Linear Models And . . . Hammerstein-Type . . . How Can We Speed . . . Need to Consider the . . . Why Hammerstein-Type Need to Minimize the . . . Block Models Are So One Stage Is Not . . . What About Two Stages? Efficient: Case Study of Are Three . . . How This Applies to . . . Financial Econometrics Home Page Thongchai Dumrongpokaphan 1 , Afshin Gholamy 2 Title Page Vladik Kreinovich 2 , and Hoang Phuong Nguyen 3 ◭◭ ◮◮ 1 Department of Mathematics, Chiang Mai University, ◭ ◮ Thailand, tcd43@hotmail.com Page 1 of 23 2 University of Texas at El Paso, El Paso, Texas 79968, USA afshingholamy@gmail.com, vladik@utep.edu Go Back 3 Division Informatics, Math-Informatics Faculty, Thang Long University, Hanoi, Vietnam, nhphuong2008@gmail.com Full Screen Close Quit
Linear Models And . . . Hammerstein-Type . . . 1. Linear Models And Need to Go Beyond Them How Can We Speed . . . • In the 1st approximation, the dynamics of an economic Need to Consider the . . . system can be often described by a linear model. Need to Minimize the . . . One Stage Is Not . . . • In a linear model, the values y 1 ( t ) , . . . , y n ( t ) of the de- What About Two Stages? sired quantities at moment t linearly depend: Are Three . . . – on the values of these quantities at the previous How This Applies to . . . moments of time, and Home Page – on the values of related quantities x 1 ( t ) , . . . , x m ( t ) Title Page at the current and previous moments of time: ◭◭ ◮◮ n S m S � � � � ◭ ◮ y i ( t ) = C ijs · y j ( t − s )+ D ips · x p ( t − s )+ y i 0 . j =1 s =1 p =1 s =0 Page 2 of 23 • In practice, however, many real-life processes are non- Go Back linear. Full Screen • It is desirable to take this non-linearity into account. Close Quit
Linear Models And . . . Hammerstein-Type . . . 2. Hammerstein-Type Block Models for Nonlin- How Can We Speed . . . ear Dynamics Are Efficient in Econometrics Need to Consider the . . . • In many econometric applications: Need to Minimize the . . . One Stage Is Not . . . – the most accurate and the most efficient models What About Two Stages? turned out to be models Are Three . . . – which in control theory are known as Hammerstein- How This Applies to . . . type block models . Home Page • These models combine linear dynamic equations with Title Page non-linear static transformations. ◭◭ ◮◮ • In such models, the transition from moment t to the ◭ ◮ next one consists of several sequential transformations. Page 3 of 23 • Some are linear dynamical transformations. Go Back • Others are static non-linear transformations, that take Full Screen into account only the current values of the quantities. Close Quit
Linear Models And . . . Hammerstein-Type . . . 3. A Toy Example of a Block Model How Can We Speed . . . • To illustrate the idea of a Hammerstein-type block Need to Consider the . . . model, let us consider the simplest case, when: Need to Minimize the . . . One Stage Is Not . . . – the state of the system is described by a single What About Two Stages? quantity y 1 , Are Three . . . – the state y 1 ( t ) at moment t is uniquely determined How This Applies to . . . only by its previous state y 1 ( t − 1), and Home Page – no other quantities affect the dynamics. Title Page • In the linear approximation, the dynamics of such a ◭◭ ◮◮ system is described by a linear dynamic equation ◭ ◮ y 1 ( t ) = C 111 · y 1 ( t − 1) + y 10 . Page 4 of 23 • The simplest possible non-linearity here will be an ad- Go Back ditional term which is quadratic in y 1 ( t ): Full Screen y 1 ( t ) = C 111 · y 1 ( t − 1) + c · ( y 1 ( t − 1)) 2 + y 10 . Close Quit
Linear Models And . . . Hammerstein-Type . . . 4. A Toy Example (cont-d) How Can We Speed . . . def = ( y 1 ( t )) 2 , the Need to Consider the . . . • In terms of an auxiliary variable s ( t ) Need to Minimize the . . . above system can be described in terms of two blocks: One Stage Is Not . . . – a linear dynamical block described by a linear dy- What About Two Stages? namic equation Are Three . . . y 1 ( t ) = C 111 · y 1 ( t − 1) + c · s ( t − 1) + y 10 , and How This Applies to . . . Home Page – a nonlinear block described by a non-linear static Title Page transformation s ( t ) = ( y ( t )) 2 . ◭◭ ◮◮ • In econometrics, non-quadratic transformations are of- ◭ ◮ ten used: e.g., logarithms and exponential functions. Page 5 of 23 • They transform a multiplicative relation z = x · y be- tween quantities into a linear relation between logs: Go Back Full Screen ln( z ) = ln( x ) + ln( y ) . Close Quit
