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Master's degree

A master's degree (from Latin magister) is a postgraduate academic degree awarded by universities or colleges upon completion of a course of study demonstrating mastery or a high-order overview of a specific field of study or area of professional practice. A master's degree normally requires previous study at the bachelor's level, either as a separate degree or as part of an integrated course. Within the area studied, master's graduates are expected to possess advanced knowledge of a specialized body of theoretical and applied topics; high order skills in analysis, critical evaluation, or professional application; and the ability to solve complex problems and think rigorously and independently.

Medieval era to 18th century
The master's degree dates back to the origin of European universities, with a Papal bull of 1233 decreeing that anyone admitted to the mastership in the University of Toulouse should be allowed to teach freely in any other university. The original meaning of the master's degree was thus that someone who had been admitted to the rank (degree) of master (i.e. teacher) in one university should be admitted to the same rank in other universities. This gradually became formalised as the licentia docendī (licence to teach). Originally, masters and doctors were not distinguished, but by the 15th century it had become customary in the English universities to refer to the teachers in the lower faculties (arts and grammar) as masters and those in the higher faculties as doctors. Initially, the Bachelor of Arts (BA) was awarded for the study of the trivium and the Master of Arts (MA) for the study of the quadrivium.

From the late Middle Ages until the 19th century, the pattern of degrees was therefore to have a bachelor's and master's degree in the lower faculties and to have bachelor's and doctorates in the higher faculties. In the United States, the first master's degrees (Magister Artium, or Master of Arts) were awarded at Harvard University soon after its foundation. In Scotland, the pre-Reformation universities (St Andrews, Glasgow, and Aberdeen) developed so that the Scottish MA became their first degree, while in Oxford, Cambridge and Trinity College Dublin, the MA was awarded to BA graduates of a certain standing without further examination from the late 17th century, its main purpose being to confer full membership of the university. At Harvard the 1700 regulations required that candidates for the master's degree had to pass a public examination, but by 1835 this was awarded Oxbridge-style three years after the BA.

19th century
The 19th century saw a great expansion in the variety of master's degrees offered. At the start of the century, the only master's degree was the MA, and this was normally awarded without any further study or examination. The Master in Surgery degree was introduced by the University of Glasgow in 1815. By 1861 this had been adopted throughout Scotland as well as by Cambridge and Durham in England and the University of Dublin in Ireland. When the Philadelphia College of Surgeons was established in 1870, it too conferred the Master of Surgery, "the same as that in Europe".

In Scotland, Edinburgh maintained separate BA and MA degrees until the mid-19th century, although there were major doubts as to the quality of the Scottish degrees of this period. In 1832 Lord Brougham, the Lord Chancellor and an alumnus of the University of Edinburgh, told the House of Lords that "In England the Universities conferred degrees after a considerable period of residence, after much labour performed, and if they were not in all respects so rigorous as the statutes of the Universities required, nevertheless it could not be said, that Masters of Arts were created at Oxford and Cambridge as they were in Scotland, without any residence, or without some kind of examination. In Scotland, all the statutes of the Universities which enforced conditions on the grant of degrees were a dead letter."

It 1837, separate examinations were reintroduced for the MA in England, at the newly established Durham University (even though, as in the ancient English universities, this was to confer full membership), to be followed in 1840 by the similarly new University of London, which was only empowered by its charter to grant degrees by examination. However, by the middle of the century the MA as an examined second degree was again under threat, with Durham moving to awarding it automatically to those who gained honours in the BA in 1857, along the lines of the Oxbridge MA, and Edinburgh following the other Scottish universities in awarding the MA as its first degree, in place of the BA, from 1858. At the same time, new universities were being established around the then British Empire along the lines of London, including examinations for the MA: the University of Sydney in Australia and the Queen's University of Ireland in 1850, and the Universities of Bombay (now the University of Mumbai), Madras and Calcutta in India in 1857.

In the US, the revival of master's degrees as an examined qualification began in 1856 at the University of North Carolina, followed by the University of Michigan in 1859, although the idea of a master's degree as an earned second degree was not well established until the 1870s, alongside the PhD as the terminal degree. Sometimes it was possible to earn an MA either by examination or by seniority in the same institution; for example, in Michigan the "in course" MA was introduced in 1848 and was last awarded in 1882, while the "on examination" MA was introduced in 1859.

Probably the most important master's degree introduced in the 19th century was the Master of Science (MS in the US, MSc in the UK). At the University of Michigan this was introduced in two forms in 1858: "in course", first awarded in 1859, and "on examination", first awarded in 1862. The "in course" MS was last awarded in 1876. In Britain, however, the degree took a while longer to arrive. When London introduced its Faculty of Sciences in 1858, the university was granted a new charter giving it the power "to confer the several Degrees of Bachelor, Master, and Doctor, in Arts, Laws, Science, Medicine, Music", but the degrees it awarded in science were the Bachelor of Science and the Doctor of Science. The same two degrees, again omitting the master's, were awarded at Edinburgh, despite the MA being the standard undergraduate degree for Arts in Scotland. In 1862, a royal commission suggested that Durham should award master's degrees in theology and science (with the suggested abbreviations MT and MS, contrary to later British practice of using MTh or MTheol and MSc for these degrees), but its recommendations were not enacted. In 1877, Oxford introduced the Master of Natural Science, along with the Bachelor of Natural Science, to stand alongside the MA and BA degrees and be awarded to students who took their degrees in the honours school of natural sciences. In 1879 a statute to actually establish the faculty of Natural Sciences at Oxford was promulgated, but in 1880 a proposal to rename the degree as a Master of Science was rejected along with a proposal to grant Masters of Natural Sciences a Master of Arts degree, in order to make them full members of the university. This scheme would appear to have then been quietly dropped, with Oxford going on to award BAs and MAs in science.

The Master of Science (MSc) degree was finally introduced in Britain in 1878 at Durham, followed by the new Victoria University in 1881. At the Victoria University both the MA and MSc followed the lead of Durham's MA in requiring a further examination for those with an ordinary bachelor's degree but not for those with honours.

20th century
At the start of the 20th century, there were four different sorts of master's degree in the UK: the Scottish MA, granted as a first degree; the Master of Arts (Oxbridge and Dublin), granted to all BA graduates a certain period after their first degree without further study; master's degrees that could be gained either by further study or by gaining an honours degree (which, at the time in the UK involved further study beyond the ordinary degree, as it still does in Scotland and some Commonwealth countries); and master's degrees that could only be obtained by further study (including all London master's degrees). In 1903, the London Daily News criticised the practice of Oxford and Cambridge, calling their MAs "the most stupendous of academic frauds" and "bogus degrees". Ensuing correspondence pointed out that "A Scotch M.A., at the most, is only the equivalent of an English B.A." and called for common standards for degrees, while defenders of the ancient universities said that "the Cambridge M.A. does not pretend to be a reward of learning" and that "it is rather absurd to describe one of their degrees as a bogus one because other modern Universities grant the same degree for different reasons".

In 1900, Dartmouth College introduced the Master of Commercial Science (MCS), first awarded in 1902. This was the first master's degree in business, the forerunner of the modern MBA. The idea quickly crossed the Atlantic, with Manchester establishing a Faculty of Commerce, awarding Bachelor and Master of Commerce degrees, in 1903. Over the first half of the century the automatic master's degrees for honours graduates vanished as honours degrees became the standard undergraduate qualification in the UK. In the 1960s, new Scottish universities (except for Dundee, which inherited the undergraduate MA from St Andrews) reintroduced the BA as their undergraduate degree in arts, restoring the MA to its position as a postgraduate qualification. Oxford and Cambridge retained their MAs, but renamed many of their postgraduate bachelor's degrees in the higher faculties as master's degrees, e.g. the Cambridge LLB became the LLM in 1982, and the Oxford BLitt, BPhil (except in philosophy) and BSc became the MLitt, MPhil and MSc.

In 1983, the Engineering Council issued a "'Statement on enhanced and extended undergraduate engineering degree courses", proposing the establishment of a four-year first degree (Master of Engineering). These were up and running by the mid-1980s and were followed in the early 1990s by the MPhys for physicists and since then integrated master's degrees in other sciences such as MChem, MMath, and MGeol, and in some institutions general or specific MSci (Master in Science) and MArts (Master in Arts) degrees. This development was noted by the Dearing Report into UK Higher Education in 1997, which called for the establishment of a national framework of qualifications and identified five different routes to master's degrees: This led to the establishment of the Quality Assurance Agency, which was charged with drawing up the framework.
 * Four year (five in Scotland) first degrees such as the MEng
 * Conversion degrees, sometimes below the standard of undergraduate degrees in the same subject
 * The honours arts degree of the ancient universities of Scotland
 * Specialist postgraduate programmes, such as the MA and MSc
 * The Oxbridge MA, awarded without additional work

21st century
In 2000, renewed pressure was put on Oxbridge MAs in the UK Parliament, with Labour MP Jackie Lawrence introducing an early day motion calling for them to be scrapped and telling the Times Higher Education it was a "discriminatory practice" and that it "devalues and undermines the efforts of students at other universities". The following month the Quality Assurance Agency announced the results of a survey of 150 major employers showing nearly two thirds mistakenly thought the Cambridge MA was a postgraduate qualification and just over half made the same error regarding the Edinburgh MA, with QAA chief executive John Randall calling the Oxbridge MA "misleading and anachronistic".

The QAA released the first "framework for higher education qualifications in England, Wales and Northern Ireland" in January 2001. This specified learning outcomes for M-level (master's) degrees and advised that the title "Master" should only be used for qualifications that met those learning outcomes in full. It addressed many of the Dearing Report's concerns, specifying that shorter courses at H-level (honours), e.g. conversion courses, should be styled Graduate Diploma or Graduate Certificate rather than as master's degrees, but confirmed that the extended undergraduate degrees were master's degrees, saying that "Some Masters degrees in science and engineering are awarded after extended undergraduate programmes that last, typically, a year longer than Honours degree programmes". It also addressed the Oxbridge MA issue, noting that "the MAs granted by the Universities of Oxford and Cambridge are not academic qualifications". The first "framework for qualifications of Higher Education Institutes in Scotland", also published in January 2001, used the same qualifications descriptors, adding in credit values that specified that a stand-alone master should be 180 credits and a "Masters (following an integrated programme from undergraduate to Masters level study)" should be 600 credits with a minimum of 120 at M-level. It was specified that the title "Master" should only be used for qualifications that met the learning outcomes and credit definitions, although it was noted that "A small number of universities in Scotland have a long tradition of labelling certain first degrees as 'MA'. Reports of Agency reviews of such provision will relate to undergraduate benchmarks and will make it clear that the title reflects Scottish custom and practice, and that any positive judgement on standards should not be taken as implying that the outcomes of the programme were at postgraduate level."

