Must see, time series analysis

In this article, we will mainly introduce ARIMA Model , This is the most commonly used time series model in practical cases .

01 What is the time series ？

Time series data is arranged in chronological order 、 A time-varying and interrelated sequence of data , By studying the changing trend of historical data , To evaluate and predict future data . Time series data often appear in the economy 、 Finance 、 In the field of business data analysis .

02 Inspection process

First of all , First, the preprocessing of time series includes two aspects of testing , Stationarity test and white noise test . The stationarity of a sequence , Generally, it is judged by sequence diagram and correlation diagram . If the sequence is nonstationary , Through differential processing of data , Then perform the stability test , Judge whether the sequence is stable . Generally in application , The order of the difference （d） No more than 2.

second , ARIMA Model (p,d,q) It is called differential autoregressive moving average model , According to whether the original sequence is stable and the parts contained in the regression ,ARIMA The model can be split into 3 term , Namely AR(p) Model 、I(d) Difference 、 and MA(q) Model , Therefore, it is necessary to determine the order of these three parameters respectively . Generally, partial ( since ) The correlation diagram is appropriate p、q Order , And the use of ADF The appropriate order of difference is obtained by testing d.

Third , In determining the order of autoregression p, The order of difference d Value and moving average order q this 3 The model can be built after three parameters .SPSSAU Of 【ARIMA prediction model 】 Can intelligently find the best model , Provide the best q、d、p Value recommendation .（ The principle is to use AIC The rule of minimum value , Traverse various possible model combinations for model construction , And combine it with AIC The minimum rule , Finally get the best model , The process of model optimization is omitted .）

Fourth , ARIMA The model results are output 3 A form , The first 1 The first table is the table of fitting model parameters , Show the results of model construction, including the value of regression coefficient ,p It's worth waiting for ; The first 2 A table is a model Q Statistics table , Used to check whether the residual is white noise ; The first 3 The first table is the predicted value of the model , Provide future 12 The model prediction value of the period .

03 case analysis

（1） background

The current is 1978~2006 Annual total 29 Data on per capita health costs in , Want to use SPSSAU The data analysis platform finds the right ARIMA The model predicts the per capita health expenditure in China .

（2） data format

The format of time series includes Time and actual analysis items There are two columns .

For example, the year in the figure below is the time item ,“ Per capita health costs ” Is the actual analysis item . There is no need to set a time item during analysis , But the data compiled by researchers must be in this format , The date increases from top to bottom . Then upload the sorted data to SPSSAU Analysis platform .

The unit of time series is generally year , such as “ Over the years in China GDP data ”、“ China's population data over the years ” etc. . Of course, if the unit is month or quarter 、 Or wait a week , It can reflect the change law of data , It can also be used as time series data .

（3） Stability test

The first problem encountered in time series analysis is the stability of data . Data stationarity can be measured by sequence diagram , Visually observe the data characteristics to judge whether it is stable . however , The graph test method is highly subjective , So I also use ADF The test is the unit root test to get a more accurate judgment .

• Scatter plot （ Sequence diagram ）

Operation steps ：SPSSAU【 visualization 】-【 Scatter plot 】.

A stable time series is often shown graphically as a continuous fluctuation around the mean , If it is a non-stationary series, it often shows that it has different mean values in different time periods . Such as continuous rise or decline .

As can be seen from the above figure , The scatter points in the figure have an obvious upward trend , Does not meet the requirements of time series . So take the logarithm of the data first , Then carry out differential processing and then carry out inspection .

Generating variables - Logarithmic processing

• ADF test

The sequence diagram tests the stability of the sequence with strong subjectivity , So I also use ADF The test is the unit root test to get a more accurate judgment .

The unit root test can be used to check whether there is a unit root in the time series , If there is a unit root, it means that it is a nonstationary sequence . If there is a unit root, that is, the time series data is unstable , Subsequent analysis is usually not possible, such as ARIMA Model .

Operation steps ：

① choice SPSSAU【 Econometric research 】-【ADF test 】.

② In the analysis box , Put in “Ln_ Per capita health costs ”. Differential order selection “ Automatically ”, Type default , Click start analysis .

ADF test

Result analysis ：

As can be seen from the above table , For per capita health costs , The time series data ADF Tested t The statistic is -0.778,p The value is 0.825,1%、5%、10% The critical values are -3.700、-2.976、-2.628.

p=0.825>0.1, The original hypothesis cannot be rejected , The sequence is unstable . Make a first-order difference to the sequence and then ADF test .
Data after first-order difference ADF The test results show that p=0.287>0.1, The original hypothesis cannot be rejected , The sequence is unstable , Perform second-order difference on the sequence and then ADF test .
Data after second-order difference ADF The test results show that p=0.000<0.01, There are more than 99% Grasp and reject the original hypothesis , At this time, the sequence is stable . Therefore, we can consider the number of differences d As the 2.

