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# Econometrics

Questions 16.8, C17.3

#### Transcript of Econometrics

Hill Question 16.8

Wooldridge C17.3 Econometrics Week 9 16.8 a) Using the estimates given in the table, calculate the probability that a student will choose no college, a 2 year college and a 4 year college if the student's grades are 6.64 and again if they are 4.905. Discuss the probability changes. Are they what you anticipated? For grades = 6.64, no college:

=Ф(µ1-βGrades)

=Ф(-2.9456-(-0.3066x6.64))

=Ф(-0.909776)

Using z-tables for -0.91

=0.1814

=18.14% For grades = 6.64, 2 years college

=Ф(µ2-βGrades)-Ф(µ1-βGrades)

=Ф(-2.0900-(0.3066x6.64))-Ф(-2.9456-(-0.3066x6.64))

=Ф(-0.054176)-Ф(-0.909776)

similarly, using z-tables

=0.2987

=29.87% For Grades = 6.64, 4 years college

=1-Ф(µ2-βGrades)

=1-Ф(-2.0900-(-0.3066x6.64))

=1-Ф(-0.0514176)

=0.5199

=51.99% For grades = 4.905, no college:

=Ф(-2.9456-(-0.3066x4.905))

=Ф(-1.441727)

Using z-tables for -1.44

=0.0749

=7.49% For grades = 4.905, 2 years college

=Ф(-2.0900-(0.3066x4.905))-Ф(-2.9456-(-0.3066x4.905))

=Ф(-0.586127)-Ф(-1.441727)

=0.20286

=20.29% For grades = 4.905, 4 years college

=1-Ф(-2.0900-(-0.3066x4.905))

=1-Ф(0.586127)

=0.7224

=72.24% Comparing the Results In general the change in grade from 6.64 to 4.905 represents an increase in mark of 1.735. The marks are taken from a 13 point scale where 1 is the highest and 13 the lowest. Hence when we examine the reults for "no college" it is clear that an increase in mark by the above margin will lead to a decrease in the percentage of people not attending college by 10.65%. This is the result that one would anticipate as a higher mark would lead to more options at college and tend to demonstrate a higher focus/ inclination to wish to proceed in futher studies.

This trend is again demonstrated for those attending 2 years of college as seen above however when looking at the 3rd option "4 years of college" the percentage increases with the increase of grade. This result is the result that one would expect both because as the other percentages decrease one must be increasing (as they must equal 100) and also because the higher mark as mentioned earlier would tend to demonstrate a higher desire to pursue more extensive study. Again looking at this 4 year bracket it is representative of people who are within the top 40% based on grades and intuitively one would assume that the top 40% of students would be likely to attend the highest option of college years. b) Expand the ordered probit model to include Faminc, Famsiz and the dummy variables Black and Parcoll. Discuss the estimates and their signs and significance In terms of further analyzing these variables for significance each variable is significant except for Famsiz based on it's p-value. d) Compute the probability that a black student from a household of 4 members, including a parent who went to college, and a household income of 52000, will attend a 4 year college. If i) Grades=6.64, ii) Grades=4.905 For this question i have adapted the model used in part a, now y* simply represents a fuller regression with the inclusion of the additional variables as seen below.

P(y=3|Grades=6.64, Faminc=52.000, Famsiz=4, Black=1, Parcoll=1)

From the notes we know: P(y=3) = P(y*>µ2)

For this example we now have: y* = βGrades + βFaminc + βFamsiz + βBlack + βParcoll + e

Hence:

=P(βGrades + βFaminc + βFamsiz + βBlack + βParcoll + e>µ2)

=P(e>µ2-(βGrades + βFaminc + βFamsiz + βBlack + βParcoll)

=1-Ф(µ2-(βGrades + βFaminc + βFamsiz + βBlack + βParcoll))

=1-Ф(-1.694591-(-0.295292(6.64)+0.005252(52)-0.024122(4)+0.713131(1)+0.423623(1))

=1-Ф(-1.04722212)

using z-tables for value -1.05

=1-0.1469

=85.31%

ii)

P(y=3|Grades=4.905, Faminc=52.000, Famsiz=4, Black=1, Parcoll=1)

Similarly,

=1-Ф(-1.694591-(-0.295292(4.905)+0.005252(52)-0.024122(4)+0.713131(1)+0.423623(1))

=1-Ф(-1.55955374)

for z=1.56

=1-0.0594

=94.06%

Interpreting these results it is demonstrated that as one would expect the lower the grade the higher the % chance of going to 4 years of college with the other variables held constant. It is clear that both these percentages are quite high and therefore that with these variables the percentage attendance of 4 years of college is very likely. e) Repeat d) for a "non-black" student and discuss the differences Much like the previous question we now do the same process with the difference that black is now 0.

