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## PowerPoint Slideshow about 'Chapter 3 – Moving Average and Exponential Smoothing' - jacob

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Installing ForecastX

- Here we go
- Link to Software

1. Install

2. Activate the plug-in

3. Activate Analysis ToolPak, if it’s not already

Note: on lab machines, you may have to install every day you use it…

Deep freeze!!!

The naïve model – assumes most recent values are the best predictors.

Alternatively:

- Moving average (MA) – assumes the best predictor is some average of past values.
- Exponential smoothing models – best predictors are both present and past values, but more distant values are of decreasing usefulness.
- You could think of both MA and ES as “smoothing models,” just with different weighting schemes, with different emphasis placed on the past.

MA & Smoothing Models

- Advantages
- -minimal data requirements
- -can produce accurate short-range forecasts
- -some models allow for a trend or seasonality
- -widely used and accepted
- Disadvantages
- -don’t allow for a cyclical pattern in the data
- -add little understanding with regard to why the
- variable to be forecast changes over time

Moving Averages-an “MA” process

- An MA assumes there is some underlying “smooth” curve that represents the path the data is actually taking.
- Sharp up and down movements are assumed to be random departures from this underlying curve.
- Estimating the MA helps “smooth out” short-term fluctuations to reveal that curve.
- A moving average is calculated just exactly like it sounds.

Forecasting with MA’s

MA(2) or a two-period Moving average

- Ft = (Xt-1 + Xt-2) / 2

MA(3) or a two-period Moving average

- Ft = (Xt-1 + Xt-2 + Xt-3) / 3

…and, so on

Question…what does an MA(1) look like???

Why use the MA in this instance?

- May supply clearer signaling of direction and momentum.
- Suppose you are following exchange rates…

You can use it to develop rules of thumb to determine which currency to hold.

If you see a few periods of decline in the MA, it might be a signal of the market momentum downward…sell off yen.

If you see a few periods of increases in the MA, it might signal an upward momentum…buy yen.

Two Real-World Applications Using Moving Averages

- Initial Unemployment Claims to forecast changes in the economy
- Buy and Sell Signals for the Stock Market

Initial Unemployment Claims

- Compute 4-week moving average of initial unemployment insurance claims for the U.S.
- When 4-week moving average < 400,000
- Indicates improving U.S. labor market conditions
- When 4-week moving average > 400,000
- Indicates deteriorating U. S. labor market conditions

Buy and Sell Signals for Stock Market

- Compute 24-month moving average of stock market index
- When index > than 24-month moving average,
- A buy signal is triggered
- When index < than 24-month moving average,
- A sell signal is triggered

Chapter 3: Un-graded Homework

- P 151-152 Problems 6 and 7 (to be covered next week on Tue)
- I’ll post the questions just in case.

Smoothing Models

-all smoothing model forecast are simply based on weighted averages of past values…

The weights are just different for different models

Smoothing Models

- Advantages
- -minimal data requirements
- -can produce accurate short-range forecasts
- -some models allow for a trend or seasonality
- -widely used and accepted
- Disadvantages
- -don’t allow for a cyclical pattern in the data, no trends or seasonality.
- -add little understanding with regard to why the
- variable to be forecast changes over time

A Specific Problem with theMA Model

- The MA model treats each of the observations used in the average equally.
- Intuitively, past data should affect the present, but the effect should decline over time or with distance.

Exponential Smoothing Models

- forecast is weighted average of values from all available previous periods where more recent periods are given more weight.
- Simple Exponential Smoothing Model -
- - the simplest model assumes no trend or seasonality in data

Simple Exponential Smoothing (SES) Model

- Ft +1 = αXt + (1 - α ) Ft where
- Ft +1 = Forecast value for period t + 1
- α = Smoothing constant ( 0 < α < 1)
- Xt = Actual value in present (period t)
- Ft = Forecast for present (period t)

i.e., the smoothed value

Re-writing the model to see one of the neat things about the SES Model

Ft +1 = αXt + (1 –α ) Ft

Ft +1 = αXt + Ft–αFt

Ft +1 = Ft + α (Xt – Ft)

It’s forecasting error

for period t…hmmm.

Each forecast “learns” from past error.

What is this???

The model “learns” from its past mistakes!!!

