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Probability

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Probability is a way of expressing knowledge or belief that an event will occur or has occurred. The concept has an exact mathematical meaning in probability theory, which is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, artificial intelligence/machine learning and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.

Contents
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1 Interpretations

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2 Etymology 3 History 4 Theory 5 Applications 6 Mathematical treatment 7 Relation to randomness 8 See also 9 Notes 10 References 11 Quotations 12 External links

[edit] Interpretations
Main article: Probability interpretations The word probability does not have a consistent direct definition. In fact, there are two broad categories of probability interpretations, whose adherents possess different (and sometimes conflicting) views about the fundamental nature of probability:
1. Frequentists talk about probabilities only when dealing with experiments that are random

and well-defined. The probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. Frequentists consider probability to be the relative frequency "in the long run" of outcomes.[1] 2. Bayesians, however, assign probabilities to any statement whatsoever, even when no random process is involved. Probability, for a Bayesian, is a way to represent an individual's degree of belief in a statement, or an objective degree of rational belief, given the evidence.

[edit] Etymology
The word Probability derives from the Latin probabilitas, which can also mean probity, a measure of the authority of a witness in a legal case in Europe, and often correlated with the witness's nobility. In a sense, this differs much from the modern meaning of probability, which, in contrast, is a measure of the weight of empirical evidence, and is arrived at from inductive reasoning and statistical inference.[2][3]

[edit] History
The scientific study of probability is a modern development. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions arose much later. There are reasons of course, for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues are still obscured by the superstitions of gamblers.[4]

James Ivory (1825. x being any error and y its probability. but it led to unmanageable equations. A probable action or opinion was one such as sensible people would undertake or hold. It is symmetric as to the y-axis. Adrien-Marie Legendre (1805) developed the method of least squares. 1826). F. and introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets). and laid down three properties of this curve: 1. The area enclosed is 1. He gave two proofs. The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable."[5] However. the probability of the error being 0. a formula for the law of facility of error (a term Lagrange used in 1774). the second being essentially the same as John Herschel's (1850). W. in the circumstances. 1856). it being certain that an error exists. 'probable' could also apply to propositions for which there was good evidence. In ignorance of Legendre's contribution. in legal contexts especially. Further proofs were given by Laplace (1810. "Before the middle of the seventeenth century. but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation. Jakob Bernoulli's Ars Conjectandi (posthumous.[7] See Ian Hacking's The Emergence of Probability and James Franklin's The Science of Conjecture for histories of the early development of the very concept of mathematical probability. Robert Adrain. univocally. an Irish-American writer. and . He also provided. Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors. first deduced the law of facility of error. Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject. 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the subject as a branch of mathematics. in 1781. to opinion and to action. the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654). He represented the law of probability of errors by a curve y = φ(x). Gauss gave the first proof that seems to have been known in Europe (the third after Adrain's) in 1809. The theory of errors may be traced back to Roger Cotes's Opera Miscellanea (posthumous. The x-axis is an asymptote. and that certain assignable limits define the range of all errors. 3. Donkin (1844. Gauss (1823). editor of "The Analyst" (1808).According to Richard Jeffrey. and was applied in that sense. Hagen (1837). Pierre-Simon Laplace (1774) first tried to deduce a rule for combining observations from the principles of the theory of probabilities.[6] Aside from elementary work by Girolamo Cardano in the 16th century. 1812). 2. and c a scale factor ensuring that the area under the curve equals 1. Simpson also discusses continuous errors and describes a probability curve. h being a constant depending on precision of observation. the term 'probable' (Latin probabilis) meant approvable. 1722). Friedrich Bessel (1838).

