisis occurred. Figure 3 indicates that trend pattern still dominates the time series
for the adjusted data from January1977 to December 1986 even though several
recessions occurred during this time period. It seems that recent recession is more
severe than the recessions in the 1970s. Because aggregate retail sales are
influenced by macroeconomic instability, we expect the forecasts for the second
time period to be more accurate because EMD can capture the nonlinear activities
well if fluctuations are big in the time series.
3.5 Accuracy Criteria of Forecasting
There are a variety of measures of forecast accuracy in the literature. In this study,
we use the mean absolute percentage error (MAPE) to compare and evaluate the
forecasting performance of single ARIMA and EMD-ARIMA models. We use MAPE
because it is not prone to changes in the magnitude of the time series to be
predicted (Alon et al., 2001).
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5
4
x 10
3.5
3
2.5
2
1.5
1999
2000
2001
2002
2003
2004
Year
2005
2006
2007
2008
Figure 1: US Unadjusted Retail Sales from January 1999 to December 2008
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5
3.6
x 10
3.4
3.2
3
2.8
2.6
2.4
2.2
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Figure 2: US Adjusted Retail Sales from January 1999 to December 2008
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4
13
x 10
12
11
10
9
8
7
6
5
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
Figure 3: US Adjusted Retail Sales from January 1977 to December 1986
4. The Findings
4.1 IMFs
Figure 4 and Figure 5 show the IMFs extracted by EMD procedure for the two time
periods. In each figure, the last one is the residue which represents the long-term
pattern in the time series. Other IMFs listed in order from the highest frequency to
the lowest frequency represent the short-term variations or cyclical patterns. In this
case, trend dominates the series, which justifies why we include ARIMA model.
According to Figure 5, although there are significant short-term variations during the
year of 1986, these changes don’t seem to affect the dominant trend significantly.
However, Figure 4 shows that the dramatic drop in the last quarter of 2008 does
impact the long-term trend pattern significantly, which may result in the change of
direction for the trend pattern. Thus, it can be concluded that a great deal of nonlinear activities and irregularities were generated in the time series due to the
financial crisis.
Resi
IMF3
IMF2
IMF1
Signal
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1999
2000
2001
2002
2003
2004
Year
2005
2006
2007
2008
Figure 4: The Decomposition of Adjusted Retail Sales from
January 1999 - December 2008
2009
Resi
IMF3
IMF2
IMF1
Signal
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1977
1978
1979
1980
1981
1982
Year
1983
1984
1985
1986
1987
Figure 5: The Decomposition of Adjusted Retail Sales from
January 1977- December 1986
Table 2 indicates that one-period forecast of EMD-ARIMA improves by 80 % in term
of forecast error reduction when compared to the forecasts generated by ARIMA
model. It is also surprising to notice that there is a 91% reduction of forecasting error
for 12-peroid out-of-sample forecasts using EMD. Thus, EMD-ARIMA should be
promoted in forecasting retail sales especially during severe recession since EMD
can significantly improve the forecasting performance by successfully capturing the
intrinsic modes of the complicated time series. In order to test the robustness of the
EMD-ARIMA model, we applied it on the retail sales data from January 1977 to
December 1986 to predict future sales in 1987. One period out-of-sample forecast
shows that EMD-ARIMA helps reduce 15.3% of the forecasting errors compared to
single ARIMA. However, the 12 period ahead out-of-sample forecasts indicate that
EMD-ARIMA underperform single ARIMA, this could be explained by the fact that
the prevailing trend still dominates the time series in the first time period.
4.2 Period 1 vs. Period 2
The MAPE is smaller in the second forecasting period than that in the first period
except for the 12-period forecasts of ARIMA model. The results confirm our
expectation that the forecasts in the second period will be more accurate due to the
more volatile macroeconomic conditions. The EMD was able to successfully capture
the nonlinear activities right after the financial crisis in 2008. Thus, EMD-ARIMA
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significantly reduces the forecasting error compared to the forecasts generated by
ARIMA.
