OLS Model

OLS MODEL

VARIABLE COEFFICIENT STDERROR T STAT P-VALUE

const 6.78312 0.124651 54.417 <0.00001 ***

EXPER 0.0193983 0.00583211 3.326 0.00178 ***

GENDER 0.235009 0.0699596 3.359 0.00162 ***

RACE 0.0604869 0.0788078 0.768 0.44687

EDUC 0.0544107 0.0168106 3.237 0.00230 ***

Log wage = 6.7 +0.23gender +0.06race +0.054education +0.019 experience

In order to test for the significance of the individual t variables we test for presence of autocorrelation and/or heteroscedasticity.

Using the DW test we test for autocorrelation and the chi square for heteroscedasticity.

Autocorrelation test

DW cal = 1.76

N = 49

K = 4

Since the calculated DW falls in the third region we conclude that there is no autocorrelation.

We test for heteroscedasticity with the obtained chi square test

Ho homoscedasticitry v Ha heteroscedasticity

X^2 calculated = 24.14

X^2 tab, with 4 degrees of freedom = 9.48773at 5%

Since X^2 cal is greater than X^2 tabulated we reject Ho and say heteroscedasticity is present.

T test

variables

coefficients

T cal

T tab

significance

Constant(Bo)

6.78

60.68

1.75305

yes

Gender (B1)

0.23

3.41

1.75305

yes

Race (B2)

0.06

0.83

1.75305

no

Educ (B3)

0.05

2.87

1.75305

yes

Exper (B4)

0.02

3.22

1.75305

yes

Significance and decision making: if the tabulated t is greater than the calculated t we accept Ho and state the variables are not statistically significant. Gender therefore plays a significant part in determining wage paid but not race as shown by their individual t test with gender being significant but not race.

Model test using the F statistics

Ho B0 to Bi = 0

Ha B0 to Bi = 0 where i= 0 to 4

Degrees of freedom is 44

Tabulated F is 2.58367

Calculated F (4, 44) =9.62 at 5% significance level

Since the calculated F is greater than the tabulated we reject Ho and say the model is statistically significant both after and before heteroscedasticity was corrected.

1) The economic significance of the models is derived from the implication of the individual variables or the independent variables on the dependent variable. A positive effect on the independent variable will mean as the variable increases or decreases the dependent variable react in the same way. The logged variables represent percentage changes, while the linear variables represent unit changes. Therefore, in the log-lin model, a unit change in the independent variable leads to a percentage change in the dependent variable.

Log wage = 6.7 +0.23gender +0.06race +0.054education +0.019 experience

According to this model if the level of education increases the wage level also goes up. This makes economical sense as higher education tends to merit higher wages. If education increases by 10units it will cause a proportionate increase in wages by 10*0.054= 0.54 or a 54%. The percentage change also represents the elasticity of change between education and wages. The changes are also rational to some extent until a higher education may not match the same level of change.

Rationally a higher level of experience leads to a higher wage level and that is depicted in the model with a positive relationship. A unit change of 10 in experience of a worker will lead to a proportionate change of 10*0.019= 0.19 or 19% in the wage level. The percentage change is rational in comparison with the educational level.

Researches have proven in the past that men on average earn more than women in the same work force. This is depicted in the regression with a gender change of 1 unit, that moving from being a woman who is 0 to a man who 1 is causes a percentage change of 23% in the wages level.

When we take two people of the same gender, same experience, same educational level but one is a white, the non-white stands at a disadvantage in the wage level because he will fall below to the white by 6%. So in effects white earn more wages to non-whites by 6% according to our regression.

2) OLS

VARIABLE COEFFICIENT STDERROR T STAT P-VALUE

const -0.733751 0.319243 -2.298 0.02544 **

GPA 0.0412225 0.123913 0.333 0.74067

PHY 0.0922320 0.0358289 2.574 0.01282 **

RED 0.0322516 0.0471275 0.684 0.49668

QNT -0.00174720 0.0429991 -0.041 0.96774

GENDER 0.109244 0.113246 0.965 0.33902

VARIABLE

COEFFICIENT

T STAT

T Tabulated

Significance

const

-0.733751

-0.733751

1.75305

NO

GPA

0.0412225

0.0412225

1.75305

NO

GENDER

0.0922320

0.0922320

1.75305

NO

RED

0.109244

0.109244

1.75305

NO

QNT

-0.00174720

-0.00174720

1.75305

NO

PHY

0.0322516

0.0322516

1.75305

NO

The constant represent the chances of every applicant being accepted having all the other variables at zero. Any applicant therefore has a 0.7 chance of not being accepted.

With a range of 1.88 - 2.9, if an individual's undergraduate GPA score increasing by 1 unit, the chances of being accepted increases by 4%.

If an applicant moves from 0 to 1 in this case from female to male, the chances of being accepted increases by 9% if they have the same GPA, age, and score in all the related Med school courses.

From 3 any extra score gained to a maximum of 7 in reading increases the chances of an applicant being accepted by 11% compared to others with the same GPA, age, gender and score in other related Med school courses.

When an applicant has an extra score in quantitative analysis compared to other applicant with the same other criteria, their chances of acceptance reduce by 0.1%. The range is between 2 to 7.

In the range of 3 to 9, the chances of an applicant being accepted rises by 3% if they gain any extra point in physics compared to other applicants with the same scores in the related Med school courses, gender, GPA and age.

Ho vs. Ha

Degrees of freedom

Tabulated F is 2.38607at 5% significance level

Calculated F-statistic (5, 54) = 5.78

Since the calculated F is greater than the tabulated F test we reject the Ho and conclude that the model is significant.

