In statistics, the **Johansen test**,^{ [1] } named after Søren Johansen, is a procedure for testing cointegration of several, say *k*, I(1) time series.^{ [2] } 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.^{ [3] }

There are two types of Johansen test, either with trace or with eigenvalue, and the inferences might be a little bit different.^{ [4] } 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** + 1 and, again, testing proceeds sequentially for *r** = 1,2,etc., with the first non-rejection used as an estimator for *r*.

Just like a unit root test, there can be a constant term, a trend term, both, or neither in the model. For a general VAR(*p*) model:

There are two possible specifications for error correction: that is, two vector error correction models (VECM):

1. The longrun VECM:

- where

2. The transitory VECM:

- where

Be aware that the two are the same. In both VECM,

Inferences are drawn on Π, and they will be the same, so is the explanatory power.^{[ citation needed ]}

In physics, the **Lorentz transformations** are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parametrized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.

In mathematical physics and mathematics, the **Pauli matrices** are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries. They are

In mechanics and geometry, the **3D rotation group**, often denoted **SO(3)**, is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.

In mathematics, **Green's theorem** gives the relationship between a line integral around a simple closed curve *C* and a double integral over the plane region *D* bounded by *C*. It is the two-dimensional special case of Stokes' theorem.

In quantum mechanics and quantum field theory, the **propagator** is a function that specifies the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. These may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called *(causal) Green's functions*.

In statistics, econometrics and signal processing, an **autoregressive** (**AR**) **model** is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term ; thus the model is in the form of a stochastic difference equation. Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable.

**Arc length** is the distance between two points along a section of a curve.

In electromagnetism, the **electromagnetic tensor** or **electromagnetic field tensor** is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely.

A **theoretical motivation for general relativity**, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation *a priori*. This provides a means to inform and verify the formalism.

The **Newman–Penrose** (**NP**) **formalism** is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.

In mathematics, the **Schur orthogonality relations**, which is proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3).

The **Liénard–Wiechert potentials** describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations, these describe the complete, relativistically correct, time-varying electromagnetic field for a point charge in arbitrary motion, but are not corrected for quantum-mechanical effects. Electromagnetic radiation in the form of waves can be obtained from these potentials. These expressions were developed in part by Alfred-Marie Liénard in 1898 and independently by Emil Wiechert in 1900.

In mathematics, **Maass forms** or **Maass wave forms** are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup of as modular forms. They are Eigenforms of the hyperbolic Laplace Operator defined on and satisfy certain growth conditions at the cusps of a fundamental domain of . In contrast to the modular forms the Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949.

An **error correction model (ECM)** belongs to a category of multiple time series models most commonly used for data where the underlying variables have a long-run stochastic trend, also known as cointegration. ECMs are a theoretically-driven approach useful for estimating both short-term and long-term effects of one time series on another. The term error-correction relates to the fact that last-period's deviation from a long-run equilibrium, the *error*, influences its short-run dynamics. Thus ECMs directly estimate the speed at which a dependent variable returns to equilibrium after a change in other variables.

In mathematics, the **Butcher group**, named after the New Zealand mathematician John C. Butcher by Hairer & Wanner (1974), is an infinite-dimensional Lie group first introduced in numerical analysis to study solutions of non-linear ordinary differential equations by the Runge–Kutta method. It arose from an algebraic formalism involving rooted trees that provides formal power series solutions of the differential equation modeling the flow of a vector field. It was Cayley (1857), prompted by the work of Sylvester on change of variables in differential calculus, who first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics.

In mathematics, **Ricci calculus** constitutes the rules of index notation and manipulation for tensors and tensor fields in a Riemannian manifold. It is also the modern name for what used to be called the **absolute differential calculus**, developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century.

In quantum field theory, and in the significant subfields of quantum electrodynamics (QED) and quantum chromodynamics (QCD), the **two-body Dirac equations (TBDE)** of constraint dynamics provide a three-dimensional yet manifestly covariant reformulation of the Bethe–Salpeter equation for two spin-1/2 particles. Such a reformulation is necessary since without it, as shown by Nakanishi, the Bethe–Salpeter equation possesses negative-norm solutions arising from the presence of an essentially relativistic degree of freedom, the relative time. These "ghost" states have spoiled the naive interpretation of the Bethe–Salpeter equation as a quantum mechanical wave equation. The two-body Dirac equations of constraint dynamics rectify this flaw. The forms of these equations can not only be derived from quantum field theory they can also be derived purely in the context of Dirac's constraint dynamics and relativistic mechanics and quantum mechanics. Their structures, unlike the more familiar two-body Dirac equation of Breit, which is a single equation, are that of two simultaneous quantum relativistic wave equations. A single two-body Dirac equation similar to the Breit equation can be derived from the TBDE. Unlike the Breit equation, it is manifestly covariant and free from the types of singularities that prevent a strictly nonperturbative treatment of the Breit equation.

In the Newman–Penrose (NP) formalism of general relativity, independent components of the Ricci tensors of a four-dimensional spacetime are encoded into seven **Ricci scalars** which consist of three real scalars , three complex scalars and the NP curvature scalar . Physically, Ricci-NP scalars are related with the energy–momentum distribution of the spacetime due to Einstein's field equation.

**Stellar aberration** is an astronomical phenomenon "which produces an apparent motion of celestial objects". It can be proven mathematically that stellar aberration is due to the change of the astronomer's inertial frame of reference. The formula is derived with the use of Lorentz transformation of the star's *coordinates*.

In theoretical physics, **Hamiltonian field theory** is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory.

- ↑ Johansen, Søren (1991). "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models".
*Econometrica*.**59**(6): 1551–1580. JSTOR 2938278. - ↑ For the presence of I(2) variables see Ch. 9 of Johansen, Søren (1995).
*Likelihood-based Inference in Cointegrated Vector Autoregressive Models*. Oxford University Press. - ↑ Davidson, James (2000).
*Econometric Theory*. Wiley. ISBN 0-631-21584-0. - ↑ Hänninen, R. (2012). "The Law of One Price in United Kingdom Soft Sawnwood Imports – A Cointegration Approach".
*Modern Time Series Analysis in Forest Products Markets*. Springer. p. 66. ISBN 978-94-011-4772-9.

- Banerjee, Anindya; et al. (1993).
*Co-Integration, Error Correction, and the Econometric Analysis of Non-Stationary Data*. New York: Oxford University Press. pp. 266–268. ISBN 0-19-828810-7. - Favero, Carlo A. (2001).
*Applied Macroeconometrics*. New York: Oxford University Press. pp. 56–71. ISBN 0-19-829685-1. - Hatanaka, Michio (1996).
*Time-Series-Based Econometrics: Unit Roots and Cointegration*. New York: Oxford University Press. pp. 219–246. ISBN 0-19-877353-6. - Maddala, G. S.; Kim, In-Moo (1998).
*Unit Roots, Cointegration, and Structural Change*. Cambridge University Press. pp. 198–248. ISBN 0-521-58782-4.

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