Although there were two recessions in the first period, those recessions seem to be
less severe than the one starting in 2008. As a result, EMD was able to help reduce
forecasting error compared to the forecasts generated by ARIMA for the first period.
The only exception is that multiple-period forecasts show that ARIMA model is better
than EMD-ARIMA, which could be explained by the capability of ARIMA to
approximate the linear relationship. Since the trend still dominates the time series in
period 1, thus ARIMA model outperforms EMD-ARIMA with regard to multiple-period
forecasts. However, the one-period forecast from EMD-ARIMA is more accurate
than that from ARIMA model. Thus, it can be concluded that EMD-ARIMA
outperforms ARIMA in most cases.
Table 1 : Forecasting Error Measures of Different Models Based
on Period 1 Data (Jan.1977 through Dec. 1986)
Out-of-sample forecast
In-sample
forecast
One-period
12-period
ahead
ahead
forecast
forecasts
MAPE
MAPE
MAPE
Model
6.15
2.49
1.21
Model I: ARIMA
5.21
7.74
0.572
Model II: EMD-ARIMA
Table 2 : Forecasting Error Measures of Different Models Based
on Period 2 Data (Jan.1999 through Dec. 2008)
Out-of-sample forecast
In-sample
forecast
One-period
12-period
ahead
ahead
forecast
forecasts
MAPE
MAPE
MAPE
Model
3.65
17.38
0.91
Model I: ARIMA
0.70
1.52
0.569
Model II: EMD-ARIMA
5. Summary and Conclusions
EMD has proven to be successful at improving the forecasting performance of
ARIMA model during economic recessions. The U.S. aggregate retail sales usually
show strong seasonal and trend patterns. Thus, the seasonal ARIMA model is good
enough to deliver viable performance during non-recessionary periods. However,
when economic recession occurs, EMD-ARIMA should be the right model to forecast
future retail sales because ARIMA fails to approximate the non-linear relationship as
shown in Figure 6. It is EMD that can capture the intrinsic modes of the raw data
series which may be very complicated. Thus, EMD-ARIMA successfully overcomes
the limitations of ARIMA model and significantly improves the forecasting
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performance. Overall, in forecasting retail sales, EMD-ARIMA model proves to be
more efficient and robust.
Our findings also indicate that linear models such as ARIMA can still deliver viable
performance when the macroeconomic conditions are relatively stable. In this study,
the EMD decomposition of retail sales from January1977 to December 1986 (Figure
5) clearly shows that the trend still dominates the time series. Thus, the out-ofsample multiple-period forecasts by ARIMA are more accurate than that of EMDARIMA. However, when the trend cannot dominate the time series for certain time
periods as shown in Figure 4, EMD-ARIMA can significantly reduce the forecasting
error since a lot of nonlinearity and irregularity was involved in the time series of
retail sales due to the recession. Moreover, one-step forecasts may be preferred
under different macroeconomic conditions. This study demonstrates that one-period
forecasts from EMD-ARIMA are more accurate under most cases. In addition,
multiple-period forecasts may not provide better results than one-period forecasts
since they cannot incorporate recently updated information.
This study highlights important managerial and practical implications. Managers and
practitioners should have a better understanding of the characteristics of the time
series and the economic conditions when selecting forecasting models. EMD-ARIMA
is a promising tool to improve the accuracy of forecasts for aggregate retail sales
especially when the economic system experiences great fluctuations such as
recession. Moreover, the implementation of EMD-ARIMA is relatively easier
compared to ANNs because ANNs require greater expertise and skills on the part of
managers and practitioners to use it properly. Furthermore, it is normally difficult to
interpret the estimates from the ANN model. Thus, EMD-ARIMA should be promoted
in forecasting retail sales to help firms reduce costs and increase revenues.
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