Why Interval Models . . . What Is Econometrics: . . . Why Interval Models . . . Towards More Realistic How Interval Data Is . . . Discussion Interval Models in How to Actually Solve . . . Econometrics Acknowledgments Home Page Songsak Sriboonchitta 1 , Thach N. Nguyen 2 , Olga Kosheleva 3 , and Vladik Kreinovich 3 Title Page ◭◭ ◮◮ 1 Chiang Mai University, Chiang Mai, Thailand 2 Banking University, Ho Chi Minh City, Vietnam ◭ ◮ 3 University of Texas at El Paso, El Paso, TX 79968, USA Page 1 of 15 songsakecon@gmail.com, Thachnn@buh.edu.vn, olgak@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit
Why Interval Models . . . 1. Why Interval Models In General: A Brief Re- What Is Econometrics: . . . minder Why Interval Models . . . • In most application areas, values of the quantities come How Interval Data Is . . . from measurements. Discussion How to Actually Solve . . . • Measurement of a physical quantity x is always approx- Acknowledgments imate: Home Page – it produces a value � x Title Page – which is, in general, different from the desired ac- ◭◭ ◮◮ tual value x . ◭ ◮ • In many case, we have no information about the prob- abilities of different values of the measurement error Page 2 of 15 def Go Back ∆ x = � x − x. Full Screen • We only know the upper bound ∆ on the absolute value Close of the measurement error: | ∆ x | ≤ ∆ . Quit
Why Interval Models . . . 2. Why Interval Models In General (cont-d) What Is Econometrics: . . . Why Interval Models . . . • Often, we only know the upper bound ∆ on the abso- lute value of the measurement error: | ∆ x | ≤ ∆ . How Interval Data Is . . . Discussion • In such cases: How to Actually Solve . . . – once we know the measurement result � x , Acknowledgments – the only information that we have about the actual Home Page (unknown) value x is that x ∈ [ � x − ∆ , � x + ∆]. Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 15 Go Back Full Screen Close Quit
Why Interval Models . . . 3. What Is Econometrics: A Brief Reminder What Is Econometrics: . . . • In econometrics – a quantitative study of economics – Why Interval Models . . . we deal with values like prices, indexes, etc. How Interval Data Is . . . Discussion • Most of these values are known exactly, there is no How to Actually Solve . . . measurement uncertainty. Acknowledgments • The stock’s prices, the amounts of stocks traded – all Home Page these numbers are known exactly. Title Page • So, at first glance, there seems to be no need for interval ◭◭ ◮◮ models in econometrics. ◭ ◮ • But, as we will show, there is such a need. Page 4 of 15 • Indeed, the main objective of econometrics is: Go Back – to use the past economic data Full Screen – to predict – and, if needed, change – the future Close economic behavior. Quit
Why Interval Models . . . 4. What Is Econometrics (cont-d) What Is Econometrics: . . . Why Interval Models . . . • The simplest – and often efficient – way to predict is to find a linear dependence between: How Interval Data Is . . . Discussion – the desired future value y and the present and How to Actually Solve . . . – past values x 1 , . . . , x n of this and related quantities: Acknowledgments n � Home Page y ≈ c 0 + c i · x i for some coefficients c i . Title Page i =1 ◭◭ ◮◮ • The coefficients can be determined from the available � n , y ( k ) � x ( k ) 1 , . . . , x ( k ) data , 1 ≤ k ≤ K . ◭ ◮ • When the approximation error is normally distributed, Page 5 of 15 we can use the Least Squares method Go Back � � �� 2 K n � � Full Screen y ( k ) − c i · x ( k ) c 0 + → min . i Close k =1 i =1 Quit
Why Interval Models . . . 5. What Is Econometrics (cont-d) What Is Econometrics: . . . Why Interval Models . . . • In general, we can use other particular cases of the Maximum Likelihood method. How Interval Data Is . . . Discussion • In the case of Least Squares, differentiation leads to an How to Actually Solve . . . easy-to-solve system of linear equations for c i . Acknowledgments • For stock trading, we have millions of records daily, Home Page corresponding to seconds and even milliseconds. Title Page • A few decades ago, it was not possible to process all ◭◭ ◮◮ this data; so: ◭ ◮ – instead of considering all second-by-second prices Page 6 of 15 of a stock, – econometricians considered only one value per day Go Back – e.g., the price at the end of the working day. Full Screen • Nowadays, with more computational power at our dis- Close posal, we can consider many more data points. Quit
