Reading 14 Forecasting with Time Series Qualitative and...

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Reading 14 Forecasting with Time Series

Qualitative and Quantitative Methods of Forecasting

Business decisions are made under conditions of uncertainty. Business professionals want methods to more accurately make better business decisions. Business forecasting and business planning are two different processes. Forecasting will provide estimates of certain variables, which will then be used in the planning process. Planning is much broader than forecasting. For example, I might want to provide my company with a five-year plan of the opportunities and challenges available. However, I might like to include a forecast of my sales during that period of time. The forecast might come from the use of the time series method or the regression analysis techniques from the previous lecture. The techniques associated with forecasting will provide the tools for quantitatively enhancing business decisions. Forecasting involves the development and use of data sets and numbers. Planning on the other hand might include the use of any one of the following three non-numeric, qualitative approaches. Qualitative approaches incorporate the judgment of those participating, whereas quantitative approaches employ statistical techniques, which are usually applied to historical data to develop answers for future time periods. One implicit assumption is necessary when one applies statistical techniques to historical data sets. The assumption is that the history of yesterday is representative of the trend of today. This assumption is often the difference between the correct use of historical quantitative approaches and too much reliance on historical quantitative approaches.

For example, in the late 1960’s and early 1970’s, the American automobile industry was on an upward sales trend which appeared to be all roses. The Arab nations failed to defeat Israel in 1967 and in partial response, OPEC in 1973 sharply curtailed the supply of oil. Prices rose and shortages of gasoline occurred. The result was disaster for Detroit. The American public turned away from the gas-guzzlers to the smaller cars of Japan. Detroit experienced a downturn in sales. The industry, which had largely relied on historical forecast to build certain levels of inventory, found themselves in a fight for their life as inventory failed to move as anticipated.

Time series forecasting (which had been used by Detroit as a guide) is the manipulation of historical numbers and as such is not capable of easily seeing the turns in the road. Had Detroit used some of the qualitative approaches discussed below to supplement their strictly historical approach, the downturn might have been better anticipated. Interestingly in the 1990’s the trend reversed itself and the American public turned once again to the larger vehicles and the SUV was born. Could the future of oil hold similar prospects as the early 1970’s?


Let’s look at three qualitative approaches (non-quantifiable), which add some qualitative validity to the quantitative values.

Qualitative Approaches:

Delphi Approach

This is a group consensus approach. Opinions are gathered from a panel of experts often managers of a company. The panel does not have to physically be in the same location. In today’s instant communication world, the process can be conducted via email especially if the corporation is large and scatter throughout the USA. A moderator, selected by upper management, will develop a questionnaire based on upper management’s identification of an important issue or issues for the future.

The questionnaire might be as simple as

(1) What are the 10 most important opportunities for XYZ Corporation in the next 5 years?

(2) What are the 10 biggest challenges for XYZ Corporation over the next 5 years?

As the moderator, you would send out the questionnaire to all of the pre-selected managers (experts). There would be a response time limit. Usually the response time is very short, perhaps a day or two at the most. The managers respond to your questions. Often all managers will not participate at the first level, but this does not void the study. As the process condenses the answers, more and more managers will offer their opinions. Late-comers do not void the study. The answers are returned to you as the moderator of the study. As the moderator, you tabulate the answers noting the number of common responses for each answer.

You then re-pose the questions giving the respondents an opportunity to adjust their answer. Again the time limit is short, usually a day. You will receive the modified answers, re-tabulate and re-submit back to the managers again. Each time you will find the answers tending to begin focusing on better and better responses. This is essentially the process of prioritizing the most recent responses through a natural process. You will probably have to administer this process up to 6 times before you have 10 to 15 good, consensus answers. These answers then form the nucleus for a moderated discussion forum in a face-to- face meeting with the managers. Out of that discussion group, you will be able to best identify the opportunities and challenges for the immediate future. Of course, the final discussion should include the development of ideas on how the company should respond to each of the issues on the table. The final outcome is a non-quantitative approach to planning.


The biggest drawback to the Delphi Approach is the expertise of the experts. This is especially true if the process only uses internal managers to respond. The internal managers might not want to express opinions that run contrary to the perceived direction of upper management. To avoid this, outside participants should also be included in the survey process.

