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DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS

Moving Averages and Smoothing Methods

ECON 504

Chapter 7

Dr. Mohammad Zainal Fall 2013

This chapter will describe three simple approaches to forecasting a time series:

Naïve

Averaging

Smoothing

Naive methods are used to develop simple models that assume that very recent data provide the best predictors of the future.

Averaging methods generate forecasts based on an average of past observations.

Smoothing methods produce forecasts by averaging past values of a series with a decreasing (exponential) series of weights.


ECON 504, CH 7 by M. Zainal

2

Forecasting Procedure

You observe Yt at time t

Then use the past data Yt-1,Yt-2, Yt-3

Obtain forecasts for different periods in the future



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Steps for Evaluating Forecasting Methods

Select an appropriate forecasting method based on the nature of the data and experience.

Divide the observations into two parts.

Apply the selected forecasting method to develop fitted model for the first part of the data.

Use the fitted model to forecast the second part of the data.

Calculate the forecasting errors and evaluate using measures of accuracy.

Make your decision to accept or modify or reject the suggested method or develop another method to compare the results.



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Types of Forecasting Methods

Three simple approaches to forecasting a time series will be discussed in this chapter:

Naïve Methods

These methods assume that the most recent information is the best predictors of the future.

For the stationary data, we use the simplest model is .

For the nonstationary (trended) data, we use the following models

The amount of change model is .

The rate of change model is .

For the strongly seasonal (quarterly) data, we use the following models

The quarterly data model is .

The monthly data model is .



5

1ˆt tY Y

1 1ˆt t t tY Y Y Y

1

1

ˆ . tt t

t

YY Y

Y

1 3ˆt tY Y

1 11ˆt tY Y

For the seasonal (quarterly) and trended data, we use

Example 4.1



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41 3

ˆ4

t tt t

Y YY Y

The figure shows the quarterly sales of saws for the Acme Tool Company. The naive technique forecasts that sales for the next quarter will be the same as the previous quarter.

The table below shows the data from 1996 to 2002

If the data from 1996 to 2001 are used as the initialization part and 2002

as the test part.

The forecast for the first quarter of 2002



7

The forecasting error is

Similarly, the forecast for period 26 is 850 with an error of -250.

But its very clear that the data have an upward trend and there appears to be a seasonal pattern (first and fourth quarters are relatively high).

A decision must be made to modify the naive model!

Our technique can be adjusted to take trend into consideration by adding the difference between this period and the last period.

The forecast equation is

The forecast for the first quarter of 2002 is



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Visual inspection of the data in indicates that seasonal variation seems to exist.

Sales in the first and fourth quarters are typically larger than those in any of the other quarters.

If the seasonal pattern is strong, an appropriate forecast equation for quarterly data should be used.

The previous equation says that next quarter the variable will take on the same value that it did in the corresponding quarter one year ago.

The major weakness of this approach is that it ignores everything that has occurred since last year and also any trend



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1

There are several ways of introducing more recent information.

For example, the analyst can combine seasonal and trend estimates and forecast the next quarter using

The Yt-3 term forecasts the seasonal pattern, and the remaining term averages the amount of change for the past four quarters and provides an estimate of the trend.

The naive forecasting models in the last equations are given for quarterly data.

Adjustments can be made for data collected over different time periods.

For monthly data, the seasonal period is 12, not 4, and the forecast for the next period (month) given by



10

2

For some purposes, the rate of change might be more appropriate than the absolute amount of change. If so, it is reasonable to generate forecasts according to

The forecasts for the first quarter of 2002 using Equations 1, 2 and 3 are



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3

Averaging methods

Frequently, management faces the situation in which forecasts need to be updated daily, weekly, or monthly for inventories containing hundreds or thousands of items.

Often it is not possible to develop sophisticated forecasting techniques for each item.

Instead, some quick, inexpensive, very simple short-term forecasting tools are needed to accomplish this task.

These tools assume that the most recent information is used to predict the future and the forecasts need to be updated daily, weekly, monthly, etc.

They use a form of weighted average of past observations to smooth short-term fluctuations.

In this case different averaging methods should be considered.



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Simple Average method

This method uses the mean of all relevant historical observations as the forecast of the next period.

Also, it used when the forces generating series have stabilized and the environment in which the series exists is generally unchanging.

The objective is to use past data to develop a forecasting model for future periods.

As with the naive methods, a decision is made to use the first t data points as the initialization part and the remaining data as a test part.

Next, the following equation is used to average the initialization part of the data and to forecast the next period.

When a new observation becomes available, the forecast for the next period (t + 2) is the average of the above equation and that data point



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Example 4.2

The Spokane Transit Authority (STA) operates a fleet of vans used to transport both the disabled and elderly. A record of the gasoline purchased for this fleet of vans is shown in table below. The actual amount of gasoline consumed by a van on a given day is determined by the random nature of the calls and the destinations.