In both cases. sets are interpreted as events and probability itself as a measure on a class of sets.Morgan Crofton (1870). which played an important role in stochastic processes theory and its applications. Watson. Hermann Laurent (1873). Adolphe Quetelet (1853). Glaisher (1872). Governments typically apply probabilistic methods in environmental regulation. Other contributors were Ellis (1844). and any results are interpreted or translated back into the problem domain." often measuring well-being using methods that are stochastic in . but those are essentially different and not compatible with the laws of probability as usually understood. The modern theory of probability based on the measure theory was developed by Andrey Kolmogorov (1931). such as the Dempster-Shafer theory or possibility theory. and Giovanni Schiaparelli (1875). Helmert (1872). Liagre. De Morgan (1864). In Cox's theorem. where it is called "pathway analysis. probability is taken as a primitive (that is. in terms that can be considered separately from their meaning. On the geometric side (see integral geometry) contributors to The Educational Times were influential (Miller. Didion. McColl. not further analyzed) and the emphasis is on constructing a consistent assignment of probability values to propositions. Crofton. Andrey Markov introduced the notion of Markov chains (1906). and Artemas Martin). Richard Dedekind (1860). In the nineteenth century authors on the general theory included Laplace. Wolstenholme. There are other methods for quantifying uncertainty. Sylvestre Lacroix (1816). Augustus De Morgan and George Boole improved the exposition of the theory. the laws of probability are the same. namely the Kolmogorov formulation and the Cox formulation. There have been at least two successful attempts to formalize probability. Peters's (1856) formula for r. Littrow (1833). Further information: History of probability Further information: History of statistics [edit] Theory Main article: Probability theory Like other theories. is well known. the theory of probability is a representation of probabilistic concepts in formal terms—that is. These formal terms are manipulated by the rules of mathematics and logic. and Karl Pearson. except for technical details. [edit] Applications Probability theory is applied in everyday life in risk assessment and in trade on commodity markets. In Kolmogorov's formulation (see probability space). the probable error of a single observation.

and choosing projects to undertake based on statistical analyses of their probable effect on the population as a whole. Many consumer products. and on peace and conflict. and even uncountable sample spaces. {3}. it may be of some importance to most citizens to understand how odds and probability assessments are made.3.[11] This mathematical definition of probability can extend to infinite sample spaces. especially in a democracy. One collection of possible results give an odd number on the die. These collections are called "events. on policy." In this case. A good example is the effect of the perceived probability of any widespread Middle East conflict on oil prices—which have ripple effects in the economy as a whole. use reliability theory in product design to reduce the probability of failure..5} is an element of the power set of the sample space of die rolls.) To qualify as a probability. the subset {1. Accordingly.P(A).g. p(A) or Pr(A). For example.[12] As an example. less likely sends prices up or down. Another significant application of probability theory in everyday life is reliability.6}) is assigned a value of one. the event of A not occurring).4. such as automobiles and consumer electronics.6}. The opposite or complement of an event A is the event [not A] (that is.3. The power set of the sample space is formed by considering all different collections of possible results. A probability is a way of assigning every event a value between zero and one. An assessment by a commodity trader that a war is more likely vs. the events {1. (In our example.nature. the chance of not .[10] The probability of an event A is written as P(A). with the requirement that the event made up of all possible results.5.5} is the event that the die falls on some odd number. the event {1. the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events (events with no common results. and signals other traders of that opinion.2. using the concept of a measure.3.4} are all mutually exclusive). Accordingly. Failure probability may influence a manufacture's decisions on a product's warranty. The theory of behavioral finance emerged to describe the effect of such groupthink on pricing. the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events. {1.[8] It can reasonably be said that the discovery of rigorous methods to assess and combine probability assessments has profoundly affected modern society. e. rolling a die can produce six possible results.[9] [edit] Mathematical treatment Consider an experiment that can produce a number of results. and {2. Thus. the event is said to have occurred. the probabilities are not assessed independently nor necessarily very rationally. its probability is given by P(not A) = 1 . The collection of all results is called the sample space of the experiment. and how they contribute to reputations and to decisions. If the results that actually occur fall in a given event.

and is read "the probability of A. given B". See If both events A and B occur on a single performance of an experiment. It is defined by [14] If P(B) = 0 then is undefined. because of the 52 cards of a deck 13 are hearts. the chance of getting a heart or a face card (J. . Summary of probabilities Probability Event A . this is called the intersection or joint probability of A and B. denoted as independent then the joint probability is . A and B are for example.K) (or one that is both) is . when drawing a single card at random from a regular deck of cards. Conditional probability is the probability of some event A.rolling a six on a six-sided die is 1 – (chance of rolling a six) Complementary event for a more complete treatment. if two coins are flipped the chance of both being heads is [13] If either event A or event B or both events occur on a single performance of an experiment this is called the union of the events A and B denoted as exclusive then the probability of either occurring is . Note that in this case A and B are independent. the chance of rolling a 1 or 2 on a six-sided die is If the events are not mutually exclusive then For example. If two events.Q. given the occurrence of some other event B. Conditional probability is written P(A|B). If two events are mutually For example. and 3 are both: here the possibilities included in the "3 that are both" are included in each of the "13 hearts" and the "12 face cards" but should only be counted once. 12 are face cards.