Scenario Writing Approach

This technique is much less structured than the Delphi Approach. A set of well-defined assumptions is developed. The assumptions act as a plausibility statement of what the future might hold. Often it is necessary to develop several sets of well-defined assumptions that correspond to a number of future scenarios. A panel of experts is asked to develop a conceptual scenario of the future based on each set of well-defined assumptions. Once the scenarios are written, management will select the scenario which seems most likely and then determine if the scenario is realistic. Often pieces of each expert’s scenario will be appropriated for a consensus result. This process is often restricted to just a few experts and involves more time to properly develop. I would be very surprised if Detroit does not include in their forecast a set of assumptions for both tightening and loosing of the oil supply.

Brainstorming Approach

This approach should be done as a very positive approach. No negative reasons or explanations should be given. No criticism of any person making a comment should be done. A group of managers will meet to discuss issues within the company and their solutions to those issues. The managers will, without commenting negatively, list all of the possible outcomes (collectively exhaustive approach). At the end of the discussion, the alternatives will be prioritized and those suggestions that are less favorable will fall to the bottom of the list. This approach, if used in the positive manner, will encourage participation from all.

During WWII, a panel of think-tankers was convened to develop options for the commander in the field to use under certain battle conditions. A battlefield situation was presented to the group. One such situation was to assume you are in a ship and the engines have gone dead. Your ship cannot move. Floating toward you is a water mine. If the mine hits you, the ship will blowup and perhaps be totally lost. What do you do?

There was much discussion of the issue. No acceptable response was reached. One individual, in shear frustration, spoke out. There are over 300 sailors on the ship. Why don’t we get them on the side of the ship where the mine is approaching and all inhale at the same time, then exhale at the same time. The air will blow the mine away from the ship.


Suddenly the solution was clear. Take the water hose and create a stream of water to gently push the mine away from the ship. The point is this, had the frustrated individual not felt comfortable to make a silly comment, the solution could have gone without discovery. Positive brain-storming can be a good thing. Don’t discourage even the simplest comment.

These three approaches are often very helpful in gaining a consensus opinion that is useful in planning the future of an organization.

Quantitative Approaches:

Time Series Analysis

Time series analysis is an important quantitative tool for analyzing your data. This approach uses much the same approach that was used in simple regression. Remember, in simple regression, I used one dependent variable (Y) and one independent variable (X). The example I used was that the sales in a pizza restaurant are linearly related to the number of students attending a college close to the restaurant. Sales was the dependent variable and number of students or student population was the independent variable. My hope was to see if the movement in student population could help predict the movement in the restaurant sales.

A similar approach is used in time series. The difference is that the X-variable is time (horizontal axis) rather and a specific variable. I am studying the movement of a single variable over time. Examples would be the movement of stock prices for a given stock over time or the movement of interest rates over time or the movement of sales over time.

There are four very important factors that influence all movements of data sets over time. The first is secular trend, seasonal variation, cyclical variation, and random movements.

These four components will be present in any time series data set (TSCR). Time series analysis involves the use of these four historical components to arrive at a future forecast for the data in question. All of you have seen charts on stock or the Dow Jones Average. These are charts associate with time thus are time series.

Definitions of T S C and R:

Linear Trend (T): Trend is long-term movement over an extended period of time. It is longer than one year. By observing the trend, you can generally detect the general direction of the movement - up, down or even. The rate of change is relatively constant.


Seasonality(S): Seasons of the year bring on different sales for example. December brings on the Christmas sales frenzy. Spring brings on the sale of yard equipment and supplies. August brings on the sale of "back to school" purchases of clothing, books and supplies. The seasonal pattern tends to occur at the same time each year. The movement is short-term and is complete in less than one year. I could not use annual data to determine seasonal movements.

Cyclical (C): This is a wavelike variation, which in general follows the general level of business activity over a relatively long period of time. Usually three years or longer is involved in the cyclical portion of time series movements. There are four phases to the cycle - the upswing or expansion, the peak when the economy tops out, the downturn or contraction when employment declines and sales decline, and the trough, where the business activity is at the lowest point. These movements take longer than one year to develop thus are considered long-term movement.

Irregular (R or I): Irregular or random movements are fluctuations caused by unusual occurrences and produce no discernible patterns. The movements are unlikely to reoccur in a similar fashion. Examples are earthquakes, wars, floods, oil embargoes, terrorist attacks etc. These movements are also considered to be long-term movements even though they will usually affect only one period of time.

