# Econometrics

1) The majority of students in the data set are male (69.7%) and are from the UK (69.8%). The large standard deviation in average expenditure on alcohol (Table 1) shows that some people drink a lot whilst others drink nothing, whilst the smaller variance on expenditure on rents shows that there is limited choice in student accommodation.

The statistics point towards the theory that UK students do not work as hard as non-UK students in their first year at University. Whilst UK students spend an average of 3.29 hours per week on statistics, non-UK students spend 4.27hours (Table 4). Non-UK students tend to achieve 1.4 percentage points more in the QT exam (Table 3). The fact that an increase in the number of A-Grades at A-Level or equivalent is more beneficial to UK students, Graph 5, shows that perhaps UK students rely more on natural ability, compared to non-UK students who work hard no matter what their natural ability is.

Looking at Table 1 it is clear that there are some observations that are highly unlikely. For example that one student was aged 1 when they took the QT exam or that one student spent 100 hours revising in the final week for the QT exam; however age and hours spent revising in the final week are not used as variables in my regression, so they will not affect my results.

Table 1 shows that attendance at lectures and classes are high, with attendance at above 80% throughout the year, whilst only 45.7% of revision lectures were attended. Table 5 shows that the majority of people who achieved between 60-80% in their QT exam attended between 40-60% of the revision lectures, whilst 57.5% attended more than 80% of their lectures. The majority of those who scored the highest, attended more lectures, but attended less than 20% of their revision lectures.

Graphs 1, 2 and 3, separating for high and low IQ, show that the affect of attending lectures and working on statistics is a lot greater for low IQ students compared to high IQ students. This can be explained by the fact that high IQ students can rely on natural ability to do well in the exam, whilst low IQ students have to work harder. Correlating IQ with lecture attendance (Table 6), it can be seen that those with a high IQ attend more lectures but less revision lectures, which can help explain the negative correlation between revision lecture attendance and QT Mark.

Sociable students can be seen to benefit slightly less from attending revision lectures, as shown by Graph 4 whereby after 40% attendance, average QT Mark falls for sociable students. Table 3 shows that Economics students do better on average (with an average mark of 66.2%) than students that study EPAIS (62.5%) or Industrial Economics and Economic History (57.7%). However, Graph 7 shows that EPAIS students have a smaller variation in marks, with the lowest mark being higher than that for Economics students.

### 2a)

the average mark (percent) achieved in QT exam when attendance at revision lectures=0.

the change in QT mark (percent) for a percentage point increase in the proportion of revision lectures attended.

If a student attends one more percent of revision lectures, they will gain 0.00223 percentage points in their QT exam. The coefficient,, is not statistically significant at the 1% level, as its p-value is more than 0.01 (p-value=0.900) therefore the hypothesis that , cannot be rejected.

### 2b)

the change in QT mark (percent) for a percentage increase in the proportion of revision lectures attended, holding IQ and average hours per week spent working on statistics during the year constant.

is no longer statistically insignificant, with a p-value of 0.000, therefore the hypothesis that 0 can be rejected. This shows that in equation (1), there were omitted relevant variables, as the exclusion of the variables iq and hrsqt caused a negative bias on the attr coefficient.

The bias of excluding iq is negative, whilst for hrsqt it is positive, but the bias of iq outweighs that of hrsqt (see Table 9). If IQ is not taken into account, attendance at revision lectures shows the effect of attending revision lectures but is also acting as a proxy for the effect of IQ. When including the variable iq, it shows that students with a high IQ gain less from attending revision lectures compared to those with a low IQ.

### 2c)

Where , ,

The default dummy is therefore when hrsqt<=1. To test for evidence of non-linearity in the effect of hrsqt on qtmark, test the joint significance of . If the hypothesis: cannot be rejected, there is no substantial evidence of non-linearity.

Excluding from equation (3), lead to the RSSR=50910.

where d=number of restrictions under the null hypothesis, DoF=degrees of freedom associated with the unrestricted model.[1]

Therefore we cannot reject the hypothesis that there is zero joint significance of the dummy variables. This shows that there is insufficient evidence of non-linearity in the effect of hrsqt on qtmark.

### 3a)

Where

The default dummy is students studying Economics.

§ mark achieved in QT exam when attendance all other variables are 0.

§ By attending 1% more revision lectures, a student would be expected to increase their QT mark by 0.0841 percentage points (holding all other variables constant).

§ A student with an IQ of one point higher would be expected to have 0.409 more percentage points in the QT exam (holding all other variables constant).

§ By spending one more hour per week working on statistics during the year, a student would be expected to gain 0.554 percentage points (holding all other variables constant)/

§ Being female reduces the QT mark by -1.51 percentage points compared to being male (holding all other variables constant).

§ Getting one more A-grade at A-Level or equivalent increases the QT mark by 1.95 percentage points (holding all other variables constant).