- The forecast value at period t+1 is increased if the actual value for period t is greater than it was forecasted to be, and vice versa.

Thus, all past forecasting error is used to adjust the model as it goes forward.

5 periods of data + 1 period forecast

Every past forecast

value is used to

produce each

current forecast.

This is a

Recursive forecast. The entire history is used to make the current forecast…at least to some degree.

Each value of F

affects all

subsequent values.

- Ft -3 = αXt-4 + (1 - α ) Ft-4
- Ft -2 = αXt-3 + (1 - α ) Ft-3
- Ft -1 = αXt-2 + (1 - α ) Ft-2
- Ft = αXt-1 + (1 - α ) Ft-1
- Ft +1 = αXt + (1 - α ) Ft

The forecast

Recursive?!…how is this different than earlier forecasts?

- Each period’s forecast contains all of the previous forecasts. So, anything that changes past forecasts (a.k.a., fitted values) will affect all subsequent forecasts.
- Previous forecasting methods only use the actual values of X in forecasts…not, past “fitted values” (those estimated by the model where data already exists for X).
- See page 105, equation (3.3) for more insight.

Simple Exponential Smoothing Model

- Ft +1 = weighted average of actual values in past
- It can be shown (trust me on this) that
- Ft +1 = αXt + (1 - α ) α Xt-1 + (1 - α )2α Xt-2 + . . .
- where
- Xt = value in present period
- Xt-1 = value in previous period
- Xt-2 = value in period before previous period

How is this accomplished in practice???

The forecast value

for the first period

is chosen somewhat

arbitrarily.

$F$1 is where I put

a.

This called

Initializing the model.

Notice the use of the

Previous period

Forecast (column C) in the formula.

How is this accomplished in practice???

Using different initialization values affects all the periods’ forecasts.

Common Initialization methods

- Using the first X
- Averaging the first three X’s
- Using the average X for the series
- Square root of mother’s SSN…

well, maybe not that one….

Here is a Little Excel Hint…

- To show the formulas instead of the values they create hold down

<CTRL> and the <`>

The <`> is usually on the same key as the <~>.

Do the same thing to toggle back to the values.

Excel hint # 2, Checking Formulas

- You can check your formulas by using the [tools]
- >[formula auditing]
- >formula auditing toolbar
- You can show the flow of the formula so you can quickly check to see if you are doing it right and using the correct cells.

This button allows you to check the

Components of your formula. These arrows

Indicate what cells are being used.

Why is it called exponential?

- It’s base on the weights and how they decline in importance with time.
- (geometrically or exponentially, rather than arithmetically or linearly)
- Small values of a (like 0.1) result in a slow decay. Past values are weighted almost as much as more recent ones.
- Large values of a (like 0.9) result in a fast decay. Recent values are weighted more than past ones.

Simple Exponential Smoothing Model

- the closerα is to 0, the more evenly the weights are spread out over more periods
- α = . 1
- Ft +1 = αXt + (1 - α ) α Xt-1 + (1 - α )2α Xt-2 + . . .
- = .1Xt + (1 - .1) .1 Xt-1 + (1 - .1 )2 .1 Xt-2 + . . .
- = .1Xt + (.9) .1 Xt-1 + (.9)2 .1 Xt-2 + . . .
- = .1Xt + .09 Xt-1 + (.81) .1 Xt-2 + . . .
- = .1Xt + .09 Xt-1 + .081 Xt-2 + . . .
- 19% of weight is given to two most recent periods

Simple Exponential Smoothing Model

- the closerα is to 1, the more weight is given to the most recent periods
- α = . 9
- Ft +1 = αXt + (1 - α ) α Xt-1 + (1 - α )2α Xt-2 + . . .
- = .9Xt + (1 - .9) .9 Xt-1 + (1 - .9 )2 .9 Xt-2 + . . .
- = .9Xt + (.1) .9 Xt-1 + (.1)2 .9 Xt-2 + . . .
- = .9Xt + .09 Xt-1 + (.01) .9 Xt-2 + . . .
- = .9Xt + .09 Xt-1 + .009 Xt-2 + . . .
- 99% of weight is given to two most recent periods

How does the choice ofa affect the present weight of past values

0.3044 of the total

weight in the first 4

periods.

0.9999 of the total

weight in the first 4

periods.

So, your choice of a has a substantial effect on the forecast.