P(y=3|Grades=6.64, Faminc=52.000, Famsiz=4, Black=0, Parcoll=1)

=P(e>µ2-(βGrades + βFaminc + βFamsiz + βBlack + βParcoll)

=1-Ф(µ2-(βGrades + βFaminc + βFamsiz + βBlack + βParcoll))

=1-Ф(-1.694591-(-0.295292(6.64)+0.005252(52)-0.024122(4)+0.713131(0)+0.423623(1))

=1-Ф(-0.33409112)

Using z-table for -0.33

=1-0.3707

=62.93%

ii)P(y=3|Grades=4.905, Faminc=52.000, Famsiz=4, Black=0, Parcoll=1)

=1-Ф(-0.84642274)

=1-0.1977

=80.23% Interpretation:

The difference in the findings for the non blacks when the grade was increased from 6.64 to 4.905 was an addition 17.3% chance of attending 4 years of college. Compared to the difference of only 8.75% for "blacks". Yet the percentage to attend 4 years of college for a "black" student was higher even with the poorer grade performance.

When looking at the results I originally was surprised that the results for "non-blacks" were so much lower. My hypothesis as to why is as follows, the "non-black" data takes into account all other races which means it is testing over a larger group and therefore more external influences can come into play. For example cultural influences may affect a certain % of peoples decisions about attending a 4 year college. Alternatively perhaps the percentage of black people who have already committed to a higher level of education are more inclined to continue to pursue that route. Perhaps even certain scholarships exist within the Black community which provides incentive to attend 4 years of college. Over view of Tobit Model:

Another important kind of limited dependent variable is a corner solution response. Such a variable is zero for a nontrivial fraction of the population but is roughly continuously distributed over positive values. The tobit model can be used to deal with these corner solution responses. Question C17.3 Wooldridge i) For what percentage of the workers in the sample is pension equal to zero? what is the range of pension for workers with nonzero pension benefits? Why is a Tobit model appropriate for modeling pension? (This question involves the data file Fringe.raw)

To do the initial part of this question i simply opened the eviews file and entered an if statement for pension = 0 which yielded 172 out of 616 possible observations. Therefore 27.92% of workers have a pension = 0 in the sample.

Next i changed the IF statement to be for pension > 0 and then used descriptive statistics to find the max and minimum values which were min: 7.28 and max: 2880.27

Based on these two results i concluded that a non trivial fraction of the sample is pension = 0 and considering the very wide range of positive pension benefits the Tobit model is well suited. ii) Estimate a tobit model explaining pension in terms of exper, age, tenure, educ, depends, married, white and male. Do whites and males have a statistically significant higher expected pension benefits? As demonstrated in the table being white or being male (or both, computed separately just as a test) increases the value of predicted pension benefits.

Checking the significance via t-statistics however shows that while both increase predicted pension only being "MALE" is statistically significant.

This is demonstrated by the t-statistic (308.1505/66.08142) =4.66(2dp) iii) use the results from part ii) to estimate the difference in expected pension benefits for a white male and a non white female, both of whom are 35 years old, are single with no dependents, have 16 years of education and have 10 years experience. Looking at this question I interpreted the given information to mean that 10 years experience would also be linked to 10 years tenure.

To solve a Tobit question like this one must first calculate the Xiβ value, which for this particular question is done as follows:

Xiβ= -1252.429 + 5.203458(10) -4.638944(35) + 36.02385(10) + 93.21262(16) + 144.0855 + 308.1505

=941.11896

where the above values are the coefficients and Variables respectively.

The next step is to find the σ term which is the coefficient under the error distribution heading, in this case 677.7383

Now we have all the information we can get from the Tobit regression and we put it into E(pension|x) where the formula is:

Ф(Xiβ/σ)(Xiβ)+σф(Xiβ/σ) where ф represents small phi.

Ф(1.388617052)(941.11896) + (677.7383)ф(1.388617052)

Solving for the appropriate values we get

=966.764453.

