In order to implement the various techniques discussed in this class, the students must be able to determine the mathematical relation between the economic variables that make up the various functions used in economicsdemand functions, production functions, cost functions, and others.
For example, a manager often must determine the total cost of producing various levels of output. As you will see later, the relation between total cost (C) and quantity (Q) can be specified as
PARAMETERS OF THE EQUATION
where a, b, c, and d are the parameters of the cost equation. Parameters are coefficients in an equation that determine the exact mathematical relation among the variables in the equation.
Once the numerical values of the parameters are determined, the manager then knows the quantitative relation between output and total cost. For example, suppose the values of the parameters of the cost equation are determined to be a = 1,262, b = 1.0, c = 0.03, and d = 0.005. The cost equation can now be expressed as:
This equation can now be used to compute the total cost of producing various levels of output.
If, for example, the manager wishes to produce 30 units of output , the total cost can be calculated as
The process of finding estimates of the numerical values of the parameters of an equation is called parameter estimation.
Although there are several techniques for estimating parameters,
the values of the parameters are often obtained by using a technique called regression analysis.
Regression analysis uses data on economic variables to determine a mathematical equation that describes the relation between the economic variables.
Regression analysis involves both:
1. the estimation of parameter values and
2. testing for statistical significance.
In this notes and the notes that will follow, we are not as much interested in your knowing the ways the various statistics are calculated, as we are in your knowing how these statistics can be interpreted and used.
THE SIMPLE LINEAR REGRESSION MODEL
Regression analysis is a technique used to determine the mathematical relation between a dependent variable and one or more explanatory variables.
The explanatory variables (independent variables) are the economic variables that are though to affect the value of the dependent variable.
In the simple linear regression model, the dependent variable Y is related to only one explanatory variable X, and the relation between Y and X is linear:
This is the equation for a straight line, with X plotted along the horizontal axis and Y along the vertical axis.
The parameter a is called the intercept parameter because it gives the value of Y at the point where the regression line crosses the Y axis. (X is equal to zero at this point.)
The parameter b is called the slope parameter because it gives the slope of the regression line.
The slope of a line measures the rate of change in Y as X changes (AY/AX); it is therefore the change in Y per unit change in X.
The simple regression model is based on a linear relation between Y and X, in large part because estimating the parameters of a linear model is relatively simple statistically. Assuming a linear relation is not overly restrictive. For one thing, many variables are actually linearly related or very nearly linearly related.
For those cases where Y and X are instead related in a curvilinear fashion, you will see that a simple transformation of the variables often makes it possible to model nonlinear relations within the framework of the linear regression model.
A Hypothetical Regression Model
To illustrate the simple regression model, consider a statistical problem facing the Tampa Travel...