From macro to micro variables: underdetermination in addition to network effects – how can

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From macro to micro variables: underdetermination in addition to network effects – how can

Blackwell, Laura, Contributing Editor has reference to this Academic Journal, PHwiki organized this Journal From macro to micro variables: underdetermination in addition to network effects – how can machine learning techniques helpSeptember 2016Alex in addition to er DenevJoint work with:Orazio AngeliniThe ContextIn certain practical applications we must assess the impact of a change of high level variables on a more granular structure of variables e.g.Calculating the effect of a shock to an index on a more granular portfolioCalculating the effect of a change of a macroeconomic variable on a network of companiesWe are facing the task of modelling both the exogenous shock effect on the network in addition to the network endogenous effects21,000 US EquitiesJoint Distribution3Example: Bivariate Gaussian Distribution

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The precision matrix – what it isThe precision matrix – example5Gaussian Markov NetworksIf we start from a multivariate Gaussian we can cast in the as long as m:And associate a graphical model in which two nodes (variables) are not connected if the corresponding precision matrix element is 0612345

Gaussian Markov Networks – Example7Estimation8GLASSO – Introduction9

Network Effects – What are theyThe presence of network links between variables may be due to:Omitted observable macro factorsOmitted non-observable factorsIdiosyncratic relationships10Chain GraphsLet G =(V,E) be a mixed graph with finite vertex set V in addition to an edge set E that may contain two types of edges, namely directed (uv) in addition to undirected (u-v) edgesThe graph G is called a chain graph if it does not contain any semi-directed cycles, that is, it contains no path from v to v with at least one directed edge such that all directed edges have the same orientation11S&P500Equity 1Equity 2Equity 3Chain GraphsA Chain Graph represents a Multivariate Gaussian which can be decomposed in recursive as long as m. For example, as long as the chain graph of the previous slide12Network effects

Network Effects – What are theyInserting an extra factor can explain some of the links away13S&P500Equity 1Equity 2Equity 3FXOmitted factorS&P500Equity 1Equity 2Equity 3Network Effects – What are theyAn unobserved factor can also remove links14S&P500Equity 1Equity 2Equity 3FXS&P500Equity 1Equity 2Equity 3FXXUnobserved factorChain Graphs – EstimationWe decompose the estimation of the Chain Graph in two stepsEstimation of the loadings on the macro factor(s)Estimation of the networkTwo steps estimation procedure (Drton (2006), McCarter (2014))15

The task1,000 US EquitiesIn the end we want to obtain a distribution Perturbations in addition to their effectPerturbing a factor that feeds in the network in addition to reading the results17We fix thisUnder-determination of the task

Under-determination of the taskFirst approach: exp in addition to the shock directly to the stocksS&P500Equity 1Equity 2Equity 3Under-determination of the taskSecond possible approach: exp in addition to the shock directly to the stocks by introducing network effectsS&P500Equity 1Equity 2Equity 3Under-determination of the taskThird possible approach: exp in addition to the shock by passing through 1 intermediate layer of industry indicesS&P500Equity 1Equity 2Equity 3Industry 1Industry 2

Under-determination of the taskForth possible approach: exp in addition to the shock by passing through 1 intermediate layer of industry indices in addition to by adding network effects in the last layerS&P500Equity 1Equity 2Equity 3Industry 1Industry 2Automatic selectionThere are roughly speaking three approaches to automatic learning: 1. Constrained based – it views a structure as a set of independence relationships. The search algorithm tests as long as conditional dependencies in addition to independencies in the data in addition to hence learns the structure that best explains it.2. Score based – a hypothesis space is defined, that is a set of c in addition to idate structures, in addition to a scoring function that measures how well the models fit the data. The learning is addressed as a model selection problem. The computational task is to find the highest-scoring structure.3. Bayesian model averaging – it does not try to learn a single structure but an ensemble of them in addition to averages their predictions i.e.23Results24

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