such as that of quantum decoherence being the cause of an apparent random collapse.[15] [edit] See also Logic portal Main article: Outline of probability • • Black Swan theory Calculus of predispositions .02·1023) that only statistical description of its properties is feasible. is so complex (with the number of molecules typically the order of magnitude of Avogadro constant 6. This means that probability theory is required to describe nature. daß der Alte nicht würfelt. A revolutionary discovery of 20th century physics was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws of quantum mechanics. The wave function itself evolves deterministically as long as no observation is made. Others never came to terms with the loss of determinism. at present there is a firm consensus among physicists that probability theory is necessary to describe quantum phenomena. there is no probability if all conditions are known. if the force of the hand and the period of that force are known.not A A or B A and B A given B [edit] Relation to randomness Main article: Randomness In a deterministic universe. A probabilistic description can thus be more useful than Newtonian mechanics for analyzing the pattern of outcomes of repeated rolls of roulette wheel. Although alternative viewpoints exist. is fundamental. this also assumes knowledge of inertia and friction of the wheel. according to the prevailing Copenhagen interpretation. Albert Einstein famously remarked in a letter to Max Born: Jedenfalls bin ich überzeugt. Of course. but. the randomness caused by the wave function collapsing when an observation is made. weight. smoothness and roundness of the ball. In the case of a roulette wheel. Physicists face the same situation in kinetic theory of gases. variations in hand speed during the turning and so forth. the number on which the ball will stop would be a certainty. (I am convinced that God does not play dice). based on Newtonian concepts. while deterministic in principle. where the system.

” Interpretations of Negative Probabilities”. 11. page 35. 54-55 . ^ Franklin. 8th Edition. Michael. ^ The Cambridge History of Seventeenth-century Philosophy. 2003 4. Cambridge University Press. Laurie. "Management Insights". p.. (2001). Sheldon. ISBN 0521685575. Daniel Garber. [edit] References . 1973. ^ Olofsson. 5. ^ Freund. 2006.• • • • • • • • • • • • • • • • • • • Chance (disambiguation) Class membership probabilities Decision theory Equiprobable Fuzzy measure theory Game theory Gaming mathematics Information theory Important publications in probability Measure theory Negative probability Probabilistic argumentation Probabilistic logic Random fields Random variable List of scientific journals in probability List of statistical topics Stochastic process Wiener process [edit] Notes 1. Probability and the Art of Judgment. page 29. Management Science. ^ The Logic of Statistical Inference. 1965 2.C.. Johns Hopkins University Press. Ian Hacking. ^ Gorman. "Quantum Leap". The Finance Professionals' Post. 22. ^ Burgi. The Science of Conjecture: Evidence and Probability Before Pascal. Page 26-27. ^ The Emergence of Probability: A Philosophical Study of Early Ideas about Probability. 2009. John. Cambridge University Press. Vladimir. Induction and Statistical Inference. 12. ^ Jeffrey. ^ Olofsson. 1. ^ Olofsson. 9. 2008. 9780521685573 3. page 9 13. ^ Olofsson. p 16 8. pp. ISBN 0-521-39459-7 6. J. 2010. R. pp. (1992). ^ Singh. 113. "Whither Efficient Markets? Efficient Market Theory and Behavioral Finance". A First course in Probability. “Introduction to Probability”. 1. Mark. ^ Ross. 14. 15. Ian Hacking. (2005) Page 8. Peter. Tijana Ivancevic. 2011. p. 127 7. ^ Ivancevic. 10.

