Now what do I do with this function?
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1 Now what do I do with this function? Enrique Pinzón StataCorp LP December 08, 2017 Sao Paulo (StataCorp LP) December 08, 2017 Sao Paulo 1 / 42
2 Initial thoughts Nonparametric regression and about effects/questions npregress Mean relation between an outcome and covariates Model birtweight : age, education level, smoked, number of prenatal visits,... Model wages: age, education level, profession, tenure,... E (y X), conditional mean Parametric models have a known functional form Linear regression: E (y X) = X β Binary: E (y X) = F(Xβ) Poisson: E (y X) = exp(x β) Nonparametric E (y X). The result of using predict (StataCorp LP) December 08, 2017 Sao Paulo 2 / 42
3 Initial thoughts Nonparametric regression and about effects/questions npregress Mean relation between an outcome and covariates Model birtweight : age, education level, smoked, number of prenatal visits,... Model wages: age, education level, profession, tenure,... E (y X), conditional mean Parametric models have a known functional form Linear regression: E (y X) = X β Binary: E (y X) = F(Xβ) Poisson: E (y X) = exp(x β) Nonparametric E (y X). The result of using predict (StataCorp LP) December 08, 2017 Sao Paulo 2 / 42
4 Initial thoughts Nonparametric regression and about effects/questions npregress Mean relation between an outcome and covariates Model birtweight : age, education level, smoked, number of prenatal visits,... Model wages: age, education level, profession, tenure,... E (y X), conditional mean Parametric models have a known functional form Linear regression: E (y X) = X β Binary: E (y X) = F(Xβ) Poisson: E (y X) = exp(x β) Nonparametric E (y X). The result of using predict (StataCorp LP) December 08, 2017 Sao Paulo 2 / 42
5 Initial thoughts Nonparametric regression and about effects/questions npregress Mean relation between an outcome and covariates Model birtweight : age, education level, smoked, number of prenatal visits,... Model wages: age, education level, profession, tenure,... E (y X), conditional mean Parametric models have a known functional form Linear regression: E (y X) = X β Binary: E (y X) = F(Xβ) Poisson: E (y X) = exp(x β) Nonparametric E (y X). The result of using predict (StataCorp LP) December 08, 2017 Sao Paulo 2 / 42
6 Initial thoughts Nonparametric regression and about effects/questions npregress Mean relation between an outcome and covariates Model birtweight : age, education level, smoked, number of prenatal visits,... Model wages: age, education level, profession, tenure,... E (y X), conditional mean Parametric models have a known functional form Linear regression: E (y X) = X β Binary: E (y X) = F(Xβ) Poisson: E (y X) = exp(x β) Nonparametric E (y X). The result of using predict (StataCorp LP) December 08, 2017 Sao Paulo 2 / 42
7 Initial thoughts Nonparametric regression and about effects/questions npregress Mean relation between an outcome and covariates Model birtweight : age, education level, smoked, number of prenatal visits,... Model wages: age, education level, profession, tenure,... E (y X), conditional mean Parametric models have a known functional form Linear regression: E (y X) = X β Binary: E (y X) = F(Xβ) Poisson: E (y X) = exp(x β) Nonparametric E (y X). The result of using predict (StataCorp LP) December 08, 2017 Sao Paulo 2 / 42
8 (StataCorp LP) December 08, 2017 Sao Paulo 3 / 42
9 But... We had nonparametric regression tools lpoly lowess (StataCorp LP) December 08, 2017 Sao Paulo 4 / 42
