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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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