Why Interval Models . . . 6. Why Interval Models in Econometrics What Is Econometrics: . . . • Practitioners expected that: Why Interval Models . . . How Interval Data Is . . . – by taking into account more price values per day – Discussion i.e., more data, How to Actually Solve . . . – then can get better predictions. Acknowledgments • Somewhat surprisingly, it turned out that predictions Home Page got worse. Title Page • Namely, it turned out that most daily price fluctuations ◭◭ ◮◮ are irrelevant for prediction purposes. ◭ ◮ • They constitute noise whose addition only makes the prediction worse. Page 7 of 15 • The same thing happened if instead of a single value Go Back x i , practitioners considered two numbers: Full Screen – the smallest price x i during the day and Close – the largest price x i during the day. Quit
Why Interval Models . . . 7. Interval Models in Econometrics (cont-d) What Is Econometrics: . . . Why Interval Models . . . • Many attempts to use extra data only made predictions worse. How Interval Data Is . . . Discussion • The only idea that helped improve the prediction ac- How to Actually Solve . . . curacy was: Acknowledgments – replacing the previous value x i Home Page – with some more relevant value from the correspond- Title Page ing interval [ x i , x i ]. ◭◭ ◮◮ ◭ ◮ Page 8 of 15 Go Back Full Screen Close Quit
Why Interval Models . . . 8. How Interval Data Is Treated Now What Is Econometrics: . . . • We consider situations when: Why Interval Models . . . How Interval Data Is . . . – instead of the exact values x ( k ) and y ( k ) , i � � Discussion � y ( k ) , y ( k ) � x ( k ) i , x ( k ) – we only know intervals and . How to Actually Solve . . . i Acknowledgments • To deal with such situations, researchers proposed to Home Page use the values y ( k ) = α · y ( k ) + (1 − α ) · y ( k ) and Title Page ◭◭ ◮◮ x ( k ) = α · x ( k ) + (1 − α ) · x ( k ) i . i i ◭ ◮ • Here, α is some special value – usually, α = 0, α = 0 . 5, Page 9 of 15 or α = 1. Go Back • This lead to some improvement in prediction accuracy. Full Screen Close Quit
Why Interval Models . . . 9. How Interval Data Is Treated Now (cont-d) What Is Econometrics: . . . Why Interval Models . . . • Even better results were obtained when they tried: How Interval Data Is . . . – instead of fixing a value α , Discussion – to find the value α for which the mean squared error How to Actually Solve . . . is the smallest Acknowledgments – (or, more generally, the Maximum Likelihood is the Home Page largest). Title Page • The optimization problem is no longer quadratic. ◭◭ ◮◮ • However, it is quadratic with respect to c i and with ◭ ◮ respect to α . Page 10 of 15 • So we can solve it by inter-changingly: Go Back – minimizing over c i and Full Screen – minimizing over α . Close Quit
Why Interval Models . . . 10. Discussion What Is Econometrics: . . . • Deviations from the typical daily value are random. Why Interval Models . . . How Interval Data Is . . . • One day, they are mostly increasing, another day, they Discussion are mostly decreasing, so: How to Actually Solve . . . – instead of fixing the same α for all i and k , Acknowledgments – it makes more sense to select possibly different points Home Page from different intervals, � � Title Page – i.e., to select values c i , x ( k ) x ( k ) i , x ( k ) ∈ , and i i � y ( k ) , y ( k ) � ◭◭ ◮◮ y ( k ) ∈ that minimize the expression ◭ ◮ � � �� 2 K n � � y ( k ) − c i · x ( k ) Page 11 of 15 c 0 + . i k =1 i =1 Go Back • It may seem that the existing α -approach is a good Full Screen first approximation for this optimization problem. Close • However, in the α -approach, we usually take α ∈ (0 , 1). Quit
Why Interval Models . . . 11. Discussion (cont-d) What Is Econometrics: . . . � � • When α ∈ (0 , 1), we always have x ( k ) x ( k ) i , x ( k ) Why Interval Models . . . ∈ , i i � y ( k ) , y ( k ) � How Interval Data Is . . . and y ( k ) ∈ . Discussion • So, the minimum is attained inside the corresponding How to Actually Solve . . . intervals. Acknowledgments • Thus, it seems like in our problem, we should also look Home Page for a minimum inside the corresponding intervals. Title Page • But then, the derivatives of the objective function with ◭◭ ◮◮ respect to y ( k ) and x ( k ) would be equal to 0. i ◭ ◮ • Thus, for all k , we would have exact equality Page 12 of 15 K � y ( k ) = c 0 + c i · x ( k ) Go Back i . k =1 Full Screen • In most practical problems, it is not possible to fit all Close the available intervals with the exact dependence. Quit
Recommend
More recommend
