Linear Dependencies . . . Linear Dependencies . . . Skedactic Functions Problems and What . . . Invariance Explains Multiplicative and Natural Invariance: . . . Exponential Skedactic Functions Shift and Shift-Invariance Scale Invariance: Main . . . Vladik Kreinovich 1 , Olga Kosheleva 1 , Shift Invariance: Main . . . Hung T. Nguyen 2 , 3 , and Songsak Sriboonchitta 3 General Case: Some . . . 1 University of Texas at El Paso, Home Page El Paso, TX 79968, USA Title Page vladik@utep.edu, olgak@utep.edu 2 Department of Mathematical Sciences ◭◭ ◮◮ New Mexico State University Las Cruces, NM 88003, USA, hunguyen@nmsu.edu ◭ ◮ 3 Faculty of Economics, Chiang Mai University Page 1 of 16 Chiang Mai, Thailand, songsakecon@gmail.com Go Back Full Screen Close Quit
Linear Dependencies . . . Linear Dependencies . . . 1. Linear Dependencies Are Ubiquitous Skedactic Functions • In many practical situations, a quantity y depends on Problems and What . . . several other quantities x 1 , . . . , x n : y = f ( x 1 , . . . , x n ). Natural Invariance: . . . Shift and Shift-Invariance • Often, the ranges of x i are narrow: x i ≈ x (0) for some i Scale Invariance: Main . . . def x (0) = x i − x (0) i , so ∆ x i are relatively small. i Shift Invariance: Main . . . • Then, we can expand the dependence of y on x i = General Case: Some . . . x (0) + ∆ x i in Taylor series and keep only linear terms: Home Page i n Title Page y = f ( x (0) 1 + ∆ x 1 , . . . , x (0) � n + ∆ x n ) ≈ a 0 + a i · ∆ x i , ◭◭ ◮◮ i =1 ◭ ◮ = ∂f � � def def x (0) 1 , . . . , x (0) where a 0 = f and a i . n ∂x i Page 2 of 16 • Substituting ∆ x i = x i − x (0) Go Back into this formula, we get i n n def a i · x (0) Full Screen y ≈ c + � a i · x i , where c = a 0 − � i . i =1 i =1 Close Quit
Linear Dependencies . . . Linear Dependencies . . . 2. Linear Dependencies Are Approximate Skedactic Functions • Usually, Problems and What . . . Natural Invariance: . . . – in addition to the quantities x 1 , . . . , x n that provide Shift and Shift-Invariance the most influence on y , Scale Invariance: Main . . . – there are also many other quantities that (slightly) Shift Invariance: Main . . . influence y , General Case: Some . . . – so many that it is not possible to take all of them Home Page into account. Title Page • Since we do not take these auxiliary quantities into ◭◭ ◮◮ account, the linear dependence is approximate. � � n ◭ ◮ def • The approximation errors ε = y − c + � a i · x i de- Page 3 of 16 i =1 pend on un-observed quantities. Go Back • So, we cannot predict ε based only on the observed Full Screen quantities x 1 , . . . , x n . Close • It is therefore reasonable to view ε as random variables. Quit
Linear Dependencies . . . Linear Dependencies . . . 3. Skedactic Functions Skedactic Functions • A natural way to describe a random variable is by its Problems and What . . . moments. Natural Invariance: . . . Shift and Shift-Invariance • If the first moment is not 0, then we can correct this Scale Invariance: Main . . . bias by appropriately updating the constant c . Shift Invariance: Main . . . • Since the mean is 0, the second moment coincides with General Case: Some . . . the variance v . Home Page • The dependence v ( x 1 , . . . , x n ) is known as the skedactic Title Page function. ◭◭ ◮◮ • In econometric applications, two major classes of ◭ ◮ skedactic functions have been empirically successful: Page 4 of 16 n | x i | γ i and � – multiplicative v ( x 1 , . . . , x n ) = c · Go Back i =1 � � n Full Screen – exponential v ( x 1 , . . . , x n ) = exp α + � γ i · x i . i =1 Close Quit
Linear Dependencies . . . Linear Dependencies . . . 4. Problems and What We Do Skedactic Functions • Problems: Problems and What . . . Natural Invariance: . . . – Neither of the empirically successful skedactic func- Shift and Shift-Invariance tions has a theoretical justification. Scale Invariance: Main . . . – In most situations, the multiplication function re- Shift Invariance: Main . . . sults in more accurate estimates. General Case: Some . . . – This fact also does not have an explanation. Home Page • What we do: we use invariance ideas to: Title Page – explain the empirical success of multiplicative and ◭◭ ◮◮ exponential skedactic functions, and ◭ ◮ – come up with a more general class of skedactic func- Page 5 of 16 tions. Go Back Full Screen Close Quit
