A mathematical model in which the outcome is exactly determined thro.pdf

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A deterministic model is one where outcomes are exactly determined through known relationships, without random variation, and a specific input always produces the same output, as seen in classical physics models. Statistical models, which econometrics is more interested in, involve stochastic relationships. Regression analysis examines the dependence of one variable on others, but dependence does not imply causation, which can only be plausibly claimed under certain conditions. Cross-sectional data involves many units observed at a single point in time, time-series data tracks a single unit over multiple times, and panel data follows multiple units over multiple times.

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A mathematical model in which the outcome is exactly determined through known
relationships between events, without any space for random variation, is called a
deterministic model. In these types of models, a specific input will always produce the
same output. Examples of these type of models are often found in fields like classical
physics, such as Newtons law of gravity. In econometrics, we are, however, more
interested in statistical/stochastic relationships. Using an example, define and discuss
statistical models. (5)
1.2) Regression analysis deals with the dependence of one variable on other variables.
However, dependence does not imply causation. When can causation be plausibly
claimed?
(2)
1.3) Explain the difference between cross-sectional, time-series and panel data? Provide
examples of each. (3

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RESEARCH ARTICLE systematic process analysis: when and how to use it1 peter a. hall Harvard University, Minda de Gunzburg Center for European Studies, 27 Kirkland Street, Cambridge, MA 02138, USA E-mail: [email protected] doi:10.1057/palgrave.eps.2210130 Abstract Challenging the contention that statistical methods applied to large numbers of cases invariably provide better grounds for causal inference, this article explores the value of a method of systematic process analysis that can be applied in a small number of cases. It distinguishes among three modes of explanation – historically specific, multivariate and theory- oriented – and argues that systematic process analysis has special value for developing theory-oriented explanations. It outlines the steps required to perform such analysis well and illustrates them with reference to Owen’s investigation of the ‘democratic peace’. Comparing the results available from this kind of method with those from statistical analysis, it examines the conditions under which each method is warranted. Against conceptions of the ‘comparative method’, which imply that small-n case-studies provide weak grounds for causal inference, it argues that the intensive examination of a small number of cases can be an appropriate research design for testing such inferences. Keywords systematic process analysis; methods; small-n; explanation F or securing causal explanations of social or political phenomena, how useful are research designs based on the intensive investigation of a small number of cases? A series of works, from Lijphart’s (1971) seminal article on the comparative method to the influential text of King et al (1994), declare them inferior to designs that apply statistical methods to a large number of cases. By contrast, this article argues that ‘small-n’ research designs can be valuable for causal inference in the field of management studies and social science more generally, espe- cially if a methodological approach that I term ‘systematic process analysis’ is applied to the cases. In order to make the case I outline several modes of explanation in social science, describe what ‘systematic process analysis’ entails, show how one analysis uses it, and consider when it might most usefully be employed. european political science: 7 2008 (304 – 317) & 2008 European Consortium for Political Research. 1680-4333/08 www.palgrave-journals.com/eps 304 MODES OF EXPLANATION IN SOCIAL SCIENCE In what does a causal explanation for a social phenomenon consist? Perhaps because this question is daunting, few empirical works consider it, but, if we are to compare methods and research designs, the issue cannot be avoided, although many aspects of it, such as the role of interpretation, cannot be covered here.2 To delimit the issues, I adopt a positivist perspective that sees the pro- blem as one of identifying a set of variables (x1y xn), understood as events or phenomena whose ‘value’ can vary acro ...

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What does it mean for something to be a \"dynamical system? What is a \"dynamical equation? What does it mean if a system is \" sensitive to initial conditions\"? Give one example of a dynamical system that has been shown to be sensitive to initial conditions. Solution Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. Dynamical systems are mathematical objects used to model physical phenomena whose state (or instantaneous description) changes over time. These models are used in financial and economic forecasting, environmental modeling, medical diagnosis, industrial equipment diagnosis, and a host of other applications. So a simple, if slightly imprecise, way of describing chaos is \"chaotic systems are distinguished by sensitive dependence on initial conditions and by having evolution through phase space that appears to be quite random.\" In particular, a chaotic dynamical system is generally characterized by 1. Having a dense collection of points with periodic orbits, 2. Being sensitive to the initial condition of the system (so that initially nearby points can evolve quickly into very different states), a property sometimes known as the butterfly effect, and 3. Being topologically transitive. Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. Dynamical systems are mathematical objects used to model physical phenomena whose state (or instantaneous description) changes over time. These models are used in financial and economic forecasting, environmental modeling, medical diagnosis, industrial equipment diagnosis, and a host of other applications. So a simple, if slightly imprecise, way of describing chaos is \"chaotic systems are distinguished by sensitive dependence on initial conditions and by having evolution through phase space that appears to be quite random.\" In particular, a chaotic dynamical system is generally characterized by 1. Having a dense collection of points with periodic orbits, 2. Being sensitive to the initial condition of the system (so that initially nearby points can evolve quickly into very different states), a property sometimes known as the butterfly effect, and 3. Being topologically transitive..

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Thermodynamics has been the foundation of many models of biological and technological systems. But thermodynamics is static and is misnamed. A more suitable name is thermostatics. Thermostatics does not include time as a variable and so has no velocity, flow or friction. Indeed, as usually formulated, thermostatics does not include boundary conditions. Devices require boundary conditions to define their input and output. They usually involve flow and friction. Thermostatics is an unsuitable foundation for understanding technological and biological devices. A time dependent generalization of thermostatics that might be called thermal dynamics is being developed by Chun Liu and collaborators to avoid these limitations. Electrodynamics is not restricted like thermostatics, but in its classical formulation involves drastic assumptions about polarization and an over-approximated dielectric constant. Once the Maxwell equations are rewritten without a dielectric constant, they are universal and exact. Conservation of total current,including displacement current, is a restatement of the Maxwell equations that leads to dramatic simplifications in the understanding of one dimensional systems, particularly those without branches, like the ion channel proteins of biological membranes and the two terminal devices of electronic systems. The Brownian fluctuations of concentrations and fluxes of ions become the spatially independent total current, because the displacement current acts as an unavoidable low pass filter, a consequence of the Maxwell equations for any material polarization. Electrodynamics and thermal dynamics together form a suitable foundation for models of technological and biological systems.

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