（4） partial ( since ) Correlation chart

According to the stability test above , The sequence after the second-order difference is a stationary sequence , Next you can draw 【 partial ( since ) Correlation chart 】 To judge p、q The order of .

Operation steps ：

① choice SPSSAU【 Econometric research 】-【 partial ( since ) Correlation chart 】

② In the analysis box , Put in “Ln_ Per capita health costs ”. Differential order selection “2 rank ”, Click start analysis .

About ARMA The general judgment criteria are described in the following table ：

• trailing ： There are always nonzero values , It will not quickly approach after being greater than a certain order 0（ But in 0 Nearby fluctuations ）, It can be simply understood as not for 0, But after a certain stage, in 0 Random changes around .
• truncation ： At greater than a certain order (k) After that, it tends to 0 by k Order truncation , It can be simply understood as directly changing from a certain order to 0.

On the simple interpretation of tailing and truncation , If the value does not approach 0, Then it's Tailing ; If the value quickly approaches 0, Is truncated .

combination ACF and PACF chart ,SPSSAU Automatic recognition , Finally, the autoregressive order is suggested p The value is 3, Moving average order q The value is 3. Combined with the differential phase determined in the previous step d, So far, we have determined p、d、q The order of the three parameters , Then you can establish ARIMA Model .

（5）ARIMA prediction model

Operation steps ：

① choice SPSSAU【 Econometric research 】-【ARIMA forecast 】

② In the time series box , Put in “Ln_ Per capita health costs ”. Differential order selection “ Second order ”, Autoregressive order p choice “3”, Moving average order q choice “3”, Click start analysis .

Result analysis ：

ARIMA The model results are output 3 A form , The first 1 The first table is the table of fitting model parameters , Show the results of model construction, including the value of regression coefficient ,p It's worth waiting for ; The first 2 A table is a model Q Statistics table , Used to check whether the residual is white noise ; The first 3 The first table is the predicted value of the model , Provide future 12 The model prediction value of the period .

First table ：ARIMA(3,2,3) Model parameter table

The first table shows the results of this model construction , Including model parameters 、 Information criteria and other indicators , It usually doesn't need too much attention .AIC and BIC Value is used for multiple analysis model comparisons ; The lower these two values, the better , If the analysis is carried out many times , The changes of these two values can be compared , Comprehensively explain the optimization process of model construction .SPSSAU It will automatically model and compare multiple potential alternative models .

Second form ： Model Q Statistics table

AIRMA After the model is built, it is generally required that the residual of the model is white noise , That is, there is no autocorrelation in the residuals , The second table shows through Q The result of white noise test by statistical test （ The null hypothesis ： The residual is white noise ）.

such as Q6 Before checking the residual 6 Whether the order autocorrelation coefficient meets the white noise , Usually it corresponds to p Greater than 0.1 It means that it meets the white noise test （ Otherwise, it means that it is not white noise ）, In common cases, it can be directly targeted at Q6 Can be analyzed .

from Q Statistics results ,Q6 Of p Greater than 0.1, It's in 0.1 The original hypothesis cannot be rejected under the significance level of , The residual of the model is white noise , The model basically meets the requirements .

Third form ： Forecast value table

The third table is the predicted value table , ARIMA The purpose of the model is to predict the future data , Therefore, this table is also the most concerned by researchers . One of them is backward 1 period , Represents the 2007 The forecast for the year ; backward 2 Period represents 2008 The forecast for the year , And so on .

Because we have done logarithmic processing on the data before , So the prediction value shown here is not the final prediction result , You also need to restore the predicted value results .（ If you have not done logarithmic processing, you can ignore this step ）

Finally, we can see by comparing the predicted value with the actual value , adopt ARIMA The model predicts , forecast 2007 The per capita health expenditure in China should be 848.93 element , And the real value 875.96 Yuan is basically close to , It shows that the prediction result of the model is reliable .

From experience ,ARIMA What the model expects backwards 1 Period and backward 2 The period is relatively reliable , If the number to the later stage is too much , The accuracy will be low .

In particular ：SPSSAU You can intelligently find the best by default ARIMA Model and predict , The principle of intelligent fitting model is to use AIC The smaller the value, the better , Compare and select the best model from many potential models . If researchers want to pass SPSSAU Automatically identify the best ARIMA Model , Direct use 【 Econometric research 】 Medium 【ARIMA Model 】 Can be analyzed .

The above is the content of this sharing , Sign in SPSSAU Official website Learn more ：