When two or more dependent (X) variables are required for a pre- diction the analysis is referred to as multiple linear regression.
The simple linear regression equation can be adapted to accommo- date multiple dependent variables in the following way:
Theoretically there is no limit to the number of independent variables that can be analysed, but within the spreadsheet the maximum is 75.
This is a far greater number than would ever actually be required in a business situation.
Y⫽A0⫹A1X1⫹A2X2⫹ ⭈ ⭈ ⭈ ⫹AnXn
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The following example in Figure 4.14 represents expenditure on advertising and sales promotion together with sales achieved, where the advertising and sales promotion are dependent variables and the sales achieved is the independent variable. The require- ment is to estimate future sales by entering advertising and promo- tion expenditure, and applying the multiple regression equation to predict the sales value.
In Excel there are no built-in functions for multiple regression and therefore the command method is required. Figure 4.15 shows the Figure 4.14 Multiple regression scenario
Figure 4.15 Results of Regression command for multiple linear regression
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results of the Excel Regression command where the range C3:C12 from Figure 4.14 was specified as the independent variable and
A3:B12 from Figure 4.14 as the dependent variables. The regression output has been placed on the same sheet commencing in cell A14.
In order to calculate the estimated sales value in cell G5 the data in cellsB30, B31 and B32 shown in Figure 4.15 are required and the formula for cell G5 is:
⫽(G6*B31)⫹(G7*B32)⫹B30
To use this multiple regression model, data must first be entered into cells G6 and G7. Figure 4.16 shows the estimated sales value if the advertising expenditure is 250 and the promotion expenditure is 125.
Figure 4.16 Results of multiple regression using the command method
Providing the XandYdata does not change, different values can be entered into cells G6 and G7 and the estimated sales value will recalculate correctly. However, because the command method has been used for the regression, if any of the X or Y data changes it will be necessary to execute the Regression com- mand again.
Selecting variables for multiple regression
A problem encountered when applying multiple regression methods in forecasting is deciding which variables to use. In the interests of obtaining as accurate a forecast as possible, it is obvious that all the relevant variables should be included.
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However, it is often not feasible to do this, because, apart from the complexity of the resulting equation, the following factors must be considered.
◆ In a forecasting situation the inclusion of variables which do not significantly contribute to the regression merely inflate the error of the resulting prediction.
◆ The cost of monitoring many variables may be very high – so it is advisable to exclude variables that are not contributing significantly.
◆ There is no point including terms that contribute less than the error variance, unless there is strong prior evidence that a particular variable should be included.
◆ For successful forecasting, an equation that is stable over a wide range of conditions is necessary. The smaller the number of vari- ables in the equation, the more stable and reliable the equation.
Summary
Regression analysis is a popular forecasting and estimating tech- nique. Although many users might find the mathematics involved quite difficult, the technique itself is relatively easy to use, especially when a model or template has previously been developed.
However, users who do not understand the underlying mathematics should obtain some assistance in the interpretation of the results.
Time Series Analysis
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Technology is the acknowledged master, the engine that pulls all the rest along and determines where the future shall be, how fast it shall be attained, what we shall do within it, and who shall pros- per, who shall languish, who shall fossilize.
– T. Levitt, Marketing in an Electronic Age, 1985.
Introduction
A time series is a set of observations recorded over a period of time. It is a sequence of data that is usually observed at regular intervals.
Time series analysis predicts the value of an item by studying the past movements of that item. It is a series that uses time as the independent or explanatory variable.
Time series data can be analysed for trend, seasonal fluctuations and residual. Trend analysis involves using a linear regression model with time as the independent variable. The data is also adjusted for seasonal fluctuations since this can significantly influ- ence the regression results. The residual is what is left over after the trend and seasonality has been taken into account, and in this case consists of a cycleand an error.
Many time series follow a cycle, with either an annual or seasonal pattern. These cyclical effects are analysed, as are the errors. This is achieved through decompositionanalysis.
Decomposition analysis is the method of reducing a set of time series data to a trend, a seasonal factor and a residual. This may be expressed as:
D⫽TSR
whereD⫽data,T⫽trend,S⫽seasonality factor and R⫽residual which includes the error.
The decomposition analysis may be either multiplicative or additive.
Additive decomposition takes the following form:
D⫽T⫹S⫹R
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and multiplicative decomposition analysis takes the form:
D⫽T*S*R
The multiplicative model is also referred to as a ration model. It is the multiplicative model that is used in this chapter.