### How do time series data such as GDP eliminate the price factor?

Divide the price index (GDP deflator) by the nominal GDP to get the real GDP.

Components: the time to which the phenomenon belongs, the value of the indicator reflecting the level of development of the phenomenon.

(A) the time series of the components

Long-term trend (T) phenomenon in a longer period of time by a fundamental factor in the formation of the overall trend

Seasonal changes (S) phenomenon within a year with the seasons of the change in the regular cyclical changes

Cyclic changes (C) phenomenon to a number of years as a cycle of regular wave pattern of the Wave pattern of regular changes

Irregular changes (I) is a kind of irregular changes, including strictly random changes and irregular changes in the impact of a large number of two types of sudden changes.

### [Help]: Questions about GDP forecasting model

The basic idea: to establish an ARIMA time series model based on China’s GDP data from 1978-2005, and to make short-term forecasts of China’s GDP, and to explore the development trend of the economy.

I. Establishment of per capita GDP time series model

Referring to the historical data of China’s per capita GDP (1978005) in China Statistical Yearbook (2006) as a sample for analysis

(I) Analysis of per capita GDP time series

In the ARIMA model, a time series is generated by a zero-mean smooth stochastic process, i.e. its process of zero-mean In ARMA model, the time series is generated by a zero-mean smooth stochastic process, i.e., the stochastic nature of the process has time invariance, which is graphically manifested as random fluctuations of all sample points under a certain level. For non-stationary time series, the time series need to be smoothed in advance.

1. Smoothness check. Utilize Eviews3.1 to plot China’s per capita GDP time series data. China’s current stage of per capita GDP series has obvious non-stationarity, showing a certain exponential trend.

2. Smoothing process

The variables are logarithmized to turn the exponential trend of the time series into a linear trend. Because the logarithmization is still non-stationary, the first-order difference is continued. The unit root method is used to test the stability of the difference series. However, it is still unstable, so the second-order differencing is performed, and the obtained ADF test value is -4.373666, which is larger than the critical values of -3.7343, -2.9907, -2.6348 corresponding to the significance level of 1%, 5%, 10%; therefore, the hypothesis of θ=0, i.e., presenting a unit root, is rejected, and the series obtained from the second-order differencing can be regarded as smooth. It does not show any trend, and is an I(0) stochastic process.

(II) Establishment of time series model

The series we study is a one-dimensional time series, and the purpose of modeling is to use its historical value and the current and past random error terms to forecast the prospects of the change of the variable, and it is usually assumed that the different moments of the random error terms are statistically independent and normally distributed random variables. For time series forecasting, the first step is to find the best fitting forecasting model with the data, so the determination of the order and the estimation of the parameters are the key to forecasting.

1. Model identification

Using Eviews 3.1 software, the 12th-order autocorrelation function and partial autocorrelation function of the time series after second difference are calculated:

It can be considered that the autocorrelation coefficients of the second-difference time series model and partial autocorrelation coefficients are trailing, and thus the ARMA model is chosen. In this paper, the AIC criterion is used to determine the order and select the optimal model from it, which can give a best estimate of the order of the model and the corresponding parameters at the same time on the basis of the great likelihood of the model. The method used in this paper is to establish the model by minimizing the AIC value, and then conduct the parameter significance test and residual randomness test on the estimation results. If it passes the test, the model can be regarded as the optimal model; if it does not pass, the second smallest AIC value is selected and the relevant statistical test is performed, and so on, until a suitable model is selected. Through the use of Eviews3.1 software repeatedly projected, the selected model is: ARMA (2,2).

Because this series is the original series to take the logarithm of the second-order difference after the results, so choose ARIMA (2,2,2) to estimate the model to take the logarithm of the time series, and then the index of the reduction. The following model parameters and tests are performed on the logarithmic time series.

2. Model parameter estimation and establishment

In this paper, the nonlinear least squares (NLS) method is chosen to estimate the parameters. The resulting ARIMA(2,2,2) model is of the form:

The parameters of the model are estimated using the econometric software Eviews 3.1

. After parameter adjustment by t-statistic,=0, the substance becomes autoregressive first-order lag. The estimation results are obtained as follows:

3. Model testing

The residual series e of the resulting model is tested for smoothness and randomness. If the residual series is white noise, the specific fit can be accepted; if not, then there may be useful information in the residual series that has not been extracted, and the model needs to be further improved. After the test, combined with the residual autocorrelation, partial autocorrelation plot and ADF test results, the residual series can be considered to be smooth. And the DW value is 1.969930, which indicates that there is no serious serial autocorrelation. Therefore, the residuals pass the white noise test.

Because this series is after taking the logarithm of the series estimation model. So take the natural index, that is, the annual per capita GDP forecasting model. As follows:

Second, China’s GDP per capita short-term forecasting and analysis

1. 2006, 2007 and 2008 forecasts using the resulting model are as follows:

From the forecasting results, we can see that the model prediction error is relatively small, and China’s GDP is expected to continue to grow in the next few years.

2. Because the time series model is smoothed by second-order differencing, and the model is fitted by a limited number of data, the obtained model reflects the short-term change relationship, not the long-term change relationship, so it is only suitable for short-term forecasting.

Note: The model’s forecasts must take into account non-systematic factors, such as the impact of this financial crisis.