Econometrics-I-22

# Part 22 semi and nonparametric estimation a

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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*| x )yi n E.g., for ols, µ(x*) =a+bx* n wi = 1/n + p We look for weighting scheme, local differences in relationship. OLS assumes a fixed slope, b. ™    13/26 2 ( ) / ( ) i i i - Σ - x x x x

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Part 22: Semi- and Nonparametric Estimation Nearest Neighbor Approach p Define a neighborhood of x*. Points near get high weight, points far away get a small or zero weight p Bandwidth, h defines the neighborhood: e.g., Silverman h =.9Min[s,(IQR/1.349)]/n.2 Neighborhood is + or – h/2 p LOWESS weighting function: (tricube) Ti = [1 – [Abs(xi – x*)/h]3]3. p Weight is wi = 1[Abs(xi – x*)/h < .5] * Ti . ™    14/26
Part 22: Semi- and Nonparametric Estimation LOWESS Regression ™    15/26

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Part 22: Semi- and Nonparametric Estimation OLS Vs. Lowess ™    16/26
Part 22: Semi- and Nonparametric Estimation Smooth Function: Kernel Regression ™    17/26 1 1 2 * 1 ˆ( *| , ) * 1 Kernel Functions: Normal: K(t) = (t) Logistic: K(t) = (t)[1- (t)] Epanechnikov: K(t)=.75(1-.2t )/ 5, if |t| 5 and 0 otherwise n i i i n i i x x K y B B x B x x K B B = = - μ = - φ Λ Λ x

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Part 22: Semi- and Nonparametric Estimation Kernel Regression vs. Lowess (Lwage vs. Educ) ™    18/26
Part 22: Semi- and Nonparametric Estimation Locally Linear Regression ™    19/26 1 1 1 ( *) ( *)' *. ( *) ( *, ) ( *, ) y ( *, ) [( * ) ( * ), ] n n i i i i i i i i i i i i i i w w w K h - = = μ = β β = = - - x x x x x x x x x x x x x x x x x

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Part 22: Semi- and Nonparametric Estimation OLS vs. LOWESS ™    20/26
Part 22: Semi- and Nonparametric Estimation Quantile Regression p Least squares based on: E[y| x ]= ’x p LAD based on: Median[y| x ]= (.5)’ x p Quantile regression: Q(y| x ,q)= (q)’ x p Does this just shift the constant?

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