Repeating the process for the non white female, making sure to remove white and male provides an Xiβ of 488.88296 and works out to be

=582.0448054

This is considerably less with a difference in pension of 384.7196476 iv) Add union to the model and comment on it's significance The addition of union demonstrates it has a large intercept however this doesn't tell us anything by itself. Testing the t-statistic shows us that it is above 7 and a p value that shows it should be statistically significant. v) Apply the Tobit model from part iv) but with peratio, then pension earnings ratio as the dependent variable. Does gender or race have an effect on the pension earnings ratio? Both white and male are independently and jointly insignificant based on there t-statistics and p-values.

For White t=0.484(3dp), p=06282

for Male t= 0.573(3dp), p=0.5670

White and Male are independently and jointly insignificant

Running a joint significant test results in a p val of about 0.75 which demonstrates that neither white people nor males appear to have any special relationship with pension as a fraction of earnings.

Though intuitively it should be clear that the reason white males have the higher pension is cause they have a higher average income and work for the longest time. The key concept in this question can be demonstrated by analyzing Famsiz. This variable has a negative coefficient as does the previously mentioned Grades. The concept here is that for these variables the probability of attending a 4 year college goes down when the variable increases and the probability of the lowest rank choice "no college" increases. The rank choices are based on what occurs above below and between certain limit points found under the "limit points" title above. Hence the opposite can be said to be true for positive coefficients, yet it is important to note that the percentage effect cannot simply be read off the table but must be processed through the formula to appear later. Overview The following problem is a ranking problem, where sentiment or how you feel about certain alternative choices comes into play and hence is unobservable. Certain factors that will be seen in the problem affect how we feel about alternative options.

Because the dependent variable is unobservable it is not a regression model but instead called an index model.

For the following question the outcomes y= 4yr college if y*i>µ2

y= 2yr college if µ1<y*i≤µ2

y= no college if y*i≤µ1

Where y*i = βGrades + ei, are important to note

no intercept term is included because it would be exactly collinear with the threshold variables.

We can represent the probabilities of these outcomes if we assume a particular probability distribution for y*i or equivalently for the random error. For the following questions probabilities can be calculated by assuming that the errors have standard normal distribution N(0,1), an assumption that defines the ordered probit model.

Full transcriptWooldridge C17.3 Econometrics Week 9 16.8 a) Using the estimates given in the table, calculate the probability that a student will choose no college, a 2 year college and a 4 year college if the student's grades are 6.64 and again if they are 4.905. Discuss the probability changes. Are they what you anticipated? For grades = 6.64, no college:

=Ф(µ1-βGrades)

=Ф(-2.9456-(-0.3066x6.64))

=Ф(-0.909776)

Using z-tables for -0.91

=0.1814

=18.14% For grades = 6.64, 2 years college

=Ф(µ2-βGrades)-Ф(µ1-βGrades)

=Ф(-2.0900-(0.3066x6.64))-Ф(-2.9456-(-0.3066x6.64))

=Ф(-0.054176)-Ф(-0.909776)

similarly, using z-tables

=0.2987

=29.87% For Grades = 6.64, 4 years college

=1-Ф(µ2-βGrades)

=1-Ф(-2.0900-(-0.3066x6.64))

=1-Ф(-0.0514176)

=0.5199

=51.99% For grades = 4.905, no college:

=Ф(-2.9456-(-0.3066x4.905))

=Ф(-1.441727)

Using z-tables for -1.44

=0.0749

=7.49% For grades = 4.905, 2 years college

=Ф(-2.0900-(0.3066x4.905))-Ф(-2.9456-(-0.3066x4.905))

=Ф(-0.586127)-Ф(-1.441727)

=0.20286

=20.29% For grades = 4.905, 4 years college

=1-Ф(-2.0900-(-0.3066x4.905))

=1-Ф(0.586127)

=0.7224

=72.24% Comparing the Results In general the change in grade from 6.64 to 4.905 represents an increase in mark of 1.735. The marks are taken from a 13 point scale where 1 is the highest and 13 the lowest. Hence when we examine the reults for "no college" it is clear that an increase in mark by the above margin will lead to a decrease in the percentage of people not attending college by 10.65%. This is the result that one would anticipate as a higher mark would lead to more options at college and tend to demonstrate a higher focus/ inclination to wish to proceed in futher studies.

