ADVANCED BUSINESS FORECASTING - NKD Group · ADVANCED BUSINESS FORECASTING . Definition Forecasting involves making the best possible judgment about some future event. In other words,

ADVANCED BUSINESS FORECASTING

Definition Forecasting involves making the best possible judgment about some future event. In other words, “forecasts are numerical estimates of an event for some future date that can be achieved with a specified level of support and are reproducible.”

"I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a very meagre and unsatisfactory kind."

William Thomson, Lord Kelvin, 1824-1907

"If we could first know where we are, then whither we are tending, we could then decide what to do and how to do it."

Abraham Lincoln, 1809-1865

The elements that come into play in all forecasting methods is the concept of the future and time; uncertainty; and reliance on historical data.

MELec6_6: Forecasting Page: 2

MBA555 Class Notes Prepared by: These notes contain copyrighted information Dr. Nina Kajiji Do not quote or copy without permission www.ninakajiji.net Last Update: Nov 9, 2009

Major Types of Forecasting Methods • Subjective Methods Sales Force Composites Customer Surveys Jury of Executive Opinions Delphi Method • Quantitative Methods

Exponential smoothing family ARIMA Artificial Neural Networks (ANN)

Elements of Forecasting • Source Of Data • Time Domain

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Components of a Time Series

• Secular Trend ◊ The growth of the economic system is tied to the growth

of business and industry. ◊ Based primarily in growth of the population. ◊ Trend can be linear t tY a bx= + + ε or nonlinear as

shown in the graph. ◊ In a nonlinear trend:

o Data that increases by a constant amount each successive time period

o Data that increases by increasing amounts at each successive time period.

o Data that increases by an equal percentage at each successive time period (easy to linearize by the use of logarithms).




• Seasonal Variation ◊ Variation in business and economic activity that results

from changing seasons. ◊ Periodic fluctuations in consumer spending ===>

Periodic Sales ===> Periodic production

• Cyclical Fluctuations

◊ Cyclical fluctuations are not very predictable. ◊ Business cycles -- Expansion and Contraction

• Erratic Fluctuations

◊ Are the fluctuations really erratic (chaotic)? Here is where our work with high-frequency data (daily level or greater frequency) and new modeling / forecasting methods will prove to be most useful. That is, time-series that appeared to have no economic value are now being modeled quite successfully with new methods like ANNs.




Quantitative Forecasting Methods

• Charting Approaches (read) • Moving average • Exponential Smoothing • Artificial Neural Networks Charting Approaches Arithmetic Charts (Scatter Plots) Definition The purpose of the arithmetic charts is to show actual movement

of the time series from one period to the next. ◊ Unless the axis markings (scaling) are the same it is not

possible to compare and interpret two different charts. ◊ A chart drawn on the arithmetic scale compares the amounts

of change. ◊ Data with a wide dispersion in values may not be accurately

reflected.




Trade Weighted Exchange Rate

120.000

121.000

122.000

123.000

124.000

125.000

126.000

127.000

128.000

129.000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

2001

Valu

e

Series1

Semi Logarithmic Charts Definition The purpose of the semi-logarithmic chart is to show the rate of change from one period to another. ◊ The ruling on the chart is such that the figures are

automatically reduced to a percentage basis. ◊ Note that the same vertical distance anywhere on the chart

shows the same percentage change. ◊ Thus, if the interest is in percentage changes in the data, the

semi-logarithmic chart is the preferred choice. ◊ If two or more series are shown on the same chart, the slope

of each line shows the percentage change in the series. ◊ By comparing the slopes of the two lines, it is possible to

compare the percentage changes in the series.




Trade Weighted Exchange Rate

1.000

10.000

100.000

1000.000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

2001

Valu

e TWER10%1%

Constant Rate of Change Reference Lines ◊ It should be noted that a straight line on a semi-logarithmic

chart represents a constant rate of change. ◊ The primary use of the constant rate of change reference line

is to permit the analyst to visually compare the slope of a known rate of change to that of the actual data series.

◊ These lines are extremely useful in time series analysis.




