1. Applied Business Forecasting and Planning
The Forecast Process, Data Considerations, and Model Selection

2. Chapter Objectives Learning Objectives
Establish framework for a successful forecasting system
Introduce the trend, cycle and seasonal factors of a time series
Introduce concept of Autocorrelation and Estimation of the Autocorrelation function.
This chapter establishes a practical guide to implementing a successful forecasting process stressing evaluation of data for trend, seasonal, and cyclical components. In addition, basic applied statistics are reviewed.

3. The Forecast Process
The overall forecasting process can be outlined as follows:
Problem Definition
Specify the objectives
Identify what to forecast
Gathering Information
Identify time dimensions
Data considerations
Choosing and fitting models
Model selection
Model evaluation
Using and evaluating a forecasting model
Forecast preparation
Forecast presentation
Tracking results
There is a variety of ways in which one could outline the overall forecasting process.

4. The Forecast Process Problem Definition Specify the objectives
How the forecast will be used in a decision context.
Determine what to forecast
Fore example to forecast sales one must decide whether to forecast unit sales or dollar sales, Total sales, or sales by region or product line.
Objectives and applications of the forecast will be discussed between the individuals involved in preparing the forecast and those who will utilize the results.

5. The Forecast Process Gathering Information Identify time dimensions
The length and periodicity of the forecast.
Is the forecast needed on an annual, quarterly, monthly daily basis, and how much time we have to develop the forecast?
Data consideration
Quantity and type of data that are available.
Where to go to get the data.

6. The Forecast Process Choosing and fitting models Model selection
This phase depends on the following criteria
The pattern exhibited by the data
The quantity of historical data available
The length of the forecast horizon
Model evaluation
Test the model on the specific series that we want to forecast.
Fit: refers to how well the model model works in the set that was used to develop it.
Accuracy refers to how well the model works in the “holdout” period.
The first one is the most important one.

7. The Forecast Process Using and evaluating a forecasting model
Forecast preparation
The result of having found model or models that you believe will produce an acceptably accurate forecast.
Forecast Presentation
It involve clear communication.
Tracking results
Over time, even the best of models are likely to deteriorate in terms of accuracy and should be adjusted or replaced with alternative methods.
It is recommended that more than one technique be used whenever possible.
When two or more methods that have different information bases are used, their combination will usually provide better forecasts.
A good forecaster can learn from its mistake. A careful review of forecast errors may be helpful in leading to a better understanding of what causes deviation between the actual and forecast .

8. Explanatory versus Time Series forecasting
Explanatory models
Assume that the variable to be forecasted exhibits an explanatory relationship with one or more independent variables
DCS = f (DPI, PR, Index, Error)
DCS = domestic car sales
DPI = Disposable income
PR = prime interest rate
Index = University of Michigan index of consumer index.

9. Explanatory versus Time Series forecasting
Time series forecasting makes no attempt to discover the factors affecting its behavior. Hence prediction is based on past values of a variable. The objective is to discover the pattern in the historical data series and extrapolate that pattern into the future.
DCS t+1 = f (DCS t , DCS t-1, DCS t-2, Error)

10. Trend, Seasonal, and Cyclical Data Patterns
The data that are used most often in forecasting are time series.
Time series data are collected over successive increments of time.
Example: Monthly unemployment rate, The quarterly gross domestic product, the number of visitors to a national park every year for a 30-year period.
Such time series data can display a variety of patterns when plotted over time.

11. Data Pattern
A time series is likely to contain some or all of the following components:
Trend
Seasonal
Cyclical
Irregular

12. Data Pattern
Trend in a time series is the long-term change in the level of the data i.e. observations grow or decline over an extended period of time.
Positive trend
When the series move upward over an extended period of time
Negative trend
When the series move downward over an extended period of time
Stationary
When there is neither positive or negative trend.

13. Data Pattern
Seasonal pattern in time series is a regular variation in the level of data that repeats itself at the same time every year.
Examples:
Retail sales for many products tend to peak in November and December.
Housing starts are stronger in spring and summer than fall and winter.