Linear Dependencies . . . Linear Dependencies . . . 5. Natural Invariance: Scaling Skedactic Functions • Many economics quantities correspond to prices, Problems and What . . . wages, etc. and are thus expressed in terms of money. Natural Invariance: . . . Shift and Shift-Invariance • The numerical value of such a quantity depends on the Scale Invariance: Main . . . choice of a monetary unit. Shift Invariance: Main . . . • For example, when a European country switches to General Case: Some . . . Euro from its original currency, Home Page – the actual incomes do not change, but Title Page – all the prices and wages get multiplied by the cor- ◭◭ ◮◮ responding exchange rate k : x i → x ′ i = k · x i . ◭ ◮ • Similarly, the numerical amount (of oil or sugar), Page 6 of 16 changes when we change units. Go Back • For example, for oil, we can use barrels or tons. Full Screen • When the numerical value of a quantity is multiplied by k , its variance gets multiplied by k 2 . Close Quit
Linear Dependencies . . . Linear Dependencies . . . 6. Scaling (cont-d) Skedactic Functions • Changing the measuring units for x 1 , . . . , x n does not Problems and What . . . change the economic situations. Natural Invariance: . . . Shift and Shift-Invariance • So, it makes sense to require that the skedactic function Scale Invariance: Main . . . also does not change under such re-scaling: namely, Shift Invariance: Main . . . – for each combination of re-scalings on inputs, General Case: Some . . . – there should be an appropriate re-scaling of the out- Home Page put after which the dependence remains the same. Title Page • In precise terms, this means that: ◭◭ ◮◮ – for every combination of numbers k 1 , . . . , k n , ◭ ◮ – there should exist a value k = k ( k 1 , . . . , k n ) with Page 7 of 16 the following property: Go Back v = v ( x 1 , . . . , x n ) if and only if v ′ = v ( x ′ 1 , . . . , x ′ n ), Full Screen where v ′ = k · v and x ′ i = k i · x i . Close Quit
Linear Dependencies . . . Linear Dependencies . . . 7. Shift and Shift-Invariance Skedactic Functions • While most economic quantities are scale-invariant, Problems and What . . . some are not. Natural Invariance: . . . Shift and Shift-Invariance • For example, the unemployment rate is measured in Scale Invariance: Main . . . percents, there is a fixed unit. Shift Invariance: Main . . . • Many such quantities can have different numerical val- General Case: Some . . . ues depending on how we define a starting point. Home Page • For example, we can measure unemployment: Title Page – either by the usual percentage x i , or ◭◭ ◮◮ – by the difference x i − k i , where k i > 0 is what ◭ ◮ economists mean by full employment. Page 8 of 16 • It is thus reasonable to consider shift-invariant skedac- tic functions: Go Back ∀ k 1 , . . . , k n ∃ k ( v = v ( x 1 , . . . , x n ) ⇔ v ′ = f ( x ′ 1 , . . . , x ′ n )) , Full Screen where v ′ = k · v and x ′ i = x i + k i . Close Quit
Linear Dependencies . . . Linear Dependencies . . . 8. Scale Invariance: Main Result Skedactic Functions • Definition. We say that a non-negative measurable Problems and What . . . function v ( x 1 , . . . , x n ) is scale-invariant if: Natural Invariance: . . . Shift and Shift-Invariance – for every n -tuple of real numbers ( k 1 , . . . , x n ), Scale Invariance: Main . . . – there exists a real number k = k ( k 1 , . . . , k n ) for Shift Invariance: Main . . . which, for every x 1 , . . . , x n and v : General Case: Some . . . v = v ( x 1 , . . . , x n ) ⇔ v ′ = v ( x ′ 1 , . . . , x ′ n ), where Home Page v ′ = k · v and x ′ i = k i · x i . Title Page • Proposition. A skedactic function is scale-invariant ◭◭ ◮◮ if and only it has the form ◭ ◮ n � | x i | γ i for some c and γ i . v ( x 1 , . . . , x n ) = c · Page 9 of 16 i =1 Go Back • Discussion. Thus, scale-invariance explains the use Full Screen of multiplicative skedactic functions. Close Quit
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