Correlation and Simple Linear Regression

Published

Last modified: 2026-11-07 21:34:29 (PST)

This page reviews two ways to relate two continuous variables: correlation coefficients, with tests of whether they differ from zero, and simple linear regression. It uses the $t$ reference distribution defined on the Statistical Inference page. This page is adapted from Vittinghoff et al. [1], Chapter 3.

1 The HERS data

The examples on this page use the HERS data, which the Comparing Means page describes. The rmb R package includes the dataset; haven::as_factor() converts its Stata value labels to factors:

hers <- rmb::hers |> haven::as_factor()
hers |> head()

2 Correlation

2.1 Testing the Pearson correlation

Definition 1 (Population correlation)

Definition 1 (Population correlation) The population correlation of two random variables $X$ and $Y$ with positive variances is

\[\rho\stackrel{\text{def}}{=}\frac{{\operatorname{Cov}\mathopen{}\left(X, Y\right)\mathclose{}}}{\sqrt{{\operatorname{Var}\mathopen{}\left(X\right)\mathclose{}} {\operatorname{Var}\mathopen{}\left(Y\right)\mathclose{}}}},\]

where ${\operatorname{Cov}\mathopen{}\left(X, Y\right)\mathclose{}}$ is their covariance.

Source: Wikipedia contributors [2].

Remark 1 (The sample correlation estimates the population correlation)

Remark 1 (The sample correlation estimates the population correlation). The Pearson correlation coefficient $r$ of a sample is the corresponding sample statistic, and estimates $\rho$.

Definition 2 (Test of zero correlation)

Definition 2 (Test of zero correlation) Let $r$ be the Pearson correlation coefficient of $n \ge 3$ pairs, with $\mathopen{}\left|r\right|\mathclose{} < 1$. The t-test of zero correlation of $H_0: \rho= 0$ against $H_1: \rho\neq 0$ (Definition 1) uses the statistic

\[t \stackrel{\text{def}}{=}\frac{r \sqrt{n - 2}}{\sqrt{1 - r^2}}.\]

Its p-value is $\Pr(\mathopen{}\left|T\right|\mathclose{} \ge \mathopen{}\left|t\right|\mathclose{})$, where $T$ has the $t_{n-2}$ distribution (t-distribution).

Theorem 1 (Null distribution of the correlation t statistic)

Theorem 1 (Null distribution of the correlation t statistic) Let the pairs $(X_1, Y_1), \ldots, (X_n, Y_n)$ be independent, and let each $Y_i$, given $X_1, \ldots, X_n$, be Gaussian with a mean and variance that do not depend on the $X$ values. Then the statistic $t$ of Definition 2 has the $t_{n-2}$ distribution [3, Sec. 9.6], pp. 472-473:

\[ t \sim t_{n-2}. \]

Remark 2 (When the conditions hold)

Remark 2 (When the conditions hold). The conditions of Theorem 1 hold, for example, when the pairs are independent draws from a bivariate Gaussian distribution with $\rho= 0$.

Example 1 (Correlation between BMI and fasting glucose in HERS)

Example 1 (Correlation between BMI and fasting glucose in HERS) Figure 1 plots baseline fasting glucose against BMI.

Show code

hers |>
  dplyr::filter(!is.na(BMI)) |>
  ggplot2::ggplot() +
  ggplot2::aes(x = BMI, y = glucose) +
  ggplot2::geom_point(alpha = 0.3) +
  ggplot2::geom_smooth(method = "lm", formula = y ~ x) +
  ggplot2::labs(
    x = "BMI (kg/m^2)",
    y = "Fasting glucose (mg/dL)"
  )

Figure 1: Baseline fasting glucose against BMI in HERS, with the least-squares line

The statistic of Definition 2, using the 2758 participants with a BMI measurement:

hers_bmi <- hers |> dplyr::filter(!is.na(BMI))
r <- cor(hers_bmi$BMI, hers_bmi$glucose)
n <- nrow(hers_bmi)
t_cor <- r * sqrt(n - 2) / sqrt(1 - r^2)
c(r = r, t = t_cor, p_value = 2 * pt(-abs(t_cor), df = n - 2))
#>           r           t     p_value 
#> 2.72664e-01 1.48779e+01 3.24201e-48

cor.test() reports the same values, and a confidence interval for $\rho$:

cor.test(hers$BMI, hers$glucose, method = "pearson")
#> 
#>  Pearson's product-moment correlation
#> 
#> data:  hers$BMI and hers$glucose
#> t = 14.88, df = 2756, p-value <2e-16
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#>  0.237760 0.306865
#> sample estimates:
#>      cor 
#> 0.272664

The correlation is positive but modest: glucose tends to be higher at higher BMI, with wide scatter around the line in Figure 1.