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Examination of the gasoline purchases plotted in following figure shows the data are very stable.

Hence, the method of simple averages is used for weeks 1 to 28 to forecast gasoline purchases for weeks 29 and 30.

The forecast for week 29 is



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The forecast for week 30 includes one more data point (302) added to the initialization period.

So the forecast is


Using the method of simple averages, the forecast of gallons of gasoline purchased for week 31 is



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Moving Averages

The method of simple averages uses the mean of all the data to forecast. What if the analyst is more concerned with recent observations?

A constant number of data points can be specified at the outset and a mean computed for the most recent observations.

A moving average of order k is the mean value of k consecutive observations. The most recent moving average value provides a forecast for the next period.

The term moving average is used to describe this approach.

As each new observation becomes available, a new mean is computed by adding the newest value and dropping the oldest.

This moving average is then used to forecast the next period.

The simple moving average forecast of the order k, MA(k), is given by



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Moving Averages

Moving average models do not handle the trend and the seasonality very well, It is useful in stationary time series data.

The method of simple averages uses the mean of all the data to forecast. What if the analyst is more concerned with recent observations?

In the moving average, equal weights are assigned to each observation.

Each new data point is included in the average as it becomes available, and the earliest data point is discarded.

The rate of response to changes in the underlying data pattern depends on the number of periods, k, included in the moving average.

A moving average of order 1, MA(1), is simply the naive forecasting.



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Example 4.3

The table below demonstrates the moving average forecasting technique

with the Spokane Transit Authority data, using a five-week moving

average.



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The moving average forecast for week 29 is


The forecast for week 31 is



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Minitab can be used to compute a five-week moving average.

The following figure shows the five-week moving average plotted

against the actual data.

Note that Minitab calls the MSE (mean squared error) MSD (mean

squared deviation).



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The analyst must use judgment when determining how many days, weeks, months, or quarters on which to base the moving average.

The smaller the number, the more weight is given to recent periods. Conversely, the greater the number, the less weight is given to more recent periods.

A small number is most desirable when there are sudden shifts in the level of the series.

A small number places heavy weight on recent history, which enables the forecasts to catch up more rapidly to the current level.

A large number is desirable when there are wide, infrequent fluctuations in the series.

Moving averages are frequently used with quarterly or monthly data to help smooth the components within a time series, as will be shown in Chapter 5.



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For quarterly data, a four-quarter moving average, MA(4), yields an average of the four quarters, and for monthly data, a 12-month moving average, MA(12), eliminates or averages out seasonal effects.

The larger the order of the moving average, the greater the smoothing effect.

In Example 4.3, the moving average technique was used with stationary data.

In Example 4.4, we show what happens when the moving average method is used with trending data.

The double moving average technique, which is designed to handle trending data, is introduced next.



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Double Moving Averages

This method used when the time series data are nonstationary or trended and this happen by computing the moving average of order k to the first set of moving averages.

Let

Then

Thus the model forecast m period is given by

where

k = number of periods in the moving average

p = number of periods ahead to be forecast


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First we compute the moving average

of order k.

Then we compute the second moving average of order k.

Remember differencing for

trended time series



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Example 4.4

The Movie Video Store operates several videotape rentals. The company is growing and needs to expand its inventory to accommodate the increasing demand for its services.

The president of the company wants to forecast rentals for the next month.

At first, Jill attempts to develop a forecast using a three-week moving average.

The MSE for this model is 133.

Because the data are obviously trending, she finds that her forecasts are consistently underestimating actual rentals.


The results for double moving average are shown in the table below


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To understand the forecast for week 16, the computations are:

First we computed the three week moving average ( column 3 ) using

To compute the double moving averages (column 4), we used

The difference between the two moving averages (column 5) is computed

The slope is adjusted by


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The forecast is done for one period into the future (column 7).

The forecast four weeks into the future is

Note that the MSE has been reduced from 133 to 63.7.

It seems reasonable that more recent observations are likely to contain more important information.

A procedure is introduced in the next section that gives more emphasis to the most recent observations.


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Smoothing Methods

These methods also assume that the most recent information is used to predict the future and the forecasts need to be updated daily, weekly, monthly, etc.

In this case different smoothing methods should be considered.

1. Exponential Smoothing

This method continually revises a forecast in the light of more recent experiences.

It is often appropriate for data with no predictable upward or downward trend.

It provides an exponentially weighted moving average of all previously observed values.

The most recent observation receives the largest weight (where 0 < < 1).


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The next most recent observation receives less weight, (1 — ), the observation two time periods in the past even less weight, (1 — )2, and so forth.