10 But... We had nonparametric regression tools lpoly lowess (StataCorp LP) December 08, 2017 Sao Paulo 4 / 42
11 What happened in the past lpoly bweight mage if (msmoke==0 & medu>12 & fedu>12), /// mcolor(%30) lineopts(lwidth(thick)) (StataCorp LP) December 08, 2017 Sao Paulo 5 / 42
12 Effects: A thought experiment I give you the true function. list y x a gx in 1/10, noobs y x a gx (StataCorp LP) December 08, 2017 Sao Paulo 6 / 42
13 Effects: A thought experiment I give you the true function. list y x a gx in 1/10, noobs y x a gx (StataCorp LP) December 08, 2017 Sao Paulo 6 / 42
14 Effects: A thought experiment I give you the true function Do we know what are the marginal effects Do we know causal/treatment effects Do we know counterfactuals It seems cosmetic We cannot use margins (StataCorp LP) December 08, 2017 Sao Paulo 7 / 42
15 Effects: A thought experiment I give you the true function Do we know what are the marginal effects Do we know causal/treatment effects Do we know counterfactuals It seems cosmetic We cannot use margins (StataCorp LP) December 08, 2017 Sao Paulo 7 / 42
16 Effects: A thought experiment I give you the true function Do we know what are the marginal effects Do we know causal/treatment effects Do we know counterfactuals It seems cosmetic We cannot use margins (StataCorp LP) December 08, 2017 Sao Paulo 7 / 42
17 Effects: A thought experiment I give you the true function Do we know what are the marginal effects Do we know causal/treatment effects Do we know counterfactuals It seems cosmetic We cannot use margins (StataCorp LP) December 08, 2017 Sao Paulo 7 / 42
18 A detour margins (StataCorp LP) December 08, 2017 Sao Paulo 8 / 42
19 Effects: outcome of interest (StataCorp LP) December 08, 2017 Sao Paulo 9 / 42
20 Data crash 1 if crash traffic Measure of vehicular traffic tickets Number of traffic tickets male 1 if male (StataCorp LP) December 08, 2017 Sao Paulo 10 / 42
21 Probit model and average marginal effects probit crash tickets traffic i.male. margins Predictive margins Number of obs = 948 Model VCE : OIM Expression : Pr(crash), predict() Delta-method Margin Std. Err. z P> z [95% Conf. Interval] _cons margins, dydx(traffic tickets) Average marginal effects Number of obs = 948 Model VCE : OIM Expression : Pr(crash), predict() dy/dx w.r.t. : tickets traffic Delta-method dy/dx Std. Err. z P> z [95% Conf. Interval] tickets traffic (StataCorp LP) December 08, 2017 Sao Paulo 11 / 42
22 Probit model and average marginal effects probit crash tickets traffic i.male. margins Predictive margins Number of obs = 948 Model VCE : OIM Expression : Pr(crash), predict() Delta-method Margin Std. Err. z P> z [95% Conf. Interval] _cons margins, dydx(traffic tickets) Average marginal effects Number of obs = 948 Model VCE : OIM Expression : Pr(crash), predict() dy/dx w.r.t. : tickets traffic Delta-method dy/dx Std. Err. z P> z [95% Conf. Interval] tickets traffic (StataCorp LP) December 08, 2017 Sao Paulo 11 / 42
23 Probit model and average marginal effects probit crash tickets traffic i.male. margins Predictive margins Number of obs = 948 Model VCE : OIM Expression : Pr(crash), predict() Delta-method Margin Std. Err. z P> z [95% Conf. Interval] _cons margins, dydx(traffic tickets) Average marginal effects Number of obs = 948 Model VCE : OIM Expression : Pr(crash), predict() dy/dx w.r.t. : tickets traffic Delta-method dy/dx Std. Err. z P> z [95% Conf. Interval] tickets traffic (StataCorp LP) December 08, 2017 Sao Paulo 11 / 42