The Bologna declaration in 1999 started the Bologna Process, leading to the creation of the European Higher Education Area (EHEA). This established a three-cycle bachelor's—master's—doctorate classification of degrees, leading to the adoption of master's degrees across the continent, often replacing older long-cycle qualifications such as the Magister (arts), Diplom (sciences) and state registration (professional) awards in Germany. As the process continued, descriptors were introduced for all three levels in 2004, and ECTS credit guidelines were developed. This led to questions as to the status of the integrated master's degrees and one-year master's degrees in the UK. However, the Framework for Higher Education Qualifications in England, Wales and Northern Ireland and the Framework for Qualifications of Higher Education Institutes in Scotland have both been aligned with the overarching framework for the EHEA with these being accepted as masters-level qualifications.

Titles
Master's degrees are commonly titled using the form 'Master of ...', where either a faculty (typically Arts or Science) or a field (Engineering, Physics, Chemistry, Business Administration, etc.) is specified. The two most common titles of master's degrees are the Master of Arts (MA/M.A./A.M) and Master of Science (MSc/M.Sc./M.S./S.M.) degrees, which normally consist of a mixture of research and taught material.

The title of Master of Philosophy (MPhil) indicates (in the same manner as Doctor of Philosophy) an extended degree with a large research component. Other generically named master's programs include the Master of Studies (MSt)/Master of Advanced Study (MASt)/Master of Advanced Studies (M.A.S.), and Professional Master's (MProf). Integrated master's degrees and postgraduate master's degrees oriented towards professional practice are often more specifically named for their field of study ("tagged degrees"), including, for example, Master of Business Administration, Master of Divinity, Master of Engineering, Master of Physics, and Master of Public Health.

The form "Master in ..." is also sometimes used, particularly where a faculty title is used for an integrated master's degree in addition to its use in a traditional postgraduate master's degree, e.g. Master in Science (MSci) and Master in Arts (MArts). This form is also sometimes used with other integrated master's degrees and occasionally for postgraduate master's degrees (e.g. Master's in Accounting). Some universities use Latin degree names; because of the flexibility of syntax in Latin, the Master of Arts and Master of Science degrees may be known in these institutions as Magister artium and Magister scientiæ or reversed from the English order to Artium magister and Scientiæ magister. Examples of the reversed usage include Harvard University and the University of Chicago, leading to the abbreviations A.M. and S.M. for these degrees. The forms "Master of Science" and "Master in Science" are indistinguishable in Latin.

In the UK, full stops (periods) are not commonly used in degree abbreviations. In the US, The Gregg Reference Manual recommends placing periods in degrees (e.g. B.S., Ph.D.), while The Chicago Manual of Style recommends writing degrees without periods (e.g. BS, PhD).

Master of Science is generally abbreviated M.S. or MS in countries following United States usage and MSc or M.Sc. in countries following British usage, where MS would refer to the degree of Master of Surgery. In Australia, some extended master's degrees use the title "doctor": Juris doctor and Doctors of Medical Practice, Physiotherapy, Dentistry, Optometry and Veterinary Practice. Despite their titles these are still master's degree and may not be referred to as doctoral degrees, nor may graduates use the title "doctor".

Types

 * Postgraduate/graduate master's degrees (MA/M.A./A.M., MPhil/M.Phil., MSc/M.S./SM, MBA/M.B.A., LLM/LL.M., MAsc, etc.) are the traditional formal form of master's degree, where the student already holds an undergraduate (bachelor's) degree on entry. Courses normally last one year in the UK and two years in the US.
 * Integrated master's degrees (MChem, MEng, MMath, MPharm, MPhys, MPsych, MSci, etc.) are UK degrees that combine an undergraduate bachelor's degree course with an extra year at master's level (i.e. a total of four years in England, Wales and Northern Ireland, and five years in Scotland). A 2011 survey of UK Higher Education Institutes found that 64% offered integrated master's course, mostly in STEM disciplines, with the most common degrees being MEng, MSci and MChem. 82% of respondents conferred only a master's degree for the course, while 9% conferred a bachelor's degree at the end of the bachelor's-level stage and a master's degree at the end of the course and a further 9% conferred both bachelor's and master's degrees at the end of the course.
 * Non-master's level master's degrees The ancient universities of the UK and Ireland have traditionally awarded MAs in a different manner to that usual today. The Scottish MA is a bachelor's-level qualification offered by the ancient universities of Scotland. The Oxbridge MA is not an academic qualification; it is granted without further examination to those who have gained a BA from Oxford or Cambridge Universities in England, and the MA of Trinity College Dublin in Ireland is granted to its graduates in a similar manner.

The UK Quality Assurance Agency defines three categories of master's degrees:

 * Research master's degrees are primarily research based, although may contain taught elements, particularly on research methods. Examples are the MLitt (usually, but not always a research degree), the Master's by Research, and the MPhil. The Master's by Research (MbyRes, ResM), which is a research degree, is distinct from the Master of Research (MRes), which is a taught degree concentrating on research methods.
 * Specialised or advanced study master's degrees are primarily taught degrees, although commonly at least a third of the course is devoted to a research project assessed by dissertation. These may be stand-alone master's courses, leading to, e.g., MSc, MA or MRes degrees, or integrated master's degrees.
 * Professional or practice master's degrees (see also professional degree) are designed to prepare students for a particular professional career and are primarily taught, although they may include work placements and independent study projects. Some may require professional experience for entry. Examples include MBA, MDiv, LLM and MSW as well as some integrated master's degrees. The name of the degree normally includes the subject name.

The United States Department of Education classifies master's degrees as research or professional. Research master's degrees in the US (e.g., M.A./A.M. or M.S.) require the completion of taught courses and examinations in a major and one or more minor subjects, as well as (normally) a research thesis. Professional master's degrees may be structured like research master's (e.g., M.E./M.Eng.) or may concentrate on a specific discipline (e.g., M.B.A.) and often substitute a project for the thesis.

The Australian Qualifications Framework classifies master's degrees as research, coursework or extended. Research master's degrees typically take one to two years, and at least two-thirds of their content consists of research, research training and independent study. Coursework master's degrees typically also last one to two years, and consist mainly of structured learning with some independent research and project work or practice-related learning. Extended master's degrees typically take three to four years and contain significant practice-related learning that must be developed in collaboration with relevant professional, statutory or regulatory bodies.

In Ireland, master's degrees may be either Taught or Research. Taught master's degrees are normally one to two year courses, rated at 60 - 120 ECTS credits, while research master's degrees are normally two year courses, either rated at 120 ECTS credits or not credit rated.

Structure
There is a range of pathways to the degree with entry based on evidence of a capacity to undertake higher level studies in a proposed field. A dissertation may or may not be required depending on the program. In general, structure and duration of a program of study leading to a master's degree will differ by country and university.

Duration
Master's programs in the US and Canada are normally two years (full-time) in length. In some fields/programs, work on a doctorate begins immediately after the bachelor's degree, but a master's degree may be granted along the way as an intermediate qualification if the student petitions for it. Some universities offer evening options so that students can work during the day and earn a master's degree in the evenings.

In the UK, postgraduate master's degrees typically take one to two years full-time or two to four years part-time. Master's degrees may be classified as either "research" or "taught", with taught degrees (those where research makes up less than half of the volume of work) being further subdivided into "specialist or advanced study" or "professional or practice". Taught degrees (of both forms) typically take a full calendar year (180 UK credits, compared to 120 for an academic year), while research degrees are not typically credit rated but may take up to two years to complete. An MPhil normally takes two calendar years (360 credits). An integrated master's degree (which is always a taught degree) combines a bachelor's degree course with an additional year of study (120 credits) at master's level for a four (England, Wales and Northern Ireland) or five (Scotland) academic year total period.

In Australia, master's degrees vary from one year for a "research" or "coursework" master's following on from an Australian honours degree in a related field, with an extra six months if following on straight from an ordinary bachelor's degree and another extra six months if following on from a degree in a different field, to four years for an "extended" master's degree. At some Australian universities, the master's degree may take up to two years.

In the Overarching Framework of Qualifications for the European Higher Education Area defined as part of the Bologna process, a "second cycle" (i.e. master's degree) programme is typically 90–120 ECTS credits, with a minimum requirement of at least 60 ECTS credits at second-cycle level. The definition of ECTS credits is that "60 ECTS credits are allocated to the learning outcomes and associated workload of a full-time academic year or its equivalent", thus European master's degrees should last for between one calendar year and two academic years, with at least one academic year of study at master's level. The Framework for Higher Education Qualification (FHEQ) in England Wales and Northern Ireland level 7 qualifications and the Framework for Qualification of Higher Education Institutes in Scotland (FQHEIS) level 11 qualifications (postgraduate and integrated master's degrees, except for MAs from the ancient universities of Scotland and Oxbridge MAs) have been certified as meeting this requirement.

Irish master's degrees are one to two years (60–120 ECTS credits) for taught degrees and two years (not credit rated) for taught and research degrees. These have also been certified as compatible with the FQ-EHEA.

Admission
Admission to a master's degree normally requires successful completion of study at bachelor's degree level either (for postgraduate degrees) as a stand-alone degree or (for integrated degrees) as part of an integrated scheme of study. In countries where the bachelor's degree with honours is the standard undergraduate degree, this is often the normal entry qualification. In addition, students will normally have to write a personal statement and, in the arts and humanities, will often have to submit a portfolio of work.

In the UK, students will normally need to have a 2:1. Students may also have to provide evidence of their ability to successfully pursue a postgraduate degree to be accepted into a taught master's course, and possibly higher for a research master's. Graduate schools in the US similarly require strong undergraduate performance, and may require students to take one or more standardised tests, such as the GRE, GMAT or LSAT.

Comparable European degrees
In some European countries, a magister is a first degree and may be considered equivalent to a modern (standardized) master's degree (e.g., the German, Austrian and Polish university Diplom/Magister, or the similar five-year Diploma awarded in several subjects in Greek, Spanish, Portuguese, and other universities and polytechnics).