Probability.• • • Kallenberg. and Truth. 2nd ed. Laurie Snell Source (GNU Free Documentation License) . Statistics. Preprint: Washington University. Probability Theory: The Logic of Science. Dover edition. [edit] Quotations • • • Damon Runyon. ISBN 0-387-25115-4 Kallenberg. and Stochastic Processes. — HTML index with links to PostScript files and PDF (first three chapters) People from the History of Probability and Statistics (Univ. (listen now) Probability and Statistics EBook Edwin Thompson Jaynes. of Southampton) Probability and Statistics on the Earliest Uses Pages (Univ. but its sources remain unclear because it lacks inline citations. Statistics. Springer Series in Statistics. Peter (2005) Probability. by Charles Grinstead. p 9. [edit] External links This article includes a list of references. New York. "It may be that the race is not always to the swift. O. 1812. 1957)." Pierre-Simon Laplace "It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge. (1996).-Huntsville) Probability on In Our Time at the BBC. ISBN 0-387-95313-2 Olofsson. WileyInterscience. 1981 (republication of second English edition. Springer -Verlag. of Ala. (2005) Probabilistic Symmetries and Invariance Principles. O.eBook. of Southampton) Earliest Uses of Symbols in Probability and Statistics on Earliest Uses of Various Mathematical Symbols A tutorial on probability and Bayes’ theorem devised for first-year Oxford University students pdf file of An Anthology of Chance Operations (1963) at UbuWeb Probability Theory Guide for Non-Mathematicians Understanding Risk and Probability with BBC raw Introduction to Probability . 510 pp. 650 pp. 504 pp ISBN 0-471-67969-0." Théorie Analytique des Probabilités. nor the battle to the strong—but that is the way to bet. related reading or external links. Please improve this article by introducing more precise citations where appropriate. (2002) Foundations of Modern Probability. Richard von Mises "The unlimited extension of the validity of the exact sciences was a characteristic feature of the exaggerated rationalism of the eighteenth century" (in reference to Laplace). (September 2008) Wikibooks has a book on the topic of Probability • • • • • • • • • • • • Virtual Laboratories in Probability and Statistics (Univ.

or survival analysis [show] Applications Category · Portal · Outline · Index Categories: Probability and statistics | Probability | Applied mathematics | Decision theory | Mathematics of computing • • • • • Log in / create account Article Discussion Read Edit . time-series. multivariate.[show]v · d · eLogic [hide]v · d · eAreas of mathematics Arithmetic · Algebra (elementary – linear – multilinear – abstract) · Geometry (Discrete geometry – Algebraic geometry – Differential geometry) · Calculus/Analysis · Set theory · Logic · Category theory · Number theory · Combinatorics · Graph theory · Topology · Lie theory · Differential equations/Dynamical systems · Mathematical physics · Numerical analysis · Computation · Information theory · Probability · Statistics · Optimization · Control theory · Game theory Areas Large Pure mathematics · Applied mathematics · Discrete mathematics · Computational divisions mathematics Category · Mathematics portal · Outline · Lists [hide]v · d · eStatistics [show] Descriptive statistics [show] Data collection [show] Statistical inference [show] Correlation and regression analysis [show] Categorical.

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The term means "differing variance" and comes from the Greek "hetero" ('different') and "skedasis" ('dispersion'). Some authors refer to this as conditional heteroscedasticity to emphasize the fact that it is the sequence of conditional variance that changes and not the unconditional variance. For example. In fact it is possible to observe conditional heteroscedasticity even when dealing with a sequence of unconditional homoscedastic random variables. the sequence {Yt}t=1n is said to be heteroskedastic if the conditional variance of Yt given Xt. White's influential paper (White 1980) used "heteroskedasticity" instead of "heteroscedasticity" whereas subsequent . a sequence of random variables is called homoscedastic if it has constant variance. Please help improve this article by adding more general information. In contrast. a sequence of random variables is heteroscedastic. such as ordinary least squares (OLS). or heteroskedastic. Suppose there is a sequence of random variables {Yt}t=1n and a sequence of vectors of random variables. the free encyclopedia This article only describes one highly specialized aspect of its associated subject. In statistics. {Xt}t=1n. changes with t. In dealing with conditional expectations of Yt given Xt. the opposite does not hold. One of these is that the error term has a constant variance. (October 2009) Plot with random data showing heteroscedasticity. The talk page may contain suggestions. if the random variables have different variances. This might not be true even if the error term is assumed to be drawn from identical distributions. When using some statistical techniques. the error term could vary or increase with each observation. Heteroscedasticity is often studied as part of econometrics. a number of assumptions are typically made. something that is often the case with cross-sectional or time series measurements. which frequently deals with data exhibiting it. however.• Disclaimers • • Heteroscedasticity From Wikipedia.