Several Observations and Definitions:

Forecasting Horizon: Often this is referred to as the forecast lead-time. This is the number of future periods covered by the forecast. Most of the time three or four periods are identified. Immediate-term refers to less than one month. Short-term refers to one to three months. Medium-term refers to three months to two years. Long-term refers to two years or more. The accuracy of the forecast will often depend on the depth of the original data with which you are working.

Forecasting Period: The forecasting period might be for days, weeks, months, quarters, semi-annual or annually. Historical data must be available in the appropriate units. For example, if you want monthly forecasts, data must be available in months.

Forecasting Interval: If the forecasting period is for weeks, I would develop a weekly forecast. The frequency of the forecast must be for the same period as my data set.

Two Models

Even though there are two models, the additive and multiplicative model, I will limit the discussion here to the multiplicative model. The multiplicative model is


known as the ratio to moving average model, which is helpful in isolating seasonal movement. Many textbooks do not cover the additive model and that will be the case in this lecture. The multiplicative model is as follows.

Yt = T x C x S x I

Where Yt is the entire time series data set.

Since this is an equation, you can mathematically manipulate the equation. For example, if you divide one side of the equation by one of the components, you would also divide the other side of the equation by the same component. The purpose of this exercise is to demonstrate how you can isolate some of the movements. Three of the components have been identified as long-term movements and one has been identified as short-term. What I would like to eventually do is to remove the short-term movements (seasonality) and then prepare a forecast of the long-term movements, then add back the seasonality to the forecast. Before I take you to this solution, I want to introduce you to three other methods of forecasting. Isolation of seasonal movements is quite tedious and there are some other quantitative processes which might be worth mentioning first.

Three Methods of Forecasting

1. Smoothing Techniques.

a. Moving Averages.

b. Weighted Moving Averages.

c. Exponential Smoothing.

2. Trend Projections

3. Decomposition – This is the process of isolating seasonal movement.

Let's look at the three methods one at a time.

Smoothing Techniques

There are three smoothing techniques - moving averages, weighted moving averages, and exponential smoothing. The purpose of a smoothing technique is to eliminate wild variations in the data set, which make it difficult to see any trend or cyclical movements in the time series. These three techniques range from the most simple to the more sophisticated. Often business operators will not want to spend the time or resources on some of the more sophisticated techniques so they will use a less sophisticated technique such as the three mentioned here. If


you cannot use the better techniques, use the less sophisticate techniques. A “guesstimated” forecast is not really a forecast in which the business manager can take a great deal of comfort. The point is this. Use some technique even if it is not the most sophisticated to aid in your business forecasting and planning.

Moving Averages: A moving average is a series of arithmetic averages over a given number of time periods. This is an estimate of the long-run average of the variable.

Example of Moving Average:

Let's say that I own a gasoline station and I want to use my current sales to forecast future sales. I do not want to be too sophisticated so I decide to use a three-week moving average to forecast the fourth week.

Week #

Actual Sales(000)

3 Week Moving Average

1 17

2 21 Plotting point if plotting the data

set. 3 19

4 Need this week’s forecast

I will do the following.

The 4th Period Forecast = (17 +21 + 19) ÷ 3 = 19.

So the forecast for my fourth week is 19,000 gallons of gasoline based on the three previous weeks.

Remember, I am using this method to forecast. Were I simply smoothing my data set, I would plot the value of 19 at the mid-point of my averaged data. See the 19 under moving average. Of course, this example is an over simplified data set. Any effective data set would consist of many more data points.

Okay now let's repeat the chart from above.


Week #Actual Sales (000)

3 Week Moving Average

1 17

2 21

3 19

4 If actual sales is 23,

how would I adjust my forecast for the 5th Wk?

19 - Forecast from above

5 What is this forecast?

Let's say that the fourth week comes along and the actual sales is 23,000 gallons of gasoline. In other words, the forecast was 19,000 using the moving average approach, but the actual gallons sold was 23,000. Now I want to forecast the gallons using the same 3-week moving average approach for the 5th week. What do I do?

Moving Average = (21 + 19 + 23) ÷ 3 = 21 which is my forecast for the 5th week.

Of course, there is nothing, except additional data, preventing me from making the moving average a 4 week or 5 or 6 week moving average. That choice is up to my own personal belief about my actual data set and desired forecast period.

