Preface
Why would you be interested in reading this book?
The university where I work is an affiliate center of the University of London (UoL). The UoL programs are known for their high level and strict exams. One of the notoriously difficult subjects is Econometrics. It is based on the book by Dougherty, C., Introduction to Econometrics, Oxford: Oxford University Press, 4th ed., 2011.
For past exams for the last year just Google ``uol past exam papers'' or go to the University of London site.
To say that the book is difficult would not be enough. Look at Boxes 2.2, 2.3 in the book (it's two pages of algebra) and at Problem A1(d) in the UoL exam of 2012, Zone B. To solve that problem, you would need to understand the calculations and ideas of the proofs in the boxes (which are developed for the homoscedastic case) and apply them for the heteroscedastic case suggested in the exam problem. A creative solution like that requires dexterity in Math that is above and beyond just familiarity with algebra. In particular, by the time you enroll in Econometrics you have to learn to handle algebraic derivations that are one or more pages long.
This requirement is not satisfied by two prerequisites: the first year Statistics, which is based on AP Statistics books (AP Statistics books are used only by international programs of UoL), and second year Statistics, which is based on the London School of Economics (LSE) Stats guides and the book by Newbold, P., Carlson, W.L., and Thorne, B., Statistics for Business and Economics, 9th ed., Pearson, 2020. From this point on I call the last book NCT, by the initials of the authors. While the LSE Stats guides are fairly high level, the main textbooks (AP Statistics and NCT) do not prepare the students to the complexities of Econometrics. This is the gap that the present manual intends to close.
In Stats, only 5 to 10% of the material can be explained without Math, and don't believe anybody who tells you otherwise. Once you start skipping algebra, instead of an orderly construction you get a pile of unrelated facts, and then rote memory is your only rescue. All introductory Stats textbooks have this drawback, including the following
- J. T. McClave, P. G. Benson and T. Sincich. Statistics for Business and Economics. Pearson, 12th edition, 2012.
- D. R. Anderson, D. J. Sweeney, T. A. Williams. Statistics for Business and Economics. Cengage Learning, 12th edition, 2013.
- D. Lind, W. Marchal, S. Wathen. Basic Statistics for Business and Economics. McGraw-Hill/Irwin, 8th edition, 2012.
Not only do they avoid the underlying mathematics but they also on many occasions do not provide correct definitions.
There are several textbooks on the market as rigorous as this. This book is different from all others in that it focuses on facts from Mathematical Statistics that are necessary in Econometrics. The material has been carefully selected taking into account the true constraints at UoL programs.
What is good and bad about NCT
All references are to the seventh edition of NCT.
On the good side, the book has a lot of intuitive explanations and plenty of exercises, which is especially handy for instructors. The coverage of statistical topics is wide, so that most instructors will not need to look for other sources. Statistics requires a lot of math. The authors have done their best to hide the mathematical complexities and make the book accessible to those who are content using formulas without having to derive them.
This last advantage, looked at from a different angle, becomes a major drawback. If you have to remember just five formulas, after applying each once or twice perhaps you will memorize them. But how about 100 formulas? This is an approximate number of equations and verbal definitions (which eventually translate to equations) contained in the first six chapters of NCT. Some formulas are given more than once, in different forms and contexts. For example, different types of means are defined in several different sections, and it takes time and effort to relate them to one another. For your information, all types of means are special cases of just one, population mean. The origin of most equations is not explained. When most formulas fall out of the blue sky, after a while you get lost. When it is necessary to systematize testing procedures scattered over several sections or chapters, the situation becomes even more complex. Look at Figure 10.5 that describes the decision rule for choosing the appropriate hypothesis test when comparing two population means. If you are comfortable using it and all the relevant formulas without understanding the links between them, then you don't need this textbook.
Now suppose that you are an ordinary human, who cannot mechanically memorize a bunch of formulas, like me and my students, who feel completely lost after the first six chapters if I follow the book's approach. Suppose you want to see a harmonious, coherent picture instead of a jigsaw puzzle of unrelated facts. Then I am going to try to convince you that studying formulas with proofs and derivations is the best option.
Why study math?
Firstly, it's not that much. You don't need to dig all the math usually given in the same level math textbooks. Just study what I give (which is directly related to the course), and then you'll see that all properties of means make great sense and fit just two pages, with all the intuition, special cases and derivations. The same goes for variances, covariances, confidence intervals etc.
