This subject aims at providing the theoretical and applied background in statistical inference needed by an economist. The requirements are a quarter covering the rudiments of Probability and Descriptive Statistics, density and probability functions, and characteristic functions, as covered by instance in the previous subject Statistics and Data Analysis. It also requires familiarity with differential and integral calculus, as covered in Mathematics I ant Mathematics II.



This subject is a requirement for all Econometrics courses taught in the School.

1) Use effectively quantitative techniques to interpret economic data, understanding the relationship

between verbal, graphical, mathematical and econometrical analysis in the study of Economics.



2) Use effectively information technologies and computer-based resources, as required in the field

of study.

1. Use of Statistics in economic decisions.



Introduction. Case studies. Data bases.





2. Poisson and binomial distributions.



Bernoulli distribution. Binomial distribution and binomial frequency. Poisson distribution: definition and first properties. Practical considerations. Use of statistical tables. Convergence of the binomial to the Poisson and Normal distributions. Convergence of the Poisson distribution to the Normal.





3. Gamma, Chi-square, F and t distributions.



Gamma distribution. Exponential distribución. Square-normal distribution. Pearson's chi-square. Snedecor's F distribution. Student's t distribution.







4. Parameter estimation. Properties of estimators.



Introduction. Random sample and statistic. Parameter estimation. Point estimation. Maximum likelihood estimator. Moment estimator. Unbiased estimators. Regular estimators. The Cramer-Rao bound. Efficiency. Convergence in probability and mean square. Consistency.





5. Hypothesis testing. Goodness-of-fit tests.



Statistical hypothesis testing. Design of statistical tests. Likelihood ratio test. Neyman-Pearson's theorem. Chi-square goodness-of-fit test to a totally or partially specified distribution. Independence and homogeneity tests. Interval estimation and hypothesis tests: mean, difference of means, variance, ratio of variances, lambda parameter in a Poisson distribution, binomial probability, difference of binomial probabilities.





6. Finite populations sampling.



Introduction. Simple random sampling. Stratified sampling. Conglomerate sampling. Two-stage conglomerate sampling. Sample units selection: the practice of random sampling.

Methodology may change across groups, on account of the different numbers of students enrolled. Both expository and practical sessions are held. Problems are handed for homework, after seminars demonstrating the necessary techniques.

• Final Assessment System
• Tools and qualification percentages:
  • Multiple-Choice Test (%): 15
  • Realization of Practical Work (exercises, cases or problems) (%): 15
  • Final exam (%): 70

a) Continuous evaluation



Continuous evaluation includes the realization of a mid-term exam. Depending on the number of registered students, some practical problems introduced by seminars may be proposed. All continuously assessed work (mid term and practical problems, if proposed) accounts for up to 30% of the final grade, with the final exam making up for the rest.



If, for instance, a student obtains 1,9 points in activities and tests along the course, his or her final exam will be worth the remaining 8,1 points. A grade in the final of, for instance, 80% of the maximum will earn him or her 8,1 x 80/100 = 6,48 points, which added to the previous 1,9 give a course grade of 8,38 points.



In the most unfavourable case, an student following the continuous evaluation path and earning 0 points would be in no worse situation than someone deciding from the very start to go to the final exam only.



Points obtained in continuous evaluation are valid throughout the academic year (i.e., ordinary and extraordinary exams, usually in May and July). They are no longer valid in subsequent years.



b) Evaluation only in a final exam



Students may choose to renounce continuous evaluation and have their grade based exclusively on the final exam. In order to do so, they just have not to sit at any of the exams or mini-exams scheduled throughout the course.



It should be understood, however, that students taking only the final may be required to answer questions about any topic covered in the course, with no exceptions whatsoever.



Students that choose not to take the final will have a grade of “no presentado”, irrespective of any use they may have made of continuous evaluation.