Weighted Moving Average

What if I believe that the most current week of actual data is of more value to me in forecasting than any other week? In other words, I believe that most current week of gasoline sales is more important that the actual sales from three weeks ago. I might want to use a weighted average of the actual gasoline sales. After studying the situation, the weights I decide to use are 1/6 for week one, 2/6 for week two, and 3/6 for week three. Notice that the sum of my weights is equal to one. This will always be true. I now multiply the actual sales for by the weights assigned to that week. Next I would sum the weighted values. Let's go back to the forecast for the 4th week.

(17 x 1/6) + (21 x 2/6) + (19 x 3/6) = 19.33.

My un-weighted forecast was 19,000 gallons of gasoline for the 4th week, but my weighted forecast was 19,333 gallons of gasoline. Actually, if you stop and think about it, the un-weighted, three week moving average forecast is a also a weighted average. The difference is that the weights in the moving average are


all equal (being 1/3), while the weights in the weighted moving average are unequal. Here again I can chose any weights I believe to be relevant. I could use 10% for week one, 20% for week two and 70% for week three. Again the weights must total 100%.

Exponential Smoothing

This is a weighted average of the current and past values and is a better smoothing technique than either the moving average or the weighted moving average. This technique is, of course, more difficult to use, but the results are often more accurate. The basic model is as follows.

F t + 1 = (Y t) + (1 - ) F t

Where is the smoothing constant (not the alpha value you have been using for CI and HT).

F t + 1 = The forecast of the time series for the period t + 1. (The next period)

Y t = Actual value of the time series in period t.

F t = Forecast of the time series in Period t.

The difficulty with this method is the selection of the . The must be selected to minimize the mean square error (MSE). In reality, this can only be done by trial and error.

MSE = (F t - Y t )2 ÷ (n – 1)

What I mean by trial and error is just that. You plug in a value for , say 0.2 and solve the equation for the predicted values. You then calculate the MSE. You then plug in another value for , say 0.3. You then calculate the MSE. You then compare the two MSE values and select the one with the least value. You may have to repeat this process several time to develop the smallest value for MSE. The Homework Solutions Manual walks you through the steps associated with developing an exponentially smoothed forecast. See that manual for full details.

Generally speaking if the data are rather volatile, a lower -value should be selected. This is because the smaller values for alpha assign less weight to more recent observations. Stable movements in the data may require a higher -value.


Trend Projections

A second method of forecasting is trend projections. This technique is identical to simple regression analysis. A trend line is fitted to the data set using the Method of Least Squares or the Ordinary Least Squares Method (both the same). This line is a single best-fit line which minimizes the variation. Since I am using time as the X-variable, I must make some minor modifications to the formulas. They are notational only and have no bearing on the solutions.

Yt = b0 + b1 X for Simple Regression Analysis.

Tt = b0 + b1 t for Trend Analysis.

Where Tt is the forecast value of the time series in period t.

t is a period of time.

bO is the intercept of the trend line.

b1 is the slope of the trend line.

Converting the regression formulas to time series formula for calculating the coefficients, I have the following adjustments.

∑ ∑

∑ ∑ ∑

−

−=

ntt

nYttY

bt

t

22

1 )(

))((

tbYb to 1−=

The solution to trend analysis formulas is the same as simple regression analysis. In simple regression the variable along the horizontal axis was the independent variable. In trend analysis, the variable along the horizontal axis is time. Using time along this axis is why I adjust the formulas above. The column headings change to match the formulas above.

Trend analysis is an integral part of decomposition. Since I will next show you the concept of decomposition, I will at that time show you the trend analysis technique.


Decomposition

Remember, I told you earlier in this lecture that it would be best if I could find some way to isolate the short-term movements in my data set, then fit a trend line to the long-term movements, and then re-insert the short-term movements? Decomposition does just that. This process will yield a more accurate forecast than if you fit a trend line to the original, unadjusted data set which includes seasonality. Of course, if your data set has no seasonal movement, this procedure is not necessary. Examples of data sets which contain no or limited seasonal movement would be the use of salt or the use of insulin. Using the multiplicative model as previously presented, the relationship is as follows: Yt = T x C x S x I

Any time series data set is composed of these four elements. The decomposition method isolates seasonality, which is the short-term movement. Let’s look at an example. Suppose that you were the manufacturer of television sets. Over the past four years, you have accumulated the following quarterly sales data.