when properly normalized and centered. This result is used to justify using a normal distribution. the OLS estimator has a normal asymptotic distribution when properly normalized and centered (even when the data does not come from a normal distribution).e. with a variance-covariance matrix that differs from the case of homoscedasticity. or a chi square distribution (depending on how the test statistic is calculated). the OLS estimator in the presence of heteroscedasticity is asymptotically normal. Biased standard errors lead to biased inference. make a type I error). when conducting a hypothesis test. but standard errors and therefore inferences obtained from data analysis are suspect. possibly above or below the true or population variance." With the advent of robust standard errors allowing for inference without specifying the conditional second moment of error term. although it can cause ordinary least squares estimates of the variance (and. An example of the consequence of biased standard error estimation which OLS will produce if heteroskedasticity is present. It is widely known that. under certain assumptions. More precisely. Basic Econometrics (2009) use "heteroscedasticity. White (1980) [1] . This holds even under heteroscedasticity. so results of hypothesis tests are possibly wrong. standard errors) of the coefficients to be biased. thus.Econometrics textbooks such as Gujarati et al. results compelling against the rejection of a null hypothesis as statistically significant when that null hypothesis was in fact uncharacteristic of the actual population (i. Thus. testing conditional homoscedasticity is not as important as in the past. Contents [hide] • • • • • • • 1 Consequences 2 Detection 3 Fixes 4 Examples 5 See also 6 References 7 Further reading [edit] Consequences Heteroscedasticity does not cause ordinary least squares coefficient estimates to be biased.[citation needed] The econometrician Robert Engle won the 2003 Nobel Memorial Prize for Economics for his studies on regression analysis in the presence of heteroscedasticity. is that a researcher may find at a selected confidence level. which led to his formulation of the Autoregressive conditional heteroskedasticity (ARCH) modeling technique. regression analysis using heteroscedastic data will still provide an unbiased estimate for the relationship between the predictor variable and the outcome..

a hypothesis that is going to be tested (the null hypothesis)." (Mankiw 1990. This validates the use of hypothesis testing using OLS estimators and White's variance-covariance estimator under heteroscedasticity. 400)[4] students in Econometrics should not overreact to heteroskedasticity. in general. in performing hypothesis tests. p. "unequal error variance is worth correcting only when the problem is severe. 400) And another word of caution from Mankiw. Heteroscedasticity is also a major practical issue encountered in ANOVA problems. 306) [5] (Cited in Gujarati et al." (Fox 1997. (Gujarati et al. "heteroscedasticity has never been a reason to throw out an otherwise good model. There are several methods to test for the presence of heteroscedasticity: • • • • • • • • • • Park test (1966)[7] Glejser test (1969) White test (1980) Breusch–Pagan test Goldfeld–Quandt test Cook–Weisberg test Harrison–McCabe test Spearman rank correlation coefficient Brown–Forsythe test Levene test These methods consist. p. as Gujarati et al.proposed a consistent estimator for the variance-covariance matrix of the asymptotic distribution of the OLS estimator. 1648)[6] (Cited in Gujarati et al.[2] The F test can still be used in some circumstances. These tests consist of a statistic (a mathematical expression). p. 2009. noted. 400) [edit] Detection Absolute value of residuals for simulated first order Heteroskedastic data. .[3] However. John Fox (the author of Applied Regression Analysis) wrote. 2009. 2009. p. p.