24 Not calculus. margins, at(traffic=generate(traffic*1.10)) at(traffic=generate(traffic)) /// > contrast(atcontrast(r) nowald) Contrasts of predictive margins Model VCE : OIM Expression : Pr(crash), predict() 1._at : traffic = traffic* _at : traffic = traffic Delta-method Contrast Std. Err. [95% Conf. Interval] _at (2 vs 1) (StataCorp LP) December 08, 2017 Sao Paulo 12 / 42
25 Probit model and counterfactuals. margins male Predictive margins Number of obs = 948 Model VCE : OIM Expression : Pr(crash), predict() Delta-method Margin Std. Err. z P> z [95% Conf. Interval] male margins, dydx(male) Average marginal effects Number of obs = 948 Model VCE : OIM Expression : Pr(crash), predict() dy/dx w.r.t. : 1.male Delta-method dy/dx Std. Err. z P> z [95% Conf. Interval] 1.male Note: dy/dx for factor levels is the discrete change from the base level. (StataCorp LP) December 08, 2017 Sao Paulo 13 / 42
26 Probit model and counterfactuals. margins male Predictive margins Number of obs = 948 Model VCE : OIM Expression : Pr(crash), predict() Delta-method Margin Std. Err. z P> z [95% Conf. Interval] male margins, dydx(male) Average marginal effects Number of obs = 948 Model VCE : OIM Expression : Pr(crash), predict() dy/dx w.r.t. : 1.male Delta-method dy/dx Std. Err. z P> z [95% Conf. Interval] 1.male Note: dy/dx for factor levels is the discrete change from the base level. (StataCorp LP) December 08, 2017 Sao Paulo 13 / 42
27 More counterfactuals. margins, dydx(tickets) Average marginal effects Number of obs = 948 Model VCE : OIM Expression : Pr(crash), predict() dy/dx w.r.t. : tickets Delta-method dy/dx Std. Err. z P> z [95% Conf. Interval] tickets margins, at(tickets=(0(1)5)) contrast(atcontrast(ar) nowald) Contrasts of predictive margins Model VCE : OIM Expression : Pr(crash), predict() 1._at : tickets = 0 2._at : tickets = 1 3._at : tickets = 2 4._at : tickets = 3 5._at : tickets = 4 6._at : tickets = 5 Delta-method Contrast Std. Err. [95% Conf. Interval] _at (2 vs 1) (3 vs 2) (4 vs 3) (5 vs 4) (6 vs 5) (StataCorp LP) December 08, 2017 Sao Paulo 14 / 42
28 More counterfactuals. margins, dydx(tickets) Average marginal effects Number of obs = 948 Model VCE : OIM Expression : Pr(crash), predict() dy/dx w.r.t. : tickets Delta-method dy/dx Std. Err. z P> z [95% Conf. Interval] tickets margins, at(tickets=(0(1)5)) contrast(atcontrast(ar) nowald) Contrasts of predictive margins Model VCE : OIM Expression : Pr(crash), predict() 1._at : tickets = 0 2._at : tickets = 1 3._at : tickets = 2 4._at : tickets = 3 5._at : tickets = 4 6._at : tickets = 5 Delta-method Contrast Std. Err. [95% Conf. Interval] _at (2 vs 1) (3 vs 2) (4 vs 3) (5 vs 4) (6 vs 5) (StataCorp LP) December 08, 2017 Sao Paulo 14 / 42
29 marginsplot margins, at(tickets=(0(1)5)) marginsplot, ciopts(recast(rarea)) (StataCorp LP) December 08, 2017 Sao Paulo 15 / 42
30 Back to nonparametric regression npregress and nonparametric regression (StataCorp LP) December 08, 2017 Sao Paulo 16 / 42
31 Nonparametric regression: discrete covariates Mean function for a discrete covariate Mean wage conditional on having a college degree. mean wage if collgrad==1 Mean estimation Number of obs = 4,795 Mean Std. Err. [95% Conf. Interval] wage regress wage collgrad, noconstant E(wage collgrad = 1), nonparametric estimate (StataCorp LP) December 08, 2017 Sao Paulo 17 / 42
32 Nonparametric regression: discrete covariates Mean function for a discrete covariate Mean wage conditional on having a college degree. mean wage if collgrad==1 Mean estimation Number of obs = 4,795 Mean Std. Err. [95% Conf. Interval] wage regress wage collgrad, noconstant E(wage collgrad = 1), nonparametric estimate (StataCorp LP) December 08, 2017 Sao Paulo 17 / 42