Read

Hogg et al. [3, Sec. 9.6], pp. 471–476 covers tests concerning regression and correlation.
Rice [4, Sec. 14.2.3], pp. 560–563 covers correlation and regression.

2.2 Spearman rank correlation

Definition 3 (Spearman rank correlation)

Definition 3 (Spearman rank correlation) Replace each $x_i$ by its rank $u_i$ among $x_1, \ldots, x_n$, and each $y_i$ by its rank $v_i$ among $y_1, \ldots, y_n$, giving tied values the average of the ranks they span. The Spearman rank correlation $r_S$ is the Pearson correlation coefficient evaluated at the rank pairs:

\[ r_S \stackrel{\text{def}}{=}r\mathopen{}\left((u_1, v_1), \ldots, (u_n, v_n)\right)\mathclose{}. \]

Source: Wikipedia contributors [5].

Remark 3 (What the Spearman correlation measures)

Remark 3 (What the Spearman correlation measures). Because ranks depend only on the ordering of the values, $r_S$ measures how close the association is to monotone, whether or not it is linear, and a single extreme value moves $r_S$ less than it moves $r$.

Example 2 (Spearman correlation between BMI and fasting glucose in HERS)

Example 2 (Spearman correlation between BMI and fasting glucose in HERS) The Pearson correlation of the ranks (Definition 3), and cor.test()’s Spearman test (exact = FALSE, because tied values rule out the exact p-value):

hers_bmi <- hers |> dplyr::filter(!is.na(BMI))
cor(rank(hers_bmi$BMI), rank(hers_bmi$glucose))
#> [1] 0.333751
cor.test(hers_bmi$BMI, hers_bmi$glucose, method = "spearman", exact = FALSE)
#> 
#>  Spearman's rank correlation rho
#> 
#> data:  hers_bmi$BMI and hers_bmi$glucose
#> S = 2.33e+09, p-value <2e-16
#> alternative hypothesis: true rho is not equal to 0
#> sample estimates:
#>      rho 
#> 0.333751

$r_S$ is larger than the Pearson $r$ of Example 1: the association is closer to monotone than to linear.

3 Simple linear regression

3.1 Model specification

Definition 4 (Conditional Gaussian model)

Definition 4 (Conditional Gaussian model) A conditional Gaussian model for outcomes $Y_1, \ldots, Y_n$ with covariate values $x_1, \ldots, x_n$ (each a single value or a vector) says that, given the covariates, the outcomes are independent and Gaussian: outcome $Y_i$ is centered on a mean function $\mu(x)$ evaluated at $x_i$, and has its own variance $\sigma^2_i$:

\[ \begin{aligned} Y_i\mid X_i= x_i&\ \sim_{\perp\!\!\!\perp}\ {\operatorname{N}\mathopen{}\left(\mu_i, \sigma^2_i\right)\mathclose{}},\\ \mu_i&\stackrel{\text{def}}{=}\mu(x_i). \end{aligned} \]

Definition 5 (Deviation from the conditional mean)

Definition 5 (Deviation from the conditional mean) In a conditional Gaussian model (Definition 4), the deviation $\mathbb{d}_i$ of outcome $Y_i$ is its deviation from its population mean, taking the mean conditional on $X_i= x_i$:

\[\mathbb{d}_i\stackrel{\text{def}}{=}Y_i- \mu(x_i).\]

Exercise 1 (Decomposing an outcome)

Exercise 1 (Decomposing an outcome) Express each outcome $Y_i$ in a conditional Gaussian model (Definition 4) in terms of its conditional mean $\mu_i$ and its deviation $\mathbb{d}_i$ (Definition 5).

Solution 1

Solution 1. Check that $Y_i= \mu_i+ \mathbb{d}_i$, starting from the right-hand side:

\[ \begin{aligned} \mu_i+ \mathbb{d}_i &= \mu(x_i) + \mathbb{d}_i && \text{(definition of $\mu_i$)}\\ &= \mu(x_i) + \mathopen{}\left(Y_i- \mu(x_i)\right)\mathclose{} && \text{(definition of $\mathbb{d}_i$)}\\ &= \mu(x_i) + Y_i- \mu(x_i) && \text{(remove the parentheses)}\\ &= Y_i+ \mu(x_i) - \mu(x_i) && \text{(reorder the terms)}\\ &= Y_i+ \mathopen{}\left(\mu(x_i) - \mu(x_i)\right)\mathclose{} && \text{(group the last two terms)}\\ &= Y_i+ 0 && \text{($a - a = 0$)}\\ &= Y_i && \text{($a + 0 = a$)} \end{aligned} \]

So $Y_i= \mu_i+ \mathbb{d}_i$.