The weight (where 0 < < 1) is given to the newly observed value, and weight (1 — ) is given to the old forecast.

New forecast = [ x (new observation)] + [(1 — ) x (old forecast)]

More formally, the exponential smoothing equation is

where

Also, it can be rewritten as


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Where is called smoothing constant or the weighting is factor and is the old smoothed value or forecast with .

Note that is recognized as an exponentially smoothed value since mathematically it is equivalent to

or

The value assigned to is the key to the analysis.

If it is desired that predictions be stable and random variations smoothed, a small value of is required.

If a rapid response to a real change in the pattern of observations is desired, a larger value of is appropriate.

One method of estimating is an iterative procedure that minimizes the mean squared error (MSE).

Forecasts are computed for, say, equal to .1, .2,..., .9, and the sum of the squared forecast errors is computed for each.


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ˆtY

1 1Y Y

1ˆtY

1

0

ˆ (1 )m

t t m

m

Y Y


The value of producing the smallest error is chosen for use in generating future forecasts.

To start the algorithm for, an initial value for the old smoothed series must be set.

One approach is to set the first smoothed value equal to the first observation.

Another method is to use the average of the first five or six observations for the initial smoothed value.

Exponential smoothing is often a good forecasting method when a nonrandom time series exhibits trending behavior.


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Example 4.5

The exponential smoothing technique is demonstrated in the following table and figure for Acme Tool Company for the years 1996 to 2002, using smoothing constants of .1 and .6. The data for the first quarter of 2002 will be used as test data to help determine the best value of (among the two considered).


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34

The exponentially smoothed series is computed by initially setting equal to 500.

1Y

If earlier data are available, it might be possible to use them to develop a smoothed series up to 1996 and use this experience as the initial value for the smoothed series.

The computations leading to the forecast for periods 3 and 4 are demonstrated next.


At time period 2, the forecast for period 3 with = .1 is

The error in this forecast is

The forecast for period 4 is

From the previous table, using smoothing constant of .1, the forecast for the first quarter of 2002 is 469 with a squared error of 145,161.

When the smoothing constant is .6, the forecast for the first quarter of 2002 is 576 with a squared error of 75,076.

On the basis of this limited evidence, exponential smoothing with = .6 performs better than exponential smoothing with = .1.


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36



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In summary

MSE and MAPE are both large and, on the basis of these summary statistics.

It is apparent that exponential smoothing does not represent these data well.

As we shall see, a smoothing method that allows for seasonality does a better job of predicting the data in the previous example.

Another factor beside that affects the values of subsequent forecasts is the choice of the initial value for the smoothed series.

In Example 4.5, was used as the initial smoothed value.

This choice gave Y1 too much weight in later forecasts.

Fortunately, the influence of the initial forecast diminishes greatly as t increases.


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1Y

1 1Y Y


Another approach to initialize the smoothing procedure is to average the first k observations. The smoothing then begins with

Often k is chosen to be a relatively small number. For example, the default approach in Minitab is to set k = 6.

Example 4.6

The computation of the initial value as an average for the Acme Tool Company data presented in Example 4.5 is shown next. If k is chosen to equal 6, then the initial value is

The MSE and MAPE for each alpha when an initial smoothed value of 383.3 is used are shown next.


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The initial value of 383.3 led to a decrease in MSE and MAPE for = .1 but did not have much effect when = .6.

Now the best model, based on the MSE and MAPE summary measures, appears to be one that uses = .1 instead of .6.

When the data are run on Minitab, the smoothing constant of = .266 was automatically selected by minimizing MSE.

The MSE is reduced to 19,447, the MAPE equals 32.2% and, although not shown, the MPE equals to - 6.4% and the forecast for the first quarter of 2002 is 534.

The autocorrelation function for the residuals of the exponential smoothing method using an alpha of .266 shows that when the Ljung-Box test is conducted for six time lags, the large value of LBQ (33.86) shows that the first six residual autocorrelations as a group are larger than would be expected if the residuals were random.


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In particular, the significantly large residual autocorrelations at lags 2 and 4 indicate that the seasonal variation in the data is not accounted for by simple exponential smoothing.

One way to measure this change is to use the tracking signal which involves computing a measure of forecast errors over time and setting limits so that, when the cumulative error goes outside those limits, the forecaster is alerted.

For example, a tracking signal might be used to determine when the size of the smoothing constant should be changed.

Since a large number of items are usually being forecast, common practice is to continue with the same value of for many periods before attempting to determine if a revision is necessary.

Unfortunately, the simplicity of using an established exponential smoothing model is a strong motivator for not making a change.

But at some point it may be necessary to update a or abandon exponential smoothing altogether.