. the standard errors are equivalent to conventional standard errors estimated by ols.a. in most cases. Those with higher incomes display a greater variability of food consumption. however. be rather stable. for pedagogical reasons. a wealthier person may occasionally buy inexpensive food and at other times eat expensive meals. however. asymptotic theory). As one's income increases. Unlogged series that are growing exponentially often appear to have increasing variability as the series rises over time. The White method corrects for heteroscedasticity without altering the values of the coefficients. [edit] Examples Heteroscedasticity often occurs when there is a large difference among the sizes of the observations. depending on the changing error variances. Use a different specification for the model (different X variables. Many introductory statistics and econometrics books. if the data is homoscedastistic.an alternative hypothesis. or perhaps non-linear transformations of the X variables). in which OLS is applied to transformed or weighted values of X and Y. this assumption can be relaxed by using asymptotic distribution theory (a.k. present these tests under the assumption that the data set in hand comes from a normal distribution. the variability of food consumption will increase. while still biased. • A classic example of heteroscedasticity is that of income versus expenditure on meals. A great misconception is the thought that this assumption is necessary. Several modifications of the White method of computing heteroscedasticity-consistent standard errors have been proposed as corrections with superior finite sample properties. Heteroscedasticity-consistent standard errors (HCSE). The variability in percentage terms may. A poorer person will spend a rather constant amount by always eating less expensive food. Apply a weighted least squares estimation method. This is done by realizing that the distributional statement about the statistic (the mathematical expression) can be approximated by using asymptotic theory[citation needed]. Most of the methods of detecting heteroscedasticity presented here can be used even when the data do not come from a normal distribution. and a distributional statement about the statistic (the mathematical expression). This method may be superior to regular OLS because if heteroscedasticity is present it corrects for it. [edit] Fixes There are three common corrections for heteroscedasticity: • • • • View Logged data. HCSE is a consistent estimator of standard errors in regression models with heteroskedasticity. improve upon OLS estimates (White 1980). The weights vary over observations. however.

However. 4.2307/1912934.1016/j.09. Using Econometrics (2nd ed. 3. "The ANOVA F test can still be used in some balanced designs with unequal variances and nonnormal data". In the first couple of seconds your measurements may be accurate to the nearest centimeter. (2009) Basic Econometrics. Halbert (1980). (devotes a chapter to heteroscedasticity) . Park (1966). [edit] Further reading Most statistics textbooks will include at least some material on heteroscedasticity. Weerahandi. doi:10. 6. J. ^ Mankiw. E. say.• Imagine you are watching a rocket take off nearby and measuring the distance it has traveled once each second. because of the increased distance. (1990) A Quick Refresher Course in Macroeconomics. "Size performance of some tests in one-way anova". Linear Models. XXVIII. atmospheric distortion and a variety of other factors. "A heteroscedasticity-consistent covariance matrix estimator and a direct test for heteroscedasticity". McGraw-Hill. ^ R. California:Sage Publications. ^ Bathke. Ninth Edition. 1. Jinadasa. JSTOR 1912934.010. ISBN 0-673-52125-7. (1997) Applied Regression Analysis. Communications in Statistics . ISBN 0919808813500. [edit] See also • • • • • • • Kurtosis (peakedness) Breusch–Pagan test of heteroscedasticity of the residuals of a linear regression Regression analysis homoscedasticity Autoregressive conditional heteroscedasticity (ARCH) White test Heteroscedasticity-consistent standard errors [edit] References ^ White. ^ Gujarati. Journal of Statistical Planning and Inference 126 (2): 413.2003.2307/1910108. "Estimation with Heteroscedastic Error Terms". Some examples are: • Studenmund AH. A (2004). Vol. December. ^ Fox. and Related Methods. The data you collect would exhibit heteroscedasticity. doi:10.jspi. Econometrica 34 (4): 888. N. Econometrica 48 (4): 817–838.). 2. G. doi:10. doi:10. 7. & Porter. 5. Sam (1998).1080/03610919808813500. Journal of Economics Literature. D. JSTOR 1910108. C.Simulation and Computation 27 (3): 625. D. ^ Gamage. the accuracy of your measurements may only be good to 100 m. 5 minutes later as the rocket recedes into space. N.

