The preceding chapter developed the theory of demand, including the concepts of price elasticity, income elasticity, and cross-elasticity of demand.
A manager who is contemplating an increase in the price of one of the firm’s products needs to know the impact of this increase on:
(1) quantity demanded
(2) total revenue
What questions should the manager answer?
- Is the demand elastic, inelastic, or unit elastic with respect to price over the range of contemplated price increase?
-What will happen to demand if consumer incomes increase or decrease as a result of an economic expansion or contraction.
Managers face these types of problems everyday whether in a profit-seeking enterprises, not-for-profit organizations or governments.
What will be the impact of cigarette taxes on the quantity demanded of my product?
What effect will a tuition increase have on local state university revenues?
These are the types of questions the empirical investigation attempt to answer
Statistical Estimation of the Demand Function
Econometrics is a collection of statistical techniques available for testing economic theories by empirically measuring relationships among economic variables.
The measurement of economic relationships is a necessary step in using economic theories and models to obtain estimates of the numerical values of variables that are of interest to the decision maker.
The estimation of a demand function using econometric techniques involves the following steps
- Identification of the variables
- Collection of the data
- Formulation of the demand model
- Estimation of the parameters
- Development of forecasts (estimates) based on the model
Identification of the variables
As discussed in the previous chapter, the demand function may be viewed as the relationship between the quantity demanded (the dependent variable) and several independent variables.
The first task in developing a statistical demand model is to identify the independent variables that are likely to influence quantity demanded.
These might include factors such as the price of the good in question, price of competing or substitute goods, population, per capita income, and advertising and promotional expenditures.
Collection of the data
Once the variables have been identified, the next step is to collect data on the variables. Data can be obtained from a number of different sources.
Formulation of the model
The next step is to specify the form of the equation, that indicates the relationship between the independent variables and the dependent variable.
1. Linear Model
The linear model, which is the most common form of estimation equation:
α , β 1, β 2, β 3, ε are the parameters of the model and ε is the error term.
The error term is included in the model to reflect the fact that the relationship is not an exact one, i.e., the observed demand value may not always be equal to the theoretical value.
Interpretation of the value of β
The value of each β coefficient provides an estimate of the change in quantity demanded associated with a one-unit change in the given independent variable, holding constant all other independent variables.
Interpretation of the value of β
β 1 = δY/ δX 1
β 2 = δY/ δX 2
β 3 = δY/ δX 3
The β coefficients are equivalent to the partial derivatives of the demand function:
Interpretation of the value of β and using β to determine elasticity
Ɛ d1 = δY/ δX 1 ( X 1 / Y)
Ɛ d1 = β 1 ( X 1 / Y)
1.1 Simple Linear Regression Model
The analysis in this section is limited to the case of one independent and one dependent variable (two-variable case), where the form of the relationship between the two variables is linear.
Estimating the simple linear regression coefficients
Sherwin-Williams company is attempting to develop a demand model for its line of exterior house paints. The company’s chief economist feels that the most important variable affecting paint sales (Y) (measured in gallons) is:
(1) promotional expenditures(X) (measured in dollars). These include expenditures on advertising (radio, TV, and newspaper), in-store displays and literature, and customer rebate programs.
The chief economist decides to collect data on the variables in a sample of ten company sales regions that are roughly equal in population. Data on paint sales and promotional expenditures were obtained from the company’s marketing department.
Y= a + b.X p
Y= 120.75 + 0.434X p
Interpretation: The coefficient of X (0.434) indicates that for one-unit increase in X ($1,000 in additional promotional expenditures), expected sales (Y) will increase by 0.434 (X1,000)= 434 gallons in a given sales region.
Using the regression equation to make predictions
A regression equation can be used to make predictions concerning the value of Y, given any particular value of X. This is done by substituting the particular value of X, namely X p , into the sample regression equation
Suppose one is interested in estimating Sherwin-William’s paint sales for a metropolitan area with promotional expenditures equal to $185,000 (i.e., X p= 185).
Y=120.75 + 0.434(185)
Y=201.045 gallons or 201,045 gallons
X p= 300 or $300,000
Remark: Xp falls outside of the series of observations for which the regression line was calculated.
Thus, because of the above remark, we cannot be certain that the prediction of paint sales based on the regression model would be reasonable.
Standard Error of the estimate
The error term e i is defined as the difference between the observed and predicted value of the dependent variable.
The standard deviation of the error terms is calculated as:
The standard error of the estimate (S e ) can be used to construct prediction intervals for Y. An approximate 95 percent prediction interval is equal to
Returning to our previous example 155.447 to 246.643 (that is, from 155,447 gallons to 246,643 gallons).
H0: β =0 ( No relationship between X and Y)
Ha: β≠ 0 ( linear relationship between X and Y)
There are two ways of doing the testing:
Calculate the t statistic and compare it to the critical value
Use the p-value technique
The correlation coefficient measures the degree to which two variables tend to vary together.
Correlation analysis is useful in explanatory studies of the relationship among economic variables. The information obtained in the correlation analysis can then be used as a guide in building descriptive models of economic phenomena that can serve as a basis for prediction and decision making.