33 Nonparametric regression: discrete covariates Mean function for a discrete covariate Mean wage conditional on having a college degree. mean wage if collgrad==1 Mean estimation Number of obs = 4,795 Mean Std. Err. [95% Conf. Interval] wage regress wage collgrad, noconstant E(wage collgrad = 1), nonparametric estimate (StataCorp LP) December 08, 2017 Sao Paulo 17 / 42
34 Nonparametric regression: continuous covariates Conditional mean for a continuous covariate Mean wage conditional on tenure, measured in years E(wage tenure = ) Take observations near the value of and then take an average tenure i h h is a small number referred to as the bandwidth (StataCorp LP) December 08, 2017 Sao Paulo 18 / 42
35 Nonparametric regression: continuous covariates Conditional mean for a continuous covariate Mean wage conditional on tenure, measured in years E(wage tenure = ) Take observations near the value of and then take an average tenure i h h is a small number referred to as the bandwidth (StataCorp LP) December 08, 2017 Sao Paulo 18 / 42
36 Nonparametric regression: continuous covariates Conditional mean for a continuous covariate Mean wage conditional on tenure, measured in years E(wage tenure = ) Take observations near the value of and then take an average tenure i h h is a small number referred to as the bandwidth (StataCorp LP) December 08, 2017 Sao Paulo 18 / 42
37 Nonparametric regression: continuous covariates Conditional mean for a continuous covariate Mean wage conditional on tenure, measured in years E(wage tenure = ) Take observations near the value of and then take an average tenure i h h is a small number referred to as the bandwidth (StataCorp LP) December 08, 2017 Sao Paulo 18 / 42
38 Nonparametric regression: continuous covariates Conditional mean for a continuous covariate Mean wage conditional on tenure, measured in years E(wage tenure = ) Take observations near the value of and then take an average tenure i h h is a small number referred to as the bandwidth (StataCorp LP) December 08, 2017 Sao Paulo 18 / 42
39 Nonparametric regression: continuous covariates Conditional mean for a continuous covariate Mean wage conditional on tenure, measured in years E(wage tenure = ) Take observations near the value of and then take an average tenure i h h is a small number referred to as the bandwidth (StataCorp LP) December 08, 2017 Sao Paulo 18 / 42
40 Graphical example (StataCorp LP) December 08, 2017 Sao Paulo 19 / 42
41 Graphical example (StataCorp LP) December 08, 2017 Sao Paulo 20 / 42
42 Graphical example continued (StataCorp LP) December 08, 2017 Sao Paulo 21 / 42
43 Two concepts 1 h 2 Definition of distance between points, x i x h 1 (StataCorp LP) December 08, 2017 Sao Paulo 22 / 42
44 Kernel weights u x i x h Kernel K (u) ( ) 1 Gaussian exp u2 ( 2π ) 2 3 Epanechnikov 4 1 u2 5 5 I ( u 5 ) ( 3 Epanechnikov2 4 1 u 2 ) I ( u 1) 1 Rectangular(Uniform) 2I ( u 1) Triangular (1 u ) I ( u 1) ( 15 Biweight 16 1 u 2 ) 2 I ( u 1) ( 35 Triweight 32 1 u 2 ) 3 I ( u 1) Cosine (1 + cos (2πu)) I ( u 1 ) ( 2 Parzen 4 3 8u2 + 8 u 3) I ( u 1 2) (1 u )3 I ( 1 2 < u 1) (StataCorp LP) December 08, 2017 Sao Paulo 23 / 42
45 Kernel weights u x i x h Kernel K (u) ( ) 1 Gaussian exp u2 ( 2π ) 2 3 Epanechnikov 4 1 u2 5 5 I ( u 5 ) ( 3 Epanechnikov2 4 1 u 2 ) I ( u 1) 1 Rectangular(Uniform) 2I ( u 1) Triangular (1 u ) I ( u 1) ( 15 Biweight 16 1 u 2 ) 2 I ( u 1) ( 35 Triweight 32 1 u 2 ) 3 I ( u 1) Cosine (1 + cos (2πu)) I ( u 1 ) ( 2 Parzen 4 3 8u2 + 8 u 3) I ( u 1 2) (1 u )3 I ( 1 2 < u 1) (StataCorp LP) December 08, 2017 Sao Paulo 23 / 42