Under the Bologna Process, countries in the European Higher Education Area (EHEA) are moving to a three-cycle (bachelor's - master's - doctorate) system of degrees. Two-thirds of EHEA countries have standardised on 120 ECTS credits for their second-cycle (master's) degrees, but 90 ECTS credits is the main form in Cyprus, Ireland and Scotland and 60-75 credits in Montenegro, Serbia and Spain. The combined length of the first and second cycle varies from "3 + 1" years (240 ECTS credits), through "3 + 2" or "4 + 1" years (300 ECTS credits), to "4 + 2" years (360 ECTS credits). As of 2015, 31 EHEA countries have integrated programmes that combine the first and second cycle and lead to a second-cycle qualification (e.g. the UK integrated master's degree), particularly in STEM subjects and subjects allied to medicine. These typically have a duration of 300 – 360 ECTS credits (five to six years), with the integrated master's degrees in England, Wales and Northern Ireland being the shortest at 240 ECTS credits (four years). A French diplôme d'ingénieur (postgraduate degree in engineering of grandes écoles) is also the equivalent of a master's degree, provided the diploma is recognised by the Commission des titres d'ingénieur, as are qualifications recognised at Level 7 of the répertoire national des certifications professionnelles (national register of professional certificates).
 * In Denmark there are two forms of master's degree. The Master's Degree or candidatus is a FQ-EHEA second-cycle qualification worth 120 ECTS credits. These degrees are research-based and offered through universities (e.g. University of Copenhagen and Copenhagen Business School). The second form is the Master Degree (no possessive) within the adult continuing education system, which is worth 60 ECTS credits and is taught part-time. The candidatus degree is abbreviated cand. and upon completion of, for instance, an engineering master's degree, a person becomes cand.polyt. (polytechnical). Similar abbreviations, inspired by Latin, apply to a large number of fields, e.g.: sociology (cand.scient.soc), economics (cand.merc., cand.polit. or cand.oecon), law (cand.jur), humanities (cand.mag) etc. Use of a cand. title requires a master's degree. Holders of a cand. degree are also entitled to use M.Sc. or M.A. titles, depending on the field of study. In Finland and Sweden, the title of kandidaatti/kandidat (abbreviated kand.) equates to a bachelor's degree.
 * In France, the master's degree (diplôme de master) takes two years and is worth 120 ECTS credits. The two years are often called the master 1 (M1) and master 2 (M2), following the Bologna Process. Depending on the goal of the student (a doctorate or a professional career) the master can also be called a "Master Recherche" (research master) or a "Master Professionnel" (professional master), each with different requirements.
 * In Italy the master's degree is equivalent to the two-year Laurea magistrale, which can be earned after a Laurea (a three-year undergraduate degree, equivalent to a bachelor's degree). In particular fields, namely law, pharmacy and medicine, this distinction is not made. University courses are therefore single and last five to six years, after which the master's degree is awarded (in this case referred to as Laurea magistrale a ciclo unico). The old Laurea degree (Vecchio Ordinamento, Old Regulations), which was the only awarded in Italy before the Bologna process, is equivalent to the current Laurea Magistrale.
 * In the Netherlands the titles ingenieur (ir.), meester (mr.) and doctorandus (drs.) may be rendered, if obtained in the Netherlands from a university, after the application of the Bologna process, as: MSc instead of ir., LLM instead of mr. and MA or MSc instead of drs. This is because a single program that led to these degree was in effect before 2002, which comprised the same course load as the bachelor and master programs put together. Those who had already started the program could, upon completing it, bear the appropriate title (MSc, LLM or MA), but alternatively still use the old-style title (ir., mr. or drs.), corresponding to their field of study. Since these graduates do not have a separate bachelor's degree (which is – in retrospect – incorporated into the program), the master's degree is their first academic degree. Bearers of foreign master's degree can use the titles ir., mr. and drs. only after obtaining a permission to bear such titles from the Dienst Uitvoering Onderwijs. Those who received their mr., ir. or drs. title after the application of the Bologna process have the option of signing as A. Jansen, MA or A. Jansen, MSc, depending on the field in which the degree was obtained, since the ir., mr. and drs. titles are similar to a master's degree, and the shortcut MA or MSc. may officially be used in order to render such title as an international title.
 * In Belgium, the higher education system is governed by the communities and have separately implemented the Bologna Process. In Flanders, a master's degree can be obtained when completing a two-year study at university (120 ECTS), although there also exist some master's studies which only require 1 year of education (60 ECTS). This discrepancy has various reasons, but the main cause is the transition from the pre-Bologna Process degrees. Degrees like industrial engineering were previously not obtained through university and only took one year, but got transferred to universities, hence why the master's degree for industrial engineering only takes one year at university currently. Similar situations arose for other one-year degrees obtained before 2013 and are therefore now converted to master's degrees through a special certificate. Other master's degrees, such as the degrees from the faculty of Arts and Literature are historically only one year, just like they were before the Bologna Process ratification. Master's degrees for medicine take 3 years to be completed. The situation for the French Community of Belgium is similar in many ways, but has implemented it differently.

Also, the five-year diploma (δίπλωμα) awarded in all Polytechnics (schools of engineering) and the Athens School of Fine Arts is considered equal to a graduate degree plus a master's degree.
 * In Switzerland, the old Licence or Diplom (4 to 5 years in duration) is considered equivalent to the master's degree.
 * In Slovenia and Croatia, during the pre-Bologna process education, all academic degrees were awarded after a minimum of four years of university studies and a successful defence of a written thesis and are considered equivalent to the master's degree. After the completion of that first cycle of the pre-Bologna higher education, the students obtained professional degrees with the titles of Professor (abbreviation "prof.") for educational studies, Engineer (abbreviation "ing.") for technical studies, or Licensed professional of their field of expertise (abbreviation "dipl." with a reference to the profession) for other studies. The title of Magister Scientiae (abbreviation "mr. sc.") was awarded to students who completed a postgraduate university programme (and therefore qualified for a doctorate programme), while the title of Scientiae Doctor (abbreviation "dr. sc.") was awarded to students who completed a postgraduate doctoral programme. Slovenia is a full member of the Bologna Process since 1999 and Croatia since 2001.
 * In Baltic countries there is a two-year education program that offers a chance to gain a master's degree in interdisciplinary issues. The system offers an education in different areas, such as humanities, environmental and social issues, whilst paying specific consideration to the Baltic Sea area. It is a joint-degree program, which is part of a team effort with four universities. There is for example the University of Tartu in Estonia, Vytautas Magnus University in Lithuania and the University of Latvia. The educational programmes allow students to be mobile within the system, for example one semester may be taken in a confederate school without paying additional membership or tuition fees. Subsequently, after passing the qualifications provided, people may procure teaching qualifications and continue their scholastic research around doctoral studies, or carry on studying within their career in the private or public sector. Graduates of the program, within the Baltic Sea area are also given the chance to continue onwards with their studies within the postgraduate system if they have studied the social sciences or humanities field.
 * In Greece, the metaptychiako (μεταπτυχιακό) which literally translates as post-degree (...programme or title), lasts normally from one to, more often, two years, and can be studied after a, at least, four-years undergraduate ptychio, which means degree.
 * In Russia master (магистр) degree can be obtained after a two-year master course (магистратура) which is available after a four-year bachelor or a five-year specialist course. A graduate may choose a master course completely different from his/her previous one. During these two years master students attend specialized lectures in chosen profile, choose a faculty advisor and prepare their master thesis which is eventually defended before certifying commission consisting mostly of professors, leading by the professor from another university.
 * In the United Kingdom, first degrees in medicine, dentistry and veterinary science are considered equivalent to master's degrees despite, for historical reasons, often having the titles of bachelor's degrees.
 * The old Spanish degrees of Licenciado (Licenciate), Arquitecto (Architect) and Ingeniero (Engineer) are also equivalent to master's degrees. They were integrated programmes of study that combined first and second cycles and led to a second cycle qualification. The Spanish government issued a royal decree in 2014 establishing the official equivalences between the Spanish pre-Bologna degrees and the European Qualifications Framework (EQF) levels. Most (if not all) Licenciado, Arquitecto and Ingeniero degrees were placed in level 7 (Master) of the EQF. These programmes have been phased out and replaced with the new Bologna programmes of Máster, to be completed after completion of a programme of Grado (Bachelor's).

Brazil
After acquiring a Bachelor's, Technologist or Licenciate Degree, students are qualified to continue their academic career through Master's Degree ("mestrado", in Portuguese, a.k.a. stricto sensu post-graduation) or Specialization Degree ("especialização", in Portuguese, a.k.a. lato sensu post-graduation) programs. At the Master's program there are 2–3 years of graduate-level studies. Usually focused on academic research, the Master's Degree requires, on any specific knowledge area, the development of a thesis to be presented and defended before a board of professors after the period of research. Conversely, the Specialization Degree, also comprehends a 1–2 years studies, but does not require a new thesis to be proposed and defended, being usually attended by professionals looking for complementary training on a specific area of their knowledge.

In addition, many Brazilian universities offer an MBA program. However, those are not the equivalent to a United States MBA degree, as it does not formally certify the student with a Master's degree (stricto sensu) but with a Specialization Degree (lato sensu) instead. A regular post-graduation course has to comply with a minimum of 360 class-hours, while a M.B.A. degree has to comply with a minimum of 400 class-hours. Master's degree (stricto sensu) does not require a set minimum of class-hours, but it is practically impossible to finish it in less than 18 months due to the workload and research required; an average time for the degree is 2.5 years. Specialization (lato sensu) and M.B.A. degrees can be also offered as distance education courses, while the master's degree (stricto-sensu) requires physical attendance. In Brazil, the degree often serves as additional qualification for those seeking to differentiate themselves in the job market, or for those who want to pursue a Ph.D. It corresponds to the European (Bologna Process) 2nd Cycle or the North American master's.

Hong Kong
M.Arch., M.L.A., M.U.D., M.A., M.Sc., M.Soc.Sc., M.S.W., M.Eng., LL.M.
 * Hong Kong requires one or two years of full-time coursework to achieve a master's degree.

For part-time study, two or three years of study are normally required to achieve a postgraduate degree.

M.Phil.
 * As in the United Kingdom, M.Phil/MPhil/Master of Philosophy is the most advanced master's degree and usually includes both a taught portion and a research portion which requires candidates to complete an extensive original research for their thesis. Regardless of subject, students in all faculties (including sciences, arts, humanities and social sciences) may be awarded the Master of Philosophy.

Pakistan
In Pakistani education system, there are two different master's degree programmes. Master’s degrees are earned after having received a bachelor’s pass degree and one year after the honours degree. The master's program generally lasts for two years.
 * 2 years master's programmes: these are mostly Master of Arts (MA) leading to MPhil
 * 4 years master's programmes: these are mostly Master of Science (MS) leading to PhD

Both MA and MS are offered in all major subjects.