Year Quarter Sales - Yt (000)

1994 1 4.8

2 4.1

3 6.0

4 6.5

1995 1 5.8

2 5.2

3 6.8

4 7.4

1996 1 6.0

2 5.6

3 7.5

4 7.8

1997 1 6.3

2 5.9

3 8.0


4 8.4

These sales as stated currently contain all of the four elements of time series data (TSCI). The first step is to review the data set. You need to ask yourself the question. Does this data set reflect any seasonality? The answer is “absolutely”. The third and fourth quarters are bigger than the first and second quarters. If the data set had no seasonality, what would you expect?

All four quarter would be essentially the same. The quarterly sales might reflect actual values of 4.9, 5.0, 5.0 and 5.1. This would mean the data set of interest is not highly seasonalized; therefore, is not a candidate for this technique (decomposition).

I am going to explain some of the steps in the process, but you will need to follow the tables inserted below at the various stages to really understand what I am doing.

#1) The first step: I must isolating the seasonal movement by taking a four-quarter moving average, which is centered in the middle of the year.

4 Qtr. Moving Average = (4.8 + 4.1 + 6.0 + 6.5) ÷ 4 = 5.350. This value is reflected in the table below.

Since it is a moving average, the next data point will be determined by dropping the 4.8 (first quarter of the first year) and adding 5.8 (first quarter of the second year.).

4 Qtr. Moving Average = (4.1 + 6.0 + 6.5 + 5.8) ÷ 4 = 5.600.

Continue this process long enough to complete the table as shown below. Try a couple of them so you can make sure you understand the process.


#2) The second step: Unfortunately the four-quarter moving average is centered in the middle of the year or the middle of the quarter. Centered here, the value is of no use to me. I need values that are centered on the quarter itself. In order to achieve this, I will re-center the four-quarter moving average. This is easily accomplished by averaging two quarters at a time.

Centered Moving Average = (5.350 + 5.600) ÷ 2 = 5.475.

Centered Moving Average = (5.600 + 5.875) ÷ 2 = 5.600.

Continue this process until you have completed this chart. Notice that 4 years of original data becomes 3 years of adjusted data. I will not have data for the 1st and 2nd quarter of the first year or the 3rd and 4th quarter of the last year.


This second step smoothes the averages and effectively eliminates S and I from the equation. In other words, the centered 4 quarter moving average is defined as T x C. This column includes only the measures of Trend and Cyclical movements. For now do not try to fully understand the definition of T x C, just accept it. After you have a chance to think about the process you will better understand the “minimization” process.

#3) The third step: Next I must calculate the S and I. Remember the equation is Yt = T x C x S x I. I can isolate S and I by the following:

Yt ÷ T C = S I

My original sales column started with all four elements in the data set. The 4.8, 4.1 etc has all four elements. A data set with all four elements is called Yt . So I can divide the 4.8 (first quarter of the first year of the original data) by the Centered Moving Average of 5.475 (accomplished in the second step). The result will be to isolate seasonal movement (S) and irregular movement (I).


Year Qtr Yt 4QMA CMA S and I

#4) The fourth step: I next need to take the S I and average them so I can remove (minimize) the irregular or random movements (I). I will average the quarterly values computed above as follows:

Qtr 1994 SI

1995 SI

1996 SI

1997 SI

Total Four

Columns

Divide By

Seasonal

Index

Adjusted

Seasonal Index

1 None 0.971 0.918 0.908 2.797 3 0.9323 0.9308

2 None 0.840 0.839 0.834 2.513 3 0.8377 0.8363

3 1.096 1.075 1.109 None 3.280 3 1.0933 1.0915

4 1.133 1.156 1.141 None 3.430 3 1.1433 1.1414

4.0066 4.0000


In step four most folks will wonder where I got the numbers. Scroll back to the last table. Look at the location of the 0.971. Now cross reference the 0.971 to the year and the quarter. You will notice the 0.971 is the first quarter of the second year (1995).

Why did you skip the first quarter of the first year, you might ask?

Simple, I respond. There is no value for the first quarter of the first year (1994). Remember, I told you that I would take four years of data and reduce it to three years of adjusted data.

One warning is appropriate at this point. If the total seasonal index (next to the last column in the table just above) does not total 4.00, then prorate up or down to force the total to be 4.00. In my case, the column totals 4.0066. To get the seasonal index to equal 4.00, I must develop a ratio of 4.00 ÷ 4.066 = 0.9984. I next multiply each of the quarterly seasonal index number by 0.9984 to develop the last column, which will add to 4.00. In this case, I must prorate my values down. The seasonal movement is the last column in the above table.

