mathematical modeling

Paper: Analyze the running trend of American GDP with statistical and probabilistic methods.

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Writing time: June 2008165438+1October.

In this paper, the quarterly change percentage of real GDP in the United States in recent decades is regarded as a discrete random variable, and it is quantified and modeled according to the drastic and slow change range by using statistical and probability methods and Markov chain model, so as to predict the change trend of real GDP in the United States in the next year or two.

Keywords: GDP；; Probability; Statistics; Markov chain; Transfer probability; economic projection

1 Introduction

Probability theory and mathematical statistics are mathematical disciplines that study the objective regularity of random phenomena. Their theories and methods have been widely used in various fields of natural science, technical science and social science, especially in weather forecasting and geological exploration. The famous economist Trygve haavelmo thinks that all economic laws can be described by probability. Various economic data can be regarded as a series of interactive or independent random variables, and the change of economic data is a complex random process. With the global economic integration and financial informatization, probability theory will play an increasingly important role in macroeconomic forecasting, regulation and statistics, providing effective reference data.

Gross domestic product (GDP) is one of the most important indicators to measure a country's economic performance. From the perspective of probability theory, this paper analyzes the changes of real GDP in the United States in recent decades since 1947. Starting from the range and time span of change, the percentage of real GDP change is transformed into a discrete random variable that changes in a finite state space. The transfer process of this random variable in the state space is also a random process of real GDP changing with time, and a Markov chain model of real GDP changing is constructed. Therefore, according to the established probability model, the next shift of random variables is predicted, and the future running trend of real GDP is obtained. The general analysis and prediction process can be described as: data processing->; Statistics and analysis-> establish mathematical model->; Draw a conclusion.

2 GDP analysis and modeling

America is the most developed economy in the world. Studying and investigating the operational indicators of American economic development can not only reveal the characteristics of the American economic cycle itself, but also play a good role in analyzing and drawing lessons from the economic operation, which is helpful to the macroeconomic operation prediction and intervention of all countries in the world. Moreover, the economic indicator system of the United States is also the most complete. As important public information, it is published and revised regularly, which ensures the reliability and sufficiency of data from theoretical analysis.

Gross domestic product (GDP): refers to the total value of all final products and services produced by a country. At present, GDP is a general index used by various countries and regions to measure the comprehensive level of national or regional economic development, and it is the most important and comprehensive barometer to reflect a country's overall economic situation. Generally speaking, GDP refers to nominal GDP, while real GDP takes into account the factors that inflation leads to rising prices, which relatively accurately reflects the economic development of a country. Various GDP indicators provided by the Bureau of Economic Analysis [1] are measured with different weights. In this analysis, the percentage change of real GDP (index, 2000 = 100) compared with the previous period is selected, and more attention is paid to the fluctuation of GDP. The GDP data of the United States is published quarterly. The survey interval is the percentage change of real GDP from the second quarter of 1947 to the third quarter of 2008 (see table 1), which is described as a discrete sequence by mathematical formula: t is the ranking number representing the quarter, starting from zero; X represents the percentage change of real GDP.

Studying the operation process of economic data is also the process of building a mathematical model, which must be based on a large number of data statistics. The percentage change of GDP in 246 quarters in 62 consecutive years can reflect the trend of GDP in the United States in a long period of time, so it can be used as the data basis for the analysis of GDP change in the future [2].

2. 1 intuitive analysis of GDP changes

Because the change of economic variables in economic phenomena is complex, there must be some random "interference", so it is necessary to make some assumptions about the distribution of random variables first. First, use Microsoft EXCEL software to draw the above-mentioned change percentage sequence in the form of scatter plot (see figure 1). It can be intuitively analyzed from the figure that the percentage change of real GDP in the United States has generally experienced the repeated characteristics of "rising-falling-rising-falling" for 62 consecutive years, but the difference is that the time span and the rising or falling range are different. Combined with the history of American economic development, the American economy has experienced-> growth; The random reciprocating characteristics of "recession-> growth-> recession". In the period of economic crisis or economic stagflation, the percentage change of real GDP will fluctuate greatly continuously, while in the stage of steady economic development, the percentage change of real GDP will fluctuate slightly. From this, the trend of economic operation can be inferred from the change range of real GDP.

2.2 build a Markov chain model of GDP changes

Markov process is a theoretical method to analyze stochastic processes. Markov processes with discrete time and state are called Markov chains. Markov chain model is usually used for modeling in statistics, and is widely used in natural biological population process, commodity market share change, weather change and so on. If the probability distribution of the system state at a certain moment is only related to the state at the previous moment, but has nothing to do with the state at the previous moment, then the system conforms to Markov property or has no aftereffect. The change percentage of real GDP is influenced by many external economic variables such as war and macro-control policies, and the change shows random characteristics. Therefore, it can be considered that the percentage change of real GDP in the short term is only related to the change of real GDP at this stage, which conforms to Markov nature.