Secondly, knowing the logic (the links between theoretical facts) significantly reduces the burden on your memory. For example, just being familiar with the general principle of linearity will allow you to deduce all properties of covariances and variances in the logical chain
$latex \textrm{means\rightarrow \textrm{covariances \rightarrow \textrm{variances$
from the properties of the first link (means). This is especially important in case of confidence intervals (and there are dozens of them) all of which are based on a few general ideas. My students love my handouts because instead of reading 30-40 pages in NCT they have to read just 4-7 pages for each chapter.
Thirdly, the knowledge cemented by logic stays in memory much longer than mechanically memorized facts. Being able to derive equations you studied in the beginning of the semester will give you confidence when applying them during the final exam.
Fourthly, because in our brain everything is linked, it has a wonderful property that can be called an icebreaker principle: as you develop one particular capability, many other brain activities enjoy a positive spillover effect. The logic and imagination you develop while studying math will give you a comparative advantage in all your current and future activities that require analytical abilities. This influence on your general skills and aptitude will be even more important than the knowledge of statistics as a subject.
Fifthly, the high school approach of studying formulas through their applications is not going to work in statistics. At high school, you had to study a small number of different applications and had enough time to solve many exercises for each application type. In statistics, the number of different typical applications is so large that you will not have time to solve more than two exercises for each.
Sixthly, some students avoid math remembering their dreadful experience struggling with it at high school. Their fear of math continues well into college years, and they think: If I failed to learn math at school, how can I learn it just in one semester (or year) at a university? I have a soothing and encouraging answer for such students. Most likely, you fell victim to a wrong teaching methodology, and I would not rely on your grades at high school to judge your math abilities. Some of your misunderstandings and problems with math may have come from your early age (when you saw for the first time, say, fractions or algebra rules). At that time your cognitive abilities were underdeveloped. Now you are an adult and, if you are reading this, you are a completely normal person with adequate cognitive abilities. You can comprehend everything much faster than when you were at elementary school. It's like with a human embryo, which in 9 months passes all the stages of evolution that took the mammals ages to become who they are now. There are a couple of rules to follow though. Forget your fears, rethink all the math rules you had problems with and look for common sense in everything you see. Relying on logic instead of memory is the best rule -- this is why I prefer to teach people with work experience. Unlike fresh high-school graduates, they don't take anything for granted and are sincere about their lack of math background.
The seventh and the most important reason has been overlooked by all those who recommend to sidestep all formula derivation and concentrate on mastering formulas through their applications. See, some of us are endowed with good logic and imagination from birth. For such people, indeed, it is enough to see a formula and a model exercise for its application to be able to apply the formula on their own in a slightly different situation. But most of us mortals are not that bright. I've seen many people who understand well the statement of the exercise and know all the required theory and yet do not see the solution. In this case there is only one verdict: such individual' s logic and imagination are not good enough. Explaining the solution to him/her usually doesn't help because that doesn't improve their logic and imagination and in a similar situation they will be lost again.
Logic and imagination are among the most advanced functions of our brain. Most people think that you either possess them or you don't and it is not possible to develop them. I can tell you a big secret that this is wrong, at least in math. Study mathematical facts and theories with the pertaining logic, and with time your own logic will start working (this is how your humble servant has become a mathematician). More about how to do this will be said later. For now let me state the main recommendation.
Some of the exercises in NCT are pretty tricky and their diversity is amazing. Don't even hope to learn the solutions of a dozen model exercises and then be able to solve the rest of them. Your sure bet is to develop your own logic and imagination, and the best way to achieve this purpose is to study mathematical facts with proofs and derivations.
OK, suppose I have convinced you that studying math is worthwhile and you are ready to give it a try. Then why not combine reading NCT with reading one of the extant mathematical texts? In what way is this manuscript different or better than the pile on the market? Well, this manuscript is not just another math text. It's the teaching philosophy that makes the difference. I talk about the learning strategy almost as much as about the subject itself. The exposition and study units are designed to develop your self-learning skills. There are comments on psychological aspects of the study process that both students and teachers will find useful.
Exposition
This manual is by no means a replacement for NCT. On my part it would be crazy trying to come up with as many application exercises as NCT has. You can use this text to study the theory but for application exercises you will have to use NCT. If I leave out some theoretical material, it means that it is not essential.