To repeat the concept, let’s suppose the next to last column totaled 3.9875. I would then prorate each of the quarterly values up to equal 4.00 by developing a ratio of 4.00 ÷ 3.9875 = 1.0031. This factor would then be multiplied by each of the seasonal index numbers to raise them so the column totaled 4.00. Just the opposite would be true if the last column totaled 4.0175. Here again I would develop a ration of 4.00 ÷ 4.0175 = 0.9956. This factor would then be multiplied by each of the seasonal index numbers to lower them so the column totaled 4.00.

Why the number 4.00, you might ask?

I respond, because my data set is in quarters. There are 4 quarters thus the total must be 4.00.

I pose a question. What if my data set was expressed in monthly values? Would the total of the last column still be 4.00?

No, you respond. If your data set were in months, the last column would total 12.

Excellent answer, I respond. However, remember the proration issue would still apply.

Another word of caution is in order. When you are making your calculations, the total of the seasonal index numbers will not be significantly different than 4.00. What I am saying is this, if you calculate a seasonal index number BEFORE you apply any proration adjustment, the value will be very close to 4.00, say 4.01076 or 3.9835 (arbitrary numbers to illustrate the point). The value before proration


WILL NEVER be 4.656 or 3.564 (arbitrary numbers to illustrate the point). These last two numbers tell me I have a mathematical error somewhere. I better go back and check my numbers to see if I can find my mistake. In essence, the proration adjustment will be very small in all instances.

This process isolates the seasonal index. The averaging of the S and I values minimizes the effects of the irregular or random component.

How do you interpret the seasonal index of 0.9308, 0.8363, 1.0915 and 1.1414?

Think about it. All these calculations have been for one purpose – to isolate seasonal movement. The last column is labeled S. This is a measurement of the seasonal movement. However, what should I use as the benchmark against which I measure S? The answer is Trend. The first quarter is approximately 7% below trend (100.00 less 93.08). The second quarter is approximately 16% below trend. The third quarter is approximately 9% above trend and the fourth quarter is approximately 14% above trend.

From this explanation, you can easily see that the original data set given to you in the very first column of the first table (scroll way up to see it) is highly seasonalized. Once again if your data set has little or no seasonality, this entire process of de-seasonalization is not necessary. The two examples I gave were the use of salt and the use of insulin. Neither of these data sets would produce seasonalized data; therefore, any forecasting using trend analysis (which is where I am going) would use the original data set and not the de-seasonalized data set.

Because I want to get all of my data on the page, I am setting up the last columns of what may be assumed to be a continuous table as if they were tacked on to the table I have already used in my examples just above. Don’t get confused by this step. Just realize that the first four steps are shown above and the last steps are shown in the newly created table below. I think you can follow what I am doing.

Let’s give it a try. I will repeat the year, quarter column and the original data set values (first three columns), but the fourth through eighth columns are new. I will describe them after the table. They should be fairly easy to follow.


Year Qtr

Orig. Data S

De-Seasonalized

Data Set Yt ÷ S = TCI

t – Re- numbered

Quarter t Yt t2

1994 1 4.8 0.9308 4.8÷0.9308 = 5.15 1 5.15 1

2 4.1 0.8363 4.1÷0.8363 = 4.90 2 9.80 4

3 6.0 1.0915 5.50 3 16.50 9 4 6.5 1.1414 5.69 4 22.76 16 1995 1 5.8 0.9308 6.23 5 31.15 25 2 5.2 0.8363 6.22 6 37.32 36 3 6.8 1.0915 6.23 7 43.61 49 4 7.4 1.1414 6.48 8 51.84 64 1996 1 6.0 0.9308 6.45 9 58.05 81 2 5.6 0.8363 6.70 10 67.00 100 3 7.5 1.0915 6.87 11 75.57 121 4 7.8 1.1414 6.83 12 81.96 144 1997 1 6.3 0.9308 6.77 13 88.01 169 2 5.9 0.8363 7.04 14 98.56 196 3 8.0 1.0915 7.33 15 109.95 225 4 8.4 1.1414 7.36 16 117.76 256

Totals 101.75 136 914.99 1,496

#5) The fifth step: The seasonal index column (S) simply takes the values developed for the seasonal index in the SI averaging table and repeats them for each of the four years.

