San José State University
Department of Economics |
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Bent Line in Regression Analysis |
Suppose y is a function of x but the slope of the relationship changes at x=k_{1}. A regression line for such a function is achieved by defining a new variable x such that
The variable x_{1} can be defined more succinctly as x_{1}=u(x-k_{1}), where u(z) is the function such that u(z)=0 if z<0 and u(z)=z for z≥0.
The regression of y on x and x_{1} gives an equation such as
The coefficient c_{1} gives the change in the slope of the relationship at k_{1}.
Thus the slopes of the relationship are:
For more than one bendpoint the procedure is analogous and the slope of the relationship is the sum of the coefficients up to the different levels of x.
This covers the case in which the slopes of the relationship over different intervals are required to be the same. For example, suppose y is a function of time such that there are bend points at k_{1} and k_{2}. Furthermore suppose the slope of the relationship in the third interval (from k_{2}<x has to be the same as the slope in the first interval, from 0<x<k_{1}.
Now consider the case in which a regression of y on variable x, z and w has to be of the form
This form is the same as
That is to say, y must be regressed on (x+z) and w. Adding two variables together forces the regression coefficients to be the same.
Likewise if the regression has to be of the form
Now consider again the previous example in which the slope in the third interval of a trend line is required to be the sames as the slope in the first interval. First two additional variables x_{1} and x_{2} need to be defined as
An unconstrained regression would yield an equation of the form
The slope of the relationship in the first interval is c_{0} and in the third interval it is c_{0}+c_{1}+c_{2}. For the slopes in the first and third interval to be equal requires that
Therefore the regression equation is of the form_{2}
Thus y must be regressed on x and (x_{1}−x_{2}).
This method can be generalized.
Suppose the relationship cyclic relationship is of the form
Then the regression would use a generated variable of the form
The other variable in the regression would just be the trend variable.
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