An Econometric Time Series Model Analysis: Textile Sales in Japan 1995-2014
School of Economics, University of Sydney
ECMT6003
Applied Business Forecasting
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Date?
Abstract
This project uses time series analysis techniques to elaborate its significance as an analysis and forecasting tool in the econometric space. It analyzes the variations and changes in seasonality in the sales output of a textile firm in Ottawa, Japan, over a period of two decades. The monthly net sales of fabric and textile are recorded for the period from January 1995 to December 2014. The data was sourced from the R Console database for analysis and forecasting needs. In this short study, three key factors are investigated, and conclusions are drawn based on their response to the available data. First off is the fluctuation of the volume of textile sales over a period of 20 years. Seasonal effects on the production and consumption of textile products were also investigated. Lastly, the most important of all is the forecast for the company’s volume of textile sales over the next four years. An econometric time series model with macroeconomic variables is conducted. Non-stationary time series was made stationary by applying appropriate statistical tests to transform the time series to stationary. After applying these tests, the time series became stationary and integrated of order Box-Jenkins procedure used to determine ARMA models. Performances by the chosen ARMA model are verified based on classical statistical tests and forecasting. By applying the predictive models proposed by this study, the textile company stands to enjoy the liberty of adjusting to future market uncertainties due to changes in seasonality and planning appropriately to maximize profits both in the short run and in the long run. In the concluding part of the study, prospective issues about sales projection and its impact on the textile industry are discussed, and a few suggestions for future studies.
Key terms
ARIMA model, Stationary, Box-Jenkins procedure.
Introduction
Japan’s renowned textile expertise, exquisite and creative fabric design, innovative mass production, and unparalleled technology advances are at a turning point in the early twenty-first century. During the past few decades, market forces such as high labor costs and exchange rates have caused extensive outsourcing and relocation of fabrication to other Eurasian countries, especially China, and increases in the importation of fibers and fabrics. This is a condition well known to other major industrialized fabric-producing countries. However, Japan’s case is particularly poignant, given its rapid rise to preeminence and its scientific advances in fiber, fabric, and apparel making. Recent generations of Japanese fashion designers and fabric creators, often in collaborations, have realigned aesthetic paradigms to wide international acclaim by showing daring new clothing construction in fabrics that seemingly defy gravity and previous performance and appearance
limits.
Like most indispensable sectors in the economy, the textile industry in Japan poses as a major income earner for an economy. In the last two decades, the fashion industry has made a drastic leap towards ensuring a steady supply of textile products to the ever-growing global market. Textile products are widely used to perform a wide range of undertakings. Through textiles, Japan’s economic boom was witnessed in the late 1900s into the 21st Century when the West has become complacent in the sector due to the heightened industrial revolution. Japan boasts a rich range of indigenous textile products, and it is noted that they consume over 60% of its textile products and raw materials locally.
The monthly textile sales data is used to enhance productivity in other related areas of specialization in the economy, such as industry, research institutions, finance, and other fields. The forecasting economic model is an essential component of any company’s economic decision-making process. The textile monthly net sales forecast is necessary for policymakers to forecast a feasible and productive economic model. For these reasons, this paper investigates the effectiveness of the Sales model for the textile industry in Japan during this period in the quest to address pertinent concerns of forecasting short term projections of sales, identifying and explaining the seasonality of monthly sales and its impact on the general profitability of the textile industry. It further aims to analyze the time series econometric model of the country’s sales variable in the 20th and 21st Century interface.
Time series are analyzed to understand the underlying structure and function that produce the observations (Box et al., 1996). Understanding the mechanisms of a time series allows a mathematical model to be developed that explains the data so that prediction, monitoring, or control can occur (Yang, 2009). Examples include prediction/forecasting, which is widely used in economics and business. Monitoring of ambient conditions, or input or an output, is common in science and industry. Time series analysis aims to identify data patterns and trends and explain data modeling and forecasting. Two principal approaches are adopted to maintain time series analysis, depending on the time or the frequency domain. Several procedures are used to analyze data within these domains. A useful common technique is the Box-Jenkins ARIMA method (Box et al., 1996), which can be used for univariate or multivariate data set analyses. The ARIMA technique uses moving averages (MA), smoothing, and regression methods to detect and remove data autocorrelation.
Many statistical tests are used in time series models to make it a stationary series and integrated; thus, the Box-Jenkins procedure is used for the determination of ARMA, and the OLS method is used to estimate the model parameters. In the following sections, the techniques that are useful for analyzing will be identified. By conducting time series analysis, we aim to achieve the following goals: Identify patterns in correlated data trends and seasonal variation, understanding and modeling the data, prediction of short-term trends from previous patterns, seeking to explain how a single event changes the time series, and finally to comprehend how deviations of a specified size indicate a problem (Brockwell and Davis, 2002)
Methodology, Collection, and Summary of data
The time series analysis methodology comprises the following steps: constructing a data model for that time series and predicting a near-future set of values. The autoregressive model is one of a group of linear prediction formulas that attempt to predict an output of a system based on the previous outputs and inputs, such as:
Y(t) = b1 + b2Y(t-1) + b3X(t-1) + et,
where X(t-1) and Y(t-1) are the actual value (inputs) and the forecast (outputs), respectively. These types of regressions are often referred to as Distributed Lag or Autoregressive Models among others.