India
In the Indian system, a master's degree is a postgraduate degree following a Bachelor's degree and preceding a Doctorate, usually requiring two years to complete. The available degrees include but are not limited to the following:
 * Master of Arts (M.A.)
 * Master of Social Work (MSW)
 * Master of Business Administration (M.B.A.)
 * Master of Engineering (M.Eng./M.E.)
 * Master of Philosophy (M.Phil.)
 * Master of Science (M.Sc.)
 * Master of Science in Information Technology (M.Sc. IT)
 * Master of Technology (M.Tech.)
 * Master of Statistics (M.Stat.)
 * Master of Laws (LL.M.)
 * Master of Commerce (M.Com.)
 * Master of Architecture (M.Arch.)
 * Master of Veterinary Science (MVSc)

Indonesia
In the Indonesian higher education system, a master's degree (magister) is a postgraduate degree following a Bachelor's degree, preceding a Doctorate and requiring a maximum of four years to complete. Master's degree students are required to submit their thesis (tesis) for examination by two or three examiners. The available degrees include but are not limited to the following:
 * Magister Administrasi Bisnis (M.A.B.) (Master of Business Administration)
 * Magister Manajemen (M.M.) (Master of Management)
 * Magister Sains (M.Si.) (Master of Science)
 * Magister Teknik (M.T.) (Master of Engineering)
 * Magister Hukum (M.H.) (Master of Laws)
 * Magister Pendidikan (M.Pd.) (Master of Education)
 * Magister Olahraga (M.Or.) (Master of Sport Science)

Israel
Postgraduate studies in Israel require the completion of a bachelor's degree and is dependent upon this title's grades; see Education in Israel. Degrees awarded are the M.A., M.Sc., M.B.A. and LLM; the Technion awards a non-thesis M.Eng. There also exists "a direct track" doctorate degree, which lasts four to five years. Taking this route, students prepare a preliminary research paper during their first year, after which they must pass an exam before being allowed to proceed, at which point they are awarded a master's degree.

Nepal
In Nepal, after completing a bachelor's degree, students must spend at least three or four years studying full-time in college and university, with an entrance test for those who wish to pursue master's, Ph.D., and doctorate degrees. All doctoral and Ph.D. degrees, as well as third cycle degrees, are research and experience oriented, with a focus on results.

After completing a successful bachelor's degree, students pursue master's degrees in engineering, education, and arts, as well as all law and medicine-related courses.

M.B.B.S. is only a medical degree with six and a half years of study resulting in a medical doctor and must finish its study in four years after master's degree with minimum education of 15 or 16 years of university bachelor's degree education.

The following are the most professional and internationalized programs in Nepal:
 * Master of Business Administration (M.B.A.)
 * Master of Computer Applications (M.C.A.)
 * Master of Engineering (M.Eng.)
 * Master of Science (M.Sc.)
 * Master of Science in Information Technology (M.Sc.I.T.)
 * Master of Business Studies (M.B.S.)
 * Master of Education (M.Ed.)
 * Master of Arts (M.A.)
 * Master of Science in Agriculture (M.Sc.Ag.)
 * Master of Laws (LL.M.)
 * Master of Management (M.M.)

Taiwan
In Taiwan, bachelor's degree courses are about four years in length, while an entrance examination is required for people who want to study for master's degrees and doctorates. The courses leading to these higher degrees are normally research-based.

Tuition is less expensive than would be the case in North America, costing as little as US$5000 for an M.B.A. As an incentive designed to increase foreign student numbers, the government and universities of Taiwan have redoubled their efforts to make a range of high-quality scholarships available in the form of university-specific scholarships that include tuition waivers of up to NT$20,000 per month. The government offers the Taiwan Scholarship amounting to NT$20,000–30,000 per month (US$654–981) for a two-year program.

title: Master's degree


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Poetry analysis

Poetry analysis is the process of investigating the form of a poem, content, structural semiotics, and history in an informed way, with the aim of heightening one's own and others' understanding and appreciation of the work.

The words poem and poetry derive from the Greek poiēma (to make) and poieo (to create). One might think of a poem as, in the words of William Carlos Williams, a "machine made of words." A reader analyzing a poem is akin to a mechanic taking apart a machine in order to figure out how it works.

There are many different reasons to analyze poetry. A teacher might analyze a poem in order to gain a more conscious understanding of how the poem achieves its effects, in order to communicate this to their students. A writer learning the craft of poetry might use the tools of poetry analysis to expand and strengthen their own mastery. A reader might use the tools and techniques of poetry analysis in order to discern all that the work has to offer, and thereby gain a fuller, more rewarding appreciation of the poem. Finally, the full context of the poem might be analyzed in order to shed further light on the text, looking at such aspects as the author's biography and declared intentions, as well as the historical and geographical contexts of the text (though Formalism would deny any significant analytical value for context).

Meter and Rhyme
One can derive pleasure from two of the most fundamental tools in the poet's toolbox—meter and rhyme—without necessarily knowing a lot of terminology. Consider, for example, the first stanza of Byron's "The Destruction of Sennacherib":

 * The Assyrian came down like the wolf on the fold,
 * And his cohorts were gleaming in purple and gold;
 * And the sheen of their spears was like stars on the sea,
 * When the blue wave rolls nightly on deep Galilee.

Byron's use of meter and rhyme is especially evident and rewarding when one reads the lines out loud. The lines have a powerful, rolling, and very evident rhythm, and they rhyme in a way that is impossible to ignore. In other words, the physicality of the language—how it sounds and feels—accounts for a large measure of the poem's effect. The poem does not have a deep, hidden, symbolic meaning. Rather, it is simply pleasurable to read, say, and hear.

Critical terminology becomes useful when one attempts to account for why the language is pleasurable, and how Byron achieved this effect. The lines are not simply rhythmic: the rhythm is regular within a line, and is the same for each line. A poem having a regular rhythm (not all poems do) is said to follow a certain meter. In "The Destruction of Sennacherib," each line has the basic pattern of two unstressed syllables followed by a third stressed syllable, with this basic pattern being repeated four times in a line. Those basic patterns are called feet, and this particular pattern (weak weak STRONG) is called an anapest. A line with four feet is said to be in tetrameter (tetra-, from the Greek for four). Therefore, this poem is written in anapestic tetrameter. (This process of analyzing a poem's rhythms is called scansion.) The poem also rhymes (not all poems do), and the rhymes follow a pattern (they do not have to). In this case, the rhymes come right next to each other, which emphasizes them, and therefore emphasizes the sound, the physical nature, of the language. The effect of the poem's language derives in part from Byron's choice of an appropriate pattern of rhyme (or rhyme scheme): these adjacent, rhyming lines are called couplets. The sound, the physical nature, of the language is also emphasized by alliteration, as in the repetition of s sounds in the third line: "And the sheen of their spears was like stars on the sea".

Poetic forms
Poems can have many forms. Some forms are strictly defined, with required line counts and rhyming patterns, such as the sonnet (mostly made of a 14-line poem with a defined rhyme scheme) or limerick (usually a 5-line free rhyme poem with an AABBA rhyme scheme). Such poems exhibit closed form, meaning they have strict rules regarding their structure and length. Others (which exhibit open form) have less structure or, indeed, almost no apparent structure at all. This appearance, though, is deceptive: successful open form poems are informed throughout by organic structure which may resist formal description but is nonetheless a crucial element of the poem's effect on the reading mind.

Closed forms
A poet writing in closed form follows a specific pattern, a specific design. Some designs have proven so durable and so suited to the English language that they survive for centuries and are renewed with each generation of poets (sonnets, sestinas, limericks, and so forth), while others come into being for the expression of one poem and are then set aside (Frost's "Stopping by Woods on a Snowy Evening" is a good example).

Of all closed forms in English prosody, none has demonstrated greater durability and range of expression than blank verse, which is verse that follows a regular meter but does not rhyme. In English, iambic pentameter is by far the most frequently employed meter. Among the many exemplary works of blank verse in English are Milton's Paradise Lost and most of the verse passages from Shakespeare's plays, such as this portion of a famous soliloquy from Hamlet:

 * To be, or not to be—that is the question.
 * Whether 'tis nobler in the mind to suffer
 * The slings and arrows of outrageous fortune,
 * Or to take arms against a sea of troubles
 * And by opposing end them. To die, to sleep —
 * No more, and by a sleep to say we end
 * The heartache and the thousand natural shocks
 * That flesh is heir to. 'Tis a consummation
 * Devoutly to be wished. To die, to sleep,
 * To sleep—perchance to dream. Aye, there's the rub.

Note that Shakespeare does not rigidly follow a pattern of five iambs per line. Rather, most lines have five strong syllables, and most are preceded by a weak syllable. The meter provides a rhythm that informs the line: it is not an invariable formula.

Rhymed pairs of iambic pentameter lines form the heroic couplet. Two masters of the form are Alexander Pope and John Dryden. The form has proven especially suited to conveying wit and sardonic humor, as in the opening of Pope's An Essay on Criticism.

 * ’Tis hard to say, if greater want of skill
 * Appear in writing or in judging ill;
 * But, of the two, less dang’rous is th’ offence,
 * To tire our patience, than mislead our sense.
 * Some few in that, but numbers err in this,
 * Ten censure wrong for one who writes amiss;
 * A fool might once himself alone expose,
 * Now one in verse makes many more in prose.

Fourteen lines of iambic pentameter arranged in a more elaborate rhyme scheme form a sonnet. There are two major variants. The form originated in Italy, and the word derives from "sonetto", which is Italian for "little song". The Italian sonnet or Petrarchan sonnet follows a rhyme scheme of ABBA ABBA CDE CDE, ABBA ABBA CD CD CD, ABBA ABBA CCE DDE, or ABBA ABBA CDD CEE. In each of these, a group of eight lines (the octave) is followed by a group of six (the sextet). Typically, the octave introduces a situation, idea, or problem to which the sestet provides a response or resolution. For example, consider Henry Wadsworth Longfellow's "The Sound of the Sea":

 * The sea awoke at midnight from its sleep,
 * And round the pebbly beaches far and wide
 * I heard the first wave of the rising tide
 * Rush onward with uninterrupted sweep;
 * A voice out of the silence of the deep,
 * A sound mysteriously multiplied
 * As of a cataract from the mountain's side,
 * Or roar of winds upon a wooded steep.
 * So comes to us at times, from the unknown
 * And inaccessible solitudes of being,
 * The rushing of the sea-tides of the soul;
 * And inspirations, that we deem our own,
 * Are some divine foreshadowing and foreseeing
 * Of things beyond our reason or control.

The octave presents the speaker's experience of the sound of the sea, coming to him from some distance. In the sestet, this experience mutates into a meditation on the nature of inspiration and man's connection to creation and his experience of the numinous.

English has (proportionally) far fewer rhyming words than Italian. Recognizing this, Shakespeare adapted the sonnet form to English by creating an alternate rhyme scheme: ABAB CDCD EFEF GG. The poet using this, the English sonnet or Shakespearean sonnet form, may use the fourteen lines as single unit of thought (as in "The Silken Tent" above), or treat the groups of four rhyming lines (the quatrains) as organizational units, as in Shakespeare's Sonnet 73:

 * That time of year thou mayst in me behold
 * When yellow leaves, or none, or few do hang
 * Upon those boughs which shake against the cold
 * Bare ruined choirs where late the sweet birds sang.
 * In me thou seest the twilight of such day
 * As after sunset fadeth in the west,
 * Which by and by black night doth steal away,
 * Death's second self, which seals up all in rest.
 * In me thou seest the glowing of such fire
 * That on the ashes of his youth doth lie,
 * As the deathbed whereon it must expire,
 * Consumed with that which it was nourished by.
 * This thou perceiv'st, which makes thy love more strong,
 * To love that well which thou must leave ere long.