Alternatively, solve the definition of $\mathbb{d}_i$ for $Y_i$:

\[ \begin{aligned} \mathbb{d}_i &= Y_i- \mu(x_i) && \text{(definition of $\mathbb{d}_i$)}\\ \mathbb{d}_i+ \mu(x_i) &= Y_i- \mu(x_i) + \mu(x_i) && \text{(add $\mu(x_i)$ to both sides)}\\ \mathbb{d}_i+ \mu(x_i) &= Y_i+ \mathopen{}\left(-\mu(x_i)\right)\mathclose{} + \mu(x_i) && \text{(subtracting is adding the negative)}\\ \mathbb{d}_i+ \mu(x_i) &= Y_i+ \mathopen{}\left(-\mu(x_i) + \mu(x_i)\right)\mathclose{} && \text{(group the last two terms)}\\ \mathbb{d}_i+ \mu(x_i) &= Y_i+ 0 && \text{($-a + a = 0$)}\\ \mathbb{d}_i+ \mu(x_i) &= Y_i && \text{($a + 0 = a$)}\\ Y_i &= \mathbb{d}_i+ \mu(x_i) && \text{(swap the two sides)}\\ Y_i &= \mu(x_i) + \mathbb{d}_i && \text{(reorder the terms)}\\ Y_i &= \mu_i+ \mathbb{d}_i && \text{(definition of $\mu_i$)} \end{aligned} \]

Theorem 2 (Additive form of the conditional Gaussian model)

Theorem 2 (Additive form of the conditional Gaussian model) In a conditional Gaussian model (Definition 4), each outcome is its conditional mean plus its deviation (Definition 5):

\[Y_i= \mu_i+ \mathbb{d}_i.\]

Proof

Proof. This is the solution to Exercise 1.

Exercise 2 (Distribution of the deviations)

Exercise 2 (Distribution of the deviations) Find the joint distribution of the deviations (Definition 5) in a conditional Gaussian model (Definition 4), given the covariate values.

Solution 2

Solution 2. Every probability, CDF and density in this solution is conditional on $X_i= x_i$; the condition is left out of the notation to keep the lines short. Write $F_{\mathbb{d}}$ and $f_{\mathbb{d}}$ for the CDF and density of $\mathbb{d}_i$, and $F_Y$ and $f_Y$ for those of $Y_i$.

First, the CDF of $\mathbb{d}_i$:

\[ \begin{aligned} F_{\mathbb{d}}(t) &= {\Pr\mathopen{}\left(\mathbb{d}_i\le t\right)\mathclose{}} && \text{(definition of a CDF)}\\ &= {\Pr\mathopen{}\left(Y_i- \mu(x_i) \le t\right)\mathclose{}} && \text{(definition of $\mathbb{d}_i$)}\\ &= {\Pr\mathopen{}\left(Y_i- \mu_i\le t\right)\mathclose{}} && \text{(definition of $\mu_i$)}\\ &= {\Pr\mathopen{}\left(Y_i\le t + \mu_i\right)\mathclose{}} && \text{(add $\mu_i$ to both sides of the inequality)}\\ &= F_Y(t + \mu_i) && \text{(definition of a CDF)} \end{aligned} \]

Differentiating gives the density:

\[ \begin{aligned} f_{\mathbb{d}}(t) &= \frac{\partial}{\partial t} F_{\mathbb{d}}(t) && \text{(a density is the derivative of its CDF)}\\ &= \frac{\partial}{\partial t} F_Y(t + \mu_i) && \text{(CDF of $\mathbb{d}_i$)}\\ &= f_Y(t + \mu_i) \frac{\partial}{\partial t} (t + \mu_i) && \text{(chain rule)} \end{aligned} \]

The inner derivative:

\[ \begin{aligned} \frac{\partial}{\partial t} (t + \mu_i) &= \frac{\partial}{\partial t} t + \frac{\partial}{\partial t} \mu_i && \text{(derivative of a sum)}\\ &= 1 + 0 && \text{($\mu_i$ does not depend on $t$)}\\ &= 1 && \text{(add)} \end{aligned} \]