The Exponential Smoothing Family




Moving Averages The moving average approach calculates an average of the sample observations and then employs that average as the forecast for the next period. The number of sample observations included in the calculation of the average is specified at the start of this process. The term MOVING average means that as a new observation becomes available a new average is calculated by dropping the oldest observation in order to include the newest one. Month Period Observed

Values 3-Month

MA 5-Month

MA Log Obs Values

Growth Series: 0.001%

Jan 1 262.8 2.420 2.410 Feb 2 262.9 2.420 2.412 Mar 3 262.6 2.419 2.415 Apr 4 263.2 262.8 2.420 2.417 May 5 263.9 262.9 2.421 2.420 Jun 6 265.4 263.2 263.1 2.424 2.422 Jul 7 266.5 264.2 263.6 2.426 2.424 Aug 8 267.1 265.3 264.3 2.427 2.427 Sep 9 268.5 266.3 265.2 2.429 2.429 Oct 10 269.7 267.4 266.3 2.431 2.432 Nov 11 270.4 268.4 267.4 2.432 2.434 Dec 12 269.4 269.5 268.4 2.430 2.437

Source: Business Forecasting Methods, by Jarrett, (Basil Blackwell, 1991).




Monthly Moving Average Sample Date

258

260

262

264

266

268

270

272

1 2 3 4 5 6 7 8 9 10 11 12

Observation

Valu

es

Actual

3 Mnth

5 Mnth

Advantages: 1. Data requirements are small. 2. Better than using a simple arithmetic mean because it can be

adjusted to reflect the observable patterns in the data. Disadvantages: 1. The past n sample observations must be available. 2. Equal weights are given to all past observations and no

weight is given to observations earlier than period t-n+1. 3. Assumes that the data has a stationary distribution (not

always true).




Single Exponential Smoothing

Single parameter exponential smoothing (Unadjusted) is an easy to implement method of smoothing that overcomes some of the problems associated with moving averages. In contrast to moving averages, exponential smoothing permits the researcher to weight observations. It is not unusual for recent observations to contain more relevant information for forecasting purposes than older ones. The method also generates self-correcting forecasts through its ability to produce forecast values which reflect adjustment for earlier errors. Advantages: 1. Simplifies forecasting calculations 2. Has small data requirements 3. Produces self-correcting forecasts with built-in adjustments

that regulate forecast values by changing them in the opposite direction of earlier errors.

4. Simple! Only the last period’s forecast must be saved.




Disadvantages: 1. Specification of the smoothing constant is a problem. Alpha

close to 1 implies that the new forecast includes a substantial adjustment for the error in the previous forecast. If alpha is close to zero, the new forecast will include only a small adjustment for error. Generally, it is suggested that if the smoothing constant is greater than 0.30 an alternative model should be used.

2. In general the forecasts trail the pattern in the sample data. Notation for Single Unadjusted Exp. Smoothing Dt := Actual value at time t

Ft+1 := Forecast value for time t+1

α := Smoothing constant

Ft+1 = αDt + (1 - α)Ft-1

where: 0.0 ≤ α ≤ 1.0 Fo = D1 or user input. From the above equations it is apparent that there are two specific data input required for the unadjusted option. These are: 1. Smoothing Constant (Alpha = α). 2. Time Series Base (D1)




Test Of Predictive Ability (Unadjusted)Time Series: B

Predicted PointSeries2

Observation335330325320315310305300295

Act

ual &

Pre

dict

ed

138137136135134133132131130129128127126125124123122121120119118117116115114113112111110109

Adaptive Rate of Response Single Exponential Smoothing ARRSES This method does not require the decision-maker to specify the alpha smoothing constant.

ARRSES automatically changes the value of the unspecified alpha by a predetermined weight on an on-going basis; that is, whenever there is a change in data pattern. The only smoothing parameter that is needed is the Beta term. The Beta term is the weighting factor. Advantages: Very useful when a large number of items have to be predicted.




Disadvantages: Unknown smoothing constant. It may not be possible to replicate the smoothing application.

