Neural Network Instrumental Variables A simple example library(AER) data("CigarettesSW") rprice <- with(CigarettesSW, price/cpi) tdiff <- with(CigarettesSW, (taxs - tax)/cpi) packs <- CigarettesSW$packs Estimate using OLS. lm(packs ~ rprice) ## ## Call: ## lm(formula = packs ~ rprice) ## ## Coefficients: ## (Intercept) rprice ## 222.209 -1.044 Now using instrumental variables. ivreg(packs ~ rprice | tdiff) ## ## Call: ## ivreg(formula = packs ~ rprice | tdiff) ## ## Coefficients: ## (Intercept) rprice ## 219.576 -1.019 Now using the lm function. # first stage lms1 <- lm(rprice ~ tdiff) # manually obtain fitted values lmXhat <- lms1$coefficients[2]*tdiff + lms1$coefficients[1] # estimate second stage using Xhat (lms2 <- lm(packs ~ lmXhat) ) ## ## Call: ## lm(formula = packs ~ lmXhat) ## ## Coefficients: ## (Intercept) lmXhat ## 219.576 -1.019 Now using a neural network library(nnet) set.seed(123) # first stage nns1 <- nnet(rprice ~ tdiff, size=0, skip=TRUE, linout=TRUE) ## # weights: 2 ## initial value 1123401.708750 ## final value 14467.562948 ## converged # manually obtain fitted values nnXhat <- nns1$fitted.values # estimate second stage using Xhat nns2 <- nnet(packs ~ nnXhat, size=0, skip=TRUE, linout=TRUE) ## # weights: 2 ## initial value 335265.176965 ## final value 48851.806790 ## converged summary(nns2) ## a 1-0-1 network with 2 weights ## options were - skip-layer connections linear output units ## b->o i1->o ## 219.58 -1.02 Compare output. lms2$coefficients - nns2$wts ## (Intercept) lmXhat ## 4.880515e-05 -4.206591e-07 Compare estimates. library(ggplot2) qplot(lms2$fitted.values - nns2$fitted.values) Previous Next

Model vs. Algorithm As discussion about Artificial Intelligence has become a mainstream topic, we hear a lot of terms used in loosely.

Using Days of the Week to Understand Modulo Today is Monday January 1st 2024, what day of the week will it be in 7 days?

BGV in R We begin by loading 2 packages, polynom for dealing with polynomials and my package HomomorophicEncryption which has several helper functions.