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Econ 533 Spring 2009 Assignment 1 Linear Probability Model

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					                           Econ 533 Spring 2009
               Assignment 1: Linear Probability Model Solution

1. The soaring cost of gasoline has led consumers to consider switching to other sources
   of transportation. One of the choices available to consumers is a hybrid car. In this
   homework, we seek to evaluate a discrete choice model that will help us understand
   the likelihood of switching to hybrid cars from a standard vehicle.
   The Data set for this homework can be found on the class website by clicking on
   “Hybrid Car Choice” under the “Data” link. This data set contains the following
   variables on whether or not series of individuals have chosen to purchase a hybrid car
   (rather than a standard vehicle):

    Variable     Description
    BUY          =1 if the individual buys a hybrid vehicle; =0 otherwise
    MALE         =1 if the individual is male; =0 otherwise
    CONS         =1 if the individual belongs to an environmental group (e.g., Nature Conservancy.
    FOREIGN      =1 if the vehicle being considered is foreign made; =0 otherwise
    AGE          =the age of the individual
    COSTD        =the additional cost of the hybrid vehicle relative to a standard vehicle ($1000s)
    GAS          =the current price of gasoline

2. (a) Describe the data and ensure that all variables are as stated in question one. Also,
       provide a table categorizing buying a hybrid car and the sex of individuals in the
       data. What does this tell you about the likelihood of buying a hybrid car by a
       males?
       To describe the data in STATA, the ‘summary’ command should be sufficient. It
       gives the mean, standard deviation, minimum, maximum and number of observa-
       tions for each variable. One of the reasons for this exercise is to verify that there
       are no outliers and that the data are coded correctly.
       A table categorizing the buying of a hybrid car by sex can also be done using the
       ‘table’ command. It can be used to calculate the proportion in each category that
       choose to buy a hybrid car.


                                     MALE
                                          0     1
                               BUY 0  0.283 0.347
                                   1  0.216 0.154

        The table shows that only about 15% of males in our sample bought hybrid cars
        compared to about 22% for females. We will expect the coefficient on male to be
        negative in our regression given the data. Males are less likely to buy hybrid cars
        than female.
   (b) Estimate a Linear Probability Model of the probability of purchasing a hybrid ve-
       hicle as a function of cost differential (i.e., costd). Is the cost differential found to


                                            1
    be a significant determinant of the purchase probability? Estimate the predicted
    probability of purchasing a hybrid vehicle for costd=0 and costd=500.

    The attached STATA output gives an estimate of the parameters for the LPM
    model. The linear probability model does indicate that the cost differential (COSTD)
    is a significant indicator of probability of purchasing a hybrid vehicle, with the co-
    efficient on COSTD estimated to be -0.28, with a corresponding t-statistic of -14.4
    and a P-value of less than (.001). Clearly this coefficient is significant at any rea-
    sonable level. The level of the coefficient indicates that each $1000 increase in the
    cost premium for a hybrid vehicle reduces the probability of purchasing the hybrid
    by 28%. The mfx command (‘mfx, at (costd=0)’) can be used to estimate the fitted
    probability when COSTD equals 0 and 500. This shows that the fitted probability
    of buying a hybrid is 98.9% when the cost is the same as a standard vehicle and
    still relatively high probability of about 84.8% when the cost differential is $500.
(c) Now estimate a LPM including the other explanatory variables. Is costd still a
    significant determinant of the hybrid purchase probability? Again, estimate the
    predicted probability of purchasing a hybrid vehicle for costd=0 and costd=500.
    To do this, you will have to hold all the other explanatory variables fixed at their
    means. Are the predicted probabilities from this regression and those from the
    simpler model comparable? Why or why not?
    The second and more complicated linear probability model still shows the cost
    differential to be a significant predictor of hybrid purchases with a coefficient of
    -.281 and a corresponding t-statistic of -14.9. The results also suggest that all
    of the other factors (except whether or not it is foreign made) impact the hybrid
    purchase decision. Specifically, we find that males and older individuals are less
    likely to buy a hybrid, while higher gas prices and belonging to an environmental
    group both increase the odds of purchasing a hybrid.
    The fitted predicted probabilities of purchasing a hybrid vehicle when there is no
    cost differential and a cost differential of $500 is 98.9% and 84.8% respectively.
    In this case there is no difference in the probability between the single and multiple
    variables case.
    The results in part c of the problem are similar to those that we obtained in part
    b. This need not be the case in general for several reasons. Typically, when you
    include only a few variables in the model, they will ‘pick-up’ the effects of those
    variables missing from the model.




