* The Delta method can be used to estimate the standard errors after a regression estimation.

* Imagine you have some parameter G = (3*b0-b1)*b2^2 = 3*b0*b2^2-b1*b2^2

* Where y = bo + b1*x1 + b2*x2 + u

* The delta method can be used to estimate the standard errors of G.

* The delta method states that var_hat(G)=(dG/db) var(b) (dG/db)

* dG/db is a gradient vector:

* dG/db = [dG/db0, dG/db1, dG/db2]

* dG/db = [b2^2, -b2^2, 2*(b0-b1)*b2]

* var_hat(G) = (3*b2^2)^2 * se(b0)^-2 + (-b2^2)^2 * se(b1)^-2 + (2*(b0-b1)*b2)^2 * se(b2)^-2

[There is an error in the code because I failed to include a covariance term for the coefficients. Please see the more recent update on the method.]

clear

set obs 1000

gen x1 = rnormal()

gen x2 = rnormal() * 4

global b0 = 1

global b1 = 1.5

global b2 = .3

local true_G = (3*${b0}-${b1})*${b2}^2

di `true_G'

gen y = ${b0} + ${b1}*x1 + ${b2}*x2 + rnormal()*8

reg y x1 x2

* G = (3*b0-b1)*b2^2 = 3*b0*b2^2 - b1*b2^2

local Ghat = (3*_b[_cons]-_b[x1])*_b[x2]^2

di "Ghat = `Ghat' is our estimate (true = `true_G')"

* Let's see if we can't use the delta method to derive a standard error.

local var_hatG = (3*_b[x2]^2)^2 * _se[_cons]^2 + (-_b[x2]^2)^2 * _se[x1]^2 + (2*(_b[_cons]-_b[x1])*_b[x2])^2 * _se[x2]^2

di "Standard error estimate is " `var_hatG'^.5

* Alternatively, let us attempt to bootstrap our standard errors.

cap program drop deltaOLS

program define deltaOLS, rclass

reg y x1 x2

return scalar Ghat = (3*_b[_cons]-_b[x1])*_b[x2]^2

end

bs Ghat=r(Ghat), rep(500): deltaOLS

* The bootstrap standard errors are similar to that of the delta method's standard errors.

cap program drop deltaMonteCarlo

program define deltaMonteCarlo, rclass

clear

set obs 1000

gen x1 = rnormal()

gen x2 = rnormal() * 4

gen y = ${b0} + ${b1}*x1 + ${b2}*x2 + rnormal()*8

reg y x1 x2

return scalar Ghat = (3*_b[_cons]-_b[x1])*_b[x2]^2

end

simulate Ghat=r(Ghat), reps(500): deltaMonteCarlo

sum

* We can see that our estimates of the standard error via the Delta method and bootstap is quite close to the monte carlo estimates on observed standard errors from 500 replications.

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