8

J.A. Cuesta-Albertos

2e+04

4e+04

6e+04

8e+04

1e+05

400

450

500

550

**Results in Mathematics and per capita GDP**

Per capita GDP

Scores

Qat

Vie

Lie

Figure 1: P.GDP and the results in Mathematics in PISA 2012. Lines are

polynomial estimators of the regression of the scores on the logarithm of the

P.GDP. Liechtenstein, Qatar and Vietnam are denoted as Lie, Qat and Vie

respectively.

Country

P.GDP log

10

(P.GDP)

Country P.GDP log

10

(P.GDP)

1 Qatar

98,814

4.995

. . .

2 Liechtenstein 89,400

4.951

58 Thailand 9,875

3.995

3 Luxembourg 78,670

4.896

59 Albania 9,506

3.998

4 Singapore 64,584

4.810

60 Jordan 6,115

3.786

5 Norway

54,947

4.740

61 Indonesia 5,214

3.717

. . .

62 Vietnam 4,012

3.603

Table 2: Five highest and lowest P.GDP’s and their logarithms of the countries

participating in PISA 2012.

should not damage the score. Even if there exist techniques which allow to

estimate

f

under shape restrictions (some references on those techniques are

Barlow et al., 1972, and Robertson et al., 1988), here, I have decided to skip this

assumption.

I have handled two possibilities for the regression function: Firstly I have

assumed that the function

f

in

(2.2)is a polynomial. Secondly I have made no

assumption on

f

and I have used a so-called non-parametric estimation of

f

.

Both possibilities are analyzed in the next two subsections.

Polynomial estimation.

Some theoretical development on polynomial regression can be found in

Draper & Smith (1998) or Sen & Srivastava (2011).