In lines 1–4, the speaker compares his time of life to autumn. In lines 5–8, the comparison is to twilight; in lines 9–12, the comparison is to the last moments of a dying fire. Each quatrain presents a shorter unit of time, creating a sense of time accelerating toward an inevitable end, the death implied in the final couplet.

At the "high end" of closed forms are the sestina and villanelle. At the "low end" are forms such as the limerick, which follows a metrical pattern of two lines of anapestic trimeter (three anapests per line), followed by two lines of anapestic dimeter (two anapests per line), followed by one line of anapestic trimeter. (The beginning of the metrical foot does not have to coincide with the beginning of the line.) Any poem following this metrical pattern would generally be considered a limerick, however most also follow an AABBA rhyme scheme. Most limericks are humorous, and many are ribald, or outright obscene (possible rhymes that could follow an opening like "There once was a man from Nantucket" are left as an exercise for the reader). Nonetheless, the form is capable of sophisticated and playful expression:

 * Titian was mixing rose madder.
 * His model posed nude on a ladder.
 * Her position to Titian
 * Suggested coition
 * So he nipped up the ladder and had her.

Open forms
In contrast, a poet using free verse (sometimes called "open form") seeks to find fresh and uniquely appropriate forms for each poem, letting the structure grow out of the poem's subject matter or inspiration. A common perception is that open form is easier and less rigorous than closed form (Frost likened it to "playing tennis with the net down", but such is not necessarily the case (skeptics should try playing tennis without a net): success with the open form requires great sensitivity to language and a particular type of adaptable understanding. In the best open form poems, the poet achieves something that is inaccessible through closed form. As X. J. Kennedy has said, "Should the poet succeed, then the discovered arrangement will seem exactly right for what the poem is saying" (582). The metre of ‘classical’ poetry is replaced in open verse by cadence in rhythm, line indentation, with pauses implied by the syntax, thus the limiting factor of one human breath was naturally incorporated in the poetry, essential to an oral art form, composed to be read aloud.

Walt Whitman was an important innovator of open form, and he demonstrates its merits in "A Noiseless Patient Spider."

 * A noiseless patient spider,
 * I marked where on a little promontory it stood isolated,
 * Marked how to explore the vacant vast surrounding,
 * It launched forth filament, filament, filament, out of itself,
 * Ever unreeling them, ever tirelessly speeding them.

 * And you O my soul where you stand,
 * Surrounded, detached, in measureless oceans of space,
 * Ceaselessly musing, venturing, throwing, seeking the spheres to connect them,
 * Till the bridge you will need to be formed, till the ductile anchor hold,
 * Till the gossamer thread you fling catch somewhere, O my soul.

The long, rolling lines—unified, held together like strong cords, by alliteration and assonance—partake of the same nature as the spider's filaments and the soul's threads. Two balanced stanzas, one describing a spider, the other the speaker's soul, perfectly frame the implicit comparison, with neither being privileged over the other. Just as the spider and the soul quest outward for significance, the two stanzas throw links to each other with subtly paired words: isolated/detached, launched/fling, tirelessly/ceaselessly, surrounding/surrounded.

In this poem, Whitman uses synonyms and antonyms to give structural integrity to a poem comprising two yoked stanzas, much like (but not exactly like) the way poets working within closed forms use meter and rhyme to give structural integrity to their poems. The poem has form, but the form was not imposed by previous conventions. It has open form.

Imagery and symbolism
Most poetry can be read on several levels. The surface is not necessarily the essence of the poem although in some cases (notably, the works of William McGonagall) there is little beyond the immediate. Allegory, connotation and metaphor are some of the subtler ways in which a poet communicates with the reader.

Before getting seduced into explorations of subtle nuance, however, the reader should establish the theme of the poem. What is the 'story' that is being told? Not the literal story but the heart of the poem. For example: "Another" tells of a buried child; "The Destruction of Sennacherib" tells of the annihilation of an Assyrian army by divine intervention; "The Silken Tent" compares a woman to a tent. Part of this involves recognising the voice of the poem (who is speaking), and the rest of Kipling's "six honest serving men": the events in the poem; when these occur; where is the 'speaker' and where do the events occur; why does the speaker speak? William Harmon has suggested that starting an analysis with: "This poem dramatizes the conflict between ..." is a key technique.

George Herbert in his poem Jordan (I) asks if poetry must be about the imaginary. The poem opens:

 * Who sayes that fictions onely and false hair
 * Become a verse? Is there in truth no beautie?
 * Is all good structure in a winding stair?
 * May no lines passe, except they do their dutie
 * Not to a true, but painted chair?

He was railing against the prevalent enthusiasm for pastoral poetry above all other forms (as becomes apparent in subsequent verses). Curiously, this verse uses metaphors to challenge the use of indirect approaches to their subject. False hair and a painted chair are decorations of the mundane. The winding stair is obstructive concealment of meaning. Herbert is criticising the overuse of allegory, symbolism or elaborate language.

For most poets—even the plain-speaking Herbert—metaphor is the fundamental means of communicating complexity succinctly. Some metaphors become so widely used that they are widely recognised symbols and these can be identified by using a specialist dictionary.

Allegorical verse uses an extended metaphor to provide the framework for the whole work. It was particularly prevalent in seventeenth century English but a more recent example is Charles Williams' The Masque of the Manuscript, in which the process of publishing is a metaphor for the search for truth. Allegories are usually readily apparent because of the heavy use of metaphor within them.

The symbolism used in a poem may not always be as overt as metaphor. Often the poet communicates emotionally by selecting words with particular connotations. For example, the word "sheen" in The Destruction of Sennacherib has stronger connotations of polishing, of human industry, than does the similar "shine". The Assyrians did not simply choose shiny metal; they worked to make it so. The word hints at a military machine.

Other tropes that may be used to increase the level of allusion include irony, litotes, simile, and metonymy (particularly synecdoche).

Meter and rhythm
English language poetic meter depends on stress, rather than the number of syllables. It thus stands in contrast to poetry in other languages, such as French, where syllabic stress is not present or recognized and syllable count is paramount. This often makes scansion (the analysis of metrical patterns) seem unduly arcane and arbitrary to English students of the craft.

In the final analysis, the terms of scansion are blunt instruments, clumsy ways of describing the infinitely nuanced rhythms of language. Nonetheless, they provide a tool for discerning and describing the underlying structure of poems (especially those employing closed form).

The terms fall into two groups: the names of the different feet, and the names of the varying line lengths.

The most common feet in poetry written in English are the iamb (weak STRONG), the anapest (weak weak STRONG), the trochee (STRONG weak), and the dactyl (STRONG weak weak). The iamb and anapest are known as rising meters (they move "up" from weak to strong syllables); the trochee and dactyl are falling meters (they move "down" from strong to weak). Less common, but frequently important for the variety and energy they bring to a line, are the monosyllabic foot (weak) and the spondee (STRONG STRONG).

The terms for line length follow a regular pattern: a Greek prefix denoting the number of feet and the root "meter" (for "measure"): monometer, dimeter, trimeter, tetrameter, pentameter, hexameter, heptameter, and octameter (lines having more than eight feet are possible but quite rare).

Another useful term is caesura, for a natural pause within a line.

Meter and line length are not formulas for successful lines of poetry. They are rough forms of notation for the many satisfying and variable rhythms of language. Slavish adherence to meter produces doggerel. Skillful poets structure their poems around a meter and line length, and then depart from it and play against it as needed in order to create effect, as Robert Browning does in the first line of "My Last Duchess":

 * That's my last Duchess painted on the wall.

The opening spondees, which throw the iambic line out of pattern, gives the Duke's words a certain virulent energy: he's spitting the words out.

Gerard Manley Hopkins took this idea of poetric energy through departure from meter to its extreme, with his theory and practice of sprung rhythm, an approach to poetry in which the language constantly frustrates the reading mind's expectation of a regular meter.

Sound, tone, diction, and connotation
Analyzing diction and connotation—the meanings of words as well as the feelings and associations they carry—is a good place to start for any poem. The use of specific words in the poem serve to create a tone—an attitude taken towards the subject. For example, consider the words "slither" and "sneak." When used in a poem, the words conjure up images of a snake. The sibilant s sound reinforces the image. The connotations of the words suggest something surreptitious and undercover. From the tone, one can infer that the author is suspicious or fearful of the subject.

A detached tone, or an opposite tone than the reader would expect, are sometimes purposely employed to elicit more of a response. In the opening lines of "The Love Song of J. Alfred Prufrock", T. S. Eliot quickly sets a certain tone, and then creates effect by juxtaposing it with a very different tone:

 * Let us go then, you and I,
 * When the evening is spread out against the sky
 * Like a patient etherized upon a table

Visual and concrete poetry
Poets such as E. E. Cummings experiment with punctuation and the words' layout on a page. In doing so, they venture into a realm of poetry that really cannot be read aloud: it can only be experienced through the eye.

 * l(a

 * le
 * af
 * fa

 * ll

 * s)
 * one
 * l

 * iness

Cummings has fractured language into its most atomic elements. To analyze the poem, the reader must first reassemble these into meaningful units: a brief, evocative image—"a leaf falls"—appearing, like an unbidden thought, in the middle of a word, "loneliness". A single falling leaf is by itself a fine image for the feeling of loneliness, but in order to fully experience the poem, the reader must then put the elements of language back into the visual form in which Cummings arranged them: a vertical line (like a leaf falling to the ground), followed by a horizontal series of letters (like a leaf laying flat on the ground). The vertical line is dominated by the word "one" (an absolutely appropriate expression of loneliness) and by the letter "l" (which, on the page, looks like the numeral 1... it looks like "one").

Numerous other poets, including George Herbert, Lewis Carroll, William Blake, Wyndham Lewis, and John Hollander have used the layout of words, letters, and images on the page to create effect in their poems. An analytic reader of poetry must attend to the eye as well as the ear and the mind.

Schools of poetry
There are many different 'schools' of poetry: oral, classical, romantic, modernist, etc. and they each vary in their use of the elements described above.

Schools of poetry may be self-identified by the poets that form them (such as Imagism ) or defined by critics who see unifying characteristics of a body of work by more than one poet (for example The Movement). To be a 'school' a group of poets must share a common style or a common ethos. A commonality of form is not in itself sufficient to define a school; for example, Edward Lear, George du Maurier and Ogden Nash do not form a school simply because they all wrote limericks.

