Need for . . . What Is Sparsity: A . . . Why Do We Need a . . . Continuity: Main . . . Commonsense Explanations Zipf’s Law: A Brief . . . of Sparsity, Zipf Law, and Main Idea Behind Our . . . Resulting Explanation Nash’s Bargaining Solution Nash’s Bargaining . . . Resulting Explanation . . . Home Page Olga Kosheleva 1 , Vladik Kreinovich 1 , and Kittawit Autchariyapanikul 2 Title Page 1 University of Texas at El Paso ◭◭ ◮◮ El Paso, Texas 79968, USA ◭ ◮ Olgak@utep.edu, vladik@utep.edu 2 Maejo University, Chiang Mai, Thailand Page 1 of 32 kittawit.a@mju.ac.th Go Back Full Screen Close Quit
Need for . . . What Is Sparsity: A . . . 1. Need for Commonsense Explanations Why Do We Need a . . . • In the 19 century, Continuity: Main . . . Zipf’s Law: A Brief . . . – from Adam Smith through Karl Marx, Main Idea Behind Our . . . – economic phenomena were described by relatively Resulting Explanation simple intuitively clear formulas. Nash’s Bargaining . . . • However, it was well understood that these formulas Resulting Explanation . . . provide, at best: Home Page – a qualitative understanding, and Title Page – qualitative predictions. ◭◭ ◮◮ • Economics and finance are complex phenomena. ◭ ◮ Page 2 of 32 • So not surprisingly, simple models do not provide a good quantitative description of these phenomena. Go Back Full Screen Close Quit
Need for . . . What Is Sparsity: A . . . 2. Need for Commonsense Explanations (cont-d) Why Do We Need a . . . • Starting with the 20 century, new models appeared Continuity: Main . . . that allow us: Zipf’s Law: A Brief . . . Main Idea Behind Our . . . – in many cases, Resulting Explanation – to make reasonable accurate predictions. Nash’s Bargaining . . . • The corresponding models more adequately describe Resulting Explanation . . . economic and financial phenomena. Home Page • However, they are often very mathematically compli- Title Page cated, far from common sense. ◭◭ ◮◮ • And here lies a problem: ◭ ◮ – since models are complicated, requiring complex Page 3 of 32 math beyond what most people know, Go Back – people in general – and politicians in particular – Full Screen do not fully trust these models. Close Quit
Need for . . . What Is Sparsity: A . . . 3. Need for Commonsense Explanations (cont-d) Why Do We Need a . . . • This lack of complete trust is understandable, since: Continuity: Main . . . Zipf’s Law: A Brief . . . – sometimes, predictions of previous models turn out Main Idea Behind Our . . . to be wrong, and Resulting Explanation – decisions based on these predictions made crisis sit- Nash’s Bargaining . . . uations worse. Resulting Explanation . . . • It is therefore desirable to come up with commonsense Home Page explanations: Title Page – for the current models, or at least ◭◭ ◮◮ – for the major assumptions behind these models and ◭ ◮ for the main features of these models. Page 4 of 32 Go Back Full Screen Close Quit
Need for . . . What Is Sparsity: A . . . 4. Need for Commonsense Explanations (cont-d) Why Do We Need a . . . • This need is similar to the well-publicized need for ex- Continuity: Main . . . plainable AI. Zipf’s Law: A Brief . . . Main Idea Behind Our . . . • A female CEO who was denied a bank loan by an AI- Resulting Explanation computer program would like to make sure that: Nash’s Bargaining . . . – this denial was based on inadequate state of her Resulting Explanation . . . company Home Page – and not on the fact that, in the past, most loans Title Page were given to men. ◭◭ ◮◮ ◭ ◮ Page 5 of 32 Go Back Full Screen Close Quit
Need for . . . What Is Sparsity: A . . . 5. What We Do in This Talk Why Do We Need a . . . • Of course, the goal of having explainable econometrics Continuity: Main . . . is complex. Zipf’s Law: A Brief . . . Main Idea Behind Our . . . • It requires joint efforts of many researchers. Resulting Explanation • In this talk, we provide just three examples of the de- Nash’s Bargaining . . . sired commonsense explanations. Resulting Explanation . . . Home Page • Specifically, we provide commonsense explanation: Title Page – for sparsity, ◭◭ ◮◮ – for Zipf law, and – for Nash’s bargaining solution. ◭ ◮ Page 6 of 32 Go Back Full Screen Close Quit
