FORECASTING WHEAT PRODUCTION FOR PAKISTAN VIA ARIMA MODEL
ARIMA model, built along the Box-Jenkins methodology has been used to forecast “Wheat production” for Pakistan for the period 2002 through 2022 on the basis of time series data for the last thirty years.
Comparison and contrast of Model Building and Forecasting from Box-Jenkins methodology and TSE-AX methodology:
Box-Jenkins Method Followed:
STEP#1: Choice of a Difference (d):
The stationarity of the series was judged on the basis of autocorrelation function through ‘correlogram’ such that , among the original series and the first, second, third etc differenced series, the one for which ACF converges to zero is selected.
STEP#2: Value of ‘q’ and ‘p’ are determined on the basis of autocorrelation function (ACF) and partial autocorrelation function (PACF) respectively.
STEP#3: ARIMA (2, 1, 2) identified in the previous step is estimated using “Minitab’ computer package.
STEP#4: Diagnostic Checking:
Following visual graphic analysis of residuals were done to check the validity of the fitted model.
Whether the scatter plot of residuals showed a rectangular pattern around zero on horizontal axis.
Residuals were tested for normality by plotting first differenced wheat production residual series against their corresponding normal scores.
Histogram of the residuals was examined for normality.
Residuals were plotted against their corresponding fitted values to see if they exhibit any pattern.
The model was established as valid on the basis that it passed all these criteria for “goodness of fit”.
STEP#5: Forecasting:
Forecasting from the model so fitted was done on the basis of the assumption that there were no shocks i.e. economic shocks, agricultural price shocks or abrupt changes in consumer preferences during thirty years of modeled period.
SHORTCOMING:
Outliers were overlooked instead of treating them appropriately in the form of incorporating into the model or eliminating them.
TSE-AX Method:
STEP#1: Choice of a Difference (d):
Among the options available, the series understudy should be tested for:
‘Seasonal difference’ as there may be seasonality in the series with low production during sowing season and high production during harvesting season.
Therefore,
“Kruskal and Wallis test” is to applied for seasonality evaluation, and
‘Partial autocorrelation function’ of the series is to be tested for ‘over differencing’.
The specific series understudy is expected to exhibit “regular seasonal cycles with large amplitude” which is expected to fulfill the two requirements of positive seasonality and no over-differencing for the application of seasonal differences.
If both of the conditions are not fulfilled e.g.
The series may displays ‘regular seasonal cycles of small amplitude’.
In that case TSE-AX proposes not to apply seasonal differences.
Other option to be invoked if both the requirements for ‘seasonal differences’ are not fulfilled is:
‘Non-Seasonal differences’: In that case the series with no evidence of over-differencing among the:
Original series
First order backward difference series
Second order backward difference series
would have to be used for estimation and forecasting purpose.
STEP#2: Choice of a Transformation:
Scale parameters for each year are calculated as difference between first and third quartile. (In the case considered, this step is not required)
Tau rank correlation coefficient is calculated between time variable and scale parameters for the corresponding years.
If tau= 0 is accepted then no transformation of the series would be carried out.
If tau is calculated to be significantly different from zero (at any chosen level of significance), then “Square root” transformation is applied.
Then second time on the transformed series tau test is applied.
If tau comes out to be significant again then logarithmic transformations are to be applied to the series.
STEP#3: Outlier Detection:
Instead of operating on the assumption of “no shocks” in TSE-AX method outliers are to be detected and decision about their treatment is made on the basis of the nature of these outliers. That is, the decision is made as to whether to eliminate these outliers and pursue further on the adjusted data or to incorporate these outliers into the model by modeling them separately through ‘intervention analysis”.
STEP#4: Intervention Analysis
Among all Intervention analysis options available the one suitable according to the nature of outliers are performed.
STEP#5: Model Specification:
Instead of deciding about p and q of ARIMA model merely on the basis of PACF and ACF, the four AR models are fitted in case of both seasonal non-seasonal data series While four additional models with both AR and seasonal component are also fitted to the series with seasonal characteristic and then the residuals of all these models are analyzed with the aid of ‘ACF’.
STEP#6: Model Adequacy Criteria:
Convergence of the optimization process of the likelihood function.
Roots of AR and MA polynomials should be stationary and invertable.
No more than two significant autocorrelation and partial autocorrelation of residual series.
The null hypothesis of Ljung-Box test of randomness of data should not be rejected.