--- title: "Econ 425T Homework 1" subtitle: "Due Jan 25, 2023 @ 11:59PM" author: "YOUR NAME and UID" date: "r format(Sys.time(), '%d %B, %Y')" format: html: theme: cosmo number-sections: true toc: true toc-depth: 4 toc-location: left code-fold: false engine: knitr knitr: opts_chunk: fig.align: 'center' # fig.width: 6 # fig.height: 4 message: FALSE cache: false --- ## Filling gaps in lecture notes (10pts) Consider the regression model $$Y = f(X) + \epsilon,$$ where $\operatorname{E}(\epsilon) = 0$. ### Optimal regression function Show that the choice $$f_{\text{opt}}(X) = \operatorname{E}(Y | X)$$ minimizes the mean squared prediction error $$\operatorname{E}[Y - f(X)]^2,$$ where the expectations averages over variations in both $X$ and $Y$. (Hint: condition on $X$.) ### Bias-variance trade-off Given an estimate $\hat f$ of $f$, show that the test error at a $x_0$ can be decomposed as $$\operatorname{E}[y_0 - \hat f(x_0)]^2 = \underbrace{\operatorname{Var}(\hat f(x_0)) + [\operatorname{Bias}(\hat f(x_0))]^2}_{\text{MSE of } \hat f(x_0) \text{ for estimating } f(x_0)} + \underbrace{\operatorname{Var}(\epsilon)}_{\text{irreducible}},$$ where the expectation averages over the variability in $y_0$ and $\hat f$. ## ISL Exercise 2.4.3 (10pts) ## ISL Exercise 2.4.4 (10pts) ## ISL Exercise 2.4.10 (30pts) Your can read in the boston data set directly from url . A documentation of the boston data set is [here](https://www.rdocumentation.org/packages/ISLR2/versions/1.3-2/topics/Boston). ::: {.panel-tabset} #### Python {python} import pandas as pd import io import requests url = "https://raw.githubusercontent.com/ucla-econ-425t/2023winter/master/slides/data/Boston.csv" s = requests.get(url).content Boston = pd.read_csv(io.StringIO(s.decode('utf-8')), index_col = 0) Boston  #### R {r} library(tidyverse) Boston <- read_csv("https://raw.githubusercontent.com/ucla-econ-425t/2023winter/master/slides/data/Boston.csv", col_select = -1) %>% print(width = Inf)  ::: ## ISL Exercise 3.7.3 (12pts) ## ISL Exercise 3.7.15 (20pts) ## Bonus question (20pts) For multiple linear regression, show that $R^2$ is equal to the correlation between the response vector $\mathbf{y} = (y_1, \ldots, y_n)^T$ and the fitted values $\hat{\mathbf{y}} = (\hat y_1, \ldots, \hat y_n)^T$. That is $$R^2 = 1 - \frac{\text{RSS}}{\text{TSS}} = [\operatorname{Cor}(\mathbf{y}, \hat{\mathbf{y}})]^2.$$