The Forecast Process Dr. Mohammed Alahmed
http://fac.ksu.edu.sa/alahmed [email protected] (011) 4674108 1
Dr. Mohammed Alahmed
Slide 2
Chapter 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. Dr. Mohammed Alahmed
2
Slide 3
The overall forecasting process can be outlined as follows:
Problem Definition Specify objectives Identify what to forecast
Gathering Information Identify time dimensions Data consider-
ations Choosing and fitting models Model selection Model evaluation
Using and evaluating a forecasting model Forecast preparation
Forecast presentation Tracking results Dr. Mohammed Alahmed 3
Slide 4
Problem Definition 1.Specify the objectives How the forecast
will be used in a decision context. 2.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. Dr. Mohammed Alahmed 4
Slide 5
Gathering Information 1.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? 2.Data consideration Quantity and type of
data that are available. Where to go to get the data. Dr. Mohammed
Alahmed 5
Slide 6
Choosing and fitting models 1.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 2.Model evaluation Test the model on the specific
series that we want to forecast. Fit: refers to how well the model
works in the set that was used to develop it. Accuracy refers to
how well the model works in the holdout period. Dr. Mohammed
Alahmed 6
Slide 7
Using and evaluating a forecasting model 1.Forecast preparation
The result of having found model or models that you believe will
produce an acceptably accurate forecast. 2.Forecast Presentation It
involve clear communication. 3.Tracking results Over time, even the
best models are likely to deteriorate in terms of accuracy and
should be adjusted or replaced with alternative methods. Dr.
Mohammed Alahmed 7
Slide 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. Dr. Mohammed Alahmed 8
Slide 9
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) Dr.
Mohammed Alahmed 9
Slide 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. Dr. Mohammed Alahmed 10
Slide 11
Data Pattern A time series is likely to contain some or all of
the following components: Trend Seasonal Cyclical Irregular Dr.
Mohammed Alahmed 11
Slide 12
Data Pattern (Trend) 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. Dr. Mohammed
Alahmed 12
Slide 13
Data Pattern (Seasonal) 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. Dr. Mohammed Alahmed
13
Slide 14
Data Pattern (Cyclical) 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.
Dr. Mohammed Alahmed 14
Slide 15
Data Pattern(Irregular ) 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 Dr. Mohammed Alahmed 15
Slide 16
Example1: GDP, in 1996 Dollars Dr. Mohammed Alahmed 16
Slide 17
Example2: Quarterly data on housing Dr. Mohammed Alahmed
17
Slide 18
Example3: U.S. billings of the Leo Burnet advertising agency
Dr. Mohammed Alahmed 18
Slide 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. Dr. Mohammed Alahmed 19
Slide 20
Forecasting techniques used for stationary time series data
are: Naive methods Simple averaging methods, Moving averages Simple
exponential smoothing Autoregressive moving average(ARMA) Dr.
Mohammed Alahmed 20
Slide 21
Methods used for time series data with trend are: Moving
averages Holts linear exponential smoothing Simple regression
Growth curve Exponential models Time series decomposition
Autoregressive integrated moving average (ARIMA) Dr. Mohammed
Alahmed 21
Slide 22
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: Winters exponential smoothing Time series
multiple regression Autoregressive integrated moving average(ARIMA)
Dr. Mohammed Alahmed 22
Slide 23
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. Dr. Mohammed Alahmed 23
Slide 24
Forecasting methods to be considered for these type of data
are: Classical decomposition methods Econometric models Multiple
regression Autoregressive integrated moving average (ARIMA) Dr.
Mohammed Alahmed 24
Slide 25
For GDP example, which has a trend and a cycle but no
seasonality, the following might be appropriate: Holts exponential
smoothing Linear regression trend Causal regression Time series
decomposition Dr. Mohammed Alahmed 25
Slide 26
Private housing starts example have a trend, seasonality, and a
cycle. The likely forecasting models are: Winters exponential
smoothing Linear regression trend with seasonal adjustment Causal
regression Time series decomposition Dr. Mohammed Alahmed 26
Slide 27
For U.S. billings of Leo Burnett advertising example, 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 Dr. Mohammed Alahmed 27
Slide 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. Dr.
Mohammed Alahmed 28
Slide 29
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: r k = autocorrelation coefficient for a k period lag. =
mean of the time series. y t = Value of the time series at period
t. y t-k = Value of time series k periods before period t. Dr.
Mohammed Alahmed 29
Slide 30
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 y t and y
t-k for any lag are close to zero. The successive values of a time
series are not related to each other Dr. Mohammed Alahmed 30
Slide 31
Is there a trend? If the series has a trend, y t 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. Dr. Mohammed Alahmed 31
Slide 32
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. 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. Dr. Mohammed Alahmed 32
Slide 33
To determine whether the autocorrelation at lag k is
significantly different from zero, the following hypothesis and
rule of thumb may be used. H 0 : k = 0,H a : k 0 For any k, reject
H 0 if Where n is the number of observations. This rule of thumb is
for = 5% Dr. Mohammed Alahmed 33
Slide 34
The hypothesis test developed to determine whether a particular
autocorrelation coefficient is significantly different from zero
is: Hypotheses: H 0 : k = 0,H a : k 0 Test Statistic: Reject H 0 if
Dr. Mohammed Alahmed 34
Slide 35
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. Dr. Mohammed
Alahmed 35
Slide 36
Example1: Mobil Home Shipment Correlograms for the mobile home
shipment Note that this is quarterly data Dr. Mohammed Alahmed
36
Slide 37
Example2: Japanese exchange Rate As the worlds 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? Dr. Mohammed
Alahmed 37
Slide 38
Dr. Mohammed Alahmed 38
Slide 39
Here is the autocorrelation structure for EXRJ. With a sample
size of 24, the critical value is This is the approximate 95%
critical value for rejecting the null hypothesis of zero
autocorrelation at lag K. Dr. Mohammed Alahmed 39
Slide 40
The Correlograms for EXRJ is given below Dr. Mohammed Alahmed
40 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.
Slide 41
To check for seasonality at =.05 The hypotheses are: H 0 ; 12 =
0H a : 12 0 Test statistic is: Reject H 0 if Since We do not reject
H 0, therefore seasonality does not appear to be an attribute of
the data Dr. Mohammed Alahmed 41
Slide 42
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 Dr. Mohammed Alahmed 42