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Sunday, November 18, 2018
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Statistics for Economics and Business

Master Program: Economics & Enterpreneurship
Credits: 10 ECTS
Start date: 21 September 2015
End date: 08 March 2016
Class calendar: Monday, 12:00 - 14:00
Tuesday, 12:00 - 14:00
Room: 0.05 / 1.05
Language: English

Statistics for Economics and Business - E & E


Aim of the course is to provide students with a broad overview of statistical methods and models which may be exploited to tackle economics and business issues starting from data. Students will learn statistics by doing, exploiting R, a popular open-source software for data analysis. Emphasis on the applications of the techniques and on the interpretation of results will help students to appreciate the relevance of the statistical tools in real life applications.

A basic knowledge of elementary calculus is required. The course will start assuming a previous knowledge of statistics at an undergraduate level: the minimal prerequisite is a undergraduate course in probability and basic statistics.

Lectures and lab sessions.

Written + oral final exam.


(for further details, see the diary of class, weekly updated with the topics covered in the classroom)

Gathering and Exploring Data: data description, graphical and numerical summaries, association, gathering data. Probability distributions and sampling distributions. Inferential statistics: confidence intervals and significance tests  about population central values, for comparing two population central values, about population variances, for comparing two variances. Analzying associations: association between categorical variables and between numerical variables. Statistical models: multiple regression, analysis of variance, analysis of covariance. Nonparametric statistics.

PREREQUISITE (basic requirements)

  • Basic Business Statistics, Thelfth edition. - FIRST SIX CHAPTERS
    Berenson, D.M. Levine, T.C. Krehbiel (2005)
    Prentice Hall.


  • An introduction to Mathematical Statistics and Its Applications, Fifth edition
    Larsen R.J., Marx M.L. (2012)
    Prentice Hall
  • Introduction to Probability and Statistics for Engineers and Scientists, Fifth edition
    Sheldon Ross (2014)
    Academic Press
  • Regression models for categorical and limited dependent variables
    J. Scott Long (2007)
    Sage Pubblications
  • Basic Econonometrics, Fifth edition
    Gujarati D.N., Porter D.C., Gunasekar S. (2015)


  • Discovering Statistics Using R
    Field A., Miles J., Field Z. (2012)
    Sage Pubblications
  • Econometrics by example
  • Gujarati D.N. (2015)

Diary of class

First term

Date Topics
26/09/2016 - morning


Introduction to the class. Basic jargon: population vs sample, parameter vs statistic, exploratory analysis vs inferential analysis. units, variables and their types. Population probability model. Random variable (r.v.), expectation, variability, skewness. The Bernoulli probability model.

26/09/2016 - afternoon
30/09/2016 - morning
30/09/2016 - afternoon

The binomial probability model. The binomial r.v. as sum of n independent Bernoulli r.v.s. Mean, variance and shape of the binomial distribution: the effect of the central limit theorem. The proportion of successes (r.v. relative frequency) as transformation of the binomial r.v. R commands to compute the binomial probabilities. The multinomial r.v.

10/10/2016 LECTURE 3
Other special distributions: the hypergeometric distribution, the Poisson distribution, the geometric distribution, the negative binomial distribution.

Relations among the binomial and the Poisson distributions. Special case of continuous r.v.: uniform distribution, negative exponential distribution and normal distribution. Transforming and combining random variables. Mean and variance of a linear transformation of a r.v. Mean and variance of a combination of random variables. The Normal distribution and the central limit theorem. Approximation of the binomial distribution through the normal law.


The inference problem: parameteter space, sample space and space of the statistic. The problem of estimation. Sufficiency of a statistic. The likelihood function and the method of maximum likelihood (MLE). The likelihood function for the Bernoulli model.


Likelihood and log-likelihood. Relative likelihood. The maximum likelihood estimation and the Fisher information criteria. The log-likelihood function for the Bernoulli model: derivation of the MLE and of the Fisher information criteria.


Likelihood, log-likelihood and Fisher observed information for the normal model: estimation of the mean (variance unknown), estimation of the variance (mean unknown), joint estimation of the two parameters (profile likelihood and conditional likelihood).