In order to describe the change range of real GDP percentage, it is necessary to quantify the seemingly random change data. The quantitative definition of range to percentage is as follows:

Status 1: greatly increased (one or more consecutive increases exceeding 7, including the boundary value);

State 2: sharp drop (one or more consecutive drops exceeding 7, including the boundary value);

State 3: slight increase (one or more continuous amplitude increases are greater than 1 less than 7);

State 4: slight decrease (one or more continuous amplitude decreases are greater than 1 less than 7);

It can be seen that distinguishing the variation range between large growth and small growth plays a decisive role in probability statistics, and different quantitative standards will produce different statistical results. In addition, in figure 1, we can see that some adjacent time points have very small change amplitude, which is called interference here. The sequence points whose amplitude of change before and after is less than 1 are regarded as interference signals, and the state of the latter sequence points is approximately considered to remain unchanged. If this small change is counted as a small increase or a small decrease, the effect of the interference signal will be amplified. In this way, the change percentage of real GDP is transformed into a discrete time series that changes in the finite state space of 1, 2, 3, 4. If you only pay attention to the trend and elapsed time of state change, you only need to record the serial number of 134 and the time point of state change. Such a new state sequence is described as: s stands for sort number, starting from zero; T represents the quarterly serial number with changed status; Y stands for state.

The change state of real GDP depicted in the form of Microsoft Excel scatter chart (see Figure 2) can more intuitively observe the change of the change range of real GDP in a limited state space:

By counting the above state sequence Y(t), the number of one-step transitions between states can be obtained, and then the one-step transition probability and one-step transition matrix p can be calculated. In addition, in order to obtain the time span of one-step state transition, it is necessary to calculate the time span corresponding to the state transition, that is, when tn reaches tn+ 1 and the state transitions from Yn to Yn+ 1, the corresponding time span is Sn+ 1-Sn, and the average time span corresponding to all one-step state transitions is obtained by simple averaging method (see Table 3).

Average time span of one-step transition probability of state transition times

State 1 to state 2 10 0.476 2.3

State 1 to state 4 1 1 0.524 2.3

State 2 to State 1 13 0.542 2.2

State 2 to State 3110.4581.9

State 3 to State 2 14 0.304 1.6

State 3 to State 4 32 0.696 1.8

State 4 to state170.1671.3

State 4 to State 3 35 0.833 1.6

A total of 133 times (table 3: statistical results of one-step transfer of real GDP status)

2.3 according to the Markov model to predict the recent changes in GDP in the United States.

The current state of real GDP change is 4. According to the above transfer matrix and the time span of each transfer, we can get the result of the recent state transition, that is, the recent change range of real GDP and the approximate time required.

Average time span of target state transition probability of current state transition step.

4 2 2 0.333 3.4

4 2 4 0.667 3.5

4 3 1 0.292 5.3

4 3 3 0.708 5.2

Table 4: Forecast results of Markov chain model on changes of real GDP

The forecast results given by the model show that the real GDP of the United States is currently in a small decline stage. After two shifts, there will be a slight decline and a sharp decline in the next 3-4 quarters, with the possibility of 66.7% and 33.3% respectively. After the third transfer, there will be a slight increase and a sharp increase in the next 5-6 quarters, with the probability of 70.8% and 29.2% respectively. From this analysis, it is concluded that the American economy will definitely decline in the next 3-4 quarters (currently 2008 1 1 month), and the possibility of a sharp decline is as high as 66.7%; Economic recovery needs to happen in the next 5-6 quarters, and the probability of slow recovery is even greater, accounting for 70.8%. It seems that the economic form of the United States will face a severe test in the next year or two.

3 abstract

Probability theory is a discipline that studies the quantitative law of random phenomena. Statistical analysis of data in financial economy through probability theory can be related to the economic orientation of various countries. It will gradually play an important role in the future economy. Markov analysis is a method to study the changing trend of random events. The change of economic operation data is often influenced by various uncertain factors and is random. If it has no aftereffect, we can use Markov analysis method to analyze its future development trend. The quarterly change percentage of real GDP is a random process with a fixed time interval, and it is more suitable for this kind of application to analyze its change trend with Markov chain model. Firstly, the state space of finite States is divided according to the percentage change of real GDP, and then the one-step transition between States is counted, and then the one-step transition probability matrix of real GDP change is calculated. From this probability matrix and the current state, we can calculate what is the next state of GDP change and what is its probability, that is, the actual GDP change trend in the future.

Any model that simulates natural data will have certain errors, and the difference is only the size of the errors. In this paper, in the data processing stage, that is, the division stage of probability state space, because different quantization standards produce different statistical results, some samples will be lost and some errors will be produced.

The probability analysis process of this paper is only one of many economic operation indicators. The actual economic operation body includes many econometric indicators, such as consumer price index, inflation rate, unemployment rate and so on. They are interrelated and influence each other. In order to get the economic trend more accurately, we can analyze many economic indicators one by one, and then make a comprehensive evaluation of each analysis and prediction result.

4 mark

[1] BEA (bureau of economic analysis): bea's main function is to analyze and synthesize a large amount of data in order to create a coherent American economic model. The Bank of East Asia also conducts budget and analysis on international, national and regional economies. Among them, the budget of gross national product (GDP) is the most famous.

[2] The quarterly change percentage of real GDP in the United States is only recorded from 1947, so the data is limited, and it is only used as a reference for forecasting the short-term GDP change in the future, which may have limitations in analyzing the long-term macroeconomic pattern in the future.

5 references

[1], Gao Hongsheng, Western Economics (Macro Part), 4th edition, Renmin University of China Press, 2007.

[2] Sui Yali, Li Hongru, Basis of Economic Mathematics-Probability Statistics (3rd Edition), Tsinghua University Publishing House.

[3] Fan Xiaozhi, Multidimensional Application of Probability Theory in Economic Life, statistics and decision, 2005, (8)

[4] Yang Zengwu, Principles of Statistical Forecasting, China Financial and Economic Publishing House, 1990.

[5] Hao, Markov Chain Theory and Market Share Analysis and Forecast, Shanghai Statistics, 2000, (1)

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