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Johansen Test
In statistics, the Johansen test, named after Søren Johansen, is a procedure for testing cointegration of several, say ''k'', I(1) time series. This test permits more than one cointegrating relationship so is more generally applicable than the Engle–Granger test which is based on the Dickey–Fuller (or the augmented) test for unit roots in the residuals from a single (estimated) cointegrating relationship. There are two types of Johansen test, either with trace or with eigenvalue, and the inferences might be a little bit different. The null hypothesis for the trace test is that the number of cointegration vectors is ''r'' = ''r''* < ''k'', vs. the alternative that ''r'' = ''k''. Testing proceeds sequentially for ''r''* = 1,2, etc. and the first non-rejection of the null is taken as an estimate of ''r''. The null hypothesis for the "maximum eigenvalue" test is as for the trace test but the alternative is ''r'' = ''r''*& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Econometrica
''Econometrica'' is a peer-reviewed academic journal of economics, publishing articles in many areas of economics, especially econometrics. It is published by Wiley-Blackwell on behalf of the Econometric Society. The current editor-in-chief is Guido Imbens. History ''Econometrica'' was established in 1933. Its first editor was Ragnar Frisch, recipient of the first Nobel Memorial Prize in Economic Sciences in 1969, who served as an editor from 1933 to 1954. Although ''Econometrica'' is currently published entirely in English, the first few issues also contained scientific articles written in French. Indexing and abstracting ''Econometrica'' is abstracted and indexed in: * Scopus * EconLit * Social Science Citation Index According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 5.844, ranking it 22/557 in the category "Economics". Awards issued The Econometric Society aims to attract high-quality applied work in economics for publication in ''Eco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Søren Johansen
Søren Johansen (born 6 November 1939) is a Danish statistician and econometrician who is known for his contributions to the theory of cointegration. He is currently a professor at the Department of Economics, University of Copenhagen and in the Center for Research in Econometric Analysis of Time Series (CREATES) of the Aarhus University. He has previously held positions at the Department of Statistics, University of Copenhagen, and the European University Institute in Florence. Biography Early life Johansen was born in 1939 in Denmark. Academic life Johansen graduated from the University of Copenhagen in mathematical statistics. He began his academic career at the University of Copenhagen, Institute of Mathematical Statistics in 1964 and was promoted to full professor in 1989. In 1967 he obtained the Gold medal from University of Copenhagen for the thesis "An application of extreme points methods in probability" and in 1974, he became dr. phil. with the thesis "The embedding p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cointegration
Cointegration is a statistical property of a collection of time series variables. First, all of the series must be integrated of order ''d'' (see Order of integration). Next, if a linear combination of this collection is integrated of order less than d, then the collection is said to be co-integrated. Formally, if (''X'',''Y'',''Z'') are each integrated of order ''d'', and there exist coefficients ''a'',''b'',''c'' such that is integrated of order less than d, then ''X'', ''Y'', and ''Z'' are cointegrated. Cointegration has become an important property in contemporary time series analysis. Time series often have trends—either deterministic or stochastic. In an influential paper, Charles Nelson and Charles Plosser (1982) provided statistical evidence that many US macroeconomic time series (like GNP, wages, employment, etc.) have stochastic trends. Introduction If two or more series are individually integrated (in the time series sense) but some linear combination of them has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Of Integration
In statistics, the order of integration, denoted ''I''(''d''), of a time series is a summary statistic, which reports the minimum number of differences required to obtain a covariance-stationary series. Integration of order ''d'' A time series is integrated of order ''d'' if :(1-L)^d X_t \ is a stationary process, where L is the lag operator and 1-L is the first difference, i.e. : (1-L) X_t = X_t - X_ = \Delta X. In other words, a process is integrated to order ''d'' if taking repeated differences ''d'' times yields a stationary process. In particular, if a series is integrated of order 0, then (1-L)^0 X_t = X_t is stationary. Constructing an integrated series An ''I''(''d'') process can be constructed by summing an ''I''(''d'' − 1) process: *Suppose X_t is ''I''(''d'' − 1) *Now construct a series Z_t = \sum_^t X_k *Show that ''Z'' is ''I''(''d'') by observing its first-differences are ''I''(''d'' − 1): :: \Delta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. A time series is very frequently plotted via a run chart (which is a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements. Time series ''analysis'' comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series ''forecasting' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dickey–Fuller Test
In statistics, the Dickey–Fuller test tests the null hypothesis that a unit root is present in an autoregressive time series model. The alternative hypothesis is different depending on which version of the test is used, but is usually stationarity or trend-stationarity. The test is named after the statisticians David Dickey and Wayne Fuller, who developed it in 1979. Explanation A simple AR(1) model is : y_=\rho y_+u_\, where y_ is the variable of interest, t is the time index, \rho is a coefficient, and u_ is the error term (assumed to be white noise). A unit root is present if \rho = 1. The model would be non-stationary in this case. The regression model can be written as : \Delta y_=(\rho-1)y_+u_=\delta y_+ u_\, where \Delta is the first difference operator and \delta \equiv \rho - 1. This model can be estimated and testing for a unit root is equivalent to testing \delta = 0. Since the test is done over the residual term rather than raw data, it is not possible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augmented Dickey–Fuller Test
In statistics, an augmented Dickey–Fuller test (ADF) tests the null hypothesis that a unit root is present in a time series sample. The alternative hypothesis is different depending on which version of the test is used, but is usually stationarity or trend-stationarity. It is an augmented version of the Dickey–Fuller test for a larger and more complicated set of time series models. The augmented Dickey–Fuller (ADF) statistic, used in the test, is a negative number. The more negative it is, the stronger the rejection of the hypothesis that there is a unit root at some level of confidence. Testing procedure The testing procedure for the ADF test is the same as for the Dickey–Fuller test but it is applied to the model :\Delta y_t = \alpha + \beta t + \gamma y_ + \delta_1 \Delta y_ + \cdots + \delta_ \Delta y_ + \varepsilon_t, where \alpha is a constant, \beta the coefficient on a time trend and p the lag order of the autoregressive process. Imposing the constraints \alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Root
In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if 1 is a root of the process's characteristic equation. Such a process is non-stationary but does not always have a trend. If the other roots of the characteristic equation lie inside the unit circle—that is, have a modulus (absolute value) less than one—then the first difference of the process will be stationary; otherwise, the process will need to be differenced multiple times to become stationary. If there are ''d'' unit roots, the process will have to be differenced ''d'' times in order to make it stationary. Due to this characteristic, unit root processes are also called difference stationary. Unit root processes may sometimes be confused with trend-stationary processes; while they share many properties, they are different in many asp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace (linear Algebra)
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proved that for any two matrices and . This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the th row and th column of . The entries of can be real numbers or (more generally) complex numbers. The trace is not de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