§ the change in QT mark (percent) if an EPAIS student compared to an Economics student, holding all other variables constant. (2.89 less percentage points in the QT exam).

§ the change in QT mark (percent) if an Industrial Economics or Economic History student compared to an Economics student, holding all other variables constant. (8.75 less percentage points in the QT exam)

Excluding from equation (4), lead to the RSSR=43480.

Therefore we can reject the null hypothesis that the dummy variables have no joint significance. This means that what degree a student is studying has an impact on their QT mark, with the best degree to study being Economics, and the worst being Economic History or Industrial Economics.

### 3b)

Where a sociable person is someone who spends more than the average £40 per week on alcohol.

additional change in qtmark (percent) for a unit increase in attr for sociable people, compared to unsociable people, holding all else constant.

Test the hypothesis that the additional change in QT mark for sociable people if you increase revision lecture attendance by 1% is insignificant.

, where standard error of We reject if

Cannot reject. This means that although by studying the coefficients it can be seen that the returns to attendance of revision lectures on QT mark are lower for sociable people, the coefficients are not significant, therefore we must reject the claim that missing revision lectures is less of a problem for sociable people.

3c)To test whether rich students, and in particular rich-female students, suffer less from missing revision lectures, add dummy variables to test the change in returns on revision lecture attendance for rich students, and for rich-female students.

A rich student is defined as a student who spends more than the average student on rent, which is more than £54 per week on rent.

additional change in QT mark (percent) for a one percent increase in the proportion of revision lectures attended for students who are both rich and female, compared to poor, male students, holding all other variables constant.

Test the hypothesis that there is significant difference for rich, female students, compared to poor, male students, in additional returns to revision lectures.

Cannot reject that additional gains to attending revision lectures for rich, female students is zero. This means that rich, female students neither need to attend more revision lectures, nor less, than poor, male students.

To test the joint significance of the additional slope dummy variables:

Cannot reject that there is zero joint significance. All the coefficients are insignificant, therefore cannot prove the validity of the argument that it is less of a problem for rich, female students to miss revision lectures.

From previous investigation it is clear that variables on UK, IQ and lectures attendance should be included in my regression. Once variables on IQ and UK were included, some variables became insignificant. For example, initially a variable on age was included, however this became insignificant when other variables, especially IQ, were taken into account. When deciding whether to include a variable in the regression I tested the significance of including the variable. To decide whether to include a variable in the regression, I tested the significance of its inclusion, and if it was less than 5% I would not include it.

The effect of the number of A-Grades at A-Level or equivalent was shown to be different for UK and non-UK students by Graph 6, so including a multiplicative dummy of uk_alevelsa into the model helped to explain how UK students benefited more than non-UK students from having more A-Grades at A-Level in terms of their QT Mark achieved.

No. of observations=309

The RSS was 36010, and when uk_alevelsa was not included, RSSR= 37170.

Therefore can reject the null hypothesis, which means that including this dummy variable helps explain the model.

Including the squares of hrsall, attr and attl separately in the equation, I tested for non-constant returns.

Non-constant returns if

Neither hrsqt12 nor hrsall2 exhibited non-constant marginal returns, however attl2 showed increasing marginal returns, as seen by Graph 8. This means that below a certain threshold, lecture attendance does not improve exam performance, but once you attend above that threshold, increasing lecture attendance increases exam performance.

The data was collected over 3 different years; therefore I included 2 intercept dummy variables to account for the differences in years.

Testing for structural change between males and females (where ), UK and non-UK students, and sociable and non-sociable students, the hypothesis of no structural change could not be rejected at the 5% significance level.

### Investigating the Regression Model

To check for multicollinearity I analysed the correlations between the variables in the regression, and ensured that none had a correlation above 0.85. To check for heteroskedasticity I ran a Breusch-Pagan test by using the Stata command estat hettest, the result was chi2 (1)=0.24, which means the null hypothesis of constant variance could not be rejected. Therefore I did not re-estimate with heteroscedastic robust standard errors. To check for incorrect functional form I ran a RESET test, with the F statistic being 1.48, therefore I could not reject the null hypothesis that there are no omitted variables.

The size of the standard errors show the strength of the variables, if the coefficient estimate is double the size of the standard error, the t-test of zero significance can be rejected. Looking at Table 14 it is clear that IQ has a major effect on QT Mark, whilst being an EPAIS student compared to an Economics student is not as detrimental as being an Economic History or Industrial Economics student.

The model shows that exam performance is determined not only by natural ability (IQ) and motivation (hrsqt), but also by which year the test was taken. This may be because the paper in 2003 was more difficult than the 2002 paper, whilst the 2004 paper was the easiest. It may instead be because the lecturer changed in 2004. The fact that being a non-UK student increases expected marks may be because non-UK students often have a higher opportunity cost of being at University (higher tuition fees) and thus work harder. The fact that the number of A-Grades at A-Level (or equivalent) affects UK students more may be because UK students rely on previous knowledge, not working hard in their first year of University.

