International Journal of Mathematics Trends and Technology (IJMTT) Volume 49 Number 5 September 2017

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1 The Suerosition o Markov Chain and the Prediction o Consumer Price Index Han Song #1, Liu Na #2 Institute o Inormation &Beijing Wuzi University & Beijing & China Abstract) Using China's consumer rice index historical data and the methods o Suerimosed Markov chain to orecast the CPI o the year 2016 is a new channel in the CPI orecast. Firstly,according to the relationshi between China's consumer rice index and inlation, divided the CPI into ive states ; Secondly, using C-K methods to comute the order transition robability matrix; Thirdly, according to the CPI o to orecast the situation o 2014,the CPI o to orecast the situation o 2015;Forthly, comared with the actual situation, it is conirmed that the method o basing on the suerosition o markov chain to orecast consumer rice index is accurate. Finally, using the data rom to orecast CPI this year to rovide theoretical basis or the government's macroeconomic regulation and the olicymaking. Keywords Consumer Price Index; markov chain; inlation Introduction I. INTRODUCTION CPI is an imortant macroeconomic indicator that relects the changes in the rice level o consumer goods and services related to the lie o the residents. It is also an imortant indicator o macroeconomic analysis and decision-making as well as national economic accounting. Generally seaking, the level o CPI directly aects the issue and the eorts o the state's macroeconomic measures, such as whether the central bank ranayama, whether or not to adjust the deosit reserve rate etc.at the same time, CPI also aect the caital market (such as stock market, utures market, caital market, inancial market) changes indirectly. At resent, in our country, there have been many scholars using dierent aroaches to redict the CPI. There are 790 records whoes theme is CPI + rediction rom CNKI academic journals.ci, EI, core journals, CSSCI source. However, most o the 790 records concentrated in time series, neural network, suort vector machine etc. I only ind 7 records which themed CPI + Markov. Five aers use Marko theory to exlain the intrinsic mechanism and eatures o the rice. For examle, Zhu Peijin (2015) [1] constructe a Markov transition autoregressive (MS-AR) model, using the CPI and PPI index rom January 1997 to December 2014 to analyze the nonlinear characteristics and the inner mechanism o rice index.liu Weijiang and Li Yingqiao (2015) [2] used the double stage Markov switching model to identiicate and analyze the internal transer mechanism about the consumer conidence and rice luctuation in the dynamic rocess, the results show that the luctuation o the growth o CCI is stronger than CPI s; Tong Di and Zhang Wenbin (2015)[3] used variable coeicient Markov regime transormation model roves that Wenzhou City inancial institutions loans volatility and inlation had obvious area o asymmetric eect to rivate lending rate. Restrictive Monetary Policy and high inlation lead the rate o olk lending rate at the high interest rate area imroved signiicantly; Liang Zhixiong and Liang Jian (2010) [4] use VAR method to study the long-term relationshi among the rice level, economic growth and monetary olicy. The results show that there is a cointegration relationshi among the rice level, monetary olicy and economic growth. In this aer, the rediction o CPI based on these internal mechanisms and methods. There are only three aers based on Markov method to redict the trend o economic. Ren Xiaotao, Teng Tengei, Yi Zhuoxi (2014) [5] use Markov chain model to analyze the monthly consumer rice index (CPI) since 2002 when China joined the World Trade Organization (WTO). They also construct the two order transition matrix with the changing trend based on the ixed base relative data o CPI, the third order transition matrix based on the CPI comared. Yangchun (2010) [6]used discreted CPI to estimate the state transer matrix and robability matrix, based on the estimation to build HMM model to redict the CPI. The exeriment results show that the resented algorithm is eective. This aer also gives some hel to macroeconomic analysis orecast. Hou Zengyan (2008) [7] use the consumer rice index (CPI) and some other rice index,based on the Markov orecast model, to make an emirical analysis about the luctuation characteristics o the consumer rice index (CPI) and the short and long term trend.the result shows that the rise o rice is a structural rise. In the near uture, monthly CPI will continue to maintain a high growth, but long-term CPI growth beneit rom the low level o growth. The biggest advantage o this aer is easy to oerate.it adot the most easily acquired data and the simlest method to give a high accuracy rediction o CPI. Markov rocess has no atereect, namely,the uture state only related to current stare, it does t have relationshi with the ast. Obviously,CPI is a random rocess that vary with time. It s a good idea to determine whether the number o CPI has the Markov mathematical nature with chi square, and then redict its uture trends by Markov chain. ISSN: htt:// Page 267

