Linear Probability Model Assignment Help

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Linear Probability Model:

The linear probability model defines

790_Linear Probability Model.png

The  linear probability  model  is computationally simple  and  familiar  to  all econometrics  students.  It  is  equivalent to modeling  the choice  problem  using standard linear regression methodology. However, it has several drawbacks.

1)  The  right hand  side variables are  a  combination of discrete  and  continuous variables  but  the  left  hand  side variable  is  discrete.  We  seek  to  equate "something discrete"  to  "something continuous". The data plot of the (xi, Yi) observations will plot a series of continoous  values against a series of ones and zeros. Fitting a regression line (or plane) by minimizing the squared distance as is  done  in  classical  regression  analysis  does  not  have  any meaningful interpretation here.

2)  The choice of values for Y,,  i.e., 0 and 1,  is arbitrary. This will however, change theb's, which means that thers will have no clear interpretation.

3)  As Y,  can take only two values, 0 and  1,  the disturbance E,  also  takes only one of two values, for each x,:

1193_Linear Probability Model1.png

Consequently,  the behavior of E,  can never be  approximated by any continuous probability distribution.

4)  Standardising E,  so  that the  expectation  of  e, conditional on  the  exogenous variables, is zero, as in classical regression analysis implies

2249_Linear Probability Model2.png

This is turn  implies that 1270_Linear Probability Model3.png  so  that the original regression (12.6)  is equivalent to1003_Linear Probability Model4.png. This gives the disturbance an interpretation as the difference between  the binary  response variable and the response probability. But, the response probability  is usually a continuously varying entity while the disturbance  term and the response variable are binary, discrete variables and  it does not make sense to equate a discrete variable to the difference between a continuous and a discrete variable.

The above analysis also implies  that the variance of the disturbance is,

451_Linear Probability Model5.png

Notice that this variance is a function of both xi  and  β.  In other words, not only is  the  .variance  heteroscedastic  but  its  variance depends  on  the slope coefficients  of the regression .

5)  A more serious and  fundamental problem  is that we cannot constrain xiβ and hence the response probability  to the interval [0,1] Consequently, the model produces nonsense probabilities and negative variances. Such predictions are clearly awkward and undesirable.  In practice researchers adjust the predicted probabilities  to  lie within the [0,  1] interval, but  such adjustments are ad hoc and the resulting estimator may have no known  sampling properties. This is a very serious  limitation of the linear model.

For these reasons, the linear model is being less frequently used, except as a basis for comparison to some other more appropriate models.

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