While analyzing the model errors, it was observed that the value of error for certain products has being changed with a seasonal behavior. Classical methods of forecast correction based on the data of the previous errors were described in details by researchers Clements and Hendry (1998), Entov and Nosco (2002). These methods allow adjusting the forecast in a real time. The standard technique includes two adjustment methods based on the information about one-step forecasting error (Turuntseva 2013). Accordingly, the forecast is usually adjusted by the forecasting error on the previous step, or the value of the average error of all the previous one-step forecasts. The first method involves recalculation of the forecast figures depending on the forecast error of the previous step (see Equation 5.1), (Clements & Hendry 1998).
(5.1)
where – corrected one-step demand forecast for the next three-month period, at the moment Т, fT,1 – one-step forecast for the future period T+1, eT-1,1 – forecast error on the previous step. In turn e T-1,1=yT - fT-1,1 – difference between actual sales volumes during the current period of three months Т and it's forecast at the moment Т, calculated at the moment Т–1.
Second classical method of error correction – forecast adjustment by the value of an average error of all known previous one-step forecasts – that means forecast correction by the value equal to (5.2), (Clements & Hendry 1996).
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(5.2)
where – actual demand at the moment T–i+1, fT-i,1 –
forecasted demand, calculated at the moment T–i for one step ahead.
Thus, the adjusted forecast is calculated by the flowing formula (5.3).
(5.3)
According to the results of research conducted by Entov and Nosco (2002) second method can achieve significantly better results compared with the first method. It was proved on the majority of Russian macroeconomic indicators that took part in the research. This paper also noted that both methods lead to the removal of systematic prediction error, if present. But unfortunately the quality of the predictions often worsens after the conducted adjustments (Mazmanova 2000), (Dubova 2004).
The main problem that occurs using these two methods of forecast correction is the practical difficulty of using them in "real time" in case of current forecasting model. Model produces the forecast for the next three-month product consumption every month. Thus, the company utilizes this three-month forecast during one month only. Therefore, to correct the predicted demand by standard methods becomes problematic. Given the fact that the forecasting model used by the company X is based on seasonality identification logic, new approach of taking into account seasonal errors was suggested. This method implements certain adjustments to the forecast figures based on all the available information about the previous forecasts and their errors.
To analyze the seasonality of errors occurred in the model, certain forecast files and all the available data of actual sales were taken in consideration. Due to the limited information available about the forecasts, assessment of the relationship between the errors was conducted with two time series, each with a length of eight months. Only those products that were sold steadily throughout the year participated in the analysis. As a result, the document was created where each product is presented by two sets of data: time series of three-month forecast figures and time series of actual volumes of sales for the corresponding three months. Sales are expressed in euros. Next, time series of coefficients et were calculated for each end
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product. These coefficients represent the ratio of the forecast to reality. With full compliance of the forecast and sales ratio e is equal to 1 (5.4).
𝑒𝑡 =∑ 𝑌𝑡𝑋
𝑡+2 𝑖
𝑡 , (5.4) where 𝑌𝑡− forecast for month 𝑡, 𝑋𝑖− actual volumes of monthly sales
If the variation of the coefficients shows seasonal behavior on the whole time period under review (add-in captured close relationship between the time series of coefficients), future forecasts for this particular product is adjusted on the basis of past mistakes. Figure 5.1 shows an example of identified seasonal behavior of coefficients еt dynamics for the particular product EFPPH4. Correlation coefficient between forecast errors was statistically significant, equal to 0.94 that proves that there is a close seasonal dependence.
Figure 5.1. Dynamics of the relative error for the product EFPPH
Forecasts made in a particular month T, was adjusted by the average forecast error of five months from the respective T-2 to T + 2 months of the previous year (5.5). For example, the forecast calculated in the beginning of January 2014, will be adjusted by the average forecast error of the period from November 2012 to March 2013.
𝑌´𝑡 =∑𝑌𝑡∗5𝑒
𝑡+2 𝑖
𝑡−2 , (5.5) where 𝑌´𝑡 − corrected forecast, 𝑌𝑡 – original forecast
Designed for the current model add-in of the forecast correction by the value of a systematic error has been tested on the baseline forecasts of January, February and March 2014. The
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accuracy of the adjusted forecasts was compared with the original model forecasts using the coefficient of determination measures and measures of absolute percentage error. Usage of this forecast correction add-in helped to increase the accuracy of three basic forecasts markedly, Table 5.1. Determination coefficient became closer to 1 by almost 10% in average, which indicates an increase in the proportion of explained variability of relevant variables. The value of mean absolute percentage error decreased by more than 25%. This model development makes it possible to achieve a significant increase in the forecast accuracy for those products that are characterized by seasonal behavior of their forecast errors.
Table 5.1. Results of error correction model testing
January 2014 February 2014 March 2014
Original forecast
It is important to note that a positive result was observed in the analysis of relatively small amount of available sales data. Increasing the size of data array on actual sales volumes, as well as number of forecasts will help to clarify the results of testing procedures. Since June 2014, it will be possible to correlate two annual series of relative error rates that was considered as is one of the potential directions of further research.