2 II. SUPERIMPOSED MARKOV THEORY Markov chain model was irst osed by Andre Marko (A.A.Markov, ) wh is a amous Russian mathematician in the aer o large numbers on the exansion o deendent variables in In the ollowing one hundred years, Markov chain model was alied to dierent ields,such as economic, education, hysics, genetics, comuter and so on. and achieved ruitul results. Comared with other models, Markov chain model dosen t have too high requirements or historical data, it is also unnecessary to look or the relationshi among the actors rom the comlex redictors. We only need to calculate the state transition robability by the history itsel state with gradient characteristics,to redict CPI. A. The Deinition o Markov chain A random sequence {X (n); state sace n = 0} S (discrete), i the n in,,,,... In S, P (X (0) X = =,... X (n) = >0 (X) = X (n+1) = X (0) =, P =,... X (n) = (=P) X (n+1) = X (n)) is called {X (n); n = 0} Markov chain. Inlation rate is a continuous variable, in this aer the use ive segments division method discretization, to obtain the state sace S={1,2,3,4,5}. In 1989 with the samle data o 27 years rom 2015 to that requency matrix o each state is obtained. B. The Deinition o Transition Matrix and Transition Probability Let {X (n); n = 0} or the Marko chain, the state sace is S. For I, J and S, called =P{X (n) (n+m) =j X (n) =i} {X (n); m n = 0} ste transition robability. I the transer robability is indeendent o N, called {X (n); n = 0} or homogeneous Markov chain. By using the transer matrix ormula, one ste transition robability matrix o CPI is obtained. P (n) = (n) or the M ste transer matrix. M ste transer matrix satisies the ollowing roerties: (n) = 0 sigma (n) =1 (I, J, S). C. Karl Kammann Kolmogoro Equation Let {x (n), n esilon} or homogeneous Markov chain, P (n) is its transition matrix, then or any I, j in s, m, R, m+r esilon s: (n) = sigma K esilon s (n) (n + m). By using the C-K equation and one ste transition robability matrix, the two ste, three ste, our ste, ive ste transition robability matrix are obtained 和. D. Markov Theorem Homogeneous Markov chain {X(n);n 0} as state sace o inite order s, i there exists a ositive integer, so that or all I, j in s, >, is called its ergodicity, and limit distribution lim=(,,, ) Among them, lim is the lim=lim*p equations, namely =,j=1,2, N to meet the conditions o >0, =1, the only solution =1, where lim is the stationary distribution o Markov chain. The use o chi square test, test inlation rate range o the data is consistent with the Marko. E. Suerosition o Mark o Theorem The nearly ive years o data and 1-5 ste robability transition matrix or the year CPI was ive times orecast, ive times orecast the robability o suerosition, the maximum robability o state is most likely to occur determined to redict the CPI, according to the state division according to the control back to the original zone CPI, which redicts the end o. III. CPI DATA SELECTION I can monitor the unction o the national economy. It s also an imortant indicator when the government make monetary olicy, iscal olicy, consumtion olicy, wage olicy, social security olicy an The consumer rice index colligates urban and rural consumer rice index, which relect the changes o rice about goods and services which is related to our lie. It relects the trend and degree o changes in rices o consumer goods and services by urban and rural households during a given eriod. It is an imortant indicator, which d account national economic. Thereore, it has imortant ractical signiicance to analyze the consumer rice index and make a shortterm orecasting. A. The Descrition o CPI Data Since our country makes reorm and oening olicy in 1979, our arty transerred her work center to the socialist modernization. The raid growth o macroeconomic, a surge o the investment scale and the tremendous ram-u o sending resulted in serious iscal deicits, the imorts escalation bring about trade deicit, oreign exchange reserves was close to zero quickly. This serious domestic deicit bring the irst currency inlation. During , the essence o aggregate demand is greater than the total social suly because o the enlargement o scale o ixed assets. Tha the wage growth is aster than labour roductivity caused cost-ush inlation. Along with the exansion o the scale o inrastructure construction, the demand o social consumtion and credit. We cannot avoid an overheating o the economy. The second inlation came to our lie; When the tightening olicy adoted by the central authorities during cannot work or the inlation comletely, every olicy muled in This is the actor that increases aggregate demand,maybe it is also the reason or large scale anic urchasing in The third inlation came to the society during In just 10 years rom 1979 to 1989, China exerienced 3 times currency inlation, or six years the inlation is higher than 6. The extreme volatility o inlation cannot be the valid number to redict CPI. Hence, this aer use the CPI rom 1898 to 2015 to redict CPI in ISSN: htt:// Page 268