Need for . . . What Is Sparsity: A . . . 6. What Is Sparsity: A Brief Reminder Why Do We Need a . . . • Econometric analysis often starts with trying to use Continuity: Main . . . linear regression. Zipf’s Law: A Brief . . . Main Idea Behind Our . . . • We try to predict the value y ( t + 1) of the desirable Resulting Explanation quantity y at the next moment of time t + 1 (e.g., Nash’s Bargaining . . . month or year). Resulting Explanation . . . • We predict y ( t + 1) as a linear combination of: Home Page – the values x 1 ( t ), . . . , x n ( t ) of different quantities at Title Page the current moment of time, ◭◭ ◮◮ – and maybe of their past values x i ( t − 1), x i ( t − 2), ◭ ◮ . . . Page 7 of 32 • Usually: Go Back – just to be on the safe side, Full Screen – we add as many quantities x i and as many past moments of time as possible. Close Quit
Need for . . . What Is Sparsity: A . . . 7. What Is Sparsity: A Brief Reminder (cont-d) Why Do We Need a . . . • We understand that some of these values may be irrel- Continuity: Main . . . evant. Zipf’s Law: A Brief . . . Main Idea Behind Our . . . • In the process of regression, it usually turns out that Resulting Explanation many of these variables are indeed irrelevant. Nash’s Bargaining . . . • We can see it by observing that the coefficients at these Resulting Explanation . . . variables are close to 0. Home Page • The problem with using the usual linear regression Title Page methods for linear regression is that: ◭◭ ◮◮ – since the data is noisy, and linear dependence is ◭ ◮ approximate, Page 8 of 32 – we do not get perfect match, and thus, do not get Go Back exactly 0 values even where there is no dependence. Full Screen • These erroneously non-zero value make the resulting predictions much less accurate. Close Quit
Need for . . . What Is Sparsity: A . . . 8. What Is Sparsity: A Brief Reminder (cont-d) Why Do We Need a . . . • Much more accurate predictions can be obtained if: Continuity: Main . . . Zipf’s Law: A Brief . . . – we explicitly require, from the very beginning, Main Idea Behind Our . . . – that a large number of the coefficients be zero, i.e., Resulting Explanation that the dependence is sparse . Nash’s Bargaining . . . • This is the main idea behind, e.g., the LASSO method. Resulting Explanation . . . Home Page • Sparsity is not limited to economics and finance, it is a useful tool in data processing in general. Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 32 Go Back Full Screen Close Quit
Need for . . . What Is Sparsity: A . . . 9. Why Do We Need a Commonsense Explanation Why Do We Need a . . . for Sparsity Continuity: Main . . . • At first glance, what we described makes sense. Zipf’s Law: A Brief . . . Main Idea Behind Our . . . • However, one can also raise a serious counter-argument: Resulting Explanation – in economics and in finance, everything is inter- Nash’s Bargaining . . . related, Resulting Explanation . . . – often, a small event across the world causes some Home Page stocks to go up or down, Title Page – so why would there be zero coefficients at all? ◭◭ ◮◮ • The previous text does not provide any commonsense ◭ ◮ explanation. Page 10 of 32 • It just states the empirical fact that the resulting best- Go Back fit formulas are often sparse. Full Screen • It would thus be nice to have some commonsense ex- planation for this empirical phenomenon of sparsity. Close Quit
Need for . . . What Is Sparsity: A . . . 10. Continuity: Main Commonsense Feature Un- Why Do We Need a . . . derlying Our Explanation Continuity: Main . . . • Our commonsense explanation is based on a reasonable Zipf’s Law: A Brief . . . feature of continuity – that: Main Idea Behind Our . . . Resulting Explanation – small changes in one or several quantities should Nash’s Bargaining . . . lead to Resulting Explanation . . . – small changes in other quantities. Home Page • Of course, there are extreme situations when there is Title Page no continuity. ◭◭ ◮◮ • For example: ◭ ◮ – if the amount of money that a company needs to Page 11 of 32 pay right away is slightly smaller than its current Go Back amount of liquid assets, – it can easily pay its creditors and suppliers. Full Screen Close Quit
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