时间序列论文(英文)time seies project(1).docx

Abstract: This article collects a series quarterly data of Chinas GDP from 1992 to 2010, and we use the method of factor decomposition to collect the long-term increasing trend and seasonality, then use ARMA model to fit the residuals, do analysis to get the final model and use it to generate a short-term GDP-forecast of china.Key words: factor decomposition; ARMA model; GDP forecast； 1. Introduction1.1Background From 1978, since the reform and opening up, chinas economy is developing rapidly and steadily. After joining the WTO, the developing speed has reached a new level. GDP (Gross Domestic Product) , which is the basis of national economic production of statistical indicators, can be used to reflect a countrys economy. It is the core of Statistical indicators in the national economy. GDP combines responses of the most basic aspects of macroeconomic, can not only measure the overall national output and income scale, but also can explore the economic fluctuations and cycles. Hence, it is of great importance to fit and analyze GDP accurately for exploring a countrys macroeconomics trend. The aim of this article is to generate a GDP forecast model and use it to predict the future GDP of china.1.2 Method A lot of methods have been used to analysis economy phenomenon, time series analysis is one of the most efficient methods. A time series is a collection of observations of well-defined data items obtained through repeated measurements over time. Time-series methods use economic theory mainly as a guide to variable selection, and rely on past patterns in the data to predict the future. An observed time series can be decomposed into three components: the trend (long term direction), the seasonal (systematic, calendar related movements) and the irregular (unsystematic, short term fluctuations). When these factors occur, we can use the method of decomposition, from which can we collect useful information of the data, we defined it as factor decomposition here. The trend component typically represents the longer term developments of the time series of interest and is often specified as a smooth function of time. The recurring but persistently changing patterns within the years are captured by the seasonal component. It is quite common in economic time series, when it occurs, we should use seasonal adjustment method. Seasonal adjustment is the process of estimating and then removing from a time series influences that are systematic and calendar related. Observed data needs to be seasonally adjusted as seasonal effects can conceal both the true underlying movement in the series, as well as certain non-seasonal characteristics which may be of interest to analysts. The irregular component represents the irregular fluctuations which are affected by causal factors. It usually defined as residual. Considering the Insufficiency of the deterministic decomposition, we should test the residuals, if there is no autocorrelation among the residual, it means that the information of the time series is totally recovered by the deterministic decomposition. In the case of the existence of autocorrelation, ARMA model can be used to fit the residuals. ARMA is a one of those most common time series model which was used to make precise estimation according to short term data. Its main idea can be concluded as a combination of several time-related components which can be used to predict the future data. The time series components from the ARMA model is a set of random variables which related to time itself, which shows uncertainty when observed individually combined with each other shows some kinds of regularity and can be expressed by corresponding statistical model. The ARMA model consists of two parts, an autoregressive (AR) part and a moving average (MA) part. The model is usually then referred to as the ARMA(p,q) model where p is the order of the autoregressive part and q is the order of the moving average part. 2. Data Analysis2.1 Dataset The data we collected contains historical GDP from 1992-2010, the reason we choose this time duration rather than the 1978-2011 which most other prediction article would like to use is that during the first 10-15 years the economic growth rate is relatively slow compared with the later years (1990-now) growth. So we would like to wipe out the interference of the early data. Another reason we use recent years data (1992-2011) is that it is hard for us to look for the quarterly GDP data before 1992 due to the imperfection of the statistical system of China in the end of 20th century. Table 1 quarter GDP data from 1992-2010 (Unit: 1000 million CNY)timeGDPtimeGDPtimeGDPTimeGDP1992.149741997.1162572002.01253762007.0154755.91992.263581997.2186972002.02279652007.02612431992.371191997.3191482002.03297162007.0364102.21992.484721997.4248712002.04372762007.0485709.21993.165001998.1175012003.0128861.82008.0166283.81993.280441998.2197222003.0231007.12008.02741941993.390481998.3203722003.0333460.42008.0376548.31993.4117421998.4268072003.0442493.52008.0497019.31994.190651999.1187902004.0133420.62009.0169816.91994.2110851999.2207652004.0236985.32009.0278386.71994.3124471999.3218592004.0339561.72009.0383099.71994.4156011999.4282632004.0449910.72009.04109599.51995.1118582000.1206472005.0139117.42010.0182613.41995.2141102000.2231012005.0242795.22010.0292265.41995.3155352000.3243402005.0344744.42010.0397747.91995.4192912000.4311272005.0458280.42010.04128886.11996.1142612001.01233002006.0145315.81996.2166012001.02256512006.0250112.71996.3176712001.03268672006.0351912.81996.4226442001.04338372006.0468973.1(Data source: National statistical database of China) We are going to use these historical GDP data as a time series, learn and analyze the data, then based on the past patterns to get a forecast model, use the model to predict the future GDP.2.2 Data Graphical Analysis Figure 1 shows a plot of the data, and we can find that there is a significant long-term trend and varying seasonality in the time series. The trend seems to be quadratic while the seasonality illustrate a strong yearly component occurring at lags that are multiples of s=4. For the purpose of demonstration, the sample ACF of the data is displayed in Figure 2, also, it appears a significant seasonality. Figure 1. Quartely chinas GDP from 1992(1) to 2010(4) Figure 2.Sample ACF of the GDP data 3. Time Series Model3.1 Factor Decomposition After the previous analysis, now we are going to use the method of factor decomposition to build a time series model, its principle is that through the decomposition method, we collect the useful information and measure the influence of the trend and seasonality. Define as GDP, x as time. We set the decomposition model as bellows:(1)(2) (3) The reason we use (2) as the trend model is because we can find that it seems to be a thrice model from the pattern in figure 2. , and in (3) is the dummy variables of seasons, and D1=c(1,0,0,0,1,0,0,0,)D2=c(0,1,0,0,0,1,0,0,.)D3=c(0,0,1,0,0,0,1,0,.)3.11 Data Transformation A significant varying seasonality is observed from figure 1, since the varying seasonality will have negative effects on the model fitting, so we should take some transformation of the GDP value to make the seasonality constant. The Box-Cox transformation is part of the family of power transformation, where the data is transformed using a power functions whilst preserving the rank of the data, so we take Box-Cox transformation of GDP. Figure 3 illustrate the box-cox plot of GDP, Looking at the Box-Cox diagram in figure 3, is near the 0.2 mark, so we use as a new response variable defined as. Figure 4 shows the GDP plot after transformation. From the plot we can find that the seasonality is almost constant. Figure 3.box-cox transformation Figure 4. Quartely chinas GDP after transformation 3.12 Build Model After transformation, now the decomposition model is. Using the “R” statistics package to analyse the time series and build model, the result of the full model is as follows:LM1: T = (110.322) (16.798) (-9.281) (10.442) (-15.340) (-11.405) (-10.418) P = (2e-16) (2e-16) (9.22e-14) (7.62e-16) (2e-16) ( data=read.table(st5209new.txt,header=T) Y=data$GDP x=data$time x2=x2 x3=x3plot(Y,ty=”l”)acf(Y)D1=c(1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0)D2=c(0,1,0,0,0,1,0,0,0,1