Schools of criticism
Poetry analysis is almost as old as poetry itself, with distinguished practitioners going back to figures such as Plato. At various times and places, groups of like-minded readers and scholars have developed, shared, and promoted specific approaches to poetry analysis.

The New Criticism dominated English and American literary criticism from the 1920s to the early 1960s. The New Critical approach insists on the value of close reading and rejects extra-textual sources. The New Critics also rejected the idea that the work of a critic or analyst is to determine an author's intended meaning (a view formalized by W.K. Wimsatt and Monroe Beardsley as the intentional fallacy). The New Critics prized ambiguity, and tended to favor works that lend themselves to multiple interpretations.

Reader-response criticism developed in Germany and the United States as a reaction to New Criticism. It emphasises the reader's role in the development of meaning.

Reception theory is a development of reader-response criticism that considers the public response to a literary work and suggests that this can inform analysis of cultural ideology at the time of the response.

Reading poetry aloud
All poetry was originally oral, it was sung or chanted; poetic form as we know it is an abstraction therefrom when writing replaced memory as a way of preserving poetic utterances, but the ghost of oral poetry never vanishes. Poems may be read silently to oneself, or may be read aloud solo or to other people. Although reading aloud to oneself raises eyebrows in many circles, few people find it surprising in the case of poetry.

In fact, many poems reveal themselves fully only when they are read aloud. The characteristics of such poems include (but are not limited to) a strong narrative, regular poetic meter, simple content and simple form. At the same time, many poems that read well aloud have none of the characteristics exhibited by T. S. Eliot's "Journey of the Magi", for example. Poems that read aloud well include:

 * "The Frog", by Jean Dao
 * "One Art", by Elizabeth Bishop
 * "The Tyger", by William Blake
 * "Meeting at Night", by Robert Browning
 * "She Walks in Beauty", by Byron
 * "The Song of the Western Men", by Robert Stephen Hawker
 * "November in England", by Thomas Hood
 * "Dream Variations", by Langston Hughes
 * "The Jackdaw of Rheims", by Thomas Ingoldsby
 * "To put one brick upon another", by Philip Larkin
 * "Paul Revere's Ride", by Henry Wadsworth Longfellow
 * "Adventures of Isabel", by Ogden Nash
 * "Nothing but Death", by Pablo Neruda translated by Robert Bly
 * "A Small Elegy", by Jirí Orten translated by Lynn Coffin
 * "Ozymandias", by Percy Bysshe Shelley
 * "The Cat in the Hat", by Dr. Seuss
 * "Sea Surface Full of Clouds", by Wallace Stevens
 * "Silver", by Walter de la Mare
 * "How to Tell a Story", by Robert Penn Warren
 * "On Westminster Bridge", by William Wordsworth

Poetry in different cultures
This article is focused on poetry written in English and reflects anglophone culture. Other cultures have other poetic forms and differ in their attitudes towards poetry.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Poetry analysis


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Desert climate

The desert climate or arid climate (in the Köppen climate classification BWh and BWk) is a dry climate sub-type in which there is a severe excess of evaporation over precipitation. The typically bald, rocky, or sandy surfaces in desert climates are dry and hold little moisture, quickly evaporating the already little rainfall they receive. Covering 14.2% of Earth's land area, hot deserts are the second most common type of climate on Earth after the polar climate.

There are two variations of a desert climate according to the Köppen climate classification: a hot desert climate (BWh), and a cold desert climate (BWk). To delineate "hot desert climates" from "cold desert climates", a mean annual temperature of 18 C is used as an isotherm so that a location with a BW type climate with the appropriate temperature above this isotherm is classified as "hot arid subtype" (BWh), and a location with the appropriate temperature below the isotherm is classified as "cold arid subtype" (BWk).

Most desert/arid climates receive between 25 and 200 mm of rainfall annually, although some of the most consistently hot areas of Central Australia, the Sahel and Guajira Peninsula can be, due to extreme potential evapotranspiration, classed as arid with the annual rainfall as high as 430 mm.

Precipitation
Although no part of Earth is known for certain to be rainless, in the Atacama Desert of northern Chile, the average annual rainfall over 17 years was only just 5 mm. Some locations in the Sahara Desert such as Kufra, Libya, record an even drier 0.86 mm of rainfall annually. The official weather station in Death Valley, United States reports 60 mm annually, but in 40 months between 1931 and 1934 a total of just 16 mm of rainfall was measured.

To determine whether a location has an arid climate, the precipitation threshold is determined. The precipitation threshold (in millimetres) involves first multiplying the average annual temperature in °C by 20, then adding 280 if 70% or more of the total precipitation is in the high-sun summer half of the year (April through September in the Northern Hemisphere, or October through March in the Southern), or 140 if 30–70% of the total precipitation is received during the applicable period, or 0 if less than 30% of the total precipitation is so received there. If the area's annual precipitation is less than half the threshold (50%), it is classified as a BW (desert climate), while 50–100% of the threshold results in a semi-arid climate.

Hot desert climates
Hot desert climates (BWh) are typically found under the subtropical ridge in the lower middle latitudes or the subtropics, often between 20° and 33° north and south latitudes. In these locations, stable descending air and high pressure aloft clear clouds and create hot, arid conditions with intense sunshine. Hot desert climates are found across vast areas of North Africa, West Asia, northwestern parts of the Indian Subcontinent, southwestern Africa, interior Australia, the Southwestern United States, northern Mexico, sections of southeastern Spain, the coast of Peru, and Chile. This makes hot deserts present in every continent except Antarctica.

At the time of high sun (summer), scorching, desiccating heat prevails. Hot-month average temperatures are normally between 29 and 35 C, and midday readings of 43–46 C are common. The world absolute heat records, over 50 C, are generally in the hot deserts, where the heat potential can be the highest on the planet. This includes the record of 56.7 C in Death Valley, which is currently considered the highest temperature recorded on Earth. Some deserts in the tropics consistently experience very high temperatures all year long, even during wintertime. These locations feature some of the highest annual average temperatures recorded on Earth, exceeding 30 C, up to nearly 35 C in Dallol, Ethiopia. This last feature is seen in sections of Africa and Arabia. During colder periods of the year, night-time temperatures can drop to freezing or below due to the exceptional radiation loss under the clear skies. However, very rarely do temperatures drop far below freezing under the hot subtype. Hot desert climates can be found in the deserts of North Africa such as the wide Sahara Desert, the Libyan Desert or the Nubian Desert; deserts of the Horn of Africa such as the Danakil Desert or the Grand Bara Desert; deserts of Southern Africa such as the Namib Desert or the Kalahari Desert; deserts of West Asia such as the Arabian Desert, or the Syrian Desert; deserts of South Asia such as Dasht-e Lut and Dasht-e Kavir of Iran or the Thar Desert of India and Pakistan; deserts of the United States and Mexico such as the Mojave Desert, the Sonoran Desert or the Chihuahuan Desert; deserts of Australia such as the Simpson Desert or the Great Victoria Desert and many other regions. In Europe, the hot desert climate can only be found on southeastern coast of Spain as well as small inland parts of southeastern, especially parts of the Tabernas Desert.

Hot deserts are lands of extremes: most of them are among the hottest, the driest and the sunniest places on Earth because of nearly constant high pressure; the nearly permanent removal of low-pressure systems, dynamic fronts and atmospheric disturbances; sinking air motion; dry atmosphere near the surface and aloft; the exacerbated exposure to the sun where solar angles are always high makes this desert inhospitable to most species.

Cold desert climates
Cold desert climates (BWk) usually feature hot (or warm in a few instances), dry summers, though summers are not typically as hot as hot desert climates. Unlike hot desert climates, cold desert climates tend to feature cold, dry winters. Snow tends to be rare in regions with this climate. The Gobi Desert in northern China and Mongolia is one example of cold desert. Though hot in the summer, it shares the very cold winters of the rest of Inner Asia. Summers in South America's Atacama Desert are mild with only slight temperature variations between seasons. Cold desert climates are typically found at higher altitudes than hot desert climates and are usually drier than hot desert climates.

Cold desert climates are typically located in temperate zones, usually in the leeward rain shadow of high mountains, which restricts precipitation from the westerly winds. An example of this is the Patagonian Desert in Argentina bounded by the Andes ranges to its west. In the case of Central Asia, mountains restrict precipitation from the eastern monsoon. The Kyzyl Kum, Taklamakan and Katpana Desert deserts of Central Asia are other major examples of BWk climates. The Ladakh region, and the city of Leh in the Great Himalayas in India also have a cold desert climate. In North America, the cold desert climate occurs in the drier parts of the Great Basin Desert and the Bighorn Basin in Big Horn and Washakie County in Wyoming. The Hautes Plaines, located in the northeastern section of Morocco and in Algeria is another major example of a cold desert climate. In Europe, this climate can only be found in some inland parts of southeastern Spain, such as in Lorca.

Arctic and Antarctic regions also receive very little precipitation during the year, owing to the exceptionally cold dry air freezing most precipitation; however, both of them are generally classified as having polar climates because they have average summer temperatures below 10 °C even if they have such desert-like features as intermittent streams, hypersaline lakes, and extremely barren terrain in unglaciated areas such as the McMurdo Dry Valleys of Antarctica.

title: Desert climate


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Gauss–Markov theorem

In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero. The errors do not need to be normal for the theorem to apply, nor do they need to be independent and identically distributed (only uncorrelated with mean zero and homoscedastic with finite variance).

The requirement for unbiasedness cannot be dropped, since biased estimators exist with lower variance and mean squared error. For example, the James–Stein estimator (which also drops linearity) and ridge regression typically outperform ordinary least squares. In fact, ordinary least squares is rarely even an admissible estimator, as Stein's phenomenon shows--when estimating more than two unknown variables, ordinary least squares will always perform worse (in mean squared error) than Stein's estimator.

Moreover, the Gauss-Markov theorem does not apply when considering more principled loss functions, such as the assigned likelihood or Kullback–Leibler divergence, except in the limited case of normally-distributed errors.

As a result of these discoveries, statisticians typically motivate ordinary least squares by the principle of maximum likelihood instead, or by considering it as a kind of approximate Bayesian inference.

The theorem is named after Carl Friedrich Gauss and Andrey Markov. Gauss provided the original proof, which was later substantially generalized by Markov.

Scalar Case Statement
Suppose we are given two random variable vectors, $$ X \text {, } Y, \text { where } X,Y \in \mathbb{R}^k $$.

Suppose want to find the best linear estimator of Y given X, such that
 * $$ \hat Y = \alpha X + \mu, \text{ where }  (   \alpha \in \mathbb{R} \text{ and } \mu\in\mathbb{R}) $$ would be the estimator,

and parameters $$ \alpha, \mu $$ such that it would be the best linear estimator of X.

Such an estimator $$ \hat Y $$ would have the same linear properties of Y, $$ \sigma _{\hat Y} = \sigma _{Y},  \mu _{\hat Y}  = \mu _{Y} $$.