Test Of Predictive Ability (ARRSE)Time Series: B


Observation340338336334332330328326324322320318316314312310308306304302300298

Act

ual &

Pre

dict

ed

133

132

131

130

129

128

127

126

125

124

123

122

121

120

119

118

117

116

Brown’s Linear Exponential Smoothing (Double) Double exponential smoothing is the application of exponential smoothing to the single exponential values. Brown’s method provides an additional correction method; an approach which resembles the application of a moving average. Brown’s method uses the difference between the single and double smoothed values as an additive factor to the single smoothed value. The method further adjusts for the pattern in the data.




Advantages:

Provides an additional correction for a data series - useful when linear trend is present in the data.

Disadvantages: The forecasts trail the pattern in the sample data.

Test Of Predictive Ability (Brown's)Time Series: B


Observation336334332330328326324322320318316314312310308306304302300298296

Act

ual &

Pre

dict

ed

137136135134133132131130129128127126125124123122121120119118117116115114113112111110109108107106




Holts’ Two Parameter Linear Exponential Smoothing The Holt method is an extension to Brown’s method. The Holt approach adds a growth factor to the smoothing equation. The method smoothes the trend values directly. When growth exists in the observed values of a time series, new observations will be greater than the previously observed values. Advantages: 1. Adds a growth factor to the smoothing equation. 2. Trend values are smoothed directly (unlike the implied

method in Brown) 3. Eliminates the lag in smoothing. Disadvantages: The forecast accuracy depends on determining the correct alpha and beta parameters.




Test Of Predictive Ability (Holt's)Time Series: B


Observation340338336334332330328326324322320318316314312310308306304302300

Act

ual &

Pre

dict

ed

130129128127126125124123122121120119118117116115114113112111110109108107106105104103102101100

99989796




Winters’ Three Parameter Model Seasonal patterns in time series data are quite common in business and economics. The pattern tends to occur consistently from year to year. Winters extended the exponential smoothing model (Holt’s method) to incorporate seasonality factors. Winters’ method is a three-parameter exponential smoothing model which is used to model time series which exhibit both a trend (Holt) and a seasonal pattern (Winters). Advantages:

Adjusts for both the trend and the seasonality component in the data set.

Disadvantages: Very sensitive to: • the initial values for slope, • deseasonalized level, • the initial seasonal factors and the sum of the seasonal

factors.




Test Of Predictive Ability (Winters')Time Series: B


Observation340335330325320315310305300295290

Act

ual &

Pre

dict

ed

145

140

135

130

125

120

115

110

105

100

95

90

85

80

75

Summary: Required Parameters By Technique Method Base Trend Alpha Beta Gamma Obs/Yr Single • • Holt’s • • • • Brown's • • ARRSE • Winters' • • • • • • For a discussion on how to use the Forecasting application in WinORSai see: http://www.nkd-group.com/winorse/forecast/fcstfrm.htm


http://www.nkd-group.com/winorse/forecast/fcstfrm.htm�



Non Linear Models & Market Microstructure Does nonlinearity exist in financial TS? Yes, especially in volatility and high-frequency data. We will mention the traditional nonlinear models that have been used to study such models and focus our discussion on the neural network models. Some Specific Models

• Bilinear Model: Uses the Taylor series expansion on a linear function but employs second-order terms to improve the approximation for the non-linear component.

• TAR Model: uses a piecewise linear model in the threshold space

to obtain a better approximation of the conditional mean. It is particularly good at modeling asymmetry in rising and declining patterns. TAR Models can be used to refine TGARCH, IGARCH, and GARCH models.

• Markov Switching Models: These models use probabilities to determine when a series will switch from State 1 to State 2 and vice-versa. They also provide a measure for the expected duration to stay in a particular State.

• Nonparametric Methods: Generally used when there is insufficient knowledge to prespecify the nonlinear structure between two variables. The essence of nonparametric methods is smoothing. Methods used are: o Kernel Regression o Local Linear Regression – weighted least squares problem




• Mixed Models – (semi-parametric) o Functional Coefficient AR Model o Nonlinear Additive AR Model o Nonlinear State-Space Model o Neural-Networks

Artificial Neural Nets (Radial Basis Function) How do you recognize a face in a crowd? How does an economist predict the direction of interest rates? The human brain uses a web of interconnected processing elements called neurons to process the information. Each neuron is autonomous and independent. But, it also works asynchronously (without any synchronization to other events taking place). Neural Network algorithms rely upon the same type of structure to solve complex problems for which you cannot develop simple solution steps. A neural network is a computational structure inspired by the study of biological neural processing. Neural networks provide a semi-parametric approach to data analysis. Structure of a Network • Output layer • Input layer • Hidden layer • Nodes (neurons)




Activation Function • Logistic • Heavyside (or threshold functions) • Gaussian • Cauchy • Etc.

