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A Study on the Extraction Method of Seasonal Variation

and Cyclical Fluctuation in Time-series Analysis

(SUMMARY)

This paper studies the method of time-series analysis and economic analysis of perishable foods. Two new analysis methods are developed after learning extensively used methods of time-series analysis. One is an extraction method of the variable seasonal index. Another one is an extraction method on the cyclical fluctuation of the variable period and amplitude. Also, the characteristic of time-series change of fresh vegetables, fruits and livestock products is studied.

In chapter 1, the situation of market circulation is analyzed. The selling price has become to be not transparent in recent years, because the circulation quantity of perishable foods outside the wholesale market has increased year by year. In wholesale market, the supply persons are decreasing because the seller groups are expanding by merger of farmer's cooperatives. On the other hand, the retailer has become larger and larger due to the spread of super markets and volume sale shops. The effect of changes mentioned above on the market price was studied. The distribution characteristic of perishable foods, i.e. non-stability and non-elasticity of perishable food prices was explained. Furthermore, all data-series were analyzed using variance analysis with two-way layout to detect whether there were the seasonal variation, secular trend, cyclical fluctuation and irregular fluctuation or not. In the end of this chapter, the supplementary way to lacked numerical numbers in time-series was explained, the four variables of time-series analysis, i.e. seasonal variation, secular trend, cyclical fluctuation and irregular fluctuation were defined and analysis procedure was established.

Chapter 2 summarizes extensively used analysis methods of time-series. In seasonal variation, the extraction method of the fixed and variable seasonal index was explained and the extraction method suitable for time-series of perishable foods was studied. There are many analysis methods of the secular trend. However, in this paper, only five important analysis methods, i.e. moving average line, regression curve of least square method, correction index curve, Gompertz curve and logistic curve were used and their characteristics and problems were studied. It was proved that the regression curve of least square method is most suitable to analyze the secular trend of time-series data of perishable foods. In cyclical fluctuation analysis, period analysis methods of the auto-correlation coefficient, the periodogram of Schuster and the power spectral density, and extraction method of cyclical fluctuation using Fourier series were explained. The effect of the resolvability of period on the lag-window index in the analysis of power spectral density was studied.

In chapter 3, the suitability of the real price and nominal price of perishable foods for time-series analysis was examined. Generally, the price data from marketing price (representative value) in statistical investigation are nominal price data. However, prices, wages, social economic situations and consumer's habits change with time, which will influence time-series data in the long run. The influence of real and nominal prices on time-series analysis was analyzed in the extraction of seasonal variation, secular trend and cyclical fluctuation. The price index, which reflects the economic change of the whole society, has its own seasonal variation, secular trend and cyclical fluctuation. Every perishable food has its own specific economic characteristics in production, circulation and consumption, which affect characteristic of price change. If the real price is used in time-series analysis, the result will not represent real status of price changes of perishable foods. Therefore, time-series analysis of perishable foods price data should be based on nominal prices.

In chapter 4, a new extraction method of the variable seasonal index is developed, which is based on learning the extraction method of the Economic Planning Agency (EPA) method of Japan, and the Census method by the U.S. Department of Commerce, Bureau of the Census. The practical value of this new method on time-series analysis was evaluated. The new variable seasonal index extraction method is called Link Relative Moving (LRM) method in this paper. It is proved by variance analysis with two-way layout and F-test (p≦1%) that the variable seasonal index calculated by LRM method does not contain secular trend, cyclical fluctuation and irregular fluctuation. Also, the variable seasonal index extracted by LRM and EPA method hasn't significant difference according to co-variance analysis. Moreover, the influence of different seasonal index extracted by Link Relative method, LRM and EPA method on cyclical and irregular fluctuation was studied. The amplitude of irregular fluctuation is smaller when using variable seasonal index of EPA method in time-series analysis. When LRM method in time-series analysis is used, variable seasonal index and cyclical fluctuation absorb a part of irregular fluctuation. The variable seasonal index of LRM method has more information of raw data. LRM method as an extraction method of variable seasonal index is promising.

