Prediction in Econometrics: Our First Result: A . . . Towards - PowerPoint PPT Presentation

Prediction is Important Need to Justify . . . First Result: . . . Exponential . . . Prediction in Econometrics: Our First Result: A . . . Towards Mathematical Price Transmission: . . . Asymmetric Price . . . Justification of Simple (and Second Example: . . . Acknowledgments Successful) Heuristics Home Page Title Page Vladik Kreinovich 1 , 2 , Hung T. Nguyen 3 , 2 , and Songsak Sriboonchitta 3 ◭◭ ◮◮ ◭ ◮ 1 Department of Computer Science, University of Texas El Paso, Texas, USA, vladik@utep.edu Page 1 of 21 2 Faculty of Economics, Chiang Mai University Go Back Chaing Mai, Thailand, songsak@econ.cmu.ac.th 3 Dept. Mathematical Sciences, New Mexico State University Full Screen Las Cruces, New Mexico, USA, hunguyen@nmsu.edu Close Quit

Prediction is Important Need to Justify . . . 1. Prediction is Important First Result: . . . • Prediction (forecasting) is of upmost importance in Exponential . . . economics and finance. Our First Result: A . . . Price Transmission: . . . • If we can accurately predict the future prices, then we Asymmetric Price . . . can get the largest return on investment. Second Example: . . . • Vice versa: Acknowledgments Home Page – if we make decisions based on the wrong predic- tions, Title Page – then our financial investments collapse, ◭◭ ◮◮ – and the manufacturing plants that we built are non- ◭ ◮ profitable and thus idle. Page 2 of 21 • Many successful (semi-)heuristic methods have been Go Back proposed to predict economic and financial processes. Full Screen Close Quit

Prediction is Important Need to Justify . . . 2. Need to Justify Heuristic Strategies First Result: . . . • The success of prediction heuristics leads to a conjec- Exponential . . . ture that these heuristics have a theor. justification. Our First Result: A . . . Price Transmission: . . . • In general, when we have a theoretical justification, it Asymmetric Price . . . helps: Second Example: . . . – we can use the corresponding theory to fine-tune Acknowledgments the method, and Home Page – we can get a clearer understanding of when the Title Page method is efficient and when it is not efficient. ◭◭ ◮◮ • In this paper, we justify two heuristics: ◭ ◮ – of an intuitive exponential smoothing procedure, Page 3 of 21 that predicts slowly changing processes, and Go Back – of a seemingly counter-intuitive idea of an increase in volatility as a predictor of trend reversal. Full Screen Close Quit

Prediction is Important Need to Justify . . . 3. First Result: Prediction of Slowly Changing First Result: . . . Processes Exponential . . . • Problem: based on the past observations x 1 , . . . , x T ( x 1 Our First Result: A . . . most recent), predict the future value x 0 . Price Transmission: . . . Asymmetric Price . . . • In other words: we need a predictor function x 0 ≈ F ( x 1 , . . . , x T ). Second Example: . . . Acknowledgments • Continuity: if x i ≈ x ′ i , then F ( x 1 , . . . , x T ) ≈ F ( x ′ 1 , . . . , x ′ T ). Home Page • Motivation: we predict based on measurement results, Title Page and they are never absolutely accurate. ◭◭ ◮◮ • Additivity: F ( x (1) 1 + x (2) 1 , . . . ) = F ( x (1) 1 , . . . )+ F ( x (2) 1 , . . . ) . ◭ ◮ • Motivation: we can predict stocks x (1) and bonds x (2) 0 , 0 Page 4 of 21 or we can predict value of the whole portfolio x (1) 0 + x (2) 0 . Go Back • Conclusion: we must consider linear predictors F ( x 1 , . . . , x T ) = T Full Screen � f t · x t . Close t =1 Quit

Prediction is Important Need to Justify . . . 4. From Finite to Infinite Time First Result: . . . • The actual number of observed values is always finite. Exponential . . . Our First Result: A . . . • However, in many cases, we have very long time series Price Transmission: . . . (e.g., daily for many years). Asymmetric Price . . . • In real life, the influence of remote events is small. Second Example: . . . • It is thus reasonable to assume that we have an infinite Acknowledgments ∞ Home Page � number of records: x 0 = f t · x t . t =1 Title Page • In practice, we only know values x 1 , . . . , x T . ◭◭ ◮◮ • Thus, we use an approximate formula ◭ ◮ T Page 5 of 21 � x 0 ≈ f t · x t . Go Back t =1 Full Screen Close Quit