Plugging the inner derivative back in, and using the Gaussian density of $Y_i$, which has mean $\mu_i= \mu(x_i)$ and variance $\sigma^2_i$ (Definition 4):

\[ \begin{aligned} f_{\mathbb{d}}(t) &= f_Y(t + \mu_i) \cdot 1 && \text{(inner derivative is 1)}\\ &= f_Y(t + \mu_i) && \text{(multiply by 1)}\\ &= \frac{1}{\sigma_i\sqrt{2 \pi}} \text{e}^{-\frac{\mathopen{}\left((t + \mu_i) - \mu_i\right)\mathclose{}^2}{2 \sigma^2_i}} && \text{(Gaussian density with mean $\mu_i$)}\\ &= \frac{1}{\sigma_i\sqrt{2 \pi}} \text{e}^{-\frac{t^2}{2 \sigma^2_i}} && \text{(subtract)} \end{aligned} \]

The last line is the density of ${\operatorname{N}\mathopen{}\left(0, \sigma^2_i\right)\mathclose{}}$, so $\mathbb{d}_i\mid X_i= x_i\sim {\operatorname{N}\mathopen{}\left(0, \sigma^2_i\right)\mathclose{}}$.

For independence, factor the joint CDF of $\mathbb{d}_1, \ldots, \mathbb{d}_n$, using the same CDF step as above for each $i$ and the independence of the $Y_i$ given the covariates (Definition 4):

\[ \begin{aligned} {\Pr\mathopen{}\left(\mathbb{d}_1 \le t_1, \ldots, \mathbb{d}_n \le t_n\right)\mathclose{}} &= {\Pr\mathopen{}\left(Y_1 \le t_1 + \mu_1, \ldots, Y_n \le t_n + \mu_n\right)\mathclose{}} && \text{(add $\mu_i$ to both sides of each inequality)}\\ &= \prod_{i=1}^n{\Pr\mathopen{}\left(Y_i\le t_i + \mu_i\right)\mathclose{}} && \text{(the $Y_i$ are independent)}\\ &= \prod_{i=1}^n{\Pr\mathopen{}\left(\mathbb{d}_i\le t_i\right)\mathclose{}} && \text{(subtract $\mu_i$ from both sides of each inequality)} \end{aligned} \]

The joint CDF is the product of the marginal CDFs, so $\mathbb{d}_1, \ldots, \mathbb{d}_n$ are independent. Given the covariate values, the deviations are therefore independent, with $\mathbb{d}_i\mid X_i= x_i\ \sim_{\perp\!\!\!\perp}\ {\operatorname{N}\mathopen{}\left(0, \sigma^2_i\right)\mathclose{}}$.

Theorem 3 (Distribution of the deviations)

Theorem 3 (Distribution of the deviations) In a conditional Gaussian model (Definition 4), given the covariates, the deviations (Definition 5) are independent and Gaussian with mean 0 and variance $\sigma^2_i$:

\[\mathbb{d}_i\mid X_i= x_i\ \sim_{\perp\!\!\!\perp}\ {\operatorname{N}\mathopen{}\left(0, \sigma^2_i\right)\mathclose{}}.\]

Proof

Proof. This is the solution to Exercise 2.

Definition 6 (Homoskedastic model)

Definition 6 (Homoskedastic model) A conditional Gaussian model (Definition 4) is homoskedastic if all outcomes share one variance $\sigma^2$, which does not depend on $i$:

\[\sigma^2_i= \sigma^2\text{ for all } i.\]

Source: Wikipedia contributors [6].

Corollary 1 (Deviations in a homoskedastic model)

Corollary 1 (Deviations in a homoskedastic model) In a homoskedastic model (Definition 6), the deviations (Definition 5) all have the same variance $\sigma^2$:

\[\mathbb{d}_i\mid X_i= x_i\ \sim_{\perp\!\!\!\perp}\ {\operatorname{N}\mathopen{}\left(0, \sigma^2\right)\mathclose{}}.\]

Proof

Proof. By Theorem 3, $\mathbb{d}_i\mid X_i= x_i\ \sim_{\perp\!\!\!\perp}\ {\operatorname{N}\mathopen{}\left(0, \sigma^2_i\right)\mathclose{}}$, and Definition 6 sets $\sigma^2_i= \sigma^2$ for all $i$.

Definition 7 (Linear regression model)

Definition 7 (Linear regression model) A linear regression model is a homoskedastic (Definition 6) conditional Gaussian model (Definition 4) whose mean function is linear in $p$ covariates, where observation $i$ has covariate values $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i= (x_{i1}, \ldots, x_{ip})$:

\[\mu(x_1, \ldots, x_p) \stackrel{\text{def}}{=}\beta_{0}+ \sum_{j=1}^p\beta_{x_j} x_j.\]

Source: Wikipedia contributors [7].