  • R commands for normal likelihood and log-likelihood function: estimation of the mean (variance as nuisance parameter)
  • R commands for normal likelihood and log-likelihood function: estimation of the variance (mean as nuisance parameter)
  • R commands for normal likelihood and log-likelihood function: joint estimation of the mean and of the variance

Taylor series. Quadratic approximation of the likelihood function: the role of the maximum likelihood estimate and of Fisher information for regular probabilistic models.

07/10/2016 LECTURE 9
Element of frequentist inference: properties of estimators. Frequentist properties of the likelihood.
08/10/2016 LECTURE 10
The sampling distributions for the main statistics: inference on the mean (case of known and unknown variance), inference on variance, inference on proportion. 
14/11/2016 LECTURE 11
Resampling methods: the bootstrap method for deriving the approximate sampling distribution of a statistic. Point estimate and interval estimate. Confidence interval for the mean (case of known and unknown variance), inference on variance, inference on proportion.
15/11/2016 LECTURE 12
Confidence interval for the variance of a normal population. Bootstrap confidence intervals: the percentile method. Introduction to hypothesis testing: type of hypothesis, the spaces involved, the decision rule, the two type of errors, test statistic.
21/11/2016 LECTURE 13
Hypothesis tests for one sample problems.
22/11/2016 LECTURE 14
Inferences on two samples: comparing means, variances and proportions.

Second term

Lezione del: Argomenti trattati
09/01/2017 LECTURE 15
10/01/2017 LECTURE 16
The simple linear regression model: the problem of estimation.
16/01/2017 LECTURE 17
The simple linear regression model: the problem of inference.
17/01/2017 LECTURE 18
The simple linear regression model: the problem of prediction.
23/01/2017 LECTURE 19
The multiple linear regression model.
24/01/2017 LECTURE 20
The multiple linear regression model: use of a dummy regressor.
30/01/2017 LECTURE 21
The multiple linear regression model: use of a nominal/ordinal regressor.
31/01/2017 LECTURE 22
A regression model with a dummy response variable: the linear probability model.
06/02/2017 LECTURE  23
A regression model with a dummy response variable: the logit and the probit model.
07/02/2017 LECTURE 24
Regression model for nominal response.
13/02/2017 LECTURE  25
Regression model for ordinal response.
14/02/2017 LECTURE 26
Regression model for count data: the Poisson regression model.
20/02/2017 LECTURE 27
Regression model for count data: the negative binomial regression model.
21/02/2017 LECTURE 28

27/02/2017 LECTURE 29

28/02/2017 LECTURE 30

06/03/2017 LECTURE 31

07/03/2017 LECTURE 32



First term

Homework n. 1 (lecture of 26/9) - to do before 04/10
Readings (Berenson & al.):
- study chapter 1, chapter 2, chapter 3, chapter 4
- study the R transcript of R short course (available in the Lectures section)

Homework related to the R class (lectures of 26/9, 27/9, 28/9, 29/9, 30/9, 03/10)
- Read the Introduction to RStudio (Data & Statistical Services, Princeton University)

Homework n. 3  (lecture of 04/10) - due on 05/10
Readings (Berenson & al.):
- Study chapter 5: sections 5.1, 5.2 (not covered during the lecture) and 5.3
- Study Multinomial distribution:
* wikipedia link, link to the online multimedia course (Rice University),
* multinomial coefficient1, multinomial distribution1
1An introduction to Mathematical Statistics and its Application, Larsen and Marx, Prentice Hall

Homework n. 4  (lectures of 10/10 and 11/10) - due on 17/10
- Study chapter 5: sections 5.4, 5.5, 5.6
- Study chapter 6: sections 6.1, 6.2, 6.3 (not covered during the lecture), 6.4, 6.5, 6.6 (partially covered during the lecture)
* geometric distribution1, negative binomial distribution1
1An introduction to Mathematical Statistics and its Application, Larsen and Marx, Prentice Hall

Homework n. 5  (lectures of 17/10 and 18/10) - due on 24/10
- Study chapter 7
* Maximum likelihood estimators1
1Introduction to Probability and Statistics for Engineerings and Scientists, Sheldon Ross, Academic Press