Therefore, if the vector X has properties of $$ \mu _x, \sigma _x $$, the best linear estimator would be

$$ \hat Y = \sigma _y \frac{(X - \mu _x )}{ \sigma _x } + \mu _y $$

since it has the same mean and variance as Y.

Statement
Suppose we have, in matrix notation, the linear relationship
 * $$ y = X \beta + \varepsilon,\quad (y,\varepsilon \in \mathbb{R}^n, \beta \in \mathbb{R}^K \text{ and } X\in\mathbb{R}^{n\times K}) $$

expanding to,
 * $$ y_i=\sum_{j=1}^{K}\beta_j X_{ij}+\varepsilon_i \quad \forall i=1,2,\ldots,n$$

where $$\beta_j$$ are non-random but unobservable parameters, $$X_{ij}$$ are non-random and observable (called the "explanatory variables"), $$\varepsilon_i$$ are random, and so $$y_i$$ are random. The random variables $$\varepsilon_i$$ are called the "disturbance", "noise" or simply "error" (will be contrasted with "residual" later in the article; see errors and residuals in statistics). Note that to include a constant in the model above, one can choose to introduce the constant as a variable $$\beta_{K+1}$$ with a newly introduced last column of X being unity i.e., $$X_{i(K+1)} = 1$$ for all $$ i $$. Note that though $$y_i,$$ as sample responses, are observable, the following statements and arguments including assumptions, proofs and the others assume under the only condition of knowing $$X_{ij},$$ but not $$y_i.$$

The Gauss–Markov assumptions concern the set of error random variables, $$\varepsilon_i$$:

 * They have mean zero: $$\operatorname{E}[\varepsilon_i]=0.$$
 * They are homoscedastic, that is all have the same finite variance: $$\operatorname{Var}(\varepsilon_i)= \sigma^2 < \infty$$ for all $$i$$ and
 * Distinct error terms are uncorrelated: $$\text{Cov}(\varepsilon_i,\varepsilon_j) = 0, \forall i \neq j.$$

A linear estimator of $$ \beta_j $$ is a linear combination

 * $$\widehat\beta_j = c_{1j}y_1+\cdots+c_{kj}y_k$$

in which the coefficients $$ c_{ij} $$ are not allowed to depend on the underlying coefficients $$\beta_j$$, since those are not observable, but are allowed to depend on the values $$ X_{ij} $$, since these data are observable. (The dependence of the coefficients on each $$X_{ij}$$ is typically nonlinear; the estimator is linear in each $$ y_i $$ and hence in each random $$ \varepsilon,$$ which is why this is "linear" regression.) The estimator is said to be unbiased if and only if

 * $$\operatorname{E}\left [\widehat\beta_j \right ]=\beta_j$$

regardless of the values of $$ X_{ij} $$. Now, let $\sum_{j=1}^K\lambda_j\beta_j$ be some linear combination of the coefficients. Then the mean squared error of the corresponding estimation is

 * $$\operatorname{E} \left [\left (\sum_{j=1}^K\lambda_j \left(\widehat\beta_j-\beta_j \right ) \right)^2\right ],$$

in other words, it is the expectation of the square of the weighted sum (across parameters) of the differences between the estimators and the corresponding parameters to be estimated. (Since we are considering the case in which all the parameter estimates are unbiased, this mean squared error is the same as the variance of the linear combination.) The best linear unbiased estimator (BLUE) of the vector $$ \beta $$ of parameters $$ \beta_j $$ is one with the smallest mean squared error for every vector $$ \lambda $$ of linear combination parameters. This is equivalent to the condition that

 * $$\operatorname{Var}\left(\widetilde\beta\right)- \operatorname{Var} \left( \widehat \beta \right)$$

is a positive semi-definite matrix for every other linear unbiased estimator $$\widetilde\beta$$.

The ordinary least squares estimator (OLS) is the function

 * $$\widehat\beta=(X^\operatorname{T}X)^{-1}X^\operatorname{T}y$$

of $$ y $$ and $$X$$ (where $$X^\operatorname{T}$$ denotes the transpose of $$ X $$) that minimizes the sum of squares of residuals (misprediction amounts):

 * $$\sum_{i=1}^n \left(y_i-\widehat{y}_i\right)^2=\sum_{i=1}^n \left(y_i-\sum_{j=1}^K \widehat\beta_j X_{ij}\right)^2.$$

The theorem now states that the OLS estimator is a best linear unbiased estimator (BLUE).

The main idea of the proof is that the least-squares estimator is uncorrelated with every linear unbiased estimator of zero, i.e., with every linear combination $$a_1y_1+\cdots+a_ny_n$$ whose coefficients do not depend upon the unobservable $$ \beta $$ but whose expected value is always zero.

Remark
Proof that the OLS indeed minimizes the sum of squares of residuals may proceed as follows with a calculation of the Hessian matrix and showing that it is positive definite.

The MSE function we want to minimize is $$f(\beta_0,\beta_1,\dots,\beta_p) = \sum_{i=1}^n (y_i-\beta_0-\beta_1x_{i1}-\dots-\beta_px_{ip})^2$$ for a multiple regression model with p variables. The first derivative is $$\begin{aligned} \frac{d}{d\boldsymbol{\beta}}f &= -2X^\operatorname{T} \left(\mathbf{y}-X\boldsymbol{\beta}\right)\\ &=-2\begin{bmatrix} \sum_{i=1}^{n} (y_i - \dots - \beta_px_{ip})\\ \sum_{i=1}^nx_{i1} (y_i-\dots-\beta_px_{ip})\\ \vdots\\ \sum_{i=1}^nx_{ip} (y_i-\dots-\beta_px_{ip}) \end{bmatrix}\\ &= \mathbf{0}_{p+1}, \end{aligned}$$ where $$X^\operatorname{T}$$ is the design matrix $$X=\begin{bmatrix} 1 & x_{11} & \cdots & x_{1p}\\ 1 & x_{21} & \cdots & x_{2p}\\ &&\vdots\\ 1 & x_{n1} & \cdots & x_{np} \end{bmatrix}\in \R^{n\times(p+1)}; \qquad n\geq p+1$$

The Hessian matrix of second derivatives is $$\mathcal{H} = 2\begin{bmatrix} n & \sum_{i=1}^n x_{i1} & \cdots & \sum_{i=1}^n x_{ip} \\ \sum_{i=1}^n x_{i1}& \sum_{i=1}^n x_{i1}^2 & \cdots & \sum_{i=1}^nx_{i1}x_{ip}\\ \vdots & \vdots &\ddots & \vdots \\ \sum_{i=1}^n x_{ip} & \sum_{i=1}^n x_{ip}x_{i1}& \cdots & \sum_{i=1}^n x_{ip}^2 \end{bmatrix} = 2X^\operatorname{T}X$$

Assuming the columns of $$X$$ are linearly independent so that $$X^\operatorname{T} X$$ is invertible, let $$X=\begin{bmatrix}\mathbf{v_1}& \mathbf{v_2}& \cdots & \mathbf{v}_{p+1}\end{bmatrix}$$, then $$k_1\mathbf{v_1} + \dots + k_{p+1} \mathbf{v}_{p+1} = \mathbf 0\iff k_1= \dots =k_{p+1}=0$$

Now let $$\mathbf{k} = (k_1,\dots,k_{p+1})^T \in \R^{(p+1)\times 1}$$ be an eigenvector of $$\mathcal{H}$$.

$$\mathbf{k} \ne \mathbf{0} \implies \left(k_1\mathbf{v_1}+\dots+k_{p+1}\mathbf{v}_{p+1}\right)^2 > 0$$

In terms of vector multiplication, this means $$\begin{bmatrix} k_1 & \cdots & k_{p+1} \end{bmatrix} \begin{bmatrix}\mathbf{v_1} \\ \vdots \\ \mathbf{v}_{p+1}\end{bmatrix} \begin{bmatrix}\mathbf{v_1} & \cdots & \mathbf{v}_{p+1}\end{bmatrix} \begin{bmatrix}k_1 \\ \vdots\\ k_{p+1}\end{bmatrix} = \mathbf{k}^\operatorname{T}\mathcal{H}\mathbf{k} = \lambda \mathbf{k}^\operatorname{T}\mathbf{k}>0$$ where $$\lambda$$ is the eigenvalue corresponding to $$\mathbf{k}$$. Moreover, $$\mathbf{k}^\operatorname{T}\mathbf{k} = \sum_{i=1}^{p+1}k_i^2 > 0 \implies \lambda > 0$$

Finally, as eigenvector $$\mathbf{k}$$ was arbitrary, it means all eigenvalues of $$\mathcal{H}$$ are positive, therefore $$\mathcal{H}$$ is positive definite. Thus, $$\boldsymbol{\beta} = \left(X^\operatorname{T}X\right)^{-1}X^\operatorname{T}Y$$ is indeed a global minimum.

Or, just see that for all vectors $$\mathbf{v}, \mathbf{v}^\operatorname{T} X^\operatorname{T} X \mathbf{v} = \|\mathbf{X}\mathbf{v}\|^2 \ge 0 $$. So the Hessian is positive definite if full rank.

Proof
Let $$\tilde\beta = Cy$$ be another linear estimator of $$ \beta $$ with $$C = (X^\operatorname{T}X)^{-1}X^\operatorname{T} + D $$ where $$D$$ is a $$K \times n$$ non-zero matrix. As we're restricting to unbiased estimators, minimum mean squared error implies minimum variance. The goal is therefore to show that such an estimator has a variance no smaller than that of $$ \widehat\beta,$$ the OLS estimator. We calculate:

\begin{align} \operatorname{E} \left[ \tilde\beta \right] &= \operatorname{E}[Cy] \\ &= \operatorname{E} \left [\left ((X^\operatorname{T}X)^{-1}X^\operatorname{T} + D \right )(X\beta + \varepsilon) \right ]\\ &= \left ((X^\operatorname{T}X)^{-1}X^\operatorname{T} + D \right )X\beta   + \left ((X^\operatorname{T}X)^{-1}X^\operatorname{T} + D \right ) \operatorname{E}[\varepsilon] \\ &= \left ((X^\operatorname{T}X)^{-1}X^\operatorname{T} + D \right )X\beta && \operatorname{E}[\varepsilon] =0 \\ &= (X^\operatorname{T}X)^{-1}X^\operatorname{T}X\beta + DX\beta \\ &= (I_K + DX)\beta. \\ \end{align} $$

Therefore, since $$\beta$$ is unobservable, $$ \tilde\beta $$ is unbiased if and only if $$ DX = 0 $$. Then:

\begin{align} \operatorname{Var}\left(\tilde\beta\right) &= \operatorname{Var}(Cy) \\ &= C \text{ Var}(y)C^\operatorname{T} \\ &= \sigma^2 CC^\operatorname{T} \\ &= \sigma^2 \left ((X^\operatorname{T}X)^{-1}X^\operatorname{T} + D \right ) \left (X(X^\operatorname{T}X)^{-1} + D^\operatorname{T} \right ) \\ &= \sigma^2 \left ((X^\operatorname{T}X)^{-1}X^\operatorname{T}X(X^\operatorname{T}X)^{-1} + (X^\operatorname{T}X)^{-1}X^\operatorname{T}D^\operatorname{T} + DX(X^\operatorname{T}X)^{-1} + DD^\operatorname{T} \right) \\ &= \sigma^2(X^\operatorname{T}X)^{-1} + \sigma^2(X^\operatorname{T}X)^{-1} (DX)^\operatorname{T} + \sigma^2 DX (X^\operatorname{T}X)^{-1} + \sigma^2DD^\operatorname{T} \\ &= \sigma^2(X^\operatorname{T}X)^{-1}+ \sigma^2DD^\operatorname{T} && DX =0 \\ &= \operatorname{Var}\left(\widehat\beta\right)  + \sigma^2DD^\operatorname{T}  && \sigma^2(X^\operatorname{T}X)^{-1} =   \operatorname{Var}\left(\widehat\beta\right) \end{align} $$

Since $$DD^\operatorname{T}$$ is a positive semidefinite matrix, $$\operatorname{Var}\left( \tilde \beta \right) $$ exceeds $$\operatorname{Var}\left(\widehat\beta\right) $$ by a positive semidefinite matrix.