Definition 8 (Simple linear regression model)

Definition 8 (Simple linear regression model) A simple linear regression model is a linear regression model (Definition 7) with $p = 1$ covariate $X$, writing $x_i$ for $x_{i1}$:

\[\mu(x) \stackrel{\text{def}}{=}\beta_{0}+ \beta_{x} x.\]

Source: Wikipedia contributors [8].

Definition 9 (Intercept)

Definition 9 (Intercept) In a simple linear regression model (Definition 8), the intercept is $\beta_{0}= \mu(0)$: the mean of $Y$ among observations with $X = 0$.

Definition 10 (Slope)

Definition 10 (Slope) In a simple linear regression model (Definition 8), the slope is $\beta_{x} = \mu(x + 1) - \mu(x)$: the difference in the mean of $Y$ between two groups whose values of $X$ differ by one unit.

Remark 4 (Interpreting the variance parameter)

Remark 4 (Interpreting the variance parameter). $\sigma^2$ is the variance of $Y$ around its mean at each value of $X$.

3.2 Ordinary least squares estimation

Definition 11 (Residual sum of squares)

Definition 11 (Residual sum of squares) For a regression model whose conditional mean $\mu(x; \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})$ depends on a parameter vector $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}$, the residual sum of squares at $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}$ is the sum of the squared residuals $\hat{\varepsilon}_i(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}) \stackrel{\text{def}}{=}y_i- \mu(x_i; \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})$ that the model has when its parameters equal $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}$:

\[\text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}) \stackrel{\text{def}}{=}\sum_{i=1}^n\hat{\varepsilon}_i(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})^2.\]

At the estimate $\hat{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}}$, $\hat{\varepsilon}_i(\hat{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}})$ is the residual $\hat{\varepsilon}_i$ of the fitted model.

Source: Wikipedia contributors [9].

Example 3 (Residual sum of squares of a simple linear regression)

Example 3 (Residual sum of squares of a simple linear regression) The mean function of a simple linear regression model has parameter vector $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}= (\beta_{0}, \beta_{x})$, so write $\text{RSS}(\beta_{0}, \beta_{x})$ for its residual sum of squares (Definition 11):

\[ \begin{aligned} \text{RSS}(\beta_{0}, \beta_{x}) &= \sum_{i=1}^n\hat{\varepsilon}_i(\beta_{0}, \beta_{x})^2 && \text{(definition of RSS)}\\ &= \sum_{i=1}^n\mathopen{}\left(y_i- \mu(x_i; \beta_{0}, \beta_{x})\right)\mathclose{}^2 && \text{(definition of $\hat{\varepsilon}_i(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})$)}\\ &= \sum_{i=1}^n\mathopen{}\left(y_i- (\beta_{0}+ \beta_{x} x_i)\right)\mathclose{}^2 && \text{(definition of $\mu(x)$)}\\ &= \sum_{i=1}^n(y_i- \beta_{0}- \beta_{x} x_i)^2 && \text{(distribute the minus sign)} \end{aligned} \]

Example 4 (Residual sum of squares of a line through three points)

Example 4 (Residual sum of squares of a line through three points) For the points $(0, 1)$, $(1, 2)$, $(2, 2)$ and the line with $\beta_{0}= 1$ and $\beta_{x} = 0.5$ (Example 3), the residuals are $1 - 1 = 0$, $2 - 1.5 = 0.5$, and $2 - 2 = 0$, so

\[ \begin{aligned} \text{RSS}(1, 0.5) &= 0^2 + 0.5^2 + 0^2 \\ &= 0.25 \end{aligned} \]

(Figure 2).

Show code

rss_points <- tibble::tibble(x = c(0, 1, 2), y = c(1, 2, 2)) |>
  dplyr::mutate(fitted = 1 + 0.5 * x)

ggplot2::ggplot(rss_points, ggplot2::aes(x = x, y = y)) +
  ggplot2::geom_abline(intercept = 1, slope = 0.5) +
  ggplot2::geom_segment(
    ggplot2::aes(xend = x, yend = fitted),
    linetype = "dashed",
    color = "firebrick"
  ) +
  ggplot2::geom_point(size = 3) +
  ggplot2::coord_cartesian(xlim = c(-0.25, 2.25), ylim = c(0.75, 2.25))

Figure 2: The three points, the line with intercept 1 and slope 0.5, and the one nonzero residual (dashed)

Exercise 3 (Sample mean squared error and residual sum of squares)

Exercise 3 (Sample mean squared error and residual sum of squares) For a regression model with conditional mean $\mu(x; \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})$, let $\hat y_i\stackrel{\text{def}}{=}\mu(x_i; \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})$ be its predictions of $n \ge 1$ outcomes $y_1, \ldots, y_n$, and let $\text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})$ (Definition 11) be taken over those same outcomes. Express the sMSE of these predictions in terms of $\text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})$, and $\text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})$ in terms of that sMSE.

