Linear Models Are . . . General Ubiquity of . . . Linear Models in . . . Why Threshold Models: Need to Go Beyond . . . The Above Idea Works . . . A Theoretical Explanation Why the Name . . . Linear Models: Reminder Thongchai Dumrongpokaphan 1 , Vladik Kreinovich 2 , Towards a More . . . and Songsak Sriboonchitta 1 Home Page Title Page 1 Chiang Mai University, Thailand tcd43@hotmail.com, songsakecon@gmail.com ◭◭ ◮◮ 2 University of Texas at El Paso, El Paso, Texas 79968, USA ◭ ◮ vladik@utep.edu Page 1 of 28 Go Back Full Screen Close Quit
Linear Models Are . . . 1. Linear Models Are Often Successful in Econo- General Ubiquity of . . . metrics Linear Models in . . . Need to Go Beyond . . . • In econometrics, often, linear models are efficient. The Above Idea Works . . . • In linear models, the values q 1 ,t , . . . , q k,t of quantities Why the Name . . . q 1 , . . . , q k at time t can be predicted as linear f-s of: Linear Models: Reminder Towards a More . . . – the values of these quantities at previous moments Home Page of time t − 1, t − 2, . . . , and Title Page – of the current (and past) values e m,t , e m,t − 1 , . . . of the external quantities e 1 , . . . , e n : ◭◭ ◮◮ ℓ 0 ℓ 0 k n ◭ ◮ � � � � q i,t = a i + a i,j,ℓ · q j,t − ℓ + b i,m,ℓ · e m,t − ℓ . Page 2 of 28 j =1 ℓ =1 m =1 ℓ =0 Go Back Full Screen Close Quit
Linear Models Are . . . 2. General Ubiquity of Linear Models in Science General Ubiquity of . . . and Engineering Linear Models in . . . Need to Go Beyond . . . • At first glance, the ubiquity of linear models in econo- The Above Idea Works . . . metrics is not surprising. Why the Name . . . • Indeed, linear models are ubiquitous in science and en- Linear Models: Reminder gineering in general. Towards a More . . . Home Page • Indeed, we can start with a general dependence Title Page q i,t = f i ( q 1 ,t , q 1 ,t − 1 , . . . , q k,t − ℓ 0 , e 1 ,t , e 1 ,t − 1 , . . . , e n,t − ℓ 0 ) . ◭◭ ◮◮ • In science and engineering, the dependencies are usu- ◭ ◮ ally smooth. Page 3 of 28 • Thus, we can expand the dependence in Taylor series Go Back and keep the first few terms in this expansion. Full Screen • In particular, in the first approximation, when we only keep linear terms, we get a linear model. Close Quit
Linear Models Are . . . 3. Linear Models in Econometrics Are Applicable General Ubiquity of . . . Way Beyond the Taylor Series Explanation Linear Models in . . . Need to Go Beyond . . . • In science and engineering, linear models are effective The Above Idea Works . . . in a small vicinity of each state, when: Why the Name . . . – the deviations from a given state are small Linear Models: Reminder – and we can therefore safely ignore terms which are Towards a More . . . quadratic (or of higher order) in them. Home Page Title Page • However, in econometrics, linear models are effective even when deviations are large. ◭◭ ◮◮ • How can we explain this unexpected efficiency? ◭ ◮ Page 4 of 28 Go Back Full Screen Close Quit
Linear Models Are . . . 4. Why Linear Models Are Ubiquitous in Econo- General Ubiquity of . . . metrics Linear Models in . . . Need to Go Beyond . . . • A possible explanation for the ubiquity of linear models The Above Idea Works . . . in econometrics was proposed in our 2015 paper. Why the Name . . . • Example: predicting the country’s Gross Domestic Prod- Linear Models: Reminder uct (GDP) q 1 ,t . Towards a More . . . Home Page • To estimate the current year’s GDP, we use: Title Page – GDP values in the past years, and ◭◭ ◮◮ – different characteristics that affect the GDP, such as the population size, the amount of trade, etc. ◭ ◮ • In many cases, the corresponding description is un- Page 5 of 28 ambiguous. Go Back • However, in many other cases, there is an ambiguity in Full Screen what to consider a country. Close Quit
Linear Models Are . . . 5. Why Linear Models Are Ubiquitous (cont-d) General Ubiquity of . . . Linear Models in . . . • Indeed, in many cases, countries form a loose federa- Need to Go Beyond . . . tion: European Union is a good example. The Above Idea Works . . . • Most of European countries have the same currency. Why the Name . . . Linear Models: Reminder • There are no barriers for trade and for movement of Towards a More . . . people between different countries. Home Page • So, from the economic viewpoint, it make sense to treat Title Page the European Union as a single country. ◭◭ ◮◮ • On the other hand, there are still differences between ◭ ◮ individual members of the European Union. Page 6 of 28 • So it is also beneficial to view each country from the European Union on its own. Go Back • Thus, we have two possible approaches to predicting Full Screen the European Union’s GDP. Close Quit
