Regression analysis is found to be the most popular method of demand estimation and/or forecasting. It combines economic theory and statistical techniques of estimation. The economic theory specifies the determinants of demand and the nature of the relationship between the demand for a product and its determinants. It helps in ascertaining the general form of demand function. Statistical techniques on the other hand are employed in estimating the values of the parameters in the estimated equation.
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= na + b∑Xi (3.1.2)
2 (3.1.3)
In regression models, the quantity to be forecast in the demand function is the dependent ariable, and the determinants of demand are the independent or explanatory variables.
In specifying the demand functions for various commodities, the forecaster may come across many commodities whose demand depends, at large, on a single independent variable. For instance, suppose the demand for sugar in a given geographical area is found to depend largely on the population, then the demand function for sugar will be referred to as a single-variable demand function. But if it is found that demand functions for fruits and vegetables depend on a number of variables such as, their own-prices, substitutes, household income, population, and the like, then such demand functions are referred to as multi-variable demand functions. The single regression equation is used for single-variable demand functions, while the multi-variable equation is used for multi- variable demand functions. The single-variable and multi-single-variable regressions are outlined below.
The Simple or Bivariate Regression Technique
As mentioned earlier, in a simple regression technique, a single independent variable is used in estimating the statistical value of the dependent variable or the variable to be forecast. This technique is similar to trend fitting, though, in trend fitting, the independent variable is time, t, while in the case of simple regression, the chosen independent variable is the single most important determinant of demand.
Suppose we want to forecast the demand for sugar, for example, for particular periods on the basis of some past data, we would estimate the regression equation of the form:
Y = a + bX (3.1.1)
where Y represents the quantity of sugar to be demanded; and, X represents the single variable, population, and a and b are constants.
The parameters a and b can be estimated, using the past data, by solving the following corresponding linear quations for a and b:
∑Yi
∑XiYi = a∑Xi + bXi
The procedures for calculating the terms in equations (3.1.2) and (3.1.3) can be illustrated by the following example. Consider the following hypothetical past data on the demand for sugar for the years 2000 to 2006:
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Table 3.1.1: Demand for Sugar
Year Population Quantity of sugar demanded (millions) (000’s)
2000 10 40
2001 12 50
2002 15 60
2003 20 70
2004 25 80
2005 30 90
2006 40 100
Using this hypothetical data, we can calculate the terms as shown in table 3.1.2 below:
Table 3.1.2: Calculation of Terms of the Linear Equations in Simple Regression
Xi 2 XiYi
2000 10 40 100 400
2001 12 50 144 600
2002 15 60 225 900
2003 20 70 400 1400
2004 25 80 625 2000
2005 30 90 900 2700
2006 40 100 1600 4000 n = 7 ∑Xi = 152 ∑Yi = 490 ∑Xi 2 = 3994 ∑XiYi = 12000
Substituting the related values from table 3.1.2 into equations (3..1.2) and (3.1.3), we get:
490 = 7a + 152b (3.1.4) 12000 = 152a + 3994b (3.1.5)
Solving simultaneously for a and b in the above equations, we obtain:
a = 27.44; b = 1.96. substituting these values into the regression equation (3.3.11), the estimated regression equation becomes:
Y = 27.44 + 1.96X (3.1.6)
With the regression equation (3.1.6), the demand for the commodity concerned can be easily forecast for any period provided that the figure for the population or any single determinant of demand is known.
Suppose the population for the year 2008 is projected to be 100 million then the demand for sugar, according to our example, would be
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estimated using the regression equation as:
Y = 27.44 + 1.96(100) = 27.44 + 1960 = 223,440 units.
The simple regression technique is based on the following assumptions:
4. the independent variable will continue to grow at the estimated growth rate, 1.96 according to regression equation (3.1.6);
5. the relationship between the dependent and independent variables will continue to remain the same in the future as in the past.
The Multi-Variate Regression Technique
The technique is used in cases where the demand for a commodity is determined to be a function of many independent variables, or where the explanatory variables are greater than one. The analysis in this technique is referred to as multiple regression analysis.
The procedure of multiple regression analysis involves the following steps:
Step One: Specification of the independent or explanatory variables, that is, the variables that explain the variations in demand for the commodity in question. These variables are identified from the determinants of demand as listed earlier.
Step Two: Collection of time-series data on the independent variable.
Here, the necessary data on both the dependent (the demand for the commodity) and independent variables (the determinants of demand) are collected.
Step Three: Specification of the Regression Equation. The reliability of the demand forecast depends to a large extent on the form of regression equation and the degree of consistency of the explanatory variables in the estimated demand function. The greater the degree of consistency, the higher will be the reliability of the estimated demand function and vice versa.
The final step is to employ the necessary statistical technique in estimating the parameters of the regression equation. Some common forms of multi-variate demand functions are as follows:
1. The Linear Function. The linear demand function is where the relationship between the demand and its determinants is formulated by a straight line. The most common type of this equation is of the form:
Qx = α – bPx + cY + dPs + JA (3.1.7)
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where Qx = quantity demanded of commodity X; Px = unit price of commodity X; Y = consumer’s income; Ps = price of substitute good; A
= advertisement expenditure; α is a constant (or the demand intercept), and b, c, d, and j are the parameters (or regression coefficients) expressing the relationship between demand and Px, Y, Ps, and A, respectively.
In linear demand functions, quantity demanded is assumed to change with changes in independent variables at a constant rate. The parameters are estimated by using the least- squares method. Having estimated the parameters, the demand can be easily forecast if data on the independent variables for the reference period are available.
2. The Power Function. In the linear functions of demand, the marginal effects on demand of independent variables are assumed to be constant and independent of changes in other variables. It is assumed, for instance, that the marginal effect of a change in own price is independent of change in income or other independent variables. There may, however, be cases in which it is intuitively or theoretically found that the marginal effect of the independent variables on demand is neither constant nor independent of the values of all other variables included in the demand function. For example, the effect of an increase in the price of sugar on demand may be neutralised by a rise in consumer’s income. In such cases, a multiplicative or ‘power’ form of the demand function, considered to be the most logical form, is used for estimating the demand of a commodity. The power form of the demand function is given by:
Qx = αPxbYcPsdAj (3.1.8)
The algebraic form of multiplicative demand function can be transformed into a log- linear form for simplicity in estimation as follows:
Log Qx = log α – b log Px + c log Y + d log Ps + j log A (3.1.9) This can be estimated using the least-squares regression technique. The estimated function can easily be used in forecasting the future demand for the given commodity.
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