A model that depends only on the system’s previous outputs is called an autoregressive model (AR). In contrast, a model that depends only on the system’s inputs is called a moving average model (MA). Of course, a model based on both inputs and outputs is an autoregressive-moving-average model (ARMA). Note that by definition, the AR model has only poles, while the MA model has only zeros. Deriving the autoregressive model (AR) involves estimating the model’s coefficients using the method of least squared error.
Autoregressive processes as their name implies, regress on themselves. If an observation made at time (t), then, p-order, [AR(p)], autoregressive model satisfies the equation:
X(t) = A0 + A1X(t-1) + A2X(t-2) + A2X(t-3) + . . . . + ApX(t-p) + et,
where et is a White-Noise series.
The present value of the series is a linear combination of the last few series’ last values, including an error term, which integrates something new in the series at time t that is not clarified by past values. It is like a multiple regression formula, except it is regressed not to independent variables but previous values.
The Box-Jenkins methodology
The Box-Jenkins methodology (Box et al., 1996) is a five-step procedure for identifying, selecting and assessing models for a type of time series data. These steps include:
Time series stationarity. A time series is stationary if both its mean and its variance remains constant through time. Classical Box-Jenkins ARMA models only work satisfactorily with stationary time series.
Identify a (stationary) conditional mean model for underlying data. The sample autocorrelation functions (ACF) and partial autocorrelation functions (PACF) can help with this selection. For an autoregressive (AR) process, the sample ACF decays gradually, but PACF cuts off after a few lags. Conversely, for a moving average process (MA), the sample ACF cuts off after a few lags, but the PACF decays gradually. If both the ACF and PACF decay gradually, consider an Auto-Regressive Moving Average (ARMA) model. (Box et al. 1994)
Model Specification stage, and estimation of the parameters required.
Model checks for goodness-of-fit using methods such as Proportion of variance explained by model or Correlation between actual and predicted. Residuals should be uncorrelated, homoscedastic, and normally distributed with constant mean and variance.
Forecasting: The model can forecast or generate simulations overtime after checking its goodness of fit and forecasting ability.
Adopting the ARIMA (auto-regressive, integrated, moving average) method iteratively, to best-fit time series data, then auto-regressive component (AR) in ARIMA is designated as p, the integrated component (I) as d, and moving average (MA) as q. The AR component represents the effects of previous data observations. The I component represents trends, including seasonality (Boshnakov,2016). Furthermore, the MA component represents the effects of previous random shocks (or error). To fit an ARIMA model to a time series, each model component’s order must be selected. Usually, a small integer value (usually 0, 1, or 2) is determined for each component.
Methodology on Collection of Textile Sales data
The total net monthly textile sale equals the total amount of revenue collected from selling textile products both finished and unfinished less the running and the production expenses. This also includes corporate tax obligations. These figures are in millions of dollars, thereby giving a vivid picture of how business in the textile industry flourished.
The annual percentage sale of textile products and other related products at market prices is based on the constant local Japanese currency. Aggregates are based on the stable U.S. dollar. As pointed out by the World Bank, the Net annual textile sales in Japan were worth 27.156 billion US dollars in 2011. It represents 0.03 percent of the world economy. Through the 1960s, Japan’s textile exports were profitable. However, its market share was already then beginning to suffer from lower-wage competition from Hong Kong, Korea, and Taiwan, the so-called Newly Industrializing Economies (NIE). The acrimonious 1969-72 Japan-U.S. textile negotiations were followed by steep increases in raw materials, labor, and fuel costs, amid growing competition from the NIE. Japanese companies responded by developing new, value-added high-technology products and equipment. In an inversion of events at the beginning of the century, they entered into agreements to share their own patents and licenses with NIE countries. Through the last quarter of the twentieth century, Japanese companies launched innovative products embracing Kansei concepts. Many have already become commercially successful, and others are mainstays in projects for sustainable development.
Data Summary
The data was obtained from the R console database, and this data was converted to time series data through a series of commands. Because of this data’s univariate nature, there was little preparation of the data as far as cleaning and filtering of the unwanted data are concerned. The data was then viewed in the R consoles output window, and a summary of the data obtained. The following is our time series data illustration from the analysis software.
Figure 1: An illustration of the monthly textile sales in Japan over a time
The data was then plotted for visualization. The diagram below shows the resulting plot.
Figure 2: A plot showing the fluctuation of Sales over a time
Figure 3: A boxplot of Sales over a time
From the foregoing, by empirical analysis, it can be noted that the data from the sales were over time series data and that it clearly shows the seasonality and variation of textile sales over time for twenty years. Meaning that the data is non-stationary. Nonetheless, from the above two illustrations, we can note a few observations before delving deeper into the advanced analysis. It is seen that the yearly trend shows that the amount of sales has been increasing without fail. This is explained by the upward trend taken by the graph. Additionally, the variance and the mean sales amounts in June and December are higher than all the other months. Although the mean value of each month differs, their variance is relatively small. Therefore, we can confirm the presence of a strong seasonal effect having a cycle of 6 months.