Although there are many different types of neural networks to study, we will limit our comments to one type of neural network: the radial basis function network (RBF).

The details of RBF are beyond the scope of our discussion. But, we do need to know that RBF networks are used with supervised training. By supervised, we mean that you are required to indicate a training set of data and a test (forecast) set of data over which the model is validated.




Analysis of High-Frequency Financial Data & Market Microstructure

Market microstructure: Why is it important?

1. Important in market design & operation, e.g. to compare different markets (NYSE vs NASDAQ)

2. To study price discovery, liquidity, volatility, etc. 3. To understand costs of trading 4. Important in learning the consequences of institutional

arrangements on observed processes, e.g. • _ Nonsynchronous trading • _ Bid-ask bounce • _ Impact of changes in tick size, after-hour trading, etc. • _ Impact of daily price limits (many foreign markets)

Nonsynchronous trading: That is different stocks having different trading frequencies, and intensities. Key implication:

• induces serial correlations even when the underlying returns are iid

• induces cross-correlations between stock returns

• in some situations negative serial correlations in a portfolio return




Bid-Ask Sread: In some stock exchanges market makers play an important role in facilitating trades. The provide market liquidity by standing ready to buy or sell whenever the public wishes to buy or sell. Bid-Ask spread introduces negative serial correlations in the series of observed price changes – called the bid-ask bounce in finance. Good point is that the bid-ask spread does not introduce any serial correlation beyond lag-1. High-Frequency Financial Data Observations taken with time intervals 24 hours or less Some example:

1. Transaction (or tick-by-tick) data 2. 5-minute returns in FX 3. 1-minute returns on index futures and cash market

Some Basic Features of the Data: 1. Irregular time intervals 2. Leptokurtic or Heavy tails 3. Discrete values, e.g. price in multiples of tick size 4. Large sample size 5. Multi-dimensional variables, e.g. price, volume, quotes,

etc. 6. Diurnal Pattern




High Frequency (HF) Considerations

When dealing with some high frequency data series, especially the log return series of financial variables, one must be very concerned about how the time unit of measurement that represents the observed data. With relatively small time units (minutes, seconds, …) observed values will vary very little. Comparing this against the behavior of the same variable when the time unit is measured in days or weeks, it becomes immediately apparent that a different model may have to be applied to the data for high frequency observations. One thing is for certain, given large sample sizes any assumption about the stationarity of the series is weak at best. Generally speaking, a stationary stochastic process is one whose characteristics do not change with time.




HF – The Leverage Effect Changes in a commodity’s value (e.g,. stock price) tend to be negatively correlated with changes in volatility. That is, volatility of the time series is higher after negative shocks than after positive shocks of the same magnitude. HF – Long-range Dependence In financial data it is known that the sample autocorrelations of the data are small whereas the sample autocorrelations of the absolute and squared values of the data are significantly different from zero even for large lags. This behavior suggests that there is some form of long-range dependence (memory) in the data. HF – Aggregational Gaussianity In financial data, the distribution of log-returns over larger time interval measurement (i.e., month, half year, a year) is closer to the normal distribution than for hourly or daily (e.g., tick data) log returns.

HF – Leptokurtic Distributions The frequency of large and small changes, relative to the range of the data, is somewhat high. This fact suggests that the data do not come from a normal distribution but, rather, from a heavy-tailed (leptokurtic) distribution. This is a distribution with a high probability for extreme values.