The value of the correlation coefficient ® ranges from +1 for the two variables with perfect positive correlation to -1 for two variables with perfect negative correlation.
The Coefficient of Determination
The Coefficient of Determination
It measures the proportion of the variation in the dependent variable that is explained by the regression line (the independent variable).
The coefficient of determination ranges from 0 ( when none of the variation in Y is explained by the regression) to 1( when all the variation in Y is explained by the regression.
If r 2 =0.519 ( from the Sherwin-William’s company example).
Interpretation: the regression equation, with promotional expenditures as the independent variable, explains about 52 percent of the variation in paint sales in the sample.
Remark: In the two-variable linear regression model, the coefficient of determination is equal to the square of the correlation coefficient, i.e., r 2 =0.519=( r) 2 =(0.72059) 2 .
It is used to test whether the estimated regression equation explains a significant proportion of the variation in the dependent variable.
The decision is to reject the null hypothesis of no relationship between X and Y ( that is, no explanatory power) at the k level of significance if the calculated F-ratio is greater than the F k,1,n-2 value obtained from the F-distribution.
The value of F 0.05,1,8 from the F-distribution (from the table) is 5.32.
We reject, at the 0.05 level of significance, the null hypothesis that there is no relationship between promotional expenditures and paint sales. In other words, we conclude that the regression model does explain a significant proportion of the variation in paint sales in the sample.
Association and Causation
The presence of association (correlation) does not necessarily imply causation.
1.2 Multiple Linear Regression
A functional relationship containing two or more independent variables is known as a multiple linear regression model.
2. Non Linear Models
Non Linear models are used when the underlying relation between Y and X plots as a curve, rather than a straight line.
In some cases, economic theory will strongly suggest that Y and X are related in a nonlinear fashion.
2.1 Quadratic Model
This can be expressed as
Y = a + bX + cX 2
This will graph as U-shaped or inverted U-shaped curve.
If b<0 and c>0, the function is U-shaped
If b>0 and c<0, the function is inverted U-shaped
The parameters can be estimated by transforming the function into its linear form by creating a new variable Z defined as Z = X 2 , thus Y = a + bX + cZ
2.2 Log-linear Model
Relationship between variables maybe expressed in logarithm such as:
Y = aX b Z c where the parameters b and c can already be interpreted as elasticities with respect to the explanatory variables X and Z respectively.
The linear transformation is
ln Y = (ln a) + b(ln X) + c(ln Z), Thus
if Y’ = ln Y, X’ = Ln X and Z’ = ln Z, hence
Y’ = a’ + bX’ + cX’ and
a = antilog a’ = e a’
Consider a simple demand equation
Q= a + bP. The law of demand implies that the coefficient b should be negative, indicating that less of the product is demanded at higher prices.
Consider a small restaurant chain specializing in fresh lobster dinner. The business has collected information on prices and the average number of meals served per day for a random sample of eight restaurants in the chain. These data are shown below. Use regression analysis to estimate the coefficients of the demand function Qd= a +bP. Based on the estimated equation, calculate the point price elasticity of demand at the mean values of the variables.
The Least-Squares regression estimation
Estimating the demand for lobsters dinners using the OLS Qi Pi Qi-Q(bar) Pi-P(bar) (Pi-P(bar)) 2 (Pi-P(bar))(Qi-Q(bar)) 100 15 0 -1 1 0 90 18 -10 2 4 -20 85 19 -15 3 9 -45 110 14 10 -2 4 -20 120 13 20 -3 9 -60 90 19 -10 3 9 -30 105 16 5 0 0 0 100 14 0 -2 4 0 40 -175 b -4.375 a 170
Estimating the demand for lobster dinners
Using the ordinary Least Squares regression, we find the estimates of the demand for lobsters.
We can now use our results to determine the point elasticity of demand for lobsters.
Problems in Regression Analysis 1. Multicollinearity
This exists if the assumption of linear independence is violated, which means two or more of the explanatory variables are linearly dependent.
This would result to higher standard errors of estimate than the true value, but parameter estimates would still be unbiased
More difficult to find statistical significance because higher standard errors lower the t-statistics
Dropping one of the explanatory variables maybe done but the risk is the possibility of dropping an important variable resulting to specification error (a more serious problem) and biased parameter estimates
This problem is encountered if the variance of the error term is not constant.
This occurs if there exists some relation between the error tem and one or more of the explanatory variables (e.g. large errors are associated with large values of X
The parameter estimates will still be unbiased but the standard errors of the coefficients are biased so the calculated t-ratios are unreliable.
This normally occurs in cross-section data and can sometimes be corrected by performing a transformation on the data or equation.
This problem is associated with time-series data, and occurs when the errors are not independent over time (e.g. high error in one period tends to promote high error in the following period.
Parameter estimates are unbiased, but the standard errors are again biased, resulting in unreliability of the calculated t-ratios
Most common test for the presence of autocorrelation is the Durbin-Watson test.