Solution 3

Solution 3. First, each squared prediction error equals the squared residual at $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}$:

\[ \begin{aligned} {\varepsilon_{\text{pred}}}_i^2 &= (\hat y_i- y_i)^2 && \text{(definition of prediction error)}\\ &= \mathopen{}\left(\mu(x_i; \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}) - y_i\right)\mathclose{}^2 && \text{(definition of $\hat y_i$)}\\ &= \mathopen{}\left(-\mathopen{}\left(y_i- \mu(x_i; \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})\right)\mathclose{}\right)\mathclose{}^2 && \text{(factor out $-1$)}\\ &= \mathopen{}\left(-\hat{\varepsilon}_i(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})\right)\mathclose{}^2 && \text{(definition of $\hat{\varepsilon}_i(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})$)}\\ &= \hat{\varepsilon}_i(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})^2 && \text{($(-a)^2 = a^2$)} \end{aligned} \]

So the sMSE is the residual sum of squares divided by $n$:

\[ \begin{aligned} {\operatorname{sMSE}\mathopen{}\left(\hat y\right)\mathclose{}} &= \frac{1}{n}\sum_{i=1}^n{\varepsilon_{\text{pred}}}_i^2 && \text{(definition of sMSE)}\\ &= \frac{1}{n}\sum_{i=1}^n\hat{\varepsilon}_i(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})^2 && \text{(${\varepsilon_{\text{pred}}}_i^2 = \hat{\varepsilon}_i(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})^2$)}\\ &= \frac{1}{n} \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}) && \text{(definition of RSS)} \end{aligned} \]

Solving for $\text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})$:

\[ \begin{aligned} {\operatorname{sMSE}\mathopen{}\left(\hat y\right)\mathclose{}} &= \frac{1}{n} \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}) && \text{(result above)}\\ n \, {\operatorname{sMSE}\mathopen{}\left(\hat y\right)\mathclose{}} &= \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}) && \text{(multiply both sides by $n$)}\\ \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}) &= n \, {\operatorname{sMSE}\mathopen{}\left(\hat y\right)\mathclose{}} && \text{(swap the two sides)} \end{aligned} \]

Theorem 4 (Sample mean squared error is the residual sum of squares divided by $n$)

Theorem 4 (Sample mean squared error is the residual sum of squares divided by $n$) For a regression model with conditional mean $\mu(x; \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})$, the sMSE of its predictions $\hat y_i= \mu(x_i; \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})$ of $n \ge 1$ outcomes $y_1, \ldots, y_n$ and the residual sum of squares $\text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}})$ (Definition 11) over those same $n$ outcomes satisfy

\[{\operatorname{sMSE}\mathopen{}\left(\hat y\right)\mathclose{}} = \frac{1}{n} \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}),\]

and equivalently

\[\text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}) = n \, {\operatorname{sMSE}\mathopen{}\left(\hat y\right)\mathclose{}}.\]

Proof

Proof. Both equations are derived in Exercise 3.

Remark 5 (The same observations on both sides)

Remark 5 (The same observations on both sides). The identities in Theorem 4 need the sMSE and the residual sum of squares to be taken over the same $n$ observations. The RSS minimized in fitting is taken over the fitting data, so dividing it by $n$ gives the sMSE on the fitting data, not the sMSE of predictions of new outcomes.

Definition 12 (Ordinary least squares)

Definition 12 (Ordinary least squares) The ordinary least squares (OLS) estimate of a model’s parameter vector $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}$ is the value of $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}$ that minimizes the residual sum of squares:

\[\hat{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}} \stackrel{\text{def}}{=}\arg \min_{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}} \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}).\]

Source: Wikipedia contributors [10].