Simple Exponential Smoothing Model(degenerate SES model)

- whenα = 1, model becomes simple naïve model
- α = 1
- Ft +1 = αXt + (1 - α ) α Xt-1 + (1 - α )2α Xt-2 + . . .
- = 1Xt + (1 - 1) 1 Xt-1 + (1 - 1)2 1 Xt-2 + . . .
- = 1Xt + (0) 1 Xt-1 + (0)2 1 Xt-2 + . . .
- = 1Xt + 0 Xt-1 + 0 Xt-2 + . . .
- = Xt
- All of weight is given to the most recent period

Simple Exponential Smoothing Model(degenerate SES model)

- asα approaches 0, simple exponential smoothing model becomes a moving average with all previous periods given equal weight…but, this is just aaaaaa

What????

Horizontal straight line!

Or, an MA(N)

where N is the total number of periods.

Choosing a

- Select values close to 0 if the series has a lot of “random variation.”

More distant past values remain important predictors

- Select values close to 1 if there appears to be a recent shift (or some other structural change) in the series.

Recent values are more important than those in the more distant past

- Use RMSE to optimize.

E.g., The UMICH Index of Consumer Sentiment

- The purpose is to measure changes in consumer attitudes and expectations.
- Questions about the decisions to save, borrow, or make discretionary purchases, and to forecast changes in aggregate consumer behavior.
- Attempts to measure consumer “confidence.”
- Started in the late 1940s (quarterly). During 1977 and thereafter, (monthly).
- Current data location: http://www.icpsr.umich.edu

Remember: SES Model Summary

- Uses past X’s and F’s.
- Learns from errors.
- assumes no trend or seasonality in data.
- produces biased forecasts when there is a trend in the data.
- With a positive trend, forecasts are too low
- With a negative trend, forecasts are too high

Using SES to Forecast the Index of Consumer Sentiment

- c3t2.xls
- By Hand
- Using ForecastX
- -Choosing a manually
- -letting ForecastX choose

Chapter 3: Homework

- P 151-152 Problems 6 and 7

Chapter 3. Problem #6

- Use a 3-month and 5-month moving average (MA) to forecast inventory for next January.
- Use RMSE to evaluate the two forecasts.

Chapter 3. Problem #6

The Spreadsheet Data

Which is better???

- In this situation, the graphs don’t say much.
- We have to rely primarily on the RMSE

Chapter 3. Problem #7

- Using data of full-service restaurant sales, calculate both the 3-month and 5-month MA for these data.
- Compare these two forecasts using RMSE.

Chapter 3. Problem #7

What do you see in this graph?

Maybe some seasonality also

Maybe a trend

For now, let’s ignore the potential patterns for this problem

Holt’s Exponential Smoothing

- Extends the SES, by adding in the ability to accurately forecast in the presence of a trend.
- It does this by adding a second smoothing constant.

Holt’s Exponential Smoothing Model

- Allows for a trend in the data
- α= smoothing constant for random fluctuations
- ( 0 < α < 1 )
- gamma = smoothing constant for trend
- ( 0 < g < 1 )
- Optimal values of α and g (gamma) are the

values that minimize RMSE

Holt’s Exp. Smoothing Model

- Ft +1 = αXt + (1 - α ) (Ft + Tt)
- Tt +1 = g(Ft +1 - Ft ) + (1 - g) Tt
- Ht +1 = Ft +1 + mTt +1

Estimate of addition to trend from t to t+1

Estimate of trend

Forecasted X

Est. of future trend

Holt’s forecast

Periods ahead you need to forecast

Holt’s Exp. Smoothing Model

Where:

Ft +1 = smoothed value for period t+1

a = smoothing constant for random fluctuations

Xt = actual value for period t (now)

Ft = smoothed/forecasted period for period t (now)

Tt +1 = estimate of the trend

g = smoothing constant for the trend

m = periods ahead to forecast

Ht + m= the Holt forecast

Holt’s Exp. Smoothing Model

- Ft +1 = αXt + (1 - α ) (Ft + Tt)

This equation adjusts Ft +1 for the growth in the previous period by adding the trend estimate to last period’s forecast.

Holt’s Exp. Smoothing Model

- Tt +1 = g(Ft +1 - Ft ) + (1 - g) Tt
- The trend is calculated as the difference in the last two smoothed values.

