Nonparametric Density Estimation

Transcription

1 Nonparametric Density Estimation Advanced Econometrics Douglas G. Steigerwald UC Santa Barbara D. Steigerwald (UCSB) Density Estimation 1 / 20

2 Overview Question of interest has wage inequality among women grown over time? focus on log wage 241B Lecture 1: skewness and thick tails reduced, mean a better measure of central tendency standard deviation of log wages (CPS).41 in in % increase (9 percentage point increase) parametric approach log wage N 0, σ 2 wage log normal D. Steigerwald (UCSB) Density Estimation 2 / 20

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4 What Caused the ncreased Dispersion? from the parametric approach, we cannot see the cause What role did the change in the minimum wage play? both densities are continuous at the minimum wage not clear the minimum wage change played a role A nonparametric estimator of the wage density may shed more light D. Steigerwald (UCSB) Density Estimation 4 / 20

5 Nonparametric Density Estimator igure D. Steigerwald (UCSB) Density Estimation 5 / 20

6 Revisiting What Caused the ncreased Dispersion? the nonparametric estimate reveals in 1979 the modal employed female earned the minimum wage the minimum wage is binding for many in 1989 there is far less bunching at the minimum wage the minimum wage has fallen in real terms and is less binding the increased dispersion results from fewer workers paid at the minimum wage bad - lower wage workers earn less in real dollars good - more opportunities for women to work D. Steigerwald (UCSB) Density Estimation 6 / 20

7 Histogram bar chart - proportion of observations at di erent values an estimator of the probability mass function 1863, Louise-Adolphe Bertillon examines heights of rench soldiers 9,002 men from the region of Doubs converts centimeters to inches nds bimodal distribution nds similar features regardless of year of conscription argues for 2 distributions: tall (upstanding) Celts, short (feckless) Burgundians D. Steigerwald (UCSB) Density Estimation 7 / 20

8 igure 2 Doubts about Doubs D. Steigerwald (UCSB) Density Estimation 8 / 20

9 Histogram Weaknesses Livi (1896) (26 years later!) notes an empirical aw even if the true data had only 1 mode, Bertillon would nd 2 the middle bin contains only 2 centimeter classes the bins on either side contain 3 centimeter classes number of modes is sensitive to bin centers depends on bin width and location of rst bin Bertillon - bin centers at 58.5, 59.5,... shift bin centers to 58, 59,... only 1 mode D. Steigerwald (UCSB) Density Estimation 9 / 20

10 Histogram Sensitivity to Bin Centers D. Steigerwald (UCSB) Density Estimation 10 / 20

11 Histogram Construction value of bin for heights between 58 and 59 inches dpmf (x o ) := n j n = 1 n n 1 i=1 bin center x 0 = 58.5 and bin width w = 1 interpretation nj E n x < x i x = P x 0 2 < x x Z x0 +1/2 = f (x) dx 2 x 0 1/2 area under density over the bin (probability mass function) density estimate assume f (x) uniform within each bin bandwidth h (equals 1/2 in this example) for c 2 (x 0 h, x 0 + h] nj E = f (c) 2h n bf (c) = 1 2h n j n D. Steigerwald (UCSB) Density Estimation 11 / 20

12 Kernel Density Estimator Modern Method Two key di erences from histogram bins: kernel - overlapping histogram - not overlapping removes dependence on bin centers weights: kernel - weights can vary histogram - weights do no vary Construction bf (x) = 1 h 1 n n xi k i=1 h x k kernel : weight-assigning (smoothing) function histogram uses a uniform function h bandwidth : determines the neighborhood of x 0 in which observations receive signi cant weight histogram bandwidth is 1/2 bin width D. Steigerwald (UCSB) Density Estimation 12 / 20

13 Naive Kernel Density Estimator Uniform Kernel uniform (naive) kernel density estimator bf (x) = 1 2h 1 n x i n 1 i=1 x h 1 n o x similar to histogram 1 i x h 1 = 1 fx h < x i x + hg kernel is xi x k = 1 h 2 1 x i x h 1 1/2 if jvj = 1 k (v) = 0 otherwise unlike histogram, kernel - distinct estimates for each value of x due to overlapping bins uniform kernel - same weight function as histogram D. Steigerwald (UCSB) Density Estimation 13 / 20

14 Kernel Density Estimator Properties s bf (x) nonnegative for all x? k (v) 0 for all v ) bf (x) 0 for all x Does bf (x) integrate to 1? R k (v) dv = R dv = 1 ) R bf (x) = 1 for the uniform kernel only bf (x) is the numerical derivative of the empirical CD b (x) = 1 n n 1 fx i xg i =1 D. Steigerwald (UCSB) Density Estimation 14 / 20

15 Kernels with Varying Weights Gaussian (Normal) Kernel Gaussian kernel density estimator bf (x) = 1 h 1 n n i=1 1 p e 1 2 ( x i x h ) 2 2π kernel is xi k h x = p 1 e 1 xi 2 2π x h 2 non-zero weight for all x i, all of the data is used to estimate bf (57) greatest weight given to individuals whose height is exactly 57 inches weights decline rapidly, if h = 1/2 individual at 57 inches gets a weight almost 3000 times as great as an individual at 55 inches does not restrict bf to look like a Gaussian density D. Steigerwald (UCSB) Density Estimation 15 / 20

16 Bandwidth Selection bandwidth alters assignment of weights for a given weighting function bandwidth (not kernel) is the key choice bandwidth selection trade-o bias against variance increasing bandwidth when estimating bf (x) more observations, reducing variance of bf (x) observations further from x, increasing bias of bf (x) D. Steigerwald (UCSB) Density Estimation 16 / 20

17 Kernel Density Estimator: Bias D. Steigerwald (UCSB) Density Estimation 17 / 20

18 Bias (Continued) D. Steigerwald (UCSB) Density Estimation 18 / 20

19 Optimal Bandwidth asymptotic variance of bf (x) asymptotic MSE 1 f (x) R (k) nh " # 2 1 f f (x) R (k) + (2) (x) h4 nh 2 h miminizes asymptotic MSE h = " # f (x) R (k) n f (2) (x) 2 rule of thumb h = n 1/5 D. Steigerwald (UCSB) Density Estimation 19 / 20

20 References DiNardo and Tobias (2001: Journal of Economic Perspectives) Henderson & Parmeter Applied Nonparametric Econometrics Chapter 1, D. Steigerwald (UCSB) Density Estimation 20 / 20