Forecast based on mononomial trend. Basic information. We can use this method when we’re analyzing time series that are characterized by tendency, seasonal fluctuation terms and random fluctuations. It can be also used to create short-term econometric forecasts.
We would obtain 12 mononomial trend models. Mononomial phase is a phase which in successive cycles “is named the same”.
The essential disadvantage of this method is the necessity of accepting the status quo rule, which consist in making presumption that expansion tendency observed in respective phases (mononomial terms) of successive cycles will be kept also in future.
Each of trend models that are created for mononomial terms successive cycles can be described using following equation of expansion tendency:
yji – value of the forecasted variable in i-th phase of the
t - time variable (in this case number of successive cycles)
α0i, α1i – structural parameters of the i-th mononomial
εji - random component
k – number of the last cycle
In order to make a forecast of being analyzed variable we need to determine which phase of the cycle is the forecasted period, e.g. if mononomial trend models are created for quarter of successive years based on data inclusive 4 cycles, then forecast for 18th quarter will be a forecast for second quarter of the successive year.
ŷji – value of the forecasted variable in i-th phase of the j-th cycle (the cycle
must belong to the future)
T - time variable (in this case number of successive cycles) that belong to the
α0i, α1i – structural parameters of the i-th mononomial trend models
h – forecast horizon
The next step is to create a forecast interval. We make it for the given level of significance using the following equation:
We should also notice that in t vector there are variables equal to 1 and number of being forecasted cycle:
Matrix is the same for equations of all quarters because each of the mononomial trend models was created basing on the same collection of the independent variables (for t = 1,2,3,4,5,6).
We can see now that scale of average errors of the forecast is determined by standard error of the estimate, typical for respective mononomial trend models.
In a certain service firm in years 2003-2006 recorded the following size of service sale.
Purpose of the analysis is to put a forecast of sale’s size in 2008.
Starting point should be a plot illustrating variability of the size of service sale in time:
So, we build (create) range for: the size of service sale in time:
● first quarters of the following cycles
● second quarters of the following cycles
● third quarters of the following cycles
● fourth quarters of the following cycles
Yj1 = 61 + 2,1t
The next step is to calculate standard equations:error of the estimate for the following trend equation.
● for model of first quarters Se1=0,592 (it means that the real values of size of service sale in first quarters of the following years differ on average from theoretical values about 0,592 units).
● for model of second quarters Se2 = 1,265
● for model of third quarters Se3 = 0,949
● for model of fourth quarters Se4 = 0,316
For the first quarter of the year 2008
We must use trend model for first quarter T=6, because year 2008 is sixth as a cycle.
Yj1 = 61 + 2,1 6 = 73,6
It means that in the first quarter of year 2008, size of service sale will be equal to 73,6 units.
For second quarter of year 200 equations:8
We use trend model for second quarter T=6
Yj2 = 44 + 2,6 6= 59,6
It means that in the second quarter of year 2008, size of service sale will be equal to 59,6 units.
(58,3 in the third and 36,9 in the fourth quarter 2008).
The next step is equations:confidence interval for forecast
Matrix is the same for all trend model (we estimated together their parameters), vector t also is the same – all four forecasted sub-periods belong toone cycle (seventh).
So we see that value of Mean Forecast Error in this case is determined by standard error of the estimate, which is characteristic for individual trend model.
Knowing equations:MFE, we may construct forecast interval.
We assume that the significant level α = 0,05