Data analysis
Time series are investigated to clarify the essence of the fundamental structure and functionality of the mechanism that generates the observations. In this section, the monthly textile sales dataset from Ottawa, Japan, which includes the average monthly sales of textiles and its products in millions, will be investigated from 1995 to 2014. Owing to the nature of the non-stationarity of our data, there is a need to remove this attribute to allow us to run a predictive analysis using an appropriate model. There are three ways this can be achieved: By Detrending- Here, the trend is removed from the time series, for instance, if the equation of my time series is:
x(t) = (mean + trend * t) + error
We ignore the portion in the bracket and formulate our model using the rest of the equation. By Differencing-This is the most popular way to remove non-stationarity. Here we try to model the differences of the terms and not the actual term. For instance,
x(t) – x(t-1) = ARMA (p, q)
This is known as the Integration part in AR(I)MA. The three important parameters are;
p: AR, d: I, q: MA
Lastly, by Seasonality. Seasonality can be integrated seamlessly into the ARIMA model directly. An autocorrelation Function plot is constructed to obtain the (p, d, q) values necessary for forecasting. Using our time series data to obtain non-stationary points, we determined the natural log of all the values in our data set the used the ACF (log(x)), where x is our time series data. The following plot was obtained.
Figure 4: Acf plot of Sales data
Clearly, the ACF chart’s decay is prolonged, which means that the population is not stationary yet. We now know that we now intend to regress on the difference of logs rather than log directly. The command does this, ACF (diff(log(Salesdatats)); this gives us a new autocorrelation plot as follows.
Figure 5: Acf plot of differenced Sales data values.
To achieve stationarity, we differenced two times, thus d = 2, p = 0 and q = 2. Finally, we fit the ARIMA model using the parameters above to obtain our coefficients for the econometric model and allow us to do forecasting of sales. We fitted a seasonal component in the ARIMA model then visualized the 4-year prediction alongside the real textile sales data for contrast. Running MA(0,2,2) using R syntax, the following output was obtained but first, we run the command:
> MA(2)_model<-arima(log(Salesdata),c(0,2,2),seasonal=list(order=c(0,2,2),period=12))
> coef(MA(2)_model)
ma1 ma2
-1.99333697 -0.99427658
The proposed econometric model for this data is thus;
Xt = εt -1.993εt-1 – 0.994εt-2
> pred<-predict (MA(2)model, n.ahead=4*12)
> pred1<-2.718^pred$pred
> data<-round (pred1, digits=1)
> data
The forecasted four-year monthly textile sales spanning between January 2015 to December 2018 is as shown below.
We obtain the figure below by extrapolating this four-year predicted sales data onto the original data for justification.
Figure 6: A plot showing an extrapolation of the forecasted textile sales for 2015, 2016,2017, and 2018 using MA(2) model
Interpretation and Analysis
We construct the Autoregressive Integrated Moving Average (ARIMA) model of the average monthly textile sales in Japan series using the Box-Jenkins technique by using a linear equation. Basing values on the parameters analyzed, it was found that the ARIMA (0, 2, 2) is the most accurate model for the dataset. This, in other words, is another form of the MA(2,2) model, which proved to be much more accurate and effective. The moving average model was selected as the ultimate model. We provide a system of estimating and prediction based on data that could be relevant and useful to government and business organizations.
Using the proposed model, it has been established that sales for textile and its products are bound to continue rising considerably in the next four years. This is an essential aspect of prediction, and econometric forecast as the information available to the industry’s policymakers now have the tools and ability to plan in terms of risk assessment and profit or loss projection in the next four years.
This study’s main focus was to seek to shed light on the nature of the time series data that we had in terms of whether there is the aspect of seasonality, a trend in the data, and whether there could be a forecast of the data in the short run. The analysis has been clear in all these aspects. The sales data has two cycles within its calendar year. This is a cycle of six months in each year. Sales evidently increase in volume during June and December. This might be explained by changes in weather and festive season purchasing power. Additionally, the trend apparent in this data is a positive one, which means that as time progresses, sales of textile products increase subsequently.
Conclusion
Econometric modeling is vital in finance and in financial time series analysis. Modeling is, rather literally, the development of depictions of reality. It is important to bear in mind that, considering the significance of the paradigm, it is just a reflection of truth and not of reality itself. Consequently, the paradigm must conform to reality; it is pointless to adapt reality to the model. Models cannot be as reliable as representations. Models say the action is taken only through considerable consideration and contemplation. This will have substantial financial repercussions. The ability to build models demonstrating financial data’s inter-relationship is a core aspect of financial planning and financial forecasting. Models demonstrating association or causation between measures can be used to enhance financial decision-making. For instance, one is more worried about the effects of a slowdown in another country on the domestic stock market if it can be seen that there is a mathematically demonstrated causative effect between another nation’s economy and the domestic economy. A previously accurate model could lose its validity due to changing circumstances, thereby being an incorrect reflection of fact and adversely affecting the decision-ability maker’s to make reasonable decisions.
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