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When the model produces forecasts containing a great deal of error, a change is appropriate.

A tracking system is a method for monitoring the need for change.

Such a system contains a range of permissible deviations of the forecast from actual values.

As long as forecasts generated by exponential smoothing fall within this range, no change in is necessary.

However, if a forecast falls outside the range, the system signals a need to update .

If things are going well, the forecasting technique should over- and

underestimate equally often.

A tracking signal based on this rationale can be developed.

Let U equal the number of underestimates (positive errors) out of the last k forecasts.


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If the process is in control, the expected value of U is k/2 but sampling variability is involved, so values close to k/2 would not be unusual.

On the other hand, values that are not close to k/2 would indicate that the technique is producing biased forecasts.

Example 4.7

Suppose that Acme Tool Company has decided to use the exponential smoothing technique with equal to .1. If the process is in control and the analyst decides to monitor the last 10 error values, U has an expected value of 5.

U values of 2, 3, 4, 6, 7, or 8 would not be unduly alarming.

However, values of 0, 1 , 9, or 10 would be of concern since the probability of obtaining such values by chance alone would be .022.

A tracking system can be developed based on the following rules.


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Assume that, out of the next 10 forecasts using this technique, only one has a positive error.

Because the probability of obtaining only one positive error out of 10 is quite low (.011)

The process is considered to be out of control (overestimating), and the value of should be changed.

Another way of tracking a forecasting technique is to determine a range that should contain the forecasting errors, which can be accomplished by using the MSE that was established when the optimally sized was determined.

If the exponential smoothing technique is reasonably accurate, the forecast error should be approximately normally distributed about a mean of zero.

Under this condition, there is about a 95% chance that the actual observation will fall within approximately 2 standard deviations.


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Example 4.8

In the previous example optimal was found to be .266 with MSE = 19,447. An estimate of the standard deviation of the forecast errors is 19,447 = 139.5

If the forecast errors are approximately normally distributed about a mean of zero, there is about a 95% chance that the actual observation will fall within 2 standard deviations of the forecast or within

For this example, the permissible absolute error is 279.

If for any future forecast the magnitude of the error is greater than 279, should be updated or a different forecasting method considered.

The preceding discussion on tracking signals also applies to the smoothing methods yet to be discussed in the rest of the chapter.

Simple exponential smoothing works well when the data vary about a level that changes infrequently.


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Whenever a sustained trend exists, exponential smoothing will lag behind the actual values over time.

Holt’s linear exponential smoothing technique, which is designed to handle data with a well-defined trend, addresses this problem and is introduced next.

2. Holt's Method

This method is designed to handle time series data with a well defined trend.

It is also called Holt's linear exponential smoothing method.

When a trend in the time series is anticipated, an estimate of the current slope as well as the current level is required.

Holt’s technique smoothes the level and slope directly by using different smoothing constants for each.

These smoothing constants provide estimates of level and slope that adapt over time as new observations become available.


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One of the advantages of Holt’s technique is that it provides a great deal of flexibility in selecting the rates at which the level and trend are tracked.

The three equations used in Holt’s method are:

The exponentially smoothed series, or current level estimate:

The trend estimate

Forecast p periods in the future


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Usually, we start by setting the initial value for the level by L1 = Y1 and by setting the initial value for the trend by T1 = 0.

Another approach is to take the first five or six observations and set the initial value for the level by the mean of these observations and to set the initial trend value by the slope of the line that is fit to those five or six observations where the variable of interest Y is regressed on the time as its independent variable X.

As with simple exponential smoothing, the smoothing constants and β can be selected subjectively or by minimizing a measure of forecast error such as the MSE.

Large weights result in more rapid changes in the component; small weights result in less rapid changes.

Therefore, the larger the weights, the more the smoothed values follow the data; the smaller the weights, the smoother the pattern in the smoothed values.


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We could develop a grid of values of and β and then select the combination that provides the lowest MSE.

Most forecasting software packages use an optimization algorithm to minimize MSE.

We might insist that = β , thus providing equal amounts of smoothing for the level and the trend.

Example 4.9

In Example 4.6 simple exponential smoothing did not produce successful forecasts of Acme Tool Company saw sales. Because the data suggest that there might be a trend in these data, Holt’s linear exponential smoothing is used to develop forecasts.

To begin the computations shown in following table, two estimated initial values are needed, namely, the initial level and the initial trend value.

The estimate of the level was set equal to the first observation.


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The trend was estimated to equal zero. The technique is demonstrated in the table for = .3 and β = 1

The computations leading to the forecast for period 3 are shown next.

