---
title: "Naive Estimation of ATE"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Naive Estimation of ATE}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

## Introduction

This vignette demonstrates the usage of the `Naive_Estimation` function in the `ATE.ERROR` package. The `Naive_Estimation` function provides a simple method for estimating ATE.

## Generating Simulated Data

First, we generate our simulated data using the `data(Simulated_data)` syntax.

```{r setup}
library(ATE.ERROR)
set.seed(1)
data(Simulated_data)
Y_star <- Simulated_data$Y_star
A <- Simulated_data$T
Z <- Simulated_data$Z
X_star <- Simulated_data$X_star
```

## Naive Estimation of ATE

Now we use the `Naive_Estimation` function.

```{r}
Naive_ATE_XY <- Naive_Estimation(Y_star, A, Z, X_star)
print(Naive_ATE_XY)
```

The `Naive_Estimation` function calculates the ATE by ignoring the difference between {X^∗, Y^∗} and {X, Y}.