In chapter 5, the stability of LRM method in seasonal index extraction was examined using Monte Carlo experiment. First, the characteristic of irregular fluctuation of perishable foods was analyzed. After the arithmetic random numbers from computer was examined by χ2-test, the random numbers of normal distribution as irregular fluctuation was produced according to Box and M･･ullar method. Next, the analysis series was composed artificially, and synthetic series was analyzed through LRM and EPA method. From the analysis series composition to the extraction of variable seasonal index 10,000 times mathematical experiments were executed to test the reliability of LRM and EPA method. Also, the variable seasonal index of Monte Carlo experiment was examined by means of variance analysis and F-test. Moreover, the average difference between the real value of seasonal index and the extraction seasonal index was analyzed to evaluate extraction power of LRM and EPA method. Result is that the average difference of LRM method is smaller than that of EPA method. The LRM method is identified to have a practical value in the extraction of variable seasonal index because of simple and easy calculation and stability in extracting variable seasonal index.

In chapter 6, the extraction method of the cyclical fluctuation of the variable cycle and amplitude was examined. The cycle and amplitude is supposed to be constant in cyclical fluctuation of Fourier series. However, in the long run, the cycle and amplitude in the cyclical fluctuation of economic time-series change with the production condition, economic policy, consumer's habit and so on. Until now few researches on the variable cyclical fluctuation have been done. In this paper, the cycle and amplitude was assumed to be the function of time, and extraction method of variable cyclical fluctuation by arithmetic mean of section cyclical fluctuations was examined. Also, the functional equation of time for the variable cycle series and the variable amplitude series was computed by the regression analysis of least square method, and introduced into Fourier series to extract the cyclical fluctuation of variable cycle and amplitude. Finally, the significance of the deviation between the variable cyclical fluctuation and the original sample series or the series removed of seasonal and trend fluctuation was tested.

In chapter 7, the seasonal variation analysis of fresh vegetables and fruits was stressed, the characteristics of quantity and price change were studied, and relationship among the different kinds of markets was examined. The production and supply of vegetables and fruits are largely affected by the season, and their market dealings are affected by freshness. In addition, the expenses are high for the storage of vegetables and fruits. In the long run, the price increases with economic growth. In cyclical fluctuation analysis, the cyclical fluctuation was confirmed in the part of vegetables and fruits, and the cycle was detected. Moreover, the fluctuation in the price of vegetables and fruits was compared between the district wholesale market and the central wholesale market, and the characteristic was analyzed.

In chapter 8, the characteristic of livestock products in seasonal variation, secular trend and cyclical fluctuation were studied by analyzing the time-series of the production quantity, wholesale price and retail price. In seasonal variation analysis, the range of monthly fluctuation was examined by calculating fixed seasonal index using Link Relative method. Because the pattern of seasonal variation was changeable, the variable seasonal index was extracted by LRM method. The wholesale price of beef is higher at the beginning and end of the year than other months, although much beef is available in the market during this period. On the contrary, the wholesale price of pork is the highest from April to September. The variable seasonal amplitude of wholesale price of livestock products has become big since 1990. This result indicates that consumption quantity of livestock products have increased at the beginning of the year and year-end in recent years. In the secular trend analysis, the regression curve of least square method was used. In the long run, the suitability of quadratic function is good. The main reason for this result is attributable to the oil shock in the 1970s and the economic depression in recent years. In cyclical fluctuation analysis, the production quantity and price of beef was focused. It was pointed out that the cycle of cyclical fluctuation of beef has been changed in recent years. The period of cyclical fluctuation of beef was detected using the periodogram of Schuster and the power spectral density. It is elucidated that the cyclical fluctuations of all kinds of beef are greatly influenced by the fattened bulls.

In this paper, Except the EPA calculation based on "Micro AGNESS" program, the other analytical methods, including the calculation of the variable seasonal index and the variable cyclical fluctuation were programmed by the author using computer language of N88BASIC.

Li WAN

FEB.26.1999.

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