1. Update the exponentially smoothed series or level:


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2. Update the trend estimate:

3. Forecast one period into the future:

4. Determine the forecast error

The forecast for period 25 is computed as follows:

1. Update the exponentially smoothed series or level:


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2. Update the trend estimate:

3. Forecast one period into the future:

On the basis of minimizing the MSE over the period 1996 to 2002, Holt’s linear smoothing (with = .3 and β = 1) does not reproduce the data any better than simple exponential smoothing that used of .266.

Forecasts for the actual sales for the first quarter of 2002 using Holt’s smoothing and simple exponential smoothing are similar.

A comparison of the MAPEs shows them to be about the same.

To summarize:


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The following figure shows the results when Holt’s method using = .3 and β = .1 is run on Minitab.

In the Minitab program the trend parameter gamma (γ) is identical to our beta (β).


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3. Winter's Method

It is considered as an extension of Holt's method but it is designed to handle time series data with a well defined trend and seasonal variation.

It is also called Winters' three parameter linear and seasonal exponential smoothing method.

The four equations used in Winters’ (multiplicative) smoothing are:

The exponentially smoothed series or level estimate:

The trend estimate:

The seasonality estimate:


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Forecast p periods into the future:

As with Holt’s method, the weights , β, and γ can be selected subjectively or by minimizing a measure of forecast error such as MSE.

The most common approach for determining these values is to use an

optimization algorithm to find the optimal smoothing constants.


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Example 4.10

Winters’ technique is demonstrated in the following table for = .4, β = .1, and γ = .3 for the Acme Tool Company data. The value for is similar to the one used for simple exponential smoothing in Example 4.6 and is used to smooth the data to create a level estimate. The smoothing constant β used to create a smoothed estimate of trend. The smoothing constant γ is used to create a smoothed estimate of the seasonal component in the data.

Minitab results are shown in the next table.

The forecast for the first quarter of 2002 is 778.2.

The computations leading to the forecast value for the first quarter of 2002, or period 25, are shown next.


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The computations leading to the forecast for period 3 are shown next.

1. The exponentially smoothed series or level estimate


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2. The trend estimate:

3. Seasonality estimate:

4. Forecast p = 1 period into the future:


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In the Minitab the trend parameter gamma (γ) is identical to our beta (β). and the seasonal parameter delta (δ) is identical to our gamma (γ)


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For the parameter values considered, the Winters’ technique is better than both of the previous smoothing procedures in terms of minimizing MSE.

When the forecasts for the actual sales for the first quarter of 2002 are compared, the Winters’ technique also appears to do a better job.

Winters’ exponential smoothing method seems to provide adequate forecasts for the Acme Tool Company data.

Winters’ method provides an easy way to account for seasonality when data have a seasonal pattern.

An alternative method consists of first deseasonalizing or seasonally adjusting the data.

Deseasonalizing is a process that removes the effects of seasonality from the raw data and will be demonstrated in Chapter 5.

Exponential smoothing is a popular technique for short-run forecasting. Its major advantages are low cost and simplicity.


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When forecasts are needed for inventory systems containing thousands of items, smoothing methods are often the only acceptable approach.

Simple moving averages and exponential smoothing base forecasts on weighted averages of past measurements.

The rationale is that past values contain information about what will

occur in the future.

Because past values include random fluctuations as well as information

concerning the underlying pattern of a variable, an attempt is made to

smooth the values.

This approach assumes that extreme fluctuations represent randomness

in a series of historical observations.

Moving averages, on the other hand, are the means of a certain number,

k, of values of a variable.

The most recent average is then the forecast for the next period.


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This approach assigns an equal weight to each past value involved in the

average.

However, a convincing argument can be made for using all the data but

emphasizing the most recent values.

Exponential smoothing methods are attractive because they generate

forecasts by assigning weights that decline exponentially as the

observations get older.


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Time series are often recorded at fixed time intervals.

For example, Y might represent sales, and the associated time series could be a sequence of annual sales figures.

Other examples of time series include quarterly earnings, monthly inventory levels, and weekly exchange rates.

In general, time series do not behave like a random sample and require special methods for their analysis.

Observations of a time series are typically related to one another (autocorrelated).

This dependence produces patterns of variability that can be used to forecast future values and assist in the management of business operations.

Consider these situations.

It is important that managers understand the past and use historical data and sound judgment to make intelligent plans to meet the demands of the future.

Time Series and Their Components


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Properly constructed time series forecasts help eliminate some of the uncertainty associated with the future and can assist management in determining alternative strategies.

Forecasting is done by a set of procedures followed by judgments.



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Decomposition

It is an approach to the analysis of time series data involves an attempt to identify the component factors that influence each of the values in a series.

The components of time series are:



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Time-Series

Cyclical Component

Random Component

Trend Component

Seasonal Component

1. Trend Component

It represents the growth and the decline in a time series, denoted by T.