Prediction is Important Need to Justify . . . 5. Case of a Constant Signal First Result: . . . • In some cases, the observed signal x t does not change Exponential . . . at all: x t = c . Our First Result: A . . . Price Transmission: . . . • In this case, it is reasonable to predict the same value Asymmetric Price . . . x 0 = c . Second Example: . . . • In other words, if x 1 = x 2 = . . . = c , then Acknowledgments ∞ Home Page � x 0 = f t · x t = c. Title Page t =1 ◭◭ ◮◮ ∞ � • In precise terms, this means that f t · c = c . ◭ ◮ t =1 Page 6 of 21 ∞ � • In particular, for c = 1, we get f t = 1. Go Back t =1 Full Screen Close Quit

Prediction is Important Need to Justify . . . 6. Exponential Smoothing: a Brief Reminder First Result: . . . ∞ Exponential . . . • General formula: x 0 = � f t · x t . t =1 Our First Result: A . . . • Question: which predictor is the best? Price Transmission: . . . Asymmetric Price . . . • Empirical fact: exponential smoothing is one of the Second Example: . . . best in econometrics: f t = α · (1 − α ) t − 1 . Acknowledgments • It is widely used: described in textbooks, used in seri- Home Page ous econometric studies. Title Page • Why exponential smoothing? there exist many expla- ◭◭ ◮◮ nations for the usefulness of exponential smoothing. ◭ ◮ • Remaining problem: these explanations are based on Page 7 of 21 complex, not very intuitively clear statistical models. Go Back • What we do: we provide a new (and rather simple) Full Screen theoretical explanation of exponential smoothing. Close Quit

Prediction is Important Need to Justify . . . 7. Definitions First Result: . . . • By a time series x , we mean an infinite sequence of Exponential . . . real numbers x 1 , . . . , x n , . . . Our First Result: A . . . Price Transmission: . . . • By a predictor function f , we mean an infinite sequence Asymmetric Price . . . of real numbers f 1 , . . . , f n , . . . for which Second Example: . . . ∞ � Acknowledgments f t = 1 . Home Page t =1 Title Page • By the prediction X 0 ( f, x ) made by the predictor func- ◭◭ ◮◮ tion f t for the time series x t , we mean the value ∞ ◭ ◮ � f t · x t . Page 8 of 21 t =1 Go Back • By a noise pattern p , we mean a finite sequence of real Full Screen numbers p 1 , . . . , p k . Close Quit

Prediction is Important Need to Justify . . . 8. Definitions (cont-d) First Result: . . . • Let c be a real number, and let m be a natural number. Exponential . . . Our First Result: A . . . • By x ( p, c, m ), we mean a time series for which x m + i = Price Transmission: . . . p i for i = 1 , . . . , k , and x t = c for all other t . Asymmetric Price . . . • We say that this time series x ( p, c, m ) corresponds to Second Example: . . . – a a constant signal plus Acknowledgments Home Page – a noise pattern p before moment m . Title Page • We say that for a predictor function f t , the effect of noise always decreases with time if: ◭◭ ◮◮ – for every noise pattern p , for every real number c ◭ ◮ and Page 9 of 21 – for every two natural numbers m > m ′ , Go Back we have Full Screen | X 0 ( f, x ( p, c, m )) − c | ≤ | X 0 ( f, x ( p, c, m ′ )) − c | . Close Quit

Prediction is Important Need to Justify . . . 9. Our First Result: A Simple Justification of Ex- First Result: . . . ponential Smoothing Exponential . . . • Result: Our First Result: A . . . Price Transmission: . . . – For every α ∈ (0 , 2), for f t = α · (1 − α ) t − 1 , the Asymmetric Price . . . effect of noise always decreases with time. Second Example: . . . – If for a function f t , the effect of noise always de- Acknowledgments creases with time, then there exists α ∈ (0 , 2) s.t.: Home Page f t = α · (1 − α ) t − 1 . Title Page • Discussion: ◭◭ ◮◮ ◭ ◮ – Exponential smoothing is the only predictor for which the effect of noise always decreases with time. Page 10 of 21 – Thus, the need to satisfy this natural property ex- Go Back plains the efficiency of exponential smoothing. Full Screen Close Quit

Prediction is Important Need to Justify . . . 10. Price Transmission: Reminder First Result: . . . • The price of a manufacturing product is determined by Exponential . . . the price of the components and the price of the labor. Our First Result: A . . . Price Transmission: . . . • If one of the component prices changes, this change Asymmetric Price . . . affects the product’s price. Second Example: . . . • This change is called price transmission . Acknowledgments Home Page • Example: Title Page – when the oil price changes, the gasoline prices change as well; ◭◭ ◮◮ – when the gasoline prices change, the transportation ◭ ◮ prices change as well; Page 11 of 21 – when the transportation prices change, the price of Go Back transported goods changes. Full Screen Close Quit