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Appendix

All results are stated to 3 significant figures, unless they are to be used in further calculations, in which case they are stated to 4 significant figures.

### Table 1. Summary Statistics.

Variable

Mean(Std. Dev)

Minimum

Maximum

Age

19.4(1.26)

1

24

No. of A-grades (or equivalent) at A-level (or equivalent)

2.66(1.43)

0

17

Proportion of lectures attended (percent)

80.6(21.8)

5

100

Proportion of classes attended (percent)

81.9(22.0)

0

100

Proportion of revision lectures attended (percent)

45.7(39.7)

0

100

Expenditure on alcohol (£ per week)

27.8(24.4)

0

150

Expenditure on rent (£ per week)

53.4(10.4)

10

95

Average hours spent per week working on statistics during the year

3.57(3.49)

0

30

Average hours spent working on statistics during week before exam

14.4(12.7)

0

100

IQ of the student based on a test

121(15.4)

82.6

164

Mark in first year statistics exam (percent)

64.7(13.0)

9

95

### Table 2.

Variable

Description

epais

=1 if EPAIS student, 0 otherwise

indeconhis

=1 if Industrial Economics or Economic History Student, 0 otherwise

sociable

=1 if expenditure on alcohol>=£40, 0 otherwise

rich

=1 if expenditure on rent>=£54, 0 otherwise

hi_iq

=1 if IQ>140, 0 otherwise

### Table 3.

Description

No. of Students

Average QT Mark. (Std. Dev)

economics

266

66.2 (12.8)

indeconhis

40

57.7(15.3)

epais

65

62.5(10.5)

sociable

142

62.7(12.6)

unsociable

229

65.8(13.1)

female

113

63.4(14.0)

male

258

65.1(12.6)

uk

245

64.6(12.8)

non-uk

104

66.0(13.5)

2002

169

64.2(13.8)

2003

80

61.8(13.0)

2004

100

69.0(10.5)

hi_iq

55

71.1(11.2)

not hi_iq

294

63.9(13.0)

Table 4.

Variable

No. of Observations

Mean(Std. Dev)

Hrsqt

349

3.566(3.486)

Hrsqt for UK Students

251

3.291(2.596)

Hrsqt for non-UK Students

98

4.268(5.052)

### Table 5. Table showing the percentage of students in each band of lecture attendance who achieved a certain level of QT Mark.

Proportion of Lectures Attended (Percent)

QT Mark<=20

20<QT Mark<=40

40<QT Mark<=60

60<QT Mark<=80

80<QT Mark<=100

attl<=20

0.00

0.00

50.00

42.86

7.14

20<attl <=40

0.00

12.5

68.75

6.25

12.50

40<attl<=60

2.56

0.00

48.72

35.90

12.82

60<attl<=80

0.00

3.41

46.59

43.18

6.82

80<attl<=100

0.00

1.45

28.02

57.49

13.04

Proportion of Revision Lectures Attended (Percent)

QT Mark<=20

20<QT Mark<=40

40<QT Mark<=60

60<QT Mark<=80

80<QT Mark<=100

attr<=20

0.00

3.20

37.60

43.20

16.00

20<attr <=40

2.33

0.00

27.91

58.14

11.63

40<attr<=60

0.00

3.70

51.85

37.04

7.41

60<attr<=80

0.00

2.33

39.53

53.49

4.65

80<attr<=100

0.00

1.08

33.33

55.91

9.68

### Table 6. Correlation between QT Mark, IQ and Attr, Attl and Hrsqt

Attr

Attl

Hrsqt

QT Mark

-0.0105

0.2309

0.1023

IQ

-0.5762

0.0689

-0.1642

Table 7.

qtmark

Coef.

Std. Err.

t

P>

[95% Conf. Interval]

attr

0.002235

0.01785

0.13

0.900

-0.3289

0.03736

_cons

64.52

1.087

59.34

0.000

62.38

66.66

Table 8.

qtmark

Coef.

Std. Err.

t

P>

[95% Conf. Interval]

attr

0.07802

0.02037

3.83

0.000

0.03796

0.1181

iq

0.3988

0.05210

7.65

0.000

0.2963

0.5013

hrsqt

0.4650

0.1915

2.43

0.016

0.08819

0.8417

_cons

10.82

6.962

1.55

0.121

-2.877

24.51

Table 9.

-353.9

36.63

1576

Bias of iq

-0.08960

Bias of hrsqt

0.01081

### Table 10.

qtmark

Coef.