Example 5 (OLS estimates of a simple linear regression)

Example 5 (OLS estimates of a simple linear regression) For a simple linear regression, the OLS estimates (Definition 12) $\hat{\beta}_{0}$ and $\hat{\beta}_{x}$ are the values of $\beta_{0}$ and $\beta_{x}$ that minimize $\text{RSS}(\beta_{0}, \beta_{x})$ from Example 3. Figure 3 shows $\text{RSS}(\beta_{0}, \beta_{x})$ for the three points of Example 4: the OLS estimates sit at the bottom of the bowl, and the line of Example 4 sits higher up.

Show code

rss_grid <- expand.grid(
  b0 = seq(0, 2.5, length.out = 101),
  bx = seq(-0.5, 1.5, length.out = 101)
)
rss_grid$rss <- mapply(
  function(b0, bx) sum((rss_points$y - b0 - bx * rss_points$x)^2),
  rss_grid$b0,
  rss_grid$bx
)
ols_coefs <- coef(lm(y ~ x, data = rss_points))

ggplot2::ggplot(rss_grid, ggplot2::aes(x = b0, y = bx, z = rss)) +
  ggplot2::geom_contour(
    ggplot2::aes(color = ggplot2::after_stat(level)),
    breaks = c(0.25, 0.5, 1, 2, 4, 8)
  ) +
  ggplot2::annotate(
    "point",
    x = ols_coefs[[1]], y = ols_coefs[[2]], size = 3
  ) +
  ggplot2::annotate("point", x = 1, y = 0.5, shape = 4, size = 3) +
  ggplot2::labs(
    x = expression("Intercept, " * beta[0]),
    y = expression("Slope, " * beta[x]),
    color = "RSS"
  )

Show code


ols_coefs
#> (Intercept)           x 
#>     1.16667     0.50000

Figure 3: Contours of $\text{RSS}(\beta_{0}, \beta_{x})$ for the three points of Example 4. The dot marks the OLS estimates; the cross marks the line of Example 4.

Show code

rss3_b0 <- unique(rss_grid$b0)
rss3_bx <- unique(rss_grid$bx)
rss3_n_bx <- length(rss3_bx)
rss3_z <- matrix(rss_grid$rss, nrow = rss3_n_bx, byrow = TRUE)
rss3_ols_resid <- rss_points$y - ols_coefs[[1]] - ols_coefs[[2]] * rss_points$x
rss3_cross_resid <- rss_points$y - 1 - 0.5 * rss_points$x
rss3_marks_b0 <- c(ols_coefs[[1]], 1)
rss3_marks_bx <- c(ols_coefs[[2]], 0.5)
rss3_ols_rss <- sum(rss3_ols_resid^2)
rss3_cross_rss <- sum(rss3_cross_resid^2)
rss3_marks_rss <- c(rss3_ols_rss, rss3_cross_rss)
rss3_marks_name <- c("OLS estimates", "line of the example")
rss3_marks_symbol <- c("circle", "x")
rss3_marks <- tibble::tibble(
  b0 = rss3_marks_b0,
  bx = rss3_marks_bx,
  rss = rss3_marks_rss,
  name = rss3_marks_name,
  symbol = rss3_marks_symbol
)
rss3_marker <- list(color = "#D55E00", size = 6, symbol = ~symbol)
rss3_stops <- c(0, 1)
rss3_cols <- c("#CCE0F0", "#0072B2")
rss3_scale <- list(rss3_stops, rss3_cols)
rss3_eye <- list(x = 1.6, y = -1.6, z = 0.9)
rss3_camera <- list(eye = rss3_eye)
rss3_xaxis <- list(title = "Intercept")
rss3_yaxis <- list(title = "Slope")
rss3_zaxis <- list(title = "RSS")
rss3_scene <- list(
  xaxis = rss3_xaxis,
  yaxis = rss3_yaxis,
  zaxis = rss3_zaxis,
  camera = rss3_camera
)
plotly::plot_ly() |>
  plotly::add_surface(
    x = rss3_b0,
    y = rss3_bx,
    z = rss3_z,
    colorscale = rss3_scale,
    showscale = FALSE,
    opacity = 0.8
  ) |>
  plotly::add_markers(
    data = rss3_marks,
    x = ~b0,
    y = ~bx,
    z = ~rss,
    text = ~name,
    hoverinfo = "text",
    showlegend = FALSE,
    marker = rss3_marker
  ) |>
  plotly::layout(scene = rss3_scene)

Figure 4: Surface of $\operatorname{RSS}(\beta_{0}, \beta_{x})$ for the three points of Example 4. The dot at the bottom of the bowl marks the OLS estimates; the cross marks the line of Example 4.

