Kliknij,aby edytowa styl wzorca tytuu,Kliknij,aby edytowa style wzorca tekstu,Drugi poziom,Trzeci poziom,Czwarty poziom,Pity poziom,*,Forecast based on mononomial trend,Basic information,We can use this method when were 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.,This method,speaking briefly,consist in estimating structural parameters of trend models for respective cycle phases distinctly.E.g.if time series,that is being analyzed,consists of monthly quotation of a variable then we can consider occurring of monthly seasonality within confines of the year cycle.,Basic information,In this case we can try to estimate structural parameters of trend models for respective months in cycle phases i.e.model for January,based on empirical observations of the variable from this month in successive cycles,model for February,March,April etc.till December.,We would obtain 12 mononomial trend models.Mononomial phase is a phase which in successive cycles“is named the same.,Basic information,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.,Basic information,Each of trend models that are created for mononomial terms successive cycles can be described using following equation of expansion tendency:,where:,y,ji,value of the forecasted variable in,i-th,phase of the,j-th,cycle,t-time variable(in this case number of successive cycles),0i,1i,structural parameters of the,i-th,mononomial,trend models,ji,-random component,k number of the last cycle,Basic information,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.,Basic information,So we will need to use trend model for second quarters.Whereas for T the forecasted period we need to put number of the successive cycle(in our case it is 5th cycle),so T=5.,In order to make a forecast starting point of the being analyzed variable we use the following equation:,where:,j,i,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,future,0i,1i,structural parameters of the,i-th,mononomial trend models,h forecast horizon,Basic information,The next step is to create a forecast interval.We make it for the given level of significance using the following equation:,MFE,MFE=,MFE,Basic information,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).,Basic information,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.,Forecasting based on mononomial trend-example,Example,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 sales size in 2021.,Starting point should be a plot illustrating variability of the size of service sale in time:,So,we build(create)range for:,first quarters of the following cycles,second quarters of the following cycles,third quarters of the following cycles,fourth quarters of the following cycles,Therefore,we receive the following estimated form of equations:,Y,j1,=61+2,1t,The value of variable in 1st quarter 2002 was 61 units,T,he value of variable(size of service sale)in first quarters 2003-2006 increases in each next cycles about 2,1 units.,Y,j2,=44+2,6t,The value of variable in 2nd,quarter,2002 was 44,units,T,he value of variable(size of service sale)in,second,quarters,2003-2006,increases in each next cycles about,2,6,units.,Y,j3,=41,5+2,8t,The value of variable in 3rd,quarter,2002 was 41,5,units,T,he value of variable(size of service sale)in,third,quarters,2003-2006,increases in each next cycles about,2,8,units.,Y,j4,=28,5+1,4t,The value of variable in 4th,quarter,2002 was 28,5,units,T,he value of variable(size of service sale)in,fourth,quarters,2003-2006,increases in each next cycles about,1,4,units.,The next step is to calculate standard error of the estimate for the following trend equation.,for model of first quarters S,e1,=0,592(i,t 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 S,e2,=1,265,for model of