-Since the two are already smoothed this is assumed to be an estimate of the trend.

- To that is added the trend from through the previous period.
- So, Tt +1 is just the weighted average of current estimated trend, going forward and the past trend, going backward.

Holt’s Exp. Smoothing Model

- Ht +1 = Ft +1 + mTt +1

Now that we have Ft +1 and Tt +1 , we can just sum the two to get the forecast.

m is just the number of periods into the future we wish to forecast.

Holt’s Exp. Smoothing Model

- The Holt model accurately forecasts in the presence of any linear trend.
- Non-linear trend are NOT handled well. There are other models that do accommodate non-linear trends, but we do not cover these in this course.

Let’s Think Back…

Naïve models:

- Simple Naïve-doesn’t handle patterns of any type in the forecast period.
- Modified Naïve-incorporates trend (somewhat), but still has problems with all patterns in the forecast after a short forecast horizon.

Let’s Think Back…

Smoothing Models

- Moving Average-is the simplest smoothing model and it’s most appropriate where you think you know the underlying path…no trends.
- Simple Exponential Smoothing-doesn’t do trends or seasonality either, but uses more intuitively correct weights.
- Holt-benefits of SES (decaying weights) + handles trends…seasonality to come…

Forecasting US Consumer DebtUsing the Holt Model

Do you detect anything more than just a trend?

Forecasting US Consumer Debt

The data

Winter’s Exponential Smoothing Model

- Allows for a trend and seasonality in the data
- α= smoothing constant for random fluctuations
- ( 0 < α < 1 )
- β= smoothing constant for seasonality
- ( 0 < β < 1 )
- g= smoothing constant for trend
- ( 0 < g < 1 )
- Optimal values of α, β, g are the values that minimize RMSE

Winter’s Exponential Smoothing Model

- Ft = αXt / St-p + (1 - α ) (Ft-1 + Tt-1)
- St = bXt / Ft + (1 –b) St-p
- Tt = g(Ft– Ft-1 ) + (1 - g) Tt-1
- Wt +m = (Ft + mTt)St+m-p

Take out seasonality to get forecasts of the trend (ala Holt)

Then put seasonality back in to get the final estimates

They changed notation on us, but the idea is the same…

Note: t+m-p makes us use the correct seasonal index…if in 2 years of monthly data starting in January you forecast three periods ahead, the index would be 24+3-12…if twe would use the index for month 15, which is March.

Added features of Winters

- We are smoothing the trend (T), the random fluct. (F), and the seasonality (S)…

Ft = αXt / St-p + (1 - α ) (Ft-1 + Tt-1)

St = bXt / Ft + (1 –b) St-p

In Ft , we divide by S to scale up or down by season (loosely speaking).

In St , we using the ratio of actual to forecast value along with last season’s estimate of seasonality

Winter’s Exponential Smoothing Model

Where:

Ft = smoothed value for period t

a = smoothing constant for random fluctuations

Xt = actual value for period t (now)

Ft-1 = smoothed/forecasted period for period t-1

Tt +1 = estimate of the trend

St = Seasonality estimate (in index form)

bt = Smoothing constant for seasonality estimate (0<b<1)

g = smoothing constant for the trend

m = periods ahead to forecast

p = Number of periods in the seasonal period

Wt + m= the Winters’ forecast for m periods into the future

Things to note…

- The Winters’ model works very similarly to the Holt model.
- Both the trend and the seasonality estimates are smoothed.
- The Winters’ model generates some additional information in the form of seasonal indices.

Using ForecastX to estimate the Winters’ model

- In ForecastX the Winters’ model is called the [Holt-Winters] model.
- The Data

The Seasonal Indices

These seasonal indices can be viewed as % of an average quarter

-The 1st quarter is 104% of the average quarter

-The 4th quarter is 92% of the average quarter.

The Seasonal Indices

- When are the “BIG QUARTERS” for truck production?

The first and second quarter…

but why might this be?

- These indices are automatically calculated in ForecastX (and in most other time-series forecasting programs where the Winters’ model is an option.
- These indices can be used to de-seasonalize data by hand if needed.

Seasonal Indices for Jewelry Sales

Remember, these seasonal indices can be viewed as % of an average quarter.

When are the big sales months?

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