Long-run increase or decrease over time (overall upward or downward movement) and they could linear or nonlinear

Data taken over a long period of time



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2. Cyclical Component

It represents a long-term wavelike fluctuations or cycles of more than one year's duration in a time series, denoted by C.

Practically it is difficult to identify and frequently regarded as part of trend.

Regularly occur but may vary in length

Often measured peak to peak



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3. Seasonal Component

It represents the seasonal variation in a time series which refers to a more or less stable pattern of change that appears short-term regular wave-like patterns and repeats itself season after season, denoted by S.

Observed within 1 year.

Often monthly or quarterly.


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4. Irregular Component

It represents the unpredictable or random fluctuations in a time series, denoted by I.

Unpredictable, random, “residual” fluctuations

Due to random variations of

Nature

Accidents or unusual events

“Noise” in the time series

To study the components of a time series, the analyst must consider how the components relate to the original series.


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Time Series Components Models

Additive Components Model

It is suggested to use when the variability are the same throughout the length of the series.

Multiplicative Components Model

It is suggested to use when the variability are increasing throughout the length of the series.

Note that it is possible to convert the multiplicative model to the additive model using logarithms. i.e.


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Time series with constant variability



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Time series with increasing variability


Estimation of Time Series Components

Estimation of Trend Component

Trends are long term movements in a time series that can be sometimes be described by a straight line or a smooth curve.

Remark

Fitting a trend curve helps us in providing some indication of the general direction of the observed series, and in getting a clear picture of the seasonality after removing the trend from the original series.

The Linear Trend

The Quadratic Trend


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The Exponential Trend

Where is the predicted value of the trend at time t , b0 , b1 and are called the model parameters.

We can forecast the trend using the above models as and so on.

Note that the Error Sum of Squares (SSE) is measured by


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ˆtT


Example 5.1

Data on annual registrations of new passenger cars in the United States

from 1960 to 1992 are shown in the following table and plotted in the later

figure.


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77

We definitely have a trend

here!


The values from 1960 to 1992 are used to develop the trend equation. Registrations is the dependent variable, and the independent variable is time t coded as 1960 = 1, 1961 = 2, and so on. The fitted trend line has the equation

The slope of the trend equation indicates that registrations are estimated to increase an average of 68,700 each year.


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The figure shows a straight-line trend fitted to the actual data. It also shows forecasts of new car registrations for the years 1993 and 1994 (t = 34 and t = 35) obtained by extrapolating the trend line.


The estimated trend values for passenger car registrations from 1960 to 1992 are shown in the table. For example, the trend equation estimates registrations in 1992 (t =33) to be

or 10,255,000 registrations.

Registrations of new passenger cars were actually 8,054,000 in 1992.

For 1992, the trend equation overestimates registrations by approximately 2.2 million.

This error and the remaining estimation errors were listed in the table.

The estimation errors were used to compute the measures of fit, MAD, MSD, and MAPE also were shown in the figure.


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Forecasting a Trend

Which trend model is appropriate?

Linear, quadratic or exponential

Linear models assume that a variable is increasing (or decreasing) by a constant amount each time period.

A quadratic curve is needed to model the trend.

Based on the accuracy measures, a quadratic trend appears to be a better representation of the general direction of the data.


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When a time series starts slowly and then appears to be increasing at an increasing rate such that the percentage difference from observation to observation is constant, an exponential trend can he fitted.

The coefficient b1 is related to the growth rate.

If the exponential trend is fit to annual data, the annual growth rate is estimated to be 100(b1 — 1)%.


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The figure next contains the number of mutual fund salespeople for several consecutive years.

The increase in the number of salespeople is not constant.

It appears as if increasingly larger numbers of people are being added in the later years.


A linear trend fit to the salespeople data would indicate a constant average crease of about nine salespeople per year.

This trend overestimates the actual increase in the earlier years and underestimates the increase in the last year.

It does not model the apparent trend in the data as well as the exponential curve.

It is clear that extrapolating an exponential trend with a 31 % growth rate will quickly result in some very big numbers.

This is a potential problem with an exponential trend model.

What happens when the economy cools off and stock prices begin to retreat?

The demand for mutual fund salespeople will decrease and the number of salespeople could even decline.

The trend forecast by the exponential curve will be much too high.


82


Growth curves of the Gompertz and logistic types reflect a situation in which sales begin low, then increase as the product catches on, and finally ease off as saturation is reached.

Judgment and common sense are very important in selecting the right approach.

As we will discuss later, the line or curve that best fits a set of data points might not make sense when projected as the trend of the future.


83


Suppose we are presently at time t = n (end of series) and we want to use a trend model to forecast the value of Y, p steps ahead.