2b) WLS

VARIABLE COEFFICIENT STDERROR T STAT P-VALUE

const -0.358760 1.17886 -0.304 0.76205

g -0.0665258 0.000832013 -79.958 <0.00001 ***

p 0.121170 0.00106394 113.888 <0.00001 ***

r -0.124488 0.00148573 -83.789 <0.00001 ***

q 0.0990150 0.000364414 271.710 <0.00001 ***

g2 -0.0761270 0.000872035 -87.298 <0.00001 ***

VARIABLE

COEFFICIENT

T STAT

T Tabulated

Significance

const

-0.358760

-0.304

1.75305

NO

g

-0.0665258

-79.958

1.75305

YES

p

0.121170

113.888

1.75305

YES

r

-0.124488

-83.789

1.75305

YES

q

-0.0761270

271.710

1.75305

YES

g2

0.0990150

-87.298

1.75305

YES

We use the WLS for the reason that, estimated coefficients are best linear unbiased (BLUE). Regression of y on zi and gives correct standard errors for coefficient estimates.

Having all the other weighted variables at zero, the probability of acceptance of each applicant falls by 36%.

With a range of 1.88 - 2.9, if an individual's weighted undergraduate GPA score increasing by 1 unit, the probability of being decreases by 7%.

If an applicant weighted gender moves from 0 to 1 in this case from female to male, the chances of being accepted increases by 12% if they have the same weighted GPA, age, and score in all the related Med school courses.

From 3 any extra score gained to a maximum of 7 in the weighted reading decreases the chances of an applicant being accepted by 12% compared to others with the same weighted GPA, age, gender and score in other related Med school courses.

When an applicant has an extra score in his or her weighted quantitative analysis compared to other applicant with the same score in other weighted criteria, their chances of acceptance reduce by 0.8%. The range is between 2 to 7.

In the range of 3 to 9, the probability of an applicant being accepted rises by 10% if they gain any extra point in the weighted physics compared to other applicants with the same weighted scores in the related Med school courses, gender, GPA and age.

Ho vs. Ha

Degrees of freedom

Tabulated F is 2.38607at 5% significance level

Calculated F-statistic (5, 54) = 996031

Since the calculated F is greater than the tabulated F test we reject the Ho and conclude that the model is significant.

2c)

Accept = - 8.44 -0.23GPA + 0.56PHY + 0.25RED + 0.1QNT + 0.53 GENDER

VARIABLE

COEFFICIENT

T STAT

T Tabulated

Significance

const

-8.43758

-3.195

1.75305

yes

GPA

0.225994

0.313

1.75305

no

PHY

0.558342

2.422

1.75305

yes

RED

0.246132

0.861

1.75305

no

QNT

0.0952386

0.367

1.75305

no

GENDER

0.527912

0.774

1.75305

no

This suggests that:

If an applicant's GPA' score goes up by 1 mark, then the change in the odds will fall by 23%.

The change in the odds will fall by 56% if the score in Physics rises by one mark.

An increase by 1 mark in reading will result in an increase by 25% in the change in the odds.

And moving from a female applicant to a male one will lead to an increase of 53% in the change in the odds.

Further suggestion is that the probability of acceptance into the medical school given the above data set is equal to 0.53 (1 / [1+exp (+0.12563)] = 0.53).

Impact of an increase in GPA.

GPA of 2 gives a probability of 0.52 obtained by:

Prob(accept) = 1 / 1+ exp(-0.090892 )

And a GPA of 3 gives a probability of 0.46 obtained by:

Prob(accept) = 1 / 1+ exp(-0.14089 )

2d) Probit model

Accept = - 4.97 -0.13GPA + 0.33PHY + 0.15RED + 0.34QNT + 0.04 GENDER

VARIABLE

COEFFICIENT

T STAT

T Tabulated

Significance

const

-4.97392

-3.402

1.75305

yes

GPA

0.127129

0.307

1.75305

no

PHY

0.331395

2.569

1.75305

yes

RED

0.153491

0.906

1.75305

no

QNT

0.338329

0.848

1.75305

no

GENDER

0.0432647

0.286

1.75305

no

By an applicants score in GPA increasing by one mark, an applicant increases his/her probability of being admitted into the medical school by 13%

If we increase an applicant's score in PHY by one mark, then we also increase her probability of acceptance by 33%.

If RED increases by 1 mark, then the probability of acceptance will increase by 15%

One less mark QNT will cause a fall of 0.34% in the probability of admission.

And going from a female applicant to a male applicant will result in a 4 percent increase in the probability of acceptance into the medical school.

Impact of an increase in GPA

If GPA=2 we can estimate the probability of that applicant being accepted as,

Z = - 5.92 - 0.09*2 + 0.16*9.183 + 0.18*8.683 + 0.24*8.966 + 0.07*7.61 - 0.009*7.41 - 0.004*7.25 + 0.0008*23.66 + 0.59*0.483 = 0.011

Using the table of cumulative normal distribution we come up with a probability of 0.5040 which is 50% chance of acceptance.

If GPA=3 we can estimate the probability of that applicant being accepted as,

Z = - 5.92 - 0.09*3 + 0.16*9.183 + 0.18*8.683 + 0.24*8.966 + 0.07*7.61 - 0.009*7.41 - 0.004*7.25 + 0.0008*23.66 + 0.59*0.483 = - 0.0855

Using the table of cumulative normal distribution we come up with a probability of 0.5319 that is a 53% chance of acceptance.

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