#6) The sixth step: Next I will de-seasonalize my original data set. I will divide the 4.8 by the seasonal index 0.9308 to get 5.15 television set sales. The 5.15 value has seasonality removed from it. The entire column has no seasonality in it. Mentally match this de-seasonalized column against the original data set. You will notice that I am raising the first and second quarter of each year (4.8 raises to 5.15 and 4.1 raises to 4.90). I am lowering the third and fourth quarter (6.0 lowers to 5.50 and 6.5 lowers to 5.69). The same concept is true for the other quarters. I am removing the seasonal movement from my original data set. The logic makes sense. If my seasonal index is 0.9305 for the first quarter this tells me I am


approximately 7% below trend. If I adjust the original data it would logically go up or increase. I need to total this column because I will use it later.

#7) The seventh step: The sixth column is a renumbering of the quarters. I have 4 years each with 4 quarters, so 4 times 4 = 16. I number the columns 1 through 16. I call this column “t”, representing the quarterly time periods. I need to total this column also for later use.

#8) The eighth step: The next column is simply a multiplication of the t-column and the Yt (as adjusted). 5.15 times 1 = 5.15 and 4.90 times 2 = 9.80 etc. I need to total this column for later use.

#9) The ninth step: The final column which is the eighth column is squaring the t-value. 1 squared is 1, 2 squared is 4, 3 squared is 9, etc. I need to total this column for later use, which is now here.

I have been referring to later and that time has finally come.

In regression, I developed a straight-line predicting equation, which enabled me to predict the movement in the dependent variable (Y) by variation in the independent variable (X). Here too I am interested in developing a straight-line predicting equation, which will enable me to forecast the value of my television set sales beyond my 16th quarter.

Wait a minute, you say, as you recall a warning about regression. You told us in regression that I should not go beyond the range of my data set, however, here you are telling us to forecast beyond our data set.

Good observation, I say. You are correct about regression, but in trend analysis I have replaced the independent variable with time. It is okay for me to go beyond my data set, but the caveat is don’t go too far beyond the time sensitive data set. There is no magic number, but for 10 years of data you might project 2 years ahead. Most bosses will want a five-year look at sales. I am sure you will be forced to make a five-year projection, but I caution you the five-year forecast (using trend analysis) is just about as good as the five-day weather forecast. You draw your own conclusions about that statement.

#10) The tenth step: Let’s now apply trend analysis to develop a predicting equation. I repeat here what I set before you early in this lecture.

Yt = b0 + b1 X1 My Predicting Equation for Simple Regression Analysis Using the Method of Least Squares.


Tt = b0 + b1 t My Predicting Equation Re-written for Trend Analysis Using the Method of Least Squares.

Where Tt is the forecast value of the time series in period t.

t is a period of time.

bO is the intercept of the trend line.

b1 is the slope of the trend line.

Re-writing the regression formulas, I have the following relationships.

∑ ∑

∑ ∑ ∑

−

−=

ntt

nYttY

bt

t

22

1 )(

))((

tbYb to 1−=

Solving for b1 and bo, I have the following results.

b1 = 914.99 - [(136)(101.75)] ÷16 1,496 - [(136)2] / 16

b1 = 914.99 - 864.875 1,496 - 1,156

b1 = 50.115 340

b1 = 0.1474

bo = 101.75÷16 - (0.1474)(136÷16)

bo = 6.3594 - (0.1474)(8.5)

bo = 5.1065

The predicting equation can now be stated.

Tt = 5.1065 + 0.1474(t)


Great, you say, but how do I use it?

Good question. Now if I am given a value for any time period (t), I can forecast or predict the value for the time period plus one. For example, let’s assume I am interested in forecasting the 17th, 18th, 19th and 20th quarters (the 5th year).

#11) The eleventh step: Next I simply substitute 17 for t into the predicting equation. This gives me the following solution.

T17 = 5.1065 + 0.1474(17) = 5.1065 + 2.5058 = 7.6123.

I would then do the same thing for the 18th, 19th, and 20th quarters. The results of the calculations for the four quarters of 1998 are shown below.

Quarter Which is 1998

De-seasonalized

Sales Forecast

17th Quarter 1 7.6123




#12) The 12th step: So can I now conclude that the above figures represent my forecast for 1998? Should I start ordering television sets based on this forecast?

The answer to that question is “NO”. What I next need to do is apply seasonality back to the forecast. If you notice my forecast is rather flat. This forecast does not at all reflect the same pattern as the original data set. The following table re-seasonalized the forecast.