"A heteroscedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroscedasticity". W. ISBN 0-13-013297-7. Time Series Analysis.D.• • • • • Verbeek. Marilena (2005). pages Greene.D. Sankhya – the Indian Journal of Statistics. (2008): Hands On Intermediate Econometrics Using R: Templates for Extending Dozens of Practical Examples. Econometrica 48 (4): 817–838. Special Issue on Quantile Regression and Related Methods 67 (2): 335–358. J. an introductory but thorough general text. 2005) heteroscedasticity in QSAR Modeling [hide]v · d · eStatistics [show] Descriptive statistics [show] Data collection [show] Statistical inference [show] Correlation and regression analysis [show] Categorical. NJ . Italy. Halbert (1980). time-series. the text of reference for historical series analysis. Chichester: John Wiley & Sons. White. 2004. Hamilton. or survival analysis [show] Applications Category · Portal · Outline · Index Categories: Statistical deviation and dispersion | Time series analysis | Econometrics • • Log in / create account Article . 2. ISBN 10-981-281-885-5 (Section 2. Princeton University Press ISBN 0-69104289-6.H. ed. H.. Vinod. Special subjects • • Glejser test: Furno. it contains an introduction to ARCH models. doi:10. "The Glejser Test and the Median Regression". JSTOR 1912934. (work done at Universita di Cassino. considered the standard for a pre-doctorate university Econometrics course.8 provides R snippets) World Scientific Publishers: Hackensack. Prentice–Hall. Econometric Analysis. Marno (2004): A Guide to Modern Econometrics. (1993). multivariate. (1994).2307/1912934.

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• • • • Contact us Privacy policy About Wikipedia Disclaimers • • Durbin–Watson statistic From Wikipedia. Later. the small sample distribution of this ratio was derived in a pathbreaking article by John von Neumann (von Neumann. Durbin and Watson (1950. 1983). 1951) applied this statistic to the residuals from least squares regressions. It is named after James Durbin and Geoffrey Watson. and developed bounds tests for the null hypothesis that the errors are serially independent (not autocorrelated) against the alternative that they follow a first order autoregressive process. 1941). the free encyclopedia In statistics. Contents [hide] • • • • • • • • 1 Computing and interpreting the Durbin–Watson statistic 2 Durbin h-statistic 3 Durbin–Watson test for panel data 4 Implementations in statistics packages 5 See also 6 Notes 7 References 8 External links [edit] Computing and interpreting the Durbin–Watson statistic . However. the Durbin–Watson statistic is a test statistic used to detect the presence of autocorrelation (a relationship between values separated from each other by a given time lag) in the residuals (prediction errors) from a regression analysis. John Denis Sargan and Alok Bhargava developed several von Neumann–Durbin–Watson type test statistics for the null hypothesis that the errors on a regression model follow a process with a unit root against the alternative hypothesis that the errors follow a stationary first order autoregression (Sargan and Bhargava.

An important note is that the Durbin–Watson statistic.α. If the Durbin–Watson statistic is substantially less than 2. If d > 2 successive error terms are.α. • If d > dU.α. there is evidence of positive serial correlation. The value of d always lies between 0 and 4. Since d is approximately equal to 2(1 − r). negatively correlated.α and dU. there may be cause for alarm.0. the test is inconclusive.α < (4 − d) < dU. then the test statistic is where T is the number of observations.α): If d < dL. vary by level of significance (α).If et is the residual associated with the observation at time t. close in value to one another. • If dL. on average. • If (4 − d) > dU. is not relevant in many situations. where r is the sample autocorrelation of the residuals. In regressions.α. this can imply an underestimation of the level of statistical significance. if Durbin–Watson is less than 1. For instance. the number of observations. while displayed by many regression analysis programs. or if the dependent variable is in a lagged form as an independent variable. there is statistical evidence that the error terms are not negatively autocorrelated.α and dU.α): If (4 − d) < dL. if the error distribution is not normal. • To test for negative autocorrelation at significance α.α. Small values of d indicate successive error terms are. • The critical values.e. there is statistical evidence that the error terms are negatively autocorrelated.. Their derivation is complex— statisticians typically obtain them from the appendices of statistical texts. much different in value to one another.[1] d = 2 indicates no autocorrelation. if there is higher-order autocorrelation. and the number of predictors in the regression equation. i. the test is inconclusive.α and dU.α < d < dU.α. the test statistic d is compared to lower and upper critical values (dL. on average. A suggested test that does not have these limitations is the Breusch–Godfrey (serial correlation LM) Test. As a rough rule of thumb. this is not an appropriate test for autocorrelation. there is statistical evidence that the error terms are not positively autocorrelated. or positively correlated. [edit] Durbin h-statistic . To test for positive autocorrelation at significance α.α. • If dL. dL. the test statistic (4 − d) is compared to lower and upper critical values (dL. there is statistical evidence that the error terms are positively autocorrelated.