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  Econ 533      Friday February 20 14:50:14 2009                    Page 1

                                                                                   ___ ____ ____ ____ ____tm
                                                                                  /__     /  ____/    /  ____/
                                                                                 ___/   /   /___/   /   /___/
                                                                                   Statistics/Data Analysis

                                                                        User: Babatunde Abidoye
                                         Project: Assignment 1: Linear Probability Model{space -17}
1 . su

      Variable                  Obs           Mean           Std. Dev.             Min          Max

              obs               1000             500.5        288.8194                    1          1000
               buy              1000               .37        .4830459                    0             1
                 c              1000                 1               0                    1             1
             male               1000              .501        .5002492                    0             1
             cons               1000              .347        .4762539                    0             1

         foreign                1000           .398           .4897304                  0            1
              age               1000         51.754           15.86497                 25           90
           costd                1000       2.201674           .7139663           1.004943     3.999613
              gas               1000       2.312593           .3086123            1.80053     2.899421

2 . table buy male


                       MALE
          BUY          0        1

            0         283      347
            1         216      154


3 . regress buy costd

           Source               SS          df           MS                       Number of obs =                 1000
                                                                                    F( 1,     998)    =         208.15
            Model           40.2272555        1     40.2272555                       Prob > F          =        0.0000
         Residual           192.872744      998     .193259263                       R-squared         =        0.1726
                                                                                    Adj R-squared     =         0.1717
            Total                233.1      999     .233333333                       Root MSE          =        .43961


             buy               Coef.      Std. Err.             t      P>|t|         [95% Conf. Interval]

           costd            -.2810605      .0194809          -14.43      0.000           -.3192888      -.2428322
           _cons             .9888035      .0450873           21.93      0.000            .9003267        1.07728


4 . mfx, at(costd=0)

  Marginal effects after regress
        y = Fitted values (predict)
           =   .98880355

  variable              dy/dx          Std. Err.         z      P>|z|        [     95% C.I.    ]            X

     costd           -.2810605            .01948     -14.43          0.000       -.319242 -.242879                   0


5 . mfx, at(costd=0.5)

  Marginal effects after regress
        y = Fitted values (predict)
           =   .84827331

  variable              dy/dx          Std. Err.         z      P>|z|        [     95% C.I.    ]            X

     costd           -.2810605            .01948     -14.43          0.000       -.319242 -.242879                  .5
  Econ 533       Friday February 20 14:50:21 2009              Page 2

6 . regress buy costd male cons foreign age gas

           Source              SS         df        MS                       Number of obs =               1000
                                                                               F( 6,     993)   =         50.27
            Model       54.3073629          6    9.05122716                     Prob > F         =       0.0000
         Residual       178.792637        993    .180053008                     R-squared        =       0.2330
                                                                               Adj R-squared    =        0.2283
            Total               233.1     999    .233333333                     Root MSE         =       .42433


             buy           Coef.        Std. Err.          t      P>|t|         [95% Conf. Interval]

           costd        -.2810943        .0188597       -14.90      0.000         -.3181037      -.2440849
            male        -.0907112        .0269327        -3.37      0.001         -.1435628      -.0378596
            cons         .1921005        .0282518         6.80      0.000          .1366604       .2475406
         foreign        -.0339533        .0274528        -1.24      0.216         -.0878255       .0199189
             age         -.001749         .000847        -2.06      0.039         -.0034111      -.0000868
             gas         .1461695        .0435929         3.35      0.001          .0606248       .2317142
           _cons         .7336644         .118982         6.17      0.000          .5001793       .9671494


7 . mfx, at(costd=0)

  warning: no value assigned in at() for variables male cons foreign age gas;
     means used for male cons foreign age gas

  Marginal effects after regress
        y = Fitted values (predict)
           =   .98887809

  variable             dy/dx        Std. Err.       z      P>|z|        [     95% C.I.   ]           X

        costd       -.2810943           .01886    -14.90        0.000       -.318059 -.24413               0
         male*      -.0907112           .02693     -3.37        0.001       -.143498 -.037924           .501
         cons*       .1921005           .02825      6.80        0.000        .136728 .247473            .347
      foreign*      -.0339533           .02745     -1.24        0.216        -.08776 .019853            .398
          age        -.001749           .00085     -2.06        0.039       -.003409 -.000089         51.754
          gas        .1461695           .04359      3.35        0.001        .060729   .23161        2.31259

  (*) dy/dx is for discrete change of dummy variable from 0 to 1

8 . mfx, at(costd=0.5)

  warning: no value assigned in at() for variables male cons foreign age gas;
     means used for male cons foreign age gas

  Marginal effects after regress
        y = Fitted values (predict)
           =   .84833093

  variable             dy/dx        Std. Err.       z      P>|z|        [     95% C.I.   ]           X

        costd       -.2810943           .01886    -14.90        0.000       -.318059 -.24413              .5
         male*      -.0907112           .02693     -3.37        0.001       -.143498 -.037924           .501
         cons*       .1921005           .02825      6.80        0.000        .136728 .247473            .347
      foreign*      -.0339533           .02745     -1.24        0.216        -.08776 .019853            .398
          age        -.001749           .00085     -2.06        0.039       -.003409 -.000089         51.754
          gas        .1461695           .04359      3.35        0.001        .060729   .23161        2.31259

  (*) dy/dx is for discrete change of dummy variable from 0 to 1

9 .

				
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