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This paper firstly introduces the significance of seasonal adjustments of the consumer price index (CPI). Then this paper focuses on the theory of seasonal adjustments and the ARIMA model with regression. Based on X-13 ARIMA-SEATS program, we develop a statistically robust method to conduct seasonal adjustment on China’s monthly CPI with respect to moving holidays, especially, Chinese Spring Festival. It is demonstrated that seasonally adjusted CPI time series are more sensitive and conducive to monitor the macro economy.

The consumer price index (CPI) mainly includes the year-on-year index, which is calculated by using the same month of previous year as the base period (previous year = 100), and the chain index by taking the previous month as the base period (previous month = 100). Although the year-on-year index could weaken the influence of seasonal factors to a certain extent, the year-on-year index includes the carryover effects and the new price-rising factor, which cannot reflect the turning point of the macro-economy timely, thus affecting the accuracy of the fluctuation calculation and the forecast of the level of consumer price and bringing difficulty for the formulating of macro-control policy. Research shows that the inflection point of the economic cycle, which is reflected by the non- seasonally adjusted year-on-year CPI lags 6 months on average [

An economic time series can be decomposed into trend, cycle, season and irregularity. Seasonal adjustment is the process of excluding seasonal factors implied in the original monthly or quarterly time series [

At present, Chinese domestic research on seasonal adjustment of CPI time series is still relatively scarce. Zhang Mingfang and others adopted X-12-ARIMA to make seasonal adjustment and analyze on CPI series [

Economic time series are usually non-stationary time series. ARIMA model is the main method on modeling non-stationary time series. The process

where

In the CPI time series modeling, we should take into account the effects induced by mobile holidays (such as the Spring Festival, Mid-Autumn Festival, Dragon Boat Festival), outliers, fixed seasonal effects, working days, trading days and other factors. A general product seasonal ARIMA model with regression term could be established as:

where, L is the lag operator, s is the seasonal cycle (for the monthly CPI data, s=12),

In order to get a fully fitted sequence in the product season model, the original sequence

After making forward prediction, backward prediction and a priori adjustment of various effects by the ARIMA model with regression term, this paper uses X-12 seasonal adjustment method to decompose the components based on the moving average method based on multiple iterations and then completes seasonal adjustment. It contains the Henderson symmetrical moving average adjustment and the Musgrave asymmetric moving average adjustment when we make seasonal adjustment with X-13-ARIMA-SEATS, and we use Henderson moving average to estimate the trend-cycle component.

The regression variable

There will be a February of 29 days for every 4-year, which will have an impact on the flow of data statistics. So we need to set a leap year variable: leapyear t = 0.75 for February in leap year; leapyear t = −0.25 for February in other years; and leapyear t = 0 for others.

If it is considered that the economic activity is different for each day of the week, since the number of occurrences of each day within a week is different, the variables considered will also be a corresponding change in the same calendar month for different years. For example, if you think that the consumption level of Sunday and Monday is different, then the economic indicator variable should be correspondingly different between months with a higher number of Sundays and months have fewer Sundays but a higher number of Mondays. In X-13- ARIMA-SEATS, the program gives selections of regression variables in trading day, and selects TD to consider the trading day effect in ARIMA model with regressions.

In the holidays, people tend to consume more and make the economic variables significantly different from non-holiday. But the effects of mobile holiday are different from the holiday with fixed gregorian dates (such as the National Day, Golden Week). For example, although the Spring Festival appears regularly, but does not necessarily appear on the same date each year. The effects of a fixed holiday are already considered in the seasonal effect, so the regression variable only needs to consider moving holidays.

We may take Luan Huide, Zhang Xiaotongs’ method on assigning the weights of regression variables for the Spring Festival holidays. Assuming the daily weights of the Spring Festival are different before, during and after the festival, the closer the Spring Festival, the greater the impact, hence greater weights should be given. During the festival, the variables follow the uniform distribution, therefore the daily weights are equal. The weight vector for time interval of

According to the specific distribution of the number of effective days in different months corresponding to before, during and after the Spring Festival, we can get the proportional variable by summing the weights of each day, and then normalize them respectively. Finally, we can get regression variables

From 2001 onwards, China began to use fixed base period calculation method to publish base CPI. Chinese first round base period is fixed in 2000, that is, making the average price level in 2000 as a comparison of fixed base period, set the base index in December 2000 was 100. Then obtained the CPI fixed base index from January 1995 to June 2014 through the chain index of forward and backward recursion.

The X-13-ARIMA-SEATS program combines the seasonal adjustment functions of the US Census Bureau X-12-ARIMA with the TRAMO-SEATS by Bank of Spain, which allows users to define regression variables in the ARIMA model with regressions and detects the three outliers, which are AO (Additive Outlier), LS (Level Shift), TC (Temporary Change) automatically. The program can also filter the regression variables automatically according to the t statistic and select the optimal ARIMA model according to the information criterion automatically. The effects of the Spring Festival are set to