The order of definitions and statements in NCT does not reflect their logical sequence. Sometimes logically interrelated facts are scattered over different sections or even chapters in NCT. I do the opposite: the material is organized in study units, which combine logically linked information. This made necessary to give some material earlier or later than in NCT. Therefore placement of a study unit in a certain chapter does not correspond to that in NCT. I want my students to be able to answer questions of certain difficulty. This desired difficulty level determines the sizes of study units. The name "exercise" is used for those facts which should be obvious, at least after you study their solutions.
Italic indicates a new term even though the word ''definition'' may not be there.
Best ways to study
First best way. Study a theoretical problem, the existing approaches to its solution, try to improve or generalize upon them and write a publishable article. This way is not accessible to most students.
Second best way. As you study, write your own book or manual, in your own words and the way you understand it. This way is as rewarding as it is time-consuming (but quite accessible to some students). When I was 32, I wrote a book with my major professor M. Otelbaev. In one year I learned more than in the previous three years.
Third best way. After reading some material, close the book and write down the main points explaining the accompanying logic. The luckiest of you may have a classmate to retell the material to (I am a stickler of team-based learning). This way is quite feasible for most students. I call it active recalling, as opposed to passive repetition, when after reading the material you just skim it to see what you remember and what you don't. This way may be time-consuming initially but after a while you'll get used and your speed will increase. Many things you did not understand before will become simple. Try to work in the evenings when you are so tired that your memory refuses to function but your logic still works.
In the end the 1980's I had to give an intensive course in functional analysis. I gave 10 hours of lectures a week, 6 hours in a row on one day and 4 hours on another. During lectures I didn't use any notes. As this all was high level stuff, the course was very useful for me (and a disaster for students).
Secret to success. Actively recall the material, in increasingly large chunks (not necessarily limited to my study units) and at an increasingly high speed.
If you don't do this, I don't guarantee anything. In my classes, I make sure that students actively repeat the material by arranging competitions among student teams and practicing quizzes of increasing difficulty.
Explanation. Nobody knows exactly what creative imagination means, in terms of neural activity. Let's talk about internal vision by which I mean the ability to see a large and complex picture at once. Internal vision can be compared to a ray of a flashlight. If the ray of your flashlight is narrow and the picture is big, you can see only a part of it. The wider the ray, the more of the picture you see. When you see the whole picture, you can solve the problem.
Active recalling of increasingly large amounts of material forces the brain to adjust and gradually increase internal vision.
Increasing the speed of recalling (or writing) is important for two reasons. In addition to widening internal vision, it also strengthens the degree of excitation of neurons. Your thoughts become clearer and the speed of formation of new synapses (intra-neuron links) increases. The second reason is that high recalling speed reduces the degree of verbalization of the thinking process.
In most cases, in math too much bla-bla-bla is harmful!
Chapter 1. How to Study Statistics?
Unit 1.1. How to Study Statistics?
About this chapter. Most of the material of the first two chapters of Newbold, Carlson, and Thorne (NCT) can be safely skipped, not only because it is unimportant but primarily because our brain tries to systematize the incoming information, and there is very little systematic content in the first two chapters. Also, some topics from Chapters 1-2 are better discussed later, when you have the necessary tools at hand. I use this space to discuss some issues deeper than in the book and at the level you will need later. High-school material or the stuff you are supposed to remember from math prerequisites is reviewed. Solve the relevant exercises in NCT only when you find my explanations insufficient.
Math consists of definitions, axioms and statements. Definitions are simply names of objects. They don't require proofs. Axioms (also called postulates) are statements that we take for granted; they don't require proofs and are in the very basement of a theory. Statements have to be proved. Depending on the situation and the tastes of the researcher, they may have different other names: theorem, lemma, proposition, criterion etc. Statements are a way to economize on space and thinking efforts. They summarize sometimes very long arguments and serve as building bricks in subsequent constructions.
In math texts definitions take up at most 5 to 10% of space. A common error of students of quantitative subjects is to jump to statements without thinking enough about definitions. (When I was young I also committed this error.) Definitions not only give names to objects but they also give direction to the theory and reflect the researcher's point of view. Often understanding definitions well allows you to guess or even prove some results.
Next: Studying a definition
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