---
title: "Two-Stock Portfolio"
author: "Trevor Hastie and Robert Tibshirani"
output:
  pdf_document: default
---

Here, I am adapting the lab associated with Chapter 5 of the textbook. 

There is no reason for us to repeatedly be doing this, but this is the first time we are using the library associated with the book. So, it's a good idea to install the package `ISLR2` and make the library accessible. 

```{r}
library(ISLR2)
#Portfolio
```

Here is the content by *Hastie and Tibshirani*:

One of the great advantages of the bootstrap approach is that it can be
applied in almost all situations. No complicated mathematical calculations
are required. Performing a bootstrap analysis in `R` entails only two
steps. First, we must create a function that computes the statistic of
interest. Second, we use the `boot()` function, which is part of the `boot` library, to perform the bootstrap by repeatedly
sampling observations from the data set with replacement.

```{r}
library(boot)
??boot
```

The `Portfolio` data set in the `ISLR2` package is simulated data of $100$ pairs of returns, generated in the fashion described in Section 5.2.
To illustrate the use of the bootstrap on this data, we must first create a function, `alpha.fn()`, which takes as input the $(X,Y)$ data as well as a vector indicating which observations should be used to estimate $\alpha$. The function then outputs the estimate for $\alpha$ based on the selected observations.

```{r chunk11}
alpha.fn <- function(data, index) {
  X <- data$X[index]
  Y <- data$Y[index]
  (var(Y) - cov(X, Y)) / (var(X) + var(Y) - 2 * cov(X, Y))
}
```

This function **returns**, or outputs, an  estimate for $\alpha$ based on applying the formula for the optimal weight from the book to the observations indexed by the argument `index`. For instance, the following command tells `R` to estimate $\alpha$ using all $100$ observations.

```{r chunk12}
alpha.fn(Portfolio, 1:100)
```

The next command  uses the `sample()` function to randomly select $100$ observations from the range $1$ to $100$, with replacement. This is equivalent to constructing a new bootstrap data set and recomputing $\hat{\alpha}$
based on the new data set.

```{r chunk13}
#we set seed for replicability
set.seed(7)
alpha.fn(Portfolio, sample(100, 100, replace = T))
```

We can implement a bootstrap analysis by performing this command many times, recording all of
the corresponding estimates for $\alpha$, and computing the resulting standard deviation.
However, the `boot()` function automates this approach. Below we produce $R=1,000$ bootstrap estimates for $\alpha$.

```{r chunk14}
boot(Portfolio, alpha.fn, R = 1000)
```


The final output shows that using the original data, $\hat{\alpha}=0.5758$,
and that the bootstrap estimate for ${\rm SE}(\hat{\alpha})$ is $0.0897$.