During the exams, students are allowed to use a pocket calculator and a handwriten summary of

formulae or notes whose extension does not exceed one DIN A4 page (both sides). The use of phones or tablets (even as

replacements for a pocket calculator) will not be allowed. Exams may take place at a computer room, where the use of the computers available in such room may be compulsory.



Exceptions can be made on account of the type of exams. If that is the case, it will be announced in advance.



We contemplate presential exams; however, should pandemic-like situations happen, we might set up non-presential exams administered through e-Gela.

Students failing to obtain a passing grade, have a second chance. The same grading policy as in the first chance prevails.

In exams and problem solving, both in class and out of class, students will need a pocket calculator and/or a computer running suitable software. Such software can be:

1) R, which can be obtained for free from:
http://cran.es.r-project.org/

2) Gretl, similarly downloadable from:
     http://gretl.sourceforge.net/

Both are installed in most computer rooms available to students. Familiarity
with such software may be required to answer questions, both in the course of continuous evaluation and in the final exams.

Basic bibliography

The following manuals essentially cover the syllabus, at the required level:



MARTIN-PLIEGO, F.J. and RUIZ-MAYA, L. (2004). Estadística I: Probabilidad, 2a Edición. AC, Madrid.



RUIZ-MAYA, F.J. and MARTIN-PLIEGO, L.(2002) Fundamentos de Inferencia Estadística, 3ª ed., Editorial AC, Madrid.



PEÑA, D. (2001) Fundamentos de Estadística, Alianza Editorial, Madrid.



In non-Spanish languages, students might want to check:





BAIN, L. and ENGELHARDT, M. (1992). Introduction to Probability and Mathematical Statistics, Second edition. Duxbury Press, Boston.



GASPERONI, IEVA, PAGANONI (2020) Eserciziario di Statistica Inferenziale, Springer Verlag Italia, Milano.



GILLARD (2020) A First Course in Statistical Inference, Springer Nature, Switzerland.



HOGG, TANISS, ZIMMERMAN (2013) Probability and Statistical Inference, 9th. edition, Pearson, Boston.



ROSS, S. (2001). Probability and Statistics for Engineers and Scientists. Academic Press, London.



ROUSAS, G.G. (2015) An Introduction to Probability and Statistical Inference, Academic Press/Elsevier, Amsterdam

In-depth bibliography

Books of problems and exercises include among many others:

ARTECHE, J. et AL.(2000) Ejercicios de Estadística II: Estadística Empresarial y para Economistas, Servicio Editorial de la UPV-EHU, Bilbao.

FERNANDEZ AGIRRE, K. et al. (1996) Estatistika I eta Estatisitka II ariketak. Probabilitate Teoria eta Inferenzia Estatistikoa, UEU, Bilbo.

GARIN, A. and TUSELL, F. (1990) Ejercicios de Probabilidad e Inferencia Estadística, Ed. Tébar Flores, Madrid.

MARTIN PLIEGO, F.J. et al. (2005) Problemas de Inferencia Estadística, 3ª ed., Paraninfo, Madrid.

MARTIN PLIEGO, F.J. et al. (2006) Problemas de Probabilidad, 3ª ed., Paraninfo, Madrid.

Journals

At the introductory level of this course there is no need to have direct recourse to the specialized journals (of which, nonetheless, the students may find an ample selection at the library, both in hard copy and available on-line).

Regarding statistical data for Spain, students may find useful the electronic journal "Índice". On applications of Statistics to a number of fields, students may want to look at the journal "Chance".

Web addresses

EUSTAT: http://www.eustat.es
INE: http://www.ine.es
"Indice": http://www.revistaindice.com
"Chance": http://www.springer.com/mathematics/probability/journal/144
Open Data Euskadi: http://opendata.euskadi.net

An alternative to statistical tables is given by

https://play.google.com/store/apps/details?id=com.mbognar.probdist&hl=es

(There are versions for iOS as well as Android.) During the exams, however, the use of all kinds of wireless devices may be prohibited.

Other resources are published in Moodle.