Remarks on the proof
As it has been stated before, the condition of $$ \operatorname{Var} \left( \tilde \beta \right)- \operatorname{Var} \left(\widehat\beta\right)$$ is a positive semidefinite matrix is equivalent to the property that the best linear unbiased estimator of $$ \ell^\operatorname{T}\beta $$ is $$ \ell^\operatorname{T}\widehat\beta $$ (best in the sense that it has minimum variance). To see this, let $$ \ell^\operatorname{T}\tilde\beta $$ another linear unbiased estimator of $$ \ell^\operatorname{T}\beta $$.

\begin{align} \operatorname{Var}\left(\ell^\operatorname{T}\tilde\beta\right) &= \ell^\operatorname{T} \operatorname{Var} \left(\tilde\beta\right) \ell \\ &=\sigma^2 \ell^\operatorname{T} (X^\operatorname{T}X)^{-1}\ell+\ell^\operatorname{T}DD^\operatorname{T}\ell \\ &= \operatorname{Var}\left(\ell^\operatorname{T}\widehat\beta\right)+(D^\operatorname{T}\ell)^t(D^\operatorname{T}\ell) && \sigma^2 \ell^\operatorname{T} (X^\operatorname{T}X)^{-1}\ell = \operatorname{Var}\left(\ell^\operatorname{T}\widehat\beta\right) \\ &= \operatorname{Var}\left(\ell^\operatorname{T}\widehat\beta\right) +\|D^\operatorname{T}\ell\|\\ & \geq \operatorname{Var}\left(\ell^\operatorname{T}\widehat\beta\right) \end{align} $$

Moreover, equality holds if and only if $$ D^\operatorname{T}\ell=0 $$. We calculate

\begin{align} \ell^\operatorname{T}\tilde\beta &= \ell^\operatorname{T} \left (((X^\operatorname{T}X)^{-1}X^\operatorname{T} + D) Y \right ) && \text{ from above}\\ &= \ell^\operatorname{T}(X^\operatorname{T}X)^{-1}X^\operatorname{T}Y + \ell^\operatorname{T}DY \\ &= \ell^\operatorname{T}\widehat\beta +(D^\operatorname{T}\ell)^t Y \\ &=\ell^\operatorname{T}\widehat\beta && D^\operatorname{T}\ell = 0 \end{align} $$

This proves that the equality holds if and only if $$ \ell^\operatorname{T}\tilde\beta=\ell^\operatorname{T}\widehat\beta $$ which gives the uniqueness of the OLS estimator as a BLUE.

Generalized least squares estimator
The generalized least squares (GLS), developed by Aitken, extends the Gauss–Markov theorem to the case where the error vector has a non-scalar covariance matrix. The Aitken estimator is also a BLUE.

Gauss–Markov theorem as stated in econometrics
In most treatments of OLS, the regressors (parameters of interest) in the design matrix $$\mathbf{X}$$ are assumed to be fixed in repeated samples. This assumption is considered inappropriate for a predominantly nonexperimental science like econometrics. Instead, the assumptions of the Gauss–Markov theorem are stated conditional on $$\mathbf{X}$$.

Linearity
The dependent variable is assumed to be a linear function of the variables specified in the model. The specification must be linear in its parameters. This does not mean that there must be a linear relationship between the independent and dependent variables. The independent variables can take non-linear forms as long as the parameters are linear. The equation $$ y = \beta_{0} + \beta_{1} x^2, $$ qualifies as linear while $$ y = \beta_{0} + \beta_{1}^2 x$$ can be transformed to be linear by replacing $$\beta_{1}^2$$ by another parameter, say $$\gamma$$. An equation with a parameter dependent on an independent variable does not qualify as linear, for example $$y = \beta_{0} + \beta_{1}(x) \cdot x$$, where $$\beta_{1}(x)$$ is a function of $$x$$.

Data transformations are often used to convert an equation into a linear form. For example, the Cobb–Douglas function—often used in economics—is nonlinear:
 * $$Y = A L^\alpha K^{1 - \alpha} e^\varepsilon $$

But it can be expressed in linear form by taking the natural logarithm of both sides:

 * $$\ln Y=\ln A + \alpha \ln L + (1 - \alpha) \ln K + \varepsilon = \beta_0 + \beta_1 \ln L + \beta_2 \ln K + \varepsilon$$

This assumption also covers specification issues: assuming that the proper functional form has been selected and there are no omitted variables.

One should be aware, however, that the parameters that minimize the residuals of the transformed equation do not necessarily minimize the residuals of the original equation.

Strict exogeneity
For all $$n$$ observations, the expectation—conditional on the regressors—of the error term is zero:
 * $$\operatorname{E}[\,\varepsilon_{i}\mid \mathbf{X} ] = \operatorname{E}[\,\varepsilon_{i}\mid \mathbf{x}_{1}, \dots, \mathbf{x}_{n} ] = 0.$$

where $$\mathbf{x}_i = \begin{bmatrix} x_{i1} & x_{i2} & \cdots & x_{ik} \end{bmatrix}^{\operatorname{T}}$$ is the data vector of regressors for the ith observation, and consequently $$\mathbf{X} = \begin{bmatrix} \mathbf{x}_{1}^{\operatorname{T}} & \mathbf{x}_{2}^{\operatorname{T}} & \cdots & \mathbf{x}_{n}^{\operatorname{T}} \end{bmatrix}^{\operatorname{T}}$$ is the data matrix or design matrix.

Geometrically, this assumption implies that $$\mathbf{x}_{i}$$ and $$\varepsilon_{i}$$ are orthogonal to each other, so that their inner product (i.e., their cross moment) is zero.
 * $$\operatorname{E}[\,\mathbf{x}_{j} \cdot \varepsilon_{i}\,] = \begin{bmatrix} \operatorname{E}[\,{x}_{j1} \cdot \varepsilon_{i}\,] \\ \operatorname{E}[\,{x}_{j2} \cdot \varepsilon_{i}\,] \\ \vdots \\ \operatorname{E}[\,{x}_{jk} \cdot \varepsilon_{i}\,] \end{bmatrix} = \mathbf{0} \quad \text{for all } i, j \in n$$

This assumption is violated if the explanatory variables are measured with error, or are endogenous. Endogeneity can be the result of simultaneity, where causality flows back and forth between both the dependent and independent variable. Instrumental variable techniques are commonly used to address this problem.

Full rank
The sample data matrix $$\mathbf{X}$$ must have full column rank.
 * $$\operatorname{rank}(\mathbf{X}) = k$$

Otherwise $$\mathbf{X}^\operatorname{T} \mathbf{X}$$ is not invertible and the OLS estimator cannot be computed.

A violation of this assumption is perfect multicollinearity, i.e. some explanatory variables are linearly dependent. One scenario in which this will occur is called "dummy variable trap," when a base dummy variable is not omitted resulting in perfect correlation between the dummy variables and the constant term.

Multicollinearity (as long as it is not "perfect") can be present resulting in a less efficient, but still unbiased estimate. The estimates will be less precise and highly sensitive to particular sets of data. Multicollinearity can be detected from condition number or the variance inflation factor, among other tests.

Spherical errors
The outer product of the error vector must be spherical.
 * $$\operatorname{E}[\,\boldsymbol{\varepsilon} \boldsymbol{\varepsilon}^{\operatorname{T}} \mid \mathbf{X} ]

= \operatorname{Var}[\,\boldsymbol{\varepsilon} \mid \mathbf{X} ] = \begin{bmatrix} \sigma^{2} & 0 & \cdots & 0 \\ 0 & \sigma^{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \sigma^{2} \end{bmatrix} = \sigma^{2} \mathbf{I} \quad \text{with } \sigma^{2} > 0$$ This implies the error term has uniform variance (homoscedasticity) and no serial correlation. If this assumption is violated, OLS is still unbiased, but inefficient. The term "spherical errors" will describe the multivariate normal distribution: if $$\operatorname{Var}[\,\boldsymbol{\varepsilon}\mid \mathbf{X} ] = \sigma^{2} \mathbf{I}$$ in the multivariate normal density, then the equation $$f(\varepsilon)=c$$ is the formula for a ball centered at μ with radius σ in n-dimensional space.

Heteroskedasticity occurs when the amount of error is correlated with an independent variable. For example, in a regression on food expenditure and income, the error is correlated with income. Low income people generally spend a similar amount on food, while high income people may spend a very large amount or as little as low income people spend. Heteroskedastic can also be caused by changes in measurement practices. For example, as statistical offices improve their data, measurement error decreases, so the error term declines over time.

This assumption is violated when there is autocorrelation. Autocorrelation can be visualized on a data plot when a given observation is more likely to lie above a fitted line if adjacent observations also lie above the fitted regression line. Autocorrelation is common in time series data where a data series may experience "inertia." If a dependent variable takes a while to fully absorb a shock. Spatial autocorrelation can also occur geographic areas are likely to have similar errors. Autocorrelation may be the result of misspecification such as choosing the wrong functional form. In these cases, correcting the specification is one possible way to deal with autocorrelation.

When the spherical errors assumption may be violated, the generalized least squares estimator can be shown to be BLUE.

Other unbiased statistics

 * Best linear unbiased prediction (BLUP)
 * Minimum-variance unbiased estimator (MVUE)

title: Gauss–Markov theorem