Quarter 1998 Sales Forecast S Re-seasonalized

Forecast 17 1 7.6123 x 0.9308 = 7.1617 18 2 7.7597 x 0.8363 = 6.4894 19 3 7.9071 x 1.0915 = 8.6306 20 4 8.0545 x 1.1414 = 9.1934

The fifth column and last column in the table just above is my re-seasonalized sales forecast. I can now begin manufacturing based on this forecast. Also


notice that this forecast resembles the pattern of the original actual data. In other words, the first quarter is higher than the second but lower than the third and fourth quarter. The original data set reflected the same pattern.

Why Do All of This or the “Who Cares” Question:

You may ask, why go to all of this trouble just to get a forecast? Would I not do just as well fitting a trend line to the original data (the one that starts with the 4.8 and 4.1, etc.)? That's a very good question and I'm glad you ask.

In order to answer that question, I have used the original data set beginning with the 4.8, 4.1, etc and using the Method of Least Square developed a new predicting equation for the original data without removing the seasonal movement. From that equation, I have determined the values for the 17th, 18th, 19th, and 20th quarter. Remember the original data is composed of all four components – TCSI. Seasonality is still in the original data set. Next I will place that forecast in the third column of the table below. In the last column, I will show the difference in the deseasonalized and reseasonalized forecast (column 2) and the original data set forecast (column 3). Gee Whiz, I hope this is clear. Maybe you will understand what I am saying by simply looking at the table.

Quarter De-seasonalized & Re-seasonalized

Forecast

Original Data

Forecast

Difference Greater or

(Less)

17th 7.084 7.910 0.826

18th 6.523 8.090 1.567

19th 8.625 8.270 (0.355)

20th 9.190 8.450 (0740)

Totals 31.422 32.720 1.298

Remember my forecast is in thousands of units. Let’s examine the results shown above based on a couple of assumptions. First, the column total of 32,720 units is based on using the unadjusted original data set. The column total of 31,422 is the forecast of the units based on the very complicated decomposition process we just completed.

To analyze the results I need to make an assumption. I will assume that the company will have orders placed with it for the exact amount shown in the de and re seasonalized forecast. The second assumption is that management did not use the deseasonalized approach but used the unadjusted original data set to develop a forecast, which is shown in the third column. Okay, let me try to say


this another way, because I am sure some of you do not yet understand what I am talking about.

I am making the assumption that my forecast was for 32,720 units (quarter by quarter) but that my actual sales are running 31,422 units (quarter by quarter). Is that better? I am also assuming that I am building product to the 32,720 unit level.

By making these assumptions, notice the last column. Assume that the first quarter sales are completed and I actually sold 7.084 unit, but my forecast was 7.910 units. This means that I have 826 units (last column) more built than I sold. This would require an inventory buildup, which can be costly. As the chief executive officer of the company, I am a bit concerned. I call the marketing and sales department together to discuss how I can move additional units. Several ideas are offered, but none seem to work. The second quarter rolls along and I have sold 6,523 units but have forecasted (and built product) to the 8,090 level. Now suddenly I have an additional 1,567 units more in production than I am able to sell.

During the last two quarters, I have now produced but I have not sold 826 + 1,567 (2,393 total) units for the first and second quarter. As the CEO, I am now beginning to be concerned. I have money tied up in inventory that is not selling. I have money tied up in warehouse space and labor to move the units around. I have money tied up in labor and management to produce the units. I now call the marketing and sales departments together and give the orders to move the product as all costs. To do this the marketing and sales department tells me they must cut the price. By cutting the price, the gross margin suffers and the profit and profit per share also suffer, but at least I have moved the units.

The third quarter rolls around and I find out that I could have sold 355 units more than I have produced. That is bad. I cannot ship enough goods to fill the orders I have received. My gross margin and profit once again suffers. By the time the 4th quarter gets here, I find out that I have an additional 740 units which I could have sold but couldn’t because I did not have them. Bummer!! Now I am really mad. Why didn’t someone use the decomposition method of forecasting? My stockholders and employees should be very upset with me as the CEO.

The point is this. By removing seasonal movement from the data set and then preparing a forecast based on long-term movement and then adding back seasonality, the forecast is much more accurate. Use this approach. It works.

Reading 14 Forecasting with Time Series Qualitative and...

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