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Forecast based on mononomial trendPowerPoint Presentation

Forecast based on mononomial trend

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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.
- 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:

yji – value of the forecasted variable in i-th phase of the

j-th cycle

t - time variable (in this case number of successive cycles)

α0i, α1i – structural parameters of the i-th mononomial

trend models

εji - 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:

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

future

α0i, α1i – 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.

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 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

Therefore, we receive the following estimated form of equations:

Yj1 = 61 + 2,1t

- The value of variable in 1st quarter 2002 was 61 units
- The value of variable (size of service sale) in first quarters 2003-2006 increases in each next cycles about 2,1 units.

- Yj2 = 44 + 2,6t
- The value of variable in 2nd quarter 2002 was 44 units
- The value of variable (size of service sale) in second quarters 2003-2006 increases in each next cycles about 2,6 units.

- Y equations:j3 = 41,5 + 2,8t
- The value of variable in 3rd quarter 2002 was 41,5 units
- The value of variable (size of service sale) in third quarters 2003-2006 increases in each next cycles about 2,8 units.

- Yj4 = 28,5 + 1,4t
- The value of variable in 4th quarter 2002 was 28,5 units
- The 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 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

Now, we can create p equations:oint forecasts for year 2008

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

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