46 Discrete bandwidths Default Cell mean { 1 if xi = x k (.) = h otherwise k (.) = { 1 if xi = x 0 otherwise (StataCorp LP) December 08, 2017 Sao Paulo 24 / 42
47 Bandwidth (bias) (StataCorp LP) December 08, 2017 Sao Paulo 25 / 42
48 Bandwidth (variance) (StataCorp LP) December 08, 2017 Sao Paulo 26 / 42
49 Estimation Choose bandwidth optimally. Minimize bias variance trade off Cross-validation (default) Improved AIC (IMAIC) Compute a regression for every point in data (local linear) Computes derivatives and derivative bandwidths Compute a mean for every point in data (local-constant) (StataCorp LP) December 08, 2017 Sao Paulo 27 / 42
50 Example citations monthly drunk driving citations taxes 1 if alcoholic beverages are taxed fines drunk driving fines in thousands csize city size (small, medium, large) college 1 if college town (StataCorp LP) December 08, 2017 Sao Paulo 28 / 42
51 Example citations monthly drunk driving citations taxes 1 if alcoholic beverages are taxed fines drunk driving fines in thousands csize city size (small, medium, large) college 1 if college town (StataCorp LP) December 08, 2017 Sao Paulo 28 / 42
52 npregress bandwidth. npregress kernel citations fines Computing mean function Minimizing cross-validation function: Iteration 0: Cross-validation criterion = Iteration 1: Cross-validation criterion = Iteration 2: Cross-validation criterion = Iteration 3: Cross-validation criterion = Iteration 4: Cross-validation criterion = Iteration 5: Cross-validation criterion = Iteration 6: Cross-validation criterion = Computing optimal derivative bandwidth Iteration 0: Cross-validation criterion = Iteration 1: Cross-validation criterion = Iteration 2: Cross-validation criterion = Iteration 3: Cross-validation criterion = Iteration 4: Cross-validation criterion = Iteration 5: Cross-validation criterion = Iteration 6: Cross-validation criterion = Iteration 7: Cross-validation criterion = Iteration 8: Cross-validation criterion = (StataCorp LP) December 08, 2017 Sao Paulo 29 / 42
53 npregress output. npregress kernel citations fines, nolog Bandwidth Mean Mean Effect fines Local-linear regression Number of obs = 500 Kernel : epanechnikov E(Kernel obs) = 282 Bandwidth: cross validation R-squared = citations Estimate Mean citations Effect fines Note: Effect estimates are averages of derivatives. Note: You may compute standard errors using vce(bootstrap) or reps(). (StataCorp LP) December 08, 2017 Sao Paulo 30 / 42
54 npregress predicted values. describe _* storage display value variable name type format label variable label _Mean_citations double %10.0g mean function _d_mean_citat ~ s double %10.0g derivative of mean function w.r.t fines (StataCorp LP) December 08, 2017 Sao Paulo 31 / 42
55 npgraph. npgraph (StataCorp LP) December 08, 2017 Sao Paulo 32 / 42
56 npregress standard errors I. quietly npregress kernel citations fines, reps(3) seed(111). estimates store A. quietly npregress kernel citations fines, vce(bootstrap, reps(3) seed(111)). estimates store B. estimates table A B, se Variable A B Mean citations Effect fines legend: b/se (StataCorp LP) December 08, 2017 Sao Paulo 33 / 42
57 npregress standard errors II (percentile C.I.). npregress Bandwidth Mean Mean Effect fines Local-linear regression Number of obs = 500 Kernel : epanechnikov E(Kernel obs) = 282 Bandwidth: cross validation R-squared = Observed Bootstrap Percentile citations Estimate Std. Err. z P> z [95% Conf. Interval] Mean citations Effect fines Note: Effect estimates are averages of derivatives. (StataCorp LP) December 08, 2017 Sao Paulo 34 / 42