Std. Err.

t

P>

[95% Conf. Interval]

attr

0.0795

0.02048

3.89

0.000

0.03932

0.1199

iq

0.3983

0.05178

7.69

0.000

0.2965

0.5002

hrsqt_2

0.09614

1.683

0.06

0.954

-3.215

3.407

hrsqt_3

-1.461

1.975

-0.74

0.460

-5.344

2.423

hrsqt_4

3.897

2.150

1.81

0.071

-0.3306

8.125

_cons

12.11

6.813

1.78

0.076

-1.294

25.51

TSS=59780

RSS=49980

Number of observations=351

### Table 11.

qtmark

Coef.

Std. Err.

t

P>

[95% Conf. Interval]

attr

0.08419

0.01975

4.26

0.000

0.04533

0.1230

iq

0.4093

0.04995

8.19

0.000

0.3111

0.5076

hrsqt

0.5535

0.1832

3.02

0.003

0.1932

0.9138

female

-1.515

1.380

-1.10

0.273

-4.240

1.201

alevelsa

1.949

0.4331

4.50

0.000

1.097

2.801

epais

-2.895

1.698

-1.70

0.089

-6.236

0.4460

indeconhis

-8.746

2.062

-4.24

0.000

-12.80

-4.690

_cons

5.676

6.770

0.84

0.402

-7.644

19.00

TSS=55340

RSS=41040

Number of observations=329

### Table 12.

qtmark

Coef.

Std. Err.

t

P>

[95% Conf. Interval]

attr

0.09240

0.02351

3.93

0.000

0.04614

0.1387

iq

0.3957

0.05022

7.88

0.000

0.2968

0.4945

hrsqt

0.5002

0.1836

2.72

0.007

0.1389

0.8615

female

-2.032

1.394

-1.46

0.146

-4.775

0.7099

alevelsa

1.911

0.4323

4.42

0.000

1.060

2.761

epais

-2.928

1.689

-1.73

0.084

-6.252

0.3952

indeconhis

-8.463

2.055

-4.12

0.000

-12.51

-4.421

sociable

-1.044

1.924

-0.54

0.588

-4.830

2.742

attr_sociable

-0.03848

0.03288

-1.17

0.243

-0.1032

0.02621

_cons

8.371

6.963

1.20

0.230

-5.329

22.07

TSS=55340

RSS=40360

Number of observations=329

### Table 13.

qtmark

Coef.

Std. Err.

t

P>

[95% Conf. Interval]

attr

0.09053

0.03007

3.01

0.003

0.03137

0.1497

iq

0.4102

0.05010

8.04

0.000

0.3099

0.5106

hrsqt

0.5614

0.1854

3.03

0.003

0.1967

0.9261

female

-1.490

2.208

-0.67

0.500

-5.835

2.855

alevelsa

1.938

0.4380

4.42

0.000

1.076

2.799

epais

-2.886

1.711

-1.69

0.093

-6.251

0.4794

indeconhis

-8.850

2.095

-4.22

0.000

-12.97

-4.728

rich

0.8982

1.946

0.46

0.645

-2.931

4.728

rich_attr

-0.01320

0.03636

-0.36

0.717

-0.08474

0.05834

female_attr

-0.004710

0.04167

-0.11

0.910

-0.08669

0.07727

rich_female_attr

0.006964

0.04356

0.16

0.873

-0.07873

0.09266

_cons

5.192

6.971

10.74

0.457

-8.523

18.91

TSS=55340

RSS=41010

Number of observations=329

### Table 14.

qtmark

Coef.

Std. Err.

t

P>

[95% Conf. Interval]

iq

0.2976

0.04189

7.10

0.000

0.2151

0.3800

hrsqt

0.4965

0.1853

2.68

0.008

0.1320

0.8611

attl

-0.2894

0.1627

-1.78

0.076

-0.6096

0.03073

attlsq

0.002946

0.001211

2.43

0.016

0.000562

0.005329

alevelsa

1.102

0.5254

2.10

0.037

0.06847

2.136

uk

-10.05

2.818

-3.57

0.000

-15.59

-4.503

epais

-1.938

1.663

-1.17

0.245

-5.210

1.335

indeconhis

-8.838

2.117

-4.18

0.000

-13.00

-4.672

year_03

-2.657

1.636

-1.62

0.105

-5.876

0.5622

year_04

3.610

1.466

2.46

0.014

0.7255

6.494

uk_alevelsa

2.407

0.9271

2.60

0.010

0.5826

4.232

_cons

29.67

7.842

3.78

0.000

14.24

45.11

TSS=52430

RSS=34420

Number of observations=309

### Bibiliography

Dougherty, C., Introduction to Econometrics, 3rd Ed, Oxford: Oxford University Press, 2007

Romer, D., “Do students go to class? Should they?”, Journal of Economic Perspectives, 7(3), 1993, pp.167-74

[1] Christopher Dougherty, Introduction to Econometrics, 3rd Ed, (Oxford University Press: 2007), p.147