the Breusch–Godfrey test. Mathematica: the Durbin–Watson (d) statistic is included as an option in the LinearModelFit function.in times series data. (1982). associated with the observation in panel i at time t. MATLAB: the dwtest function in the Statistics Toolbox.The Durbin–Watson statistic is biased for autoregressive moving average models. 2. SAS: Is a standard output when using proc model and is an option (dw) when using proc reg. K (number of regressors) and N (number of individuals in the panel). But for large samples one can easily compute the unbiased normally distributed h-statistic: using the Durbin–Watson statistic d and the estimated variance of the regression coefficient of the lagged dependent variable. 3. Stata: the command -estat dwatson-. [edit] Implementations in statistics packages 1. (1982). so that autocorrelation is underestimated. 5. (1982): If ei. These values are calculated dependent on T (length of the balanced panel—time periods the individuals were surveyed). then the test statistic is This statistic can be compared with tabulated rejection values [see Alok Bhargava et al. provided [edit] Durbin–Watson test for panel data For panel data this statistic was generalized as follows by Alok Bhargava et al. following -regress. t is the residual from an OLS regression with fixed effects for each panel i. R: the dwtest function in the lmtest package. Engle's LM test for autoregressive conditional heteroskedasticity (ARCH). This test statistic can also be used for testing the null hypothesis of a unit root against stationary alternatives in fixed effects models using another set of bounds (Tables V and VI) tabulated by Alok Bhargava et al. and Durbin's alternative test for serial correlation are . a test for time-dependent volatility. page 537]. 4.

(1951) "Testing for Serial Correlation in Least Squares Regression. D. 367–395. and Alok Bhargava (1983). (1941). J. and Watson. Damodar N.N.D. (1982): "Serial Correlation and the Fixed Effects Model". 51. II. page 605f. ed.. 12. Verbeek. McGraw–Hill Gujarati. S. ^ Gujarati (2003) p. p. EXCEL: although Microsoft Excel 2007 does not have a specific Durbin–Watson function. 159–179. 49. 6. I. 2004. (1995): Basic Econometrics. Franzini. "Distribution of the ratio of the mean square successive difference to the variance". 3. 2. Durbin.y_array)/SUMSQ(array)" [edit] See also • • Time-series regression ACF / PACF [edit] Notes 1... 153–174. 1995. G. Seite 102f. von Neumann. 409–428. Sargan.: McGraw– Hill." Biometrika 38. L.. (2003) Basic econometrics. J. Econometrica. [edit] External links • Table for high n and k Categories: Econometrics | Statistical tests | Time series analysis • • • Log in / create account Article Discussion . Gujarati. Chichester: John Wiley & Sons.also available. John. 533–549. G. ed.. W. The Breusch–Godfrey test and Durbin's alternative test also allow regressors that are not strictly exogenous. (1950) "Testing for Serial Correlation in Least Squares Regression.. and Watson." Biometrika 37. All (except -dwatson-) tests separately for higher-order serial correlations. p. the d-statistic may be calculated using "=SUMXMY2(x_array. Narendranathan. 4th ed. "Testing residuals from least squares regression for being generated by the Gaussian random walk". 469 [edit] References • • • • • • • • Bhargava. Alok. Durbin. J. Review of Economic Studies. Boston. Annals of Mathematical Statistics. New York et al. Marno (2004): A Guide to Modern Econometrics. S.

Contact us Privacy policy About Wikipedia Disclaimers • • • • . See Terms of Use for details.. a non-profit organization. Inc. Wikipedia® is a registered trademark of the Wikimedia Foundation. additional terms may apply.• • • þÿ Read Edit View history • • • • • • Main page Contents Featured content Current events Random article Donate to Wikipedia Interaction • • • • • Toolbox Print/export Languages • • • • • • • • • Help About Wikipedia Community portal Recent changes Contact Wikipedia Deutsch Italiano Norsk (bokmål) Русский Türkçe Українська 中文 This page was last modified on 28 April 2011 at 17:09. Text is available under the Creative Commons Attribution-ShareAlike License.

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