3 B. Selection o CPI Data We collected annual consumer rice index (%) rom 1989 to 2015 rom the national data network. As shown in table 1. IV. CPI FORECAST EMPIRICAL PROCESS The radical idea o this aer can be divided into the ollowing our stes: First, CPI is related to inlation,it is used to realize the classiications, which dividecpi into ive classiications; Second, determine whether the number o CPI has the Markov mathematical nature by chi square test; Third, build u the transition matrix ; Forth, calculate One-ste, Twoste, Three-ste, Four-ste and Five-ste transition robability matrix by C-K method; Fanally, Predicting CPI o 2014 and 2015 to veriy the new research method. Using suerimosed Markov chain to carry on the orecast o the CPI, you irst need to according to the relationshi between CPI and inlation CPI divided into ive states; secondly, the data o chi square test that consumer rice index series has the Markov roerty; and then ith order transer matrix is constructed to the state ater the division o the data, using c-k equation obtained in one ste, two stes, three, our, ive ste robability transer matrix, inally, the suerosition method to orecast to 2014, 2015, conirm the method easible to 2016 CPI robably range orecasts. A. State Division First, according to the relationshi between the consumer rice index data and the inlation rate, the inlation rate is calculated, and the ive sections o the [8] division rincile is used or reerence, and the interval o the inlation rate is divided. 1) Inlation Rate Calculation The severity o the inlation rate is relected by the inlation rate, which shows the range o the rice o the goods in a certain eriod o time. Inlation rates are generally exressed in the consumer rice index. According to the above table 1 in our country in consumer rice index data, the conversion relationshi between consumer rice index rate o change and the rate o inlation, 1989 to 2015 the 27 years the annual inlation rate, shown in the ollowing table2: 2) State Partitioning Method Drawing on the division method o the state in the historical evolution and rediction o the Chinese consumer rice index by Chen Liang and scholars, according to the rincile o Table 3: 3) Results o The State Division The results o the state division is ollowing Table 4 B. Markov Test According to the corresonding state o the consumer rice index given by Table 4, we can draw the corresonding transition requency matrix o the system state: And then we can get a ste state transition robability matrix o the system: 1/ 4 1/ 2 1/ / 5 2 / 5 1/ 5 1/ 5 0 1/ 7 2 / 7 1/ 7 2 / 7 1/ 7 1/ 4 0 1/ 2 1/ / / 3 Using the ormula, the calculation statistics, as shown in table 5: Calculation o =49.349, at the signiicance level is 0.05, the look-u table derived rom 9 in degrees o reedom when the critical value is Here >16.919, so consumer rice series with Markov and Markov sequence analysis. C. C-K Equation to Solve the Probability Transer Matrix According to table 5, we can calculate the transition robability matrix o the consumer rice index, the ollowing are two stes, three stes, our stes, ive ste transition robability matrix: 1/ 5 2 / 5 1/ 5 1/ 6 0 1/ 5 1/ 3 1/ 4 1/ 5 0 ( 2) 1/ 5 2 / 9 1/ 3 1/ 6 1/ 9 1/ 5 1/ 4 1/ 4 1/ / 4 0 1/ 2 ij ( 3) / 5 1/ 3 1/ 4 1/ 6 0 1/ 5 1/ 3 1/ 4 1/ 5 0 1/ 6 1/ 4 1/ 4 1/ 6 1/ 8 0 1/ 7 2 / 7 1/ 8 3 / 8 1/ 6 1/ 4 1/ 4 1/ 6 1/ 9 1/ 7 1/ 5 1/ 4 1/ 7 2 / 9 ISSN: htt:// Page 269

