Part $3$: Writing your own regression function $(30$pts total$)$

For this question, you will write and test your own substitute for the $)$ function. Your

function will take in a dataframe with columns labeled $y$ and $x$ and fit the regression of the

form:

$y_i={}_0+{}_1{x}_i+{}_i$

Your function should return a list$()$ object with named components:

data, which is a dataframe that contains the original data used to estimate the regression.

params, which is a dataframe with the estimates widehat$({)}_0,$widehat$({)}_1$ and their corresponding standard

errors hat$({)}_{{}_0},$hat$({)}_{{}_1},$ which can be computed as follows: hat$({)}_1=\frac{{\displaystyle \underset{i}{\overset{?}{}}}(x_i-(\stackrel{}{x}))(y_i-(\frac{?}{\text{b}\text{a}\text{r}}(y)))}{{\displaystyle \underset{i}{\overset{?}{}}}(x_i-(\stackrel{}{x}){)}^2}$

hat$({)}_0\frac{?}{b}=ar(y)-$hat$({)}_1\stackrel{}{x}$

hat$({)}_{{}_1}=$hat$({)}_{}\sqrt[\phantom{2}]{\frac{1}{{\displaystyle \underset{i}{\overset{?}{}}}(x_i-(\stackrel{}{x}){)}^2}}$

hat$({)}_{{}_0}=$hat$({)}_{}\sqrt[\phantom{2}]{\frac{1}{n}+\frac{{\stackrel{}{x}}^2}{{\displaystyle \underset{i}{\overset{?}{}}}(x_i-(\stackrel{}{x}){)}^2}}=$hat$({)}_{{}_1}\sqrt[\phantom{2}]{\frac{{\displaystyle \underset{i}{\overset{?}{}}}{x}_i^2}{n}}$

where $n$ is the number of observations.

pred, which is a vector of the predicted $y$ values, hat$(y{)}_i,$ where

hat$(y{)}_i=$hat$({)}_0+$hat$({)}_1{x}_i$

resid, which is a vector of residuals:

$e_i=y_i-$hat$(y{)}_i$

mse which is the estimate of the error variance:

hat$({)}_{}^2=\frac{1}{n-2}{\displaystyle \underset{i}{\overset{?}{}}}{e}_i^2$

rsq which is the $R^2,$ or coefficient of determination, which is the proportion of variance

explained by the regression:

$R^2=1-\frac{{\displaystyle \underset{i}{\overset{?}{}}}{e}_i^2}{{\displaystyle \underset{i}{\overset{?}{}}}(y_i-(\frac{?}{\text{b}\text{a}\text{r}}(y)){)}^2}$

plot which is a ggplot object that contains a scatterplot of the data with a line of

best fit superimposed. To add the line, use geom$\_$abline$()$ and your estimates of hat$({)}_0,$hat$({)}_1.$

Make sure the line is a fun color!

I suggest beginning by writing individual functions for each of these tasks and testing these

functions repeatedly against the actual output of $)$ to be sure you're getting similar results.

A good way to simulate univariate linear regression data to compare to is:

N

d$\_$reg data.frame$($x $=$ rnorm$($N$))$

d$\_$regyegx $+$ rnorm$($N$)$

m lm$($y x$,$ d$\_$reg$)$

summary$($m$)$ Call:

lm$($formula $=$ y ~ x$,$ data $=$ d$\_$reg$)$

Residuals:

Min rrrrr

Coefficients:

Estimate Std$.$ Error t value Pr$(>|$t$|)$Signif$.$ codes: $0\text{'}\text{'}0\mathrm{.}001\text{'}\text{'}0\mathrm{.}01\text{'}\text{'}0\mathrm{.}05\text{'}.\text{'}0\mathrm{.}1\text{'}\text{'}1$

Residual standard error: $0\mathrm{.}9838$ on $98$ degrees of freedom

Multiple R$-$squared: $0\mathrm{.}8649,$ Adjusted R$-$squared: $0\mathrm{.}8635$

F$-$statistic: $627\mathrm{.}3$ on $1$ and $98$ DF$,$ p$-$value: $<mtext></mtext><mn>2</mn><mn>.</mn><mn>2</mn>$e$-16$

Note that you can vary the coefficients, and the means$/$variances$/$distributions used to generate

$x$ and the noise in $y.$

Your function should:

Take only a dataframe with columns labeled $x$ and $y$

Output a list object with named entities

Reproduce the corresponding output of the $)$ function exactly

Write your function, unpack the contents of the list$()$ object you produce, and show that

they produce a reasonable plot and reproduce the results from $)$ for some simulated data

$($it need not be the data I produced above$).$

Hint: If you can't find something you need in the summary$(),$ read the documentation for

$lm()!$

# TODO: write your function, fit it to some simulated data, demonstrate that

# it reproduces the output from $)$ by unpacking the contents of the model

# object you build and comparing to the results from lm$()$

MyLM function$($d$)\{$

$\}$