This trend is again demonstrated for those attending 2 years of college as seen above however when looking at the 3rd option "4 years of college" the percentage increases with the increase of grade. This result is the result that one would expect both because as the other percentages decrease one must be increasing (as they must equal 100) and also because the higher mark as mentioned earlier would tend to demonstrate a higher desire to pursue more extensive study. Again looking at this 4 year bracket it is representative of people who are within the top 40% based on grades and intuitively one would assume that the top 40% of students would be likely to attend the highest option of college years. b) Expand the ordered probit model to include Faminc, Famsiz and the dummy variables Black and Parcoll. Discuss the estimates and their signs and significance In terms of further analyzing these variables for significance each variable is significant except for Famsiz based on it's p-value. d) Compute the probability that a black student from a household of 4 members, including a parent who went to college, and a household income of 52000, will attend a 4 year college. If i) Grades=6.64, ii) Grades=4.905 For this question i have adapted the model used in part a, now y* simply represents a fuller regression with the inclusion of the additional variables as seen below.

P(y=3|Grades=6.64, Faminc=52.000, Famsiz=4, Black=1, Parcoll=1)

From the notes we know: P(y=3) = P(y*>µ2)

For this example we now have: y* = βGrades + βFaminc + βFamsiz + βBlack + βParcoll + e

Hence:

=P(βGrades + βFaminc + βFamsiz + βBlack + βParcoll + e>µ2)

=P(e>µ2-(βGrades + βFaminc + βFamsiz + βBlack + βParcoll)

=1-Ф(µ2-(βGrades + βFaminc + βFamsiz + βBlack + βParcoll))

=1-Ф(-1.694591-(-0.295292(6.64)+0.005252(52)-0.024122(4)+0.713131(1)+0.423623(1))

=1-Ф(-1.04722212)

using z-tables for value -1.05

=1-0.1469

=85.31%

ii)

P(y=3|Grades=4.905, Faminc=52.000, Famsiz=4, Black=1, Parcoll=1)

Similarly,

=1-Ф(-1.694591-(-0.295292(4.905)+0.005252(52)-0.024122(4)+0.713131(1)+0.423623(1))

=1-Ф(-1.55955374)

for z=1.56

=1-0.0594

=94.06%

Interpreting these results it is demonstrated that as one would expect the lower the grade the higher the % chance of going to 4 years of college with the other variables held constant. It is clear that both these percentages are quite high and therefore that with these variables the percentage attendance of 4 years of college is very likely. e) Repeat d) for a "non-black" student and discuss the differences Much like the previous question we now do the same process with the difference that black is now 0.

P(y=3|Grades=6.64, Faminc=52.000, Famsiz=4, Black=0, Parcoll=1)

=P(e>µ2-(βGrades + βFaminc + βFamsiz + βBlack + βParcoll)

=1-Ф(µ2-(βGrades + βFaminc + βFamsiz + βBlack + βParcoll))

=1-Ф(-1.694591-(-0.295292(6.64)+0.005252(52)-0.024122(4)+0.713131(0)+0.423623(1))

=1-Ф(-0.33409112)

Using z-table for -0.33

=1-0.3707

=62.93%

ii)P(y=3|Grades=4.905, Faminc=52.000, Famsiz=4, Black=0, Parcoll=1)

=1-Ф(-0.84642274)

=1-0.1977

=80.23% Interpretation:

The difference in the findings for the non blacks when the grade was increased from 6.64 to 4.905 was an addition 17.3% chance of attending 4 years of college. Compared to the difference of only 8.75% for "blacks". Yet the percentage to attend 4 years of college for a "black" student was higher even with the poorer grade performance.

When looking at the results I originally was surprised that the results for "non-blacks" were so much lower. My hypothesis as to why is as follows, the "non-black" data takes into account all other races which means it is testing over a larger group and therefore more external influences can come into play. For example cultural influences may affect a certain % of peoples decisions about attending a 4 year college. Alternatively perhaps the percentage of black people who have already committed to a higher level of education are more inclined to continue to pursue that route. Perhaps even certain scholarships exist within the Black community which provides incentive to attend 4 years of college. Over view of Tobit Model:

Another important kind of limited dependent variable is a corner solution response. Such a variable is zero for a nontrivial fraction of the population but is roughly continuously distributed over positive values. The tobit model can be used to deal with these corner solution responses. Question C17.3 Wooldridge i) For what percentage of the workers in the sample is pension equal to zero? what is the range of pension for workers with nonzero pension benefits? Why is a Tobit model appropriate for modeling pension? (This question involves the data file Fringe.raw)

To do the initial part of this question i simply opened the eviews file and entered an if statement for pension = 0 which yielded 172 out of 616 possible observations. Therefore 27.92% of workers have a pension = 0 in the sample.