The time period at which we make the forecast, n in this case, is called the forecast origin.

The value p is called the lead time.

For the linear trend model, we can produce a forecast by evaluating

Using the trend line fitted to the car registration data in Example 5.1 , a forecast of the trend for 1993 (t = 34) made in 1992 (t = n = 33) would be the p = 1 step ahead forecast

Similarly, the p = 2 step ahead forecast (1994) is given by


84


Using the quadratic trend curve for the car registration data, we can calculate forecasts of the trend for 1993 and 1994 by setting t = 33 + 1 = 34 and t = 33 + 2 = 35.

The forecasts are = 8.690 and = 8.470 (respectively)

Recalling that car registrations are measured in millions, the two forecasts of trend produced from the quadratic curve are quite different from the forecasts produced by the linear trend equation.

Moreover, they are headed in the opposite direction.

If we were to extrapolate the linear and quadratic trends for additional time periods, their differences would be magnified.

This example illustrates why great care must be exercised in using fitted trend curves for the purpose of forecasting future trends.

The differences can be substantial for large lead times (long-run forecasting).


85


Trend curve models are based on the following assumptions:

The correct trend curve has been selected.

The curve that fits the past is indicative of the future.

We must be able to argue that the correct trend has been selected the future will be like the past.

There are objective criteria for selecting a trend curve. We will discuss two of these criteria, the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), in later chapters.

However, although these and other criteria help to determine an appropriate model, they do not replace good judgment.


86


Estimation of Seasonal Component

A seasonal pattern is one that repeats itself year after year.

For annual data, seasonality is not an issue because there is no chance to model a within year pattern with data recorded once per year.

Time series consisting of weekly, monthly, or quarterly observations often exhibit seasonality.

The analysis of the seasonal component of a time series has direct short- term implications and is of greatest importance to mid- and lower-level management.

Marketing plans have to take into consideration expected seasonal patterns in consumer purchases.

Several methods for measuring seasonal variation have been developed.

The basic idea in all of these methods is to first estimate and remove the trend from the original series and then smooth out the irregular component.


87


The seasonal values are collected and summarized to produce a number (generally an index number) for each observed interval of the year (week, month, quarter, and so on).

The identification of the seasonal component in a time series differs from trend analysis in at least two ways:

1 . The trend is determined directly from the original data, but the seasonal component is determined indirectly after eliminating the other components from the data so that only the seasonality remains.

2. The trend is represented by one best-fitting curve, or equation, but a separate seasonal value has to be computed for each observed interval (week, month, quarter) of the year and is often in the form of an index number.

Always we estimate the seasonality in form of index numbers, percentages that show changes over time, are called seasonal index.

If an additive decomposition is used, estimates of the trend, seasonal, and irregular components are added together to produce the original series.


88


If a multiplicative decomposition is used, the individual components must be multiplied together to reconstruct the original series, and in this formulation, the seasonal component is represented by a collection of index numbers.

These numbers show which periods within the year are relatively low and which periods are relatively high.

The seasonal indices trace out the seasonal pattern.

Index numbers are percentages that show changes over time.

Remark

In this chapter we study the multiplicative model and leave the additive model to chapter 8 if we have time.

In multiplicative decomposition model, the ratio to moving average is a popular method for measuring seasonal variation.


89


Finding Seasonal Indexes

Ratio-to-moving average method:

Begin by removing the seasonal and irregular components (St and It), leaving the trend and cyclical components (Tt and Ct)

Example: Four-quarter moving average

First average:

Second average:

etc…


90

1

Q1 Q2 Q3 Q4Moving average

4

2

Q2 Q3 Q4 Q5Moving average

4



91

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 10 11

Sale

s

Quarter

Quarterly SalesQuarter Sales

1

2

3

4

5

6

7

8

9

10

11

etc…

23

40

25

27

32

48

33

37

37

50

40

etc…



92

Each moving average is for a consecutive block of 4 quarters

Quarter Sales

1 23

2 40

3 25

4 27

5 32

6 48

7 33

8 37

9 37

10 50

11 40

Average

Period

4-Quarter

Moving

Average

2.5 28.75

3.5 31.00

4.5 33.00

5.5 35.00

6.5 37.50

7.5 38.75

8.5 39.25

9.5 41.00

4

43212.5

4

2725402328.75

etc…

Centered Seasonal Index


Average periods of 2.5 or 3.5 don’t match the original quarters, so we average two consecutive moving averages to get centered moving averages

Now estimate the St x It value by dividing the actual sales value by the centered moving average for that quarter.


