Approximation of Functions

Transcription

1 Approximation of Functions Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign Nov 7th, 217 C. Hurtado (UIUC - Economics) Numerical Methods

2 On the Agenda 1 Approximation of Functions 2 C. Hurtado (UIUC - Economics) Numerical Methods

3 Approximation of Functions On the Agenda 1 Approximation of Functions 2 C. Hurtado (UIUC - Economics) Numerical Methods

4 Approximation of Functions Approximation of Functions General Objective: Given data about f (x) construct simpler g(x) to approximate f (x) Questions: - What data should be used? - What is a simpler function? - What is the notion of approximation? This is similar to statistical regression - Both approximate an unknown function with data - Statistical data is noisy but we assume data errors are small - What is the notion of approximation? C. Hurtado (UIUC - Economics) Numerical Methods 1 / 19

5 Approximation of Functions Approximation of Functions Taylor Series Approximation: f (x) = f (x )+(x x )f (x )+ (x x ) 2 f (x )+ (x x ) n f (n) (x ) 2 n! Log-linearization Interpolation Ordinary Regression Splines min a 1,,a m Orthogonal Polynomials n (a 1 φ 1 (x i ) + a 2 φ 2 (x i ) + + a m φ m (x 2 ) y i ) 2 i=1 C. Hurtado (UIUC - Economics) Numerical Methods 2 / 19

6 Approximation of Functions Interpolation Interpolation: find g(x) from an n-dimensional family of functions to exactly fit n data points. Lagrange polynomial interpolation: - Data: (x i, y i ), i = 1,, n. - Objective: Find a polynomial of degree n 1, p n (x), which agrees with the data - Result: If the x i are distinct, there is a unique interpolating polynomial Does p n (x) converge to f (x) as we use more points? No C. Hurtado (UIUC - Economics) Numerical Methods 3 / 19

7 Approximation of Functions Interpolation C. Hurtado (UIUC - Economics) Numerical Methods 4 / 19

8 On the Agenda 1 Approximation of Functions 2 C. Hurtado (UIUC - Economics) Numerical Methods

9 Introduction Source: ORG of CPS. Author s calculations. C. Hurtado (UIUC - Economics) Numerical Methods 5 / 19

10 Many Explanations for the Changes in the Wage Structure Skill-biased Technical Change Demand for more educated workers have increased the earnings of the more educated relative to the less educated. Review in Goldin and Katz (29). Relation for the relative earnings of younger college-educated in Card and Lemieux (21) Institutions The increase in Minimum Wage has little or no effect for the fast-food employments in NJ and Penn. according to Card and Krueger(1995) The nonunion labor market in U.S. can explain the differences in wage according to Blau and Kahn(1996) Using data for firms and workers representative of the French private sector Abowd et al. (1999) find that person effects tend to be more important than firm effects in explaining wage differences. Differences between workers, wage premiums at different firms, and similarities in the assignment of workers to plants explain the wage dispersion in West German according to Card et al. (213) C. Hurtado (UIUC - Economics) Numerical Methods 6 / 19

11 Going Beyond the Analysis of the Mean Skill-biased Technical Change Counterfactual Decomposition of Changes in Wage Distributions Using Quantile Regression Machado and Mata (25) Institutions Labor Market Institutions and the Distribution of Wages, : A Semiparametric Approach DiNardo et al. (1996) Measurements of Inequality Unconditional Quantile Regressions Firpo et al. (29) - Uses Influence Function of the unconditional quantile on the explanatory variables. The estimated coefficients can be interpreted as the effect of increasing the mean value of the characteristics on the unconditional quantile 1. Modeling Inequality and Spread in Multiple Regression Aaberge et al. (26) - Assuming that there is a known baseline distribution that does not depend on the covariates x, and further assuming that the variable of interest has conditional distribution that depends on x only through a known (up to scale) function, the authors manage to define a regression setup to explaing measurements of inequality in terms of covariates variables. 1 It assumes that the conditional distribution is unaffected by small variations of the distributions of the explanatory C. Hurtado (UIUC - Economics) Numerical Methods 7 / 19

12 1% Cumulative share of income or wages earned The Lorenz Curve and the Gini Coefficient The relation between the index and curve: Gini coefficient is twice the area between the line of equality (45 ) and the Lorenz curve Line of Equality (45 ) Lorenz Curve Cumulative share of people from lowest to highest income or wages C. Hurtado (UIUC - Economics) Numerical Methods 8 / 19 1%

13 The Lorenz Curve and the Gini Coefficient Recall the definition of quantile function Q Y (t) = inf{y : F Y (y) t} = F 1 Y (t). Following Koenker (25), the Lorentz curve can be express as τ L(τ) = Q Y (t)dt 1 Q Y (t)dt where, Y is a positive random variable, e.g. income or wage, with mean µ <. Setting s = F 1 (t) we get: Y L(τ) = τ Q Y (t)dt sf (s)ds = 1 µ τ Q Y (t)dt A monotone transformation h(y ), with µ h = E [h(y )] <, leads us to L h (τ) = 1 µ h τ Q h(y ) (t)dt = 1 µ h τ h (Q Y (t)) dt C. Hurtado (UIUC - Economics) Numerical Methods 9 / 19