Next i changed the IF statement to be for pension > 0 and then used descriptive statistics to find the max and minimum values which were min: 7.28 and max: 2880.27

Based on these two results i concluded that a non trivial fraction of the sample is pension = 0 and considering the very wide range of positive pension benefits the Tobit model is well suited. ii) Estimate a tobit model explaining pension in terms of exper, age, tenure, educ, depends, married, white and male. Do whites and males have a statistically significant higher expected pension benefits? As demonstrated in the table being white or being male (or both, computed separately just as a test) increases the value of predicted pension benefits.

Checking the significance via t-statistics however shows that while both increase predicted pension only being "MALE" is statistically significant.

This is demonstrated by the t-statistic (308.1505/66.08142) =4.66(2dp) iii) use the results from part ii) to estimate the difference in expected pension benefits for a white male and a non white female, both of whom are 35 years old, are single with no dependents, have 16 years of education and have 10 years experience. Looking at this question I interpreted the given information to mean that 10 years experience would also be linked to 10 years tenure.

To solve a Tobit question like this one must first calculate the Xiβ value, which for this particular question is done as follows:

Xiβ= -1252.429 + 5.203458(10) -4.638944(35) + 36.02385(10) + 93.21262(16) + 144.0855 + 308.1505

=941.11896

where the above values are the coefficients and Variables respectively.

The next step is to find the σ term which is the coefficient under the error distribution heading, in this case 677.7383

Now we have all the information we can get from the Tobit regression and we put it into E(pension|x) where the formula is:

Ф(Xiβ/σ)(Xiβ)+σф(Xiβ/σ) where ф represents small phi.

Ф(1.388617052)(941.11896) + (677.7383)ф(1.388617052)

Solving for the appropriate values we get

=966.764453.

Repeating the process for the non white female, making sure to remove white and male provides an Xiβ of 488.88296 and works out to be

=582.0448054

This is considerably less with a difference in pension of 384.7196476 iv) Add union to the model and comment on it's significance The addition of union demonstrates it has a large intercept however this doesn't tell us anything by itself. Testing the t-statistic shows us that it is above 7 and a p value that shows it should be statistically significant. v) Apply the Tobit model from part iv) but with peratio, then pension earnings ratio as the dependent variable. Does gender or race have an effect on the pension earnings ratio? Both white and male are independently and jointly insignificant based on there t-statistics and p-values.

For White t=0.484(3dp), p=06282

for Male t= 0.573(3dp), p=0.5670

White and Male are independently and jointly insignificant

Running a joint significant test results in a p val of about 0.75 which demonstrates that neither white people nor males appear to have any special relationship with pension as a fraction of earnings.

Though intuitively it should be clear that the reason white males have the higher pension is cause they have a higher average income and work for the longest time. The key concept in this question can be demonstrated by analyzing Famsiz. This variable has a negative coefficient as does the previously mentioned Grades. The concept here is that for these variables the probability of attending a 4 year college goes down when the variable increases and the probability of the lowest rank choice "no college" increases. The rank choices are based on what occurs above below and between certain limit points found under the "limit points" title above. Hence the opposite can be said to be true for positive coefficients, yet it is important to note that the percentage effect cannot simply be read off the table but must be processed through the formula to appear later. Overview The following problem is a ranking problem, where sentiment or how you feel about certain alternative choices comes into play and hence is unobservable. Certain factors that will be seen in the problem affect how we feel about alternative options.

Because the dependent variable is unobservable it is not a regression model but instead called an index model.

For the following question the outcomes y= 4yr college if y*i>µ2

y= 2yr college if µ1<y*i≤µ2

y= no college if y*i≤µ1

Where y*i = βGrades + ei, are important to note

no intercept term is included because it would be exactly collinear with the threshold variables.

We can represent the probabilities of these outcomes if we assume a particular probability distribution for y*i or equivalently for the random error. For the following questions probabilities can be calculated by assuming that the errors have standard normal distribution N(0,1), an assumption that defines the ordered probit model.