Linear Models Are . . . 6. Why Linear Models Are Ubiquitous (cont-d) General Ubiquity of . . . Linear Models in . . . • We can treat the whole European Union as a single Need to Go Beyond . . . country, and apply the general formula to it. The Above Idea Works . . . • We can also apply the general formula to each country Why the Name . . . c independently, and add the predictions: Linear Models: Reminder � � q ( c ) q ( c ) 1 ,t , q ( c ) 1 ,t − 1 , . . . , q ( c ) k,t − ℓ 0 , e ( c ) 1 ,t , e ( c ) 1 ,t − 1 , . . . , e ( c ) i,t = f i . Towards a More . . . n,t − ℓ 0 Home Page • The overall GDP q 1 ,t is the sum of GDPs of all the Title Page countries: q 1 ,t = q (1) 1 ,t + . . . + q ( C ) 1 ,t . ◭◭ ◮◮ • Similarly, the overall population, etc., can be computed ◭ ◮ as the sum of the values from individual countries: e m,t = e (1) m,t + . . . + e ( C ) Page 7 of 28 m,t . Go Back • Thus, the prediction of q 1 ,t based on applying the for- mula to the whole European Union takes the form Full Screen � � q (1) 1 ,t + . . . + q ( C ) 1 ,t , . . . , e (1) n,t − ℓ 0 + . . . + e ( C ) f i . Close n,t − ℓ 0 Quit
Linear Models Are . . . 7. Why Linear Models Are Ubiquitous (cont-d) General Ubiquity of . . . Linear Models in . . . • The sum of individual predictions takes the form Need to Go Beyond . . . � � � � q (1) 1 ,t , . . . , e (1) q ( C ) 1 ,t , . . . , e ( C ) The Above Idea Works . . . + . . . + f i f i . n,t − ℓ 0 n,t − ℓ 0 Why the Name . . . • We require that these two predictions return the same Linear Models: Reminder result: Towards a More . . . Home Page � � q (1) 1 ,t + . . . + q ( C ) 1 ,t , . . . , e (1) n,t − ℓ 0 + . . . + e ( C ) = f i n,t − ℓ 0 Title Page � � � � q (1) 1 ,t , . . . , e (1) q ( C ) 1 ,t , . . . , e ( C ) + . . . + f i f i . ◭◭ ◮◮ n,t − ℓ 0 n,t − ℓ 0 ◭ ◮ • In mathematical terms, this means that the function f i should be additive . Page 8 of 28 • It also makes sense to require that very small changes Go Back in q i and e m lead to small changes in the predictions. Full Screen • So, the function f i are continuous. Close Quit
Linear Models Are . . . 8. Why Linear Models Are Ubiquitous (cont-d) General Ubiquity of . . . Linear Models in . . . • It is known that every continuous additive function is Need to Go Beyond . . . linear. The Above Idea Works . . . • Thus the above requirement explains the ubiquity of Why the Name . . . linear econometric models. Linear Models: Reminder Towards a More . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 28 Go Back Full Screen Close Quit
Linear Models Are . . . 9. Need to Go Beyond Linear Models General Ubiquity of . . . Linear Models in . . . • While linear models are reasonably accurate, the actual Need to Go Beyond . . . econometric processes are often non-linear. The Above Idea Works . . . • Thus, to get more accurate predictions, we need to go Why the Name . . . beyond linear models. Linear Models: Reminder Towards a More . . . • Linear models correspond to the case when we: Home Page – expand the original dependence in Taylor series and Title Page – keep only linear terms in this expansion. ◭◭ ◮◮ • So, to get a more accurate model, a natural idea is: ◭ ◮ – to take into account next order terms in the Taylor Page 10 of 28 expansion, Go Back – i.e., quadratic terms. Full Screen Close Quit
Linear Models Are . . . 10. The Above Idea Works Well in Science and General Ubiquity of . . . Engineering, But Not in Econometrics Linear Models in . . . Need to Go Beyond . . . • Quadratic models are indeed very helpful in science The Above Idea Works . . . and engineering. Why the Name . . . • However, surprisingly, in econometrics, different types Linear Models: Reminder of models turn out to be more empirically successful. Towards a More . . . Home Page • Namely, so-called threshold models in which the expres- sion f i is piece-wise linear. Title Page • In this talk, explain the surprising efficiency of piecewise- ◭◭ ◮◮ linear models in econometrics. ◭ ◮ Page 11 of 28 Go Back Full Screen Close Quit
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