58 A more interesting model. npregress kernel citations fines i.taxes i.csize i.college, reps(200) seed(10) Bandwidth Mean Mean Effect fines taxes csize college Local-linear regression Number of obs = 500 Continuous kernel : epanechnikov E(Kernel obs) = 224 Discrete kernel : liracine R-squared = Bandwidth : cross validation Observed Bootstrap Percentile citations Estimate Std. Err. z P> z [95% Conf. Interval] Mean citations Effect fines taxes (tax vs no tax) csize (medium vs small) (large vs small) college (college vs not coll..) Note: Effect estimates are averages of derivatives for continuous covariates and averages of contrasts for factor covariates. (StataCorp LP) December 08, 2017 Sao Paulo 35 / 42
59 margins. margins, at(fines=generate(fines)) at(fines=generate(fines*1.15)) /// > contrast(atcontrast(r) nowald) reps(200) seed(12) (running margins on estimation sample) Bootstrap replications (200) Contrasts of predictive margins Number of obs = 500 Replications = 200 Expression : mean function, predict() 1._at : fines = fines 2._at : fines = fines*1.15 Observed Bootstrap Percentile Contrast Std. Err. [95% Conf. Interval] _at (2 vs 1) (StataCorp LP) December 08, 2017 Sao Paulo 36 / 42
60 Another example with margins 10 + x 3 + ε if a = 0 y = 10 + x 3 10x + ε if a = x 3 + 3x + ε if a = 2 (StataCorp LP) December 08, 2017 Sao Paulo 37 / 42
61 Mean and marginal effects. quietly regress y (c.x#c.x#c.x)#i.a c.x#i.a. margins Predictive margins Number of obs = 1,000 Model VCE : OLS Expression : Linear prediction, predict() Delta-method Margin Std. Err. t P> t [95% Conf. Interval] _cons margins, dydx(*) Average marginal effects Number of obs = 1,000 Model VCE : OLS Expression : Linear prediction, predict() dy/dx w.r.t. : 1.a 2.a x Delta-method dy/dx Std. Err. t P> t [95% Conf. Interval] a x Note: dy/dx for factor levels is the discrete change from the base level. (StataCorp LP) December 08, 2017 Sao Paulo 38 / 42
62 Mean and marginal effects. quietly regress y (c.x#c.x#c.x)#i.a c.x#i.a. margins Predictive margins Number of obs = 1,000 Model VCE : OLS Expression : Linear prediction, predict() Delta-method Margin Std. Err. t P> t [95% Conf. Interval] _cons margins, dydx(*) Average marginal effects Number of obs = 1,000 Model VCE : OLS Expression : Linear prediction, predict() dy/dx w.r.t. : 1.a 2.a x Delta-method dy/dx Std. Err. t P> t [95% Conf. Interval] a x Note: dy/dx for factor levels is the discrete change from the base level. (StataCorp LP) December 08, 2017 Sao Paulo 38 / 42
63 npregress estimates. npregress kernel y x i.a, vce(bootstrap, reps(100) seed(111)) (running npregress on estimation sample) Bootstrap replications (100) Bandwidth Mean Mean Effect x a 3.05e e-06 Local-linear regression Number of obs = 1,000 Continuous kernel : epanechnikov E(Kernel obs) = 363 Discrete kernel : liracine R-squared = Bandwidth : cross validation Mean Effect Observed Bootstrap Percentile y Estimate Std. Err. z P> z [95% Conf. Interval] y x a (1 vs 0) (2 vs 0) Note: Effect estimates are averages of derivatives for continuous covariates and averages of contrasts for factor covariates. (StataCorp LP) December 08, 2017 Sao Paulo 39 / 42
64 Function for different values of x. margins, at(x=(1(.5)3)) reps(100) seed(111) (StataCorp LP) December 08, 2017 Sao Paulo 40 / 42
65 Funtion at different values of x for all a. margins a, at(x=(-1(1)3)) reps(100) seed(111) (StataCorp LP) December 08, 2017 Sao Paulo 41 / 42
66 Conclusion Intuition about nonparametric regression Details about how npregress Importance of being able to ask questions to your model (StataCorp LP) December 08, 2017 Sao Paulo 42 / 42
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