14 The Lorenz Curve and the Gini Coefficient Assuming Q h(y ) (t x) = x T β (t) = P x jβ j(t)dt leads us to a conditional Lorentz curve defined by j=1 L h (τ x) = 1 µ h τ Q h(y ) (t x)dt = 1 µ h P x j τ β j(t)dt where there are P covariates. By the definition of the Gini coefficient we have: G (x) = L h (τ x)dτ j=1 = 1 1 µ h P x j 1 τ 2β j(t)dtdτ. (1) j=1 C. Hurtado (UIUC - Economics) Numerical Methods 1 / 19

15 Why is this useful? This is an additive decomposition of the Gini Coefficient for h(y ) in terms of the characteristics x: G (x) = 1 1 µ h P x j j=1 1 τ 2β j(t)dtdτ. τ 1 2βj(t)dtdτ measures the contribution of the j th characteristic to the Gini Coefficient of h(y ). If the double integral is positive, the Gini Index is smaller and vise versa. C. Hurtado (UIUC - Economics) Numerical Methods 11 / 19

16 The nice linear decomposition of the Gini Index is computed for the variable h(y ), but we are really interested in Y. What should you use for the characteristics x = (x 1, x 2,, x p)? Counterfactual C. Hurtado distribution? (UIUC - Economics) Can you use representative Numerical Methods agent? 12 / 19 What (Where) is the Problem? For a fixed vector of characteristics x = (x 1, x 2,, x P ) Ĝ (x) = 1 1ˆµ h The first issue is to compute 1 τ p x j j=1 1 τ 2 ˆβ j(t)dtdτ 2 ˆβ j(t)dtdτ The term ˆµ h is a normalization to guarantee that the Gini Coefficient is between and 1. Hence, it must be true that: P 1 τ 2 x j ˆβ j(t)dt ˆµ h j=1

17 Divide and Conquer Consider the model for wages defined as: Q ln(w) (τ x) = x T β(τ) Suppose we have the characteristic gender as explanatory variable for ln(w) C. Hurtado (UIUC - Economics) Numerical Methods 13 / 19

18 Divide and Conquer Consider the model for wages defined as: Q ln(w) (τ x) = x T β(τ) Suppose we have the characteristic gender as explanatory variable for ln(w) C. Hurtado (UIUC - Economics) Numerical Methods 13 / 19

19 Smooth the Function What would be the first approach? Yes, plug in the OLS estimator in the double integral. The implicit assumption of OLS is that the effect of the covariate is the same across quantiles. 2 1 τ ˆβ j(t)dtdτ = 2 1 τ ˆβ j,ols dtdτ = 2 ˆβ j,ols 1 τdτ = ˆβ j,ols If we believe that the effect of the covariate is the same across quantiles or if we don t think that the tail behavior is important for the analysis, we can use OLS. When analyzing inequality we are interested in the tails. It make sense to use this methodology. C. Hurtado (UIUC - Economics) Numerical Methods 14 / 19

20 Smooth the Function We can consider ˆβ j(τ) as a function of the quantile Idea: Least-Square Approximations of a Function Using Monomial Polynomials Given a function f (t), continuous on [a, b], find a polynomial P K (t) of degree at most K: P K (t) = a + a 1t + a 2t a K t K such that the integral of the square of the error is minimized. That is, is minimized. E = b a [f (t) P K (t)] 2 dt Problem: If the interval of analysis is [, 1], the matrix that solves for the system of equations is a Hilbert Matrix, which is well-known to be ill-conditioned C. Hurtado (UIUC - Economics) Numerical Methods 15 / 19

21 Smooth the Function Def: The set of functions p, p 1,, p K in [a, b] is called a set of orthogonal functions, with respect to a weight function w(t), if b a w(t)p i(t)p j(t)dt = { C j if i j if i = j where C j is a real positive number. Furthermore, if C j = 1, j =, 1,, K, then the orthogonal set is called an orthonormal set. Idea: Find a least-squares approximation of f (t) on [a, b] by means of a polynomial of the form P K (t) = a p (t) + a 1p 1(t) + + a K p K (t) where {p k } K k= is a set of orthogonal polynomials. That is, the basis for generating P K (t) in this case is a set of orthogonal polynomials. C. Hurtado (UIUC - Economics) Numerical Methods 16 / 19

22 Smooth the Function Many families of orthogonal polynomials. Which one to choose? C. Hurtado (UIUC - Economics) Numerical Methods 17 / 19

23 Smooth the Function Are we ready? Not yet. C. Hurtado (UIUC - Economics) Numerical Methods 18 / 19

24 Smooth the Function If the function is well-behaved the approximation by a polynomial very good. C. Hurtado (UIUC - Economics) Numerical Methods 19 / 19