93

Average

Period

4-Quarter

Moving

Average

2.5 28.75

3.5 31.00

4.5 33.00

5.5 35.00

6.5 37.50

7.5 38.75

8.5 39.25

9.5 41.00

Centered

Period

Centered

Moving

Average

3 29.88

4 32.00

5 34.00

6 36.25

7 38.13

8 39.00

9 40.13

etc…


Ratio-to-Moving Average formula:


94

tt t

t t

YS I

T C

Quarter Sales

Centered

Moving

Average

Ratio-to-

Moving

Average

1

2

3

4

5

6

7

8

9

10

11

…

23

40

25

27

32

48

33

37

37

50

40

…

29.88

32.00

34.00

36.25

38.13

39.00

40.13

etc…

…

…

0.837

0.844

0.941

1.324

0.865

0.949

0.922

etc…

…

…

Example

250.837

29.88



95

Quarter Sales

Centered

Moving

Average

Ratio-to-

Moving

Average

1

2

3

4

5

6

7

8

9

10

11

…

23

40

25

27

32

48

33

37

37

50

40

…

29.88

32.00

34.00

36.25

38.13

39.00

40.13

etc…

…

…

0.837

0.844

0.941

1.324

0.865

0.949

0.922

etc…

…

…

Fall

Fall

Fall

Do the same for the other three seasons to get the other seasonal indexes

Average all of the Fall values to get Fall’s seasonal index



96

Suppose we get these seasonal indices:

Season Seasonal

Index

Spring 0.825

Summer 1.310

Fall 0.920

Winter 0.945

= 4.000 -- four seasons, so must sum to 4

Spring sales average 82.5% of the annual average sales

Summer sales are 31.0% higher than the annual average sales

etc…

Interpretation:


The data is deseasonalized by dividing the observed value by its seasonal index

This smoothes the data by removing seasonal variation


97

tt t t

t

YT C I

S

Quarter Sales Seasonal Index Deseasonalized Sales

1

2

3

4

5

6

7

8

9

10

11

…

23

40

25

27

32

48

33

37

37

50

40

0.825

1.310

0.920

0.945

0.825

1.310

0.920

0.945

0.825

1.310

0.920

…

27.88

30.53

27.17

28.57

38.79

36.64

35.87

39.15

44.85

38.17

43.48

…

0.825

2327.88



98


Example 5.3

In Example 3.5 the analyst for the Outboard Marine Corporation, used autocorrelation analysis to determine that sales were seasonal on a quarterly basis. Now, he uses decomposition to understand the quarterly sales variable. Minitab was used to produce the following table and figure. To keep the seasonal pattern current, only the last seven years (1990 to 1996) of sales data (Y), were analyzed.

The trend is computed using the linear model:


99



100

1ˆ 253.742 1.284(1) 255.026T 232.7 255.026 .912SCI Y T (.912 .788 ... .812) 7 0.780S

232.7 .7796 298.486TCI Y S 232.7 (255.026 .7796)CI Y TS (1.170 1.187 1.080) 3 1.146C

/

1.187 /1.146

1.036

I CI C


The cyclical indices can be used to answer the following questions:

The series cycle?

How extreme is the cycle?

The series follow the general state of the economy (business cycle)?

One way to investigate cyclical patterns is through the study of business indicators.

A business indicator is a business-related time series that is used to help assess the general state of the economy.

The most important list of statistical indicators originated during the sharp business setback of 1937 to 1938.

Leading indicators.

Coincident indicators.

Lagging indicators.


101


Forecasting A Seasonal Time Series

In forecasting a seasonal time series, the decomposition process is reversed.

Instead of separating the series into individual components for examination, the components are recombined to develop the forecasts for future periods.

Example 5.4

Forecasts of Outboard Marine Corporation sales for the four quarters of 1997

can he developed using the previous table.

1. Trend. The quarterly trend equation is: T = 253.742 + 1.284t.

The forecast origin is the fourth quarter of 1996, or time period t = n = 28.

Sales for the first quarter of 1997 occurred in time period t = 28 + 1 = 29.

This notation shows we are forecasting p = 1 period ahead from the end of the time series.

Setting t = 29, the trend projection is then T29 = 253.742 + 1.284(29) = 290.978


102


2. Seasonal. The seasonal index for the first quarter is .7796.

3. Cyclical. The cyclical projection must be determined from the estimated cyclical

pattern (if any) and any other information generated by indicators of the general

economy for 1997.

Projecting the cyclical pattern for future time periods is fraught with uncertainty, and as

we indicated earlier, is generally assumed for forecasting purposes to be included in the

trend.

To demonstrate the completion of this example, we set the cyclical index to 1.0.

4. Irregular. Irregular fluctuations represent random variation that can’t be

explained by the other components.

For forecasting, the irregular component is set to the average value 1.0

The forecast for the first quarter of 1997 is

The forecasts for the rest of 1997 are


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