14. Data Pattern
Cyclical patterns in a time series is presented by wavelike upward and downward movements of the data around the long-term trend.
They are of longer duration and are less regular than seasonal fluctuations.
The causes of cyclical fluctuations are usually less apparent than seasonal variations.

15. Data Pattern
Irregular pattern in a time series data are the fluctuations that are not part of the other three components
These are the most difficult to capture in a forecasting model

16. Example:GDP, in 1996 Dollars

17. Example:Quarterly data on private housing starts
This is quarterly data on housing start from 1980 to 2000.
There is seasonal pattern: regular sharp upward and downward movement that repeat year after year.
There also appear to be some upward trend to the data and some cyclical movement.
The straight line shows the long term trend.
The third line, which moves above and below the long-term trend but is smoother than the plot of PHS, is what the PHS series looks like after the seasonality has been removed.
Such series is said to be “deseasonalized” or “ “seasonally adjusted” and it is represented by PHSSA.
By comparing PHSSA with the trend, the cyclical nature of private housing starts becomes clearer.

18. Example:U.S. billings of the Leo Burnet advertising agency
The data are shown on an annual basis from 1950 through 1995.
There is an upward trend in data, and it appears to be accelerating
This an example of non-linear trend.

19. Data Patterns and Model Selection
The pattern that exist in the data is an important consideration in determining which forecasting techniques are appropriate.
To forecast stationary data; use the available history to estimate its mean value, this is the forecast for future period.
The estimate can be updated as new information becomes available.
The updating techniques are useful when initial estimates are unreliable or the stability of the average is in question.

20. Data Patterns and Model Selection
Forecasting techniques used for stationary time series data are:
Naive methods
Simple averaging methods,
Moving averages
Simple exponential smoothing
autoregressive moving average(ARMA)

21. Data Patterns and Model Selection
Methods used for time series data with trend are:
Moving averages
Holt’s linear exponential smoothing
Simple regression
Growth curve
Exponential models
Time series decomposition
Autoregressive integrated moving average(ARIMA)

22. Data Patterns and Model Selection
For time series data with seasonal component the goal is to estimate seasonal indexes from historical data.
These indexes are used to include seasonality in forecast or remove such effect from the observed value.
Forecasting methods to be considered for these type of data are:
Winter’s exponential smoothing
Time series multiple regression
Autoregressive integrated moving average(ARIMA)

23. Data Patterns and Model Selection
Cyclical time series data show wavelike fluctuation around the trend that tend to repeat.
Difficult to model because their patterns are not stable.
Because of the irregular behavior of cycles, analyzing these type data requires finding coincidental or leading economic indicators.

24. Data Patterns and Model Selection
Forecasting methods to be considered for these type of data are:
Classical decomposition methods
Econometric models
Multiple regression
Autoregressive integrated moving average (ARIMA)

25. Example:GDP, in 1996 Dollars
For GDP, which has a trend and a cycle but no seasonality, the following might be appropriate:
Holt’s exponential smoothing
Linear regression trend
Causal regression
Time series decomposition
Because of the cycle component the last two may be better that the first two

26. Example:Quarterly data on private housing starts
Private housing starts have a trend, seasonality, and a cycle. The likely forecasting models are:
Winter’s exponential smoothing
Linear regression trend with seasonal adjustment
Causal regression
Time series decomposition
Because of the cycle the last two may be a better candidate.

27. Example:U.S. billings of the Leo Burnet advertising agency
For U.S. billings of Leo Burnett advertising, There is a non-linear trend, with no seasonality and no cycle, therefore the models appropriate for this data set are:
Non-linear regression trend
Causal regression

28. Autocorrelation
Correlation coefficient is a summary statistic that measures the extent of linear relationship between two variables. As such they can be used to identify explanatory relationships.
Autocorrelation is comparable measure that serves the same purpose for a single variable measured over time.