HF – Volatility Clustering Large and small values in a log return sample tend to occur in clusters. This indicates that there is dependence in the tails. Mandelbrot (1963): “…large changes tend to be followed by large changes of either sign; or small changes by small changes…” For a discussion on how to use the ANN techniques in WinORSai see: ftp://ftp.cba.uri.edu/Classes/Dash/MBA570/RBFmstr_0606.pdf


ftp://ftp.cba.uri.edu/Classes/Dash/MBA570/RBFmstr_0606.pdf�



Accuracy of Forecasts In general most forecasting methods produce forecasts that tend to lag behind the turning points of the actual time series data. Thus, the question arises how do you know which forecast to recommend. There are two commonly used forecast analysis methods in use today – Graphical Analysis and Error Measure Analysis. When evaluating forecasts based on error measures, the rule: smaller error value within the error measure is better than larger error value. Error Measures

Average Error This is the arithmetic average of the forecast error.

Mean Percent Error (MPE %) The mean percent error shows the error as a percentage of the actual series. This measure is generally more informative than average error when the original series have large differences in their actual values.

Standard Deviation This is the standard deviation of forecast errors.

Mean Squared Error (MSE) And Root MSE (RMSE) This is a very common measure for evaluating the accuracy of the forecast. The square root of this measure is also used (RMSE).




Both MSE and RMSE tend to be overly affected by outliers. However, it is implicitly assumed that the MSE measure is zero. Stated differently, when the MSE is zero, then the forecasting model is unbiased.

Mean Absolute Deviation (MAD) MAD is less sensitive to outliers than MSE. Other than that, it is similar in concept to MSE; the primary difference is in its use of absolute deviations. The use of absolute deviations is more effective if the economic impact of forecast errors is proportional to the amount of the errors.

Mean Absolute Percentage Error (MAPE) MAPE is similar in concept to MAD. The major difference is that the error terms are converted into percentage format. This allows for direct comparison between different forecasting methods. The table that follows provides a summary of the forecast simulations for each of the methods defined above. Note that no special attention was given to the optimization procedure. That is and by way of example, in the case of the Winters’ technique default simulation parameters were accepted. Hence, the Winters’ method focused on location of an optimal Alpha without regard to the current parameter settings for either beta or gamma. The table is for demonstration purposes only. Each simulation focused on location of the smallest MSE as a measurement of forecast error.




Technique Alpha Beta Gamma Base Trend Error Error

Name ARRSE 0.580 0.290 0.000 94.097 . 0.433 MSE Unadjusted 0.290 0.000 0.000 94.097 . 1.007 MSE Brown's 0.290 0.000 0.000 94.097 . 0.517 MSE Holt's 0.290 0.010 0.000 94.097 0.097 0.910 MSE Winters' 0.290 0.010 0.010 84.599 1.359 27.003 MSE




Graphical Analysis After each execution of the forecasting method, the actual, predicted and residual terms can be compared graphically. Graphical analysis is used to augment the interpretation of the forecast. Overall quality of fit can be judged from the graphical analysis.

Constant Variance

What you are looking for: Random scatter of data point between two horizontal lines. What you are seeing: The chart indicates that a serial trend still exists in the residuals. Thus, a variable that would capture the trend would be a welcome addition to the model.




Error Bars

What you are looking for: Errors should be evenly distributed above and below the actual observations.

What you are seeing: The chart indicates that the forecast is consistently below the actual values.




Predictive Ability

What you are looking for: Actual and Predicted values should be as close to each other as possible.

What you are seeing: The actual is significantly above the predicted values. Only around some of the early observations and around observation 40 does the predicted values line up with the actual values.




Residual Analysis

What you are looking for: They should be no pattern in the residual plot.

What you are seeing: The residual plot indicates that the forecast method is not picking up the cyclical trend. May be a different method – one that can model cycle should be used. Or you may have to use a transformation to remove the trend before subjecting the time series to a “linear” forecasting method.




How to Communicate Forecasts

1. Purpose or usefulness of the forecast (including its time frame)

2. Key underlying assumptions 3. Input data 4. Forecast values 5. Graphic display of history with predictions 6. Any other comments or stipulations that are needed to

place the forecast in proper perspective – include a discussion on the error measure to support your forecast.

7. Reporting on the past forecasting performance record – if possible.


Documents

ADVANCED BUSINESS FORECASTING - NKD Group · ADVANCED BUSINESS FORECASTING . Definition Forecasting involves making the best possible judgment about some future event. In other words,