Linear Models And . . . Hammerstein-Type . . . 5. Formulation of the Problem How Can We Speed . . . • In many cases, a non-linear dynamical system can be Need to Consider the . . . represented in the Hammerstein-type block form. Need to Minimize the . . . One Stage Is Not . . . • However, the question remains why necessarily such What About Two Stages? models often work the best in econometrics. Are Three . . . • Indeed, there are many other techniques for describing How This Applies to . . . non-linear dynamical systems, such as: Home Page – Wiener models, in which y i ( t ) are described as Tay- Title Page lor series in terms of y j ( t − s ) and x p ( t − s ), ◭◭ ◮◮ – models that describe the dynamics of wavelet coef- ◭ ◮ ficients, Page 6 of 23 – models that formulate the non-linear dynamics in terms of fuzzy rules, etc. Go Back • In this talk, we explain why such models are efficient Full Screen in econometrics, especially in financial econometrics. Close Quit
Linear Models And . . . Hammerstein-Type . . . 6. Specifics of Computations Related to Econo- How Can We Speed . . . metrics, Especially to Financial Econometrics Need to Consider the . . . • In many economics-related problems, it is important: Need to Minimize the . . . One Stage Is Not . . . – not only to predict future values of the correspond- What About Two Stages? ing quantities, Are Three . . . – but also to predict them as fast as possible. How This Applies to . . . • This need for speed is easy to explain; for example: Home Page – an investor who is the first to finish computation Title Page of the future stock price ◭◭ ◮◮ – will have an advantage of knowing in what direction ◭ ◮ this price will go. Page 7 of 23 • If his/her computations show that the price will go up, Go Back the investor will buy the stock at the current price. Full Screen • Thus, the investor will gain a lot. Close Quit
Linear Models And . . . Hammerstein-Type . . . 7. Computations in Econometrics (cont-d) How Can We Speed . . . • If the computations show that the price will go down, Need to Consider the . . . the investor will sell his/her stock at the current price. Need to Minimize the . . . One Stage Is Not . . . • Thus, the investor will avoid losing money. What About Two Stages? • Similarly, an investor who is the first to predict the Are Three . . . change in the ratio of two currencies will gain a lot. How This Applies to . . . Home Page • In all these cases, fast computations are extremely im- portant. Title Page • Thus, the nonlinear models that we use in these pre- ◭◭ ◮◮ dictions must be the fastest to compute. ◭ ◮ Page 8 of 23 Go Back Full Screen Close Quit
Linear Models And . . . Hammerstein-Type . . . 8. How Can We Speed Up Computations: Need How Can We Speed . . . for Parallel Computations Need to Consider the . . . • If a task takes a lot of time for a single person, a natural Need to Minimize the . . . way to speed it up is: One Stage Is Not . . . What About Two Stages? – to have someone else help, Are Three . . . – so that several people can perform this task in par- How This Applies to . . . allel. Home Page • Similarly, Title Page – if a task takes too much time on a single computer ◭◭ ◮◮ processor, ◭ ◮ – a natural way to speed it up is to have several pro- Page 9 of 23 cessors work in parallel on different subtasks. Go Back Full Screen Close Quit
Linear Models And . . . Hammerstein-Type . . . 9. Need to Consider the Simplest Possible Com- How Can We Speed . . . putational Tasks for Each Processor Need to Consider the . . . • The overall computation time is determined by the Need to Minimize the . . . time during which each processor finishes its task; so: One Stage Is Not . . . What About Two Stages? – to make the overall computations as fast as possi- ble, Are Three . . . How This Applies to . . . – it is necessary to make the elementary tasks as- Home Page signed to each processor as fast as possible, – thus, as simple as possible. Title Page • Each computational task involves processing numbers. ◭◭ ◮◮ • We are talking about the transition from linear to non- ◭ ◮ linear models. Page 10 of 23 • So, it makes sense to consider linear versus nonlinear Go Back transformations. Full Screen • Clearly, linear transformations are much faster than Close nonlinear ones. Quit
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