29. Autocorrelation
In evaluating time series data, it is useful to look at the correlation between successive observations over time.
This measure of correlation is called autocorrelation and may be calculated as follows:
rk = autocorrelation coefficient for a k period lag.
mean of the time series.
yt = Value of the time series at period t.
y t-k = Value of time series k periods before period t.

30. Autocorrelation
Autocorrelation coefficient for different time lags can be used to answer the following questions about a time series data.
Are the data random?
In this case the autocorrelations between yt and y t-k for any lag are close to zero. The successive values of a time series are not related to each other.

31. Correlograms: An Alternative Method of Data Exploration
Is there a trend?
If the series has a trend, yt and y t-k are highly correlated
The autocorrelation coefficients are significantly different from zero for the first few lags and then gradually drops toward zero.
The autocorrelation coefficient for the lag 1 is often very large (close to 1).
A series that contains a trend is said to be non-stationary.

32. Correlograms: An Alternative Method of Data Exploration
Is there seasonal pattern?
If a series has a seasonal pattern, there will be a significant autocorrelation coefficient at the seasonal time lag or multiples of the seasonal lag.
The seasonal lag is 4 for quarterly data and 12 for monthly data.

33. Correlograms: An Alternative Method of Data Exploration
Is it stationary?
A stationary time series is one whose basic statistical properties, such as the mean and variance, remain constant over time.
Autocorrelation coefficients for a stationary series decline to zero fairly rapidly, generally after the second or third time lag.

34. Correlograms: An Alternative Method of Data Exploration
To determine whether the autocorrelation at lag k is significantly different from zero, the following hypothesis and rule of thumb may be used.
H0: k= 0, Ha: k  0
For any k, reject H0 if
Where n is the number of observations.
This rule of thumb is for  = 5%

35. Correlograms: An Alternative Method of Data Exploration
The hypothesis test developed to determine whether a particular autocorrelation coefficient is significantly different from zero is:
Hypotheses
H0: k= 0, Ha: k  0
Test Statistic:

36. Correlograms: An Alternative Method of Data Exploration
Reject H0 if

37. Correlograms: An Alternative Method of Data Exploration
The plot of the autocorrelation Function (ACF) versus time lag is called Correlogram.
The horizontal scale is the time lag
The vertical axis is the autocorrelation coefficient.
Patterns in a Correlogram are used to analyze key features of data.

38. Example:Mobil Home Shipment
Correlograms for the mobile home shipment
Note that this is quarterly data

39. Example:Japanese exchange Rate
As the world’s economy becomes increasingly interdependent, various exchange rates between currencies have become important in making business decisions. For many U.S. businesses, The Japanese exchange rate (in yen per U.S. dollar) is an important decision variable. A time series plot of the Japanese-yen U.S.-dollar exchange rate is shown below. On the basis of this plot, would you say the data is stationary? Is there any seasonal component to this time series plot?

40. Example:Japanese exchange Rate

41. Example:Japanese exchange Rate
Here is the autocorrelation structure for EXRJ.
With a sample size of 12, the critical value is
This is the approximate 95% critical value for rejecting the null hypothesis of zero autocorrelation at lag K.

42. Example:Japanese exchange Rate
The Correlograms for EXRJ is given below

43. Example:Japanese exchange Rate
Since the autocorrelation coefficients fall to below the critical value after just two periods, we can conclude that there is no trend in the data.

44. Example:Japanese exchange Rate
To check for seasonality at  = .05
The hypotheses are:
H0; 12 = 0 Ha:12  0
Test statistic is:
Reject H0 if

45. Example:Japanese exchange Rate
Since
We do not reject H0 , therefore seasonality does not appear to be an attribute of the data.

46. ACF of Forecast Error
The autocorrelation function of the forecast errors is very useful in determining if there is any remaining pattern in the errors (residuals) after a forecasting model has been applied.
This is not a measure of accuracy, but rather can be used to indicate if the forecasting method could be improved.