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Econometrics-I-22

This is essentially a histogram with small bins part

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Unformatted text preview: This is essentially a histogram with small bins. Part 22: Semi- and Nonparametric Estimation Computing the KDE ˜™™ ™ 7/26 1 2 n * * 1 Given the sample observations: x , x , ..., x (x , 1,..., ) Choose a set of points x ,...,x These may be the original data if n is not very large Otherwise, choose an equally spaced se i M i n • = = • ( 29 min max * * * 1 t of points in [x ,x ] x x 1 1 ˆ For each point x , f x K[t] is the kernel function: common choices are the normal pdf, K[t] = (t) Epanechniko n i m m x m i K n B B = - • = φ ∑ ( 29 2 1/5 * v kernel, K[t] = .75(1-.2t ) / 5, if |t| 5, 0 else B is the bandwidth: e.g., Silverman's Rule of Thumb = .9w/n w = Min(s , /1.349) ˆ Plot f x ag x x m IQR ≤ • * ainst x and connect points. m Part 22: Semi- and Nonparametric Estimation Kernel Density Estimator ˜™™ ™ 8/26 = - = = = = ∑ n i m m m i 1 * * * x x 1 1 ˆ f(x ) K , for a set of points x n B B B "bandwidth" K the kernel function x* the point at which the density is approximated. ˆ f(x*) is an estimator of f(x*) 1 The curse of dimensionality = = ≠ × = ∑ n i i 1 3/5 Q(x | x*) Q(x*). n 1 1 But, Var[Q(x*)] Something. Rather, Var[Q(x*)] * something N N ˆ I.e.,f(x*) does not converge to f(x*) at the same rate as a mean converges to a population mean. Part 22: Semi- and Nonparametric Estimation Kernel Estimator for LWAGE ˜˜™™ ™ 9/26 Part 22: Semi- and Nonparametric Estimation Application: Stochastic Frontier Model Production Function Regression: logY = b’x + v - u where u is “inefficiency.” u > 0. v is normally distributed. Save for the constant term, the model is consistently estimated by OLS. If the theory is right, the OLS residuals will be skewed to the left, rather than symmetrically distributed if they were normally distributed. Application: Spanish dairy data used in Assignment 2 yit = log of milk production x1 = log cows, x2 = log land, x3 = log feed, x4 = log labor ˜˜™™ ™ 10/26 Part 22: Semi- and Nonparametric Estimation Regression Results ˜˜™™ ™ 11/26 Part 22: Semi- and Nonparametric Estimation Distribution of OLS Residuals ˜˜™™ ™ 12/26 Part 22: Semi- and Nonparametric Estimation A Nonparametric Regression p y = µ(x) +ε p Smoothing methods to approximate µ(x) at specific points, x* p For a particular x*, µ(x*) = ∑i wi(x*|...
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This is essentially a histogram with small bins Part 22...

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