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THE                      JOURNAL OF ECONOMIC SCIENCES: THEORY AND PRACTICE, V.80, # 2, 2023, pp. 4-13

                       The paper is organized as following: introduction is followed by methodology that
                    covers  ARIMA  process,  then  comes  forecasting  of  Engel  curve  components  and
                    conclusions.

                    METHODOLOGY
                    A more important aspect of time series forecasting is whether it is stationary or not.
                    In a broad sense, a stochastic process is considered stationary when its expected value
                    and variance remain constant over time, and the covariance depends not on the time
                    at which it is calculated, but on the difference between two consecutive times. Most
                    stochastic  processes  are  considered  weakly  stationary.  Determining  stationarity  is
                    important because if a time series is non-stationary, its study is only relevant at the
                    current time. For this reason, non-stationary time series are brought to stationary series
                    (integration or differencing) and forecasted with ARIMA (autoregressive integrated
                    moving average) model. Thus, ARIMA is a genaralization of ARMA (autoregressive
                    moving average) model. The model has both autoregression (AR) and moving average
                                                   ′
                    (MA) properties. The ARIMA(   ,   ) model is given by:

                                                                                         (1)
                               =         + ⋯ +    ′     −   ′ +    +        +      
                                                                                −  
                               
                                                               
                                                                  1   −1
                                                   
                                   1   −1
                    Or:
                                             ′                   

                                                                       
                                                  
                                   (1 − ∑       )    = (1 + ∑       )                    (2)
                                                                          
                                                
                                                                     
                                                      
                                           =1                   =1

                    Where,       –  are  forecast  values  at  time  t,  L  –  is  the  lag  operator,       –  are
                                
                                                                                                
                    autoregregression part parameters,     – are moving average parameters,     – is white
                                                         
                                                                                            
                    noise, p – is the order of autoregression part built on its own lagged values, and q – is
                    the order of moving average part respectively.

                                                                        
                    Now, let to assume that polynomial  (1 − ∑    ′        ) has a unit root  (a factor (1 −
                                                                 =1    
                    L)) raised to d-th power. Then it can be expressed as:

                                                          ′
                                          ′                 −  

                                (1 − ∑       ) = (1 + ∑       ) (1 −   )                 (3)
                                                                             
                                               
                                                                   
                                                                 
                                             
                                        =1                 =1


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