4 1/ 6 1/ 4 1/ 4 1/ 6 1/ 9 1/ 7 1/ 5 1/ 4 1/ 7 2 / 9 D Year CPI Forecast By 2013, 2012 and 2011 and 2010 and 2009 CPI (consumer rice index), and ront solved ste, two stes, three stes, our ste and ive ste transition robability matrix and overlay Markov chain rediction method o thinking, to 2014 rice reers to the data to redict, as seciied in Table 6 shows: According to table 6, (max) =1.59, in state 3 achieved that in 2014 the consumer rice index or state 3, namely 2014 consumer rice index should be in between is the normal state rom table 1 know, consumer rice index, rediction is correct. E Year CPI Forecast By 2014, 2013, 2012, 2011 and 2010 consumer rice index, as well as the irst to solve the irst ste, the two ste, Table 7 F Year CPI Forecast By 2015, 2014, 2013, 2012 and 2011 CPI (consumer rice index), and ront solved ste, two stes, three stes, our ste and ive ste transition robability matrix and overlay Markov chain rediction method o thinking, to 2016 the rice index number according to the rediction, as seciied in Table 8 shows: method is simle, strong oerability. It is roved that this method has better rediction eect. However, overall model still exist some deects, the redicted results rom the oint o view, it is a range o ossible values, and is not seciic to the CPI value, how to redict CPI recise numerical is still a toic that needs to continue to exlore. Reerence [1]. Zhu Peijin. Ms-ar model based on China's rice index emirical analysis [J]. Zhejiang Financial and 2015,08: author, author. The article [J]. Name o the journal name, year, Vol volume number (issue): the starting age number age termination. [2]. Wei Jiang Liu, Li Yingqiao, Sui Jianli. Network consumer conidence index and the rice luctuation correlation [J]. Journal o Shandong University (PHILOSOPHY AND SOCIAL SCIENCES EDITION), 2015,06: [3]. Tong Di, Zhang Wenbin. The asymmetric resonse o the rivate lending rate to macroeconomic shocks: a case study o Wenzhou [J]. China's economic roblems, 2013,04: [4]. research grou, Huangshi Central Branch o the eole's Bank o China, Cai Zhixiong, Liang Jian. Emirical analysis o the dynamic relationshi between rice level and economic growth, monetary olicy [J]. Wuhan inance, 2012,07: [5]. Teng ly to Zhuoxi. Based on Markov chain model o China's consumer rice index analysis and rediction [J]. Inormation technology, 2014,12: [6]. Xiao Tao Ren, Teng Yang Chun, Liu da. [J]. Modern commercial rediction based on Hidden Markov's consumer rice index, 2010,03: [7]. Chen Liang. The historical evolution and rediction o China's consumer rice index [D]. Jilin University, 2013 [8]. Hou Zengyan. Markov chain in China CPI [J]. Chinese rice trend analysis based on 2008,11: [9]. Lu Yingchao. Study on the dynamic characteristics o inlation and the identiication o the cause o ormation in the oen economy [D]. Jilin University, 2015 According to table 7. (max) =1.46, in state 2 achieved that in 2016 the consumer rice index or hase 2, namely 2016 consumer rice index should be in between , is the normal state, but the need to revent delation. V. CONCLUSIONS Through the orecast o the CPI to or the government to imlement olicy o exanding domestic demand, stimulate consumtion and sur economic growth rovides the basis and enable the government to adjust the macro-control olicy according to the actual situation, the better to romote the sound and raid develoment o the national economy. By suerosition o the Markov chain to carry on the orecast o the CPI, only need to CPI data, the data in the national data network can be directly ound, according to the rincile o suerosition, make ull use o the short-term CPI data, according to the characteristics o randomness o CPI, the discretization using Markov chain analysis, the ISSN: htt:// Page 270

5 Table 1 China consumer rice index (%) year CPI year CPI year CPI data sources : htt://data.stats.gov.cn/ Table 2 The inlation rate o China during Year 1989-Year 2015(%) 年份 IR 年份 IR 年份 IR Table 3 the division o the CPI 状态 级别 区间 1 通货紧缩 x < 0 2 正常 ( 需预防通货紧缩 ) 0 x < 2 3 正常 2 x < 4 4 温和通货膨胀 4 x < 6 5 严重通货膨胀 x 6 ISSN: htt:// Page 271

6 Table 4 The state o CPI 年份 IR 段 Table 5 statistic calculation table o 2 State i1 i1. 1 i2 i2. 2 i3 i3. 3 i4 i4. 4 i5 i5. 5 Total Total Table consumer rice index status orecast Start original state Sace State1 State2 State3 State4 State5 Source /4 0 1/2 1/ /5 2/9 1/3 1/6 1/ /5 2/7 1/4 1/ /6 1/4 1/4 1/6 1/ /5 2/7 1/4 1/6 0 (2) (3) P. i ISSN: htt:// Page 272

7 Table consumer rice index status orecast Start original state Sace State1 State2 State3 State4 State5 Source /7 2/7 1/7 2/7 1/ /5 1/4 1/4 1/ /6 1/4 1/4 1/6 1/ /5 2/7 1/4 1/ /6 1/4 1/4 1/6 1/9 (2) (3) P. i Table consumer rice index status orecast Start original state Sace State1 State2 State3 State4 State5 Source /5 2/5 1/5 1/ /5 2/9 1/3 1/6 1/ /5 2/7 1/4 1/ /6 1/4 1/4 1/6 1/ /5 2/7 1/4 1/6 0 (2) (3) P. i ISSN: htt:// Page 273