Homework n. 6  (lectures of 24/10 and 25/10) - due on 07/11
- Study (again and in detail) the chapter on Maximum likelihood estimators (see homework #5)
- Study Taylor series: Wikipedia link
* Quadratic approximation of the log-likelihood function1
* Score function and Fisher information1
1Applied Statistics and Inference, Held and Sabanés Bové, Springer

Homework n. 7  (lectures of 07/11 and 08/11) - due on 14/11
- Study chapter 7
- Study chapter 12: section 12.5

Homework n. 8  (lectures of 14/11 and 15/11) - due on 21/11
* Bootstrap methods (not considering the Matlab code)1 - Study chapter 8
- Study chapter 9: section 9.1 1Computational Statistics Handbook with Matlab, Martinez and Martinez, Chapman and Hall/CRC

Homework n. 9  (lectures of 21/11 and 22/11) - to discuss during office hours
- Study chapter 9: sections 9.2, 9.3, 9.4, 9.5 and 9.6 (on-line topic)
- Study chapter 10
- Study chapter 12: section 12.5

Second term

Readings (Gujarati and Porter):
- Chapters 1 and 2

Homework n. 11  (lectures of 16/01 and 17/01) - due on 23/01
Readings (Gujarati and Porter):
- Chapter 3

Homework n. 12   (lectures of 23/01 and 24/01) - due on 30/01
Readings (Gujarati and Porter):
- Chapter 4
- download here the dataset for the exercise: Grade Point Average, using the command:
gpa <- read.table("")

Homework n. 13   (lectures of 30/01 and 31/02) - due on 06/02
Readings (Gujarati and Porter):
- Chapter 5
- download here the dataset for the exercise: Grade Point Average, using the command:
gpa <- read.table("")

Homework n. 14   (lectures of 06/02 and 07/02) - due on 13/02
Readings (Gujarati and Porter):
- Study again chapters 1, 2, 3, 4 and 5
- Study chapter 6: sections 2 and 3

Homework n. 15   (lectures of 13/02 and 14/02) - due on 20/02
Readings (Gujarati and Porter):
- Chapter 7 and 8
- download here the dataset for the exercise: Brand Preference, using the command:
brand <- read.table("", header = TRUE)

Homework n. 16   (lectures of 20/02 and 21/02) - due on 27/2
- Chapter 9

Homework n. 17   (lectures of 27/02 and 28/02) - due on 06/03
- Chapter 15

Homework n. 18   (lectures of 06/03 and 07/03) - to discuss during office hours
- Chapter 15



Office hours

 Office hours

First and second term:
Monday, 3:00 - 5:00 pm
Tuesday, 3:00 - 5:00 pm

III trimestre (and during the week when there is no teaching):
see the schedule weekly announced
(see yellow box "Next office hours")

You can make a date at other times writing to the teacher:

Next office hours

Next office hours 

 Wednesday 28th June, I will meet students at 9:00 - 10:30 a.m.

Final exam - a.a. 2017/18

Results for the exam of 2 October 2017 

Student ID Student Mark Oral timetable
(4-th October)
0047951 Boateng Owsu Ose 16 3:00 p.m.

Dao Khanh Ly

17 3:00 p.m.

Do Hong Duong

17 3:00 p.m.
0006975 Grossi Fabiola Failed ---

The oral exams will be held on October 4-th (room 9.24).

Academic integrity (and cheating policy)

I have absolutely no tolerance for cheating in any form. Students who are caught cheating will be given the strongest possible consequences allowed by the university. Students who cheat and are not caught will be haunted by the memory of their misdeeds for the rest of their miserable lives.

(a famous Professor of Philosophy from Ohio State University)

Midterm exam - a.a. 2016/17

Midterm exam results

 I am sorry but none of the midterm exams were sufficient.
 In your interest, I solved the midterm exam so to show the very basic underlying reasoning. See here the solutions 
I encourage you in doing your weekly homework and above all in focusing on understading what you are doing (do not just mechanically apply the formulas).
 If you think that what you studied was enough to be sufficient, you should work harder and/or differently.

You have time to adequately prepare for the full exam that will be held in March.