Definition 13 (Normal equations)

Definition 13 (Normal equations) For a model fitted by least squares with parameter vector $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}$, the normal equations set the gradient of the residual sum of squares to zero:

\[\frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}} \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}) = \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{0}}\vphantom{0}}.\]

Example 6 (Normal equations of a simple linear regression)

Example 6 (Normal equations of a simple linear regression) For a simple linear regression, the parameter vector is $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\stackrel{\text{def}}{=}{\mathopen{}\left(\beta_{0}, \beta_{x}\right)\mathclose{}}^{\top}$, and $\text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}})$ is $\text{RSS}(\beta_{0}, \beta_{x})$ from Example 3. Its gradient has one entry per coefficient, so the normal equations (Definition 13) are two scalar equations:

\[ \frac{\partial \text{RSS}}{\partial \beta_{0}} = 0, \qquad \frac{\partial \text{RSS}}{\partial \beta_{x}} = 0. \]

Definition 14 (Centered sums of squares)

Definition 14 (Centered sums of squares) For data $(x_1, y_1), \ldots, (x_n, y_n)$, the centered sums of squares of the $x_i$ and of the $y_i$ are

\[ \begin{aligned} S_{xx}&\stackrel{\text{def}}{=}\sum_{i=1}^n(x_i- {\bar{x}})^2, & S_{yy}&\stackrel{\text{def}}{=}\sum_{i=1}^n(y_i- {\bar{y}})^2. \end{aligned} \]

Definition 15 (Centered sum of cross-products)

Definition 15 (Centered sum of cross-products) For the data of Definition 14, the centered sum of cross-products is

\[ S_{xy}\stackrel{\text{def}}{=}\sum_{i=1}^n(x_i- {\bar{x}})(y_i- {\bar{y}}). \]

Exercise 4 (Summing the deviations from the sample mean)

Exercise 4 (Summing the deviations from the sample mean) Find $\sum_{i=1}^n(x_i- {\bar{x}})$.

Solution 4

Solution 4. \[ \begin{aligned} \sum_{i=1}^n(x_i- {\bar{x}}) &= \sum_{i=1}^nx_i- \sum_{i=1}^n{\bar{x}} && \text{(distribute the sum over the terms)}\\ &= \sum_{i=1}^nx_i- n {\bar{x}} && \text{(sum of a constant)}\\ &= n {\bar{x}}- n {\bar{x}} && \text{($\sum_{i=1}^nx_i= n {\bar{x}}$)}\\ &= 0 && \text{(subtract)} \end{aligned} \]

The same steps with $y_i$ in place of $x_i$ show that $\sum_{i=1}^n(y_i- {\bar{y}}) = 0$.

Lemma 1 (Deviations from the sample mean sum to zero)

Lemma 1 (Deviations from the sample mean sum to zero) For any data $z_1, \ldots, z_n$ with sample mean $\bar{z}$ (for example, the $x_i$ or the $y_i$), the deviations from the sample mean sum to zero:

\[\sum_{i=1}^n(z_i - \bar{z}) = 0.\]

Proof

Proof. This is the solution to Exercise 4.

Exercise 5 (Expanding the centered sum of squares)

Exercise 5 (Expanding the centered sum of squares) Expand the centered sum of squares $S_{xx}$ (Definition 14) into an expression in terms of $n$, $\sum_{i=1}^nx_i^2$ and ${\bar{x}}$.

Solution 5

Solution 5. \[ \begin{aligned} S_{xx} &= \sum_{i=1}^n(x_i- {\bar{x}})^2 && \text{(definition of $S_{xx}$)}\\ &= \sum_{i=1}^n\mathopen{}\left(x_i^2 - 2 {\bar{x}}x_i+ {\bar{x}}^2\right)\mathclose{} && \text{(expand the square)}\\ &= \sum_{i=1}^nx_i^2 - \sum_{i=1}^n2 {\bar{x}}x_i+ \sum_{i=1}^n{\bar{x}}^2 && \text{(distribute the sum over the terms)}\\ &= \sum_{i=1}^nx_i^2 - 2 {\bar{x}}\sum_{i=1}^nx_i+ \sum_{i=1}^n{\bar{x}}^2 && \text{(constant factor out of the sum)}\\ &= \sum_{i=1}^nx_i^2 - 2 {\bar{x}}\sum_{i=1}^nx_i+ n {\bar{x}}^2 && \text{(sum of a constant)}\\ &= \sum_{i=1}^nx_i^2 - 2 {\bar{x}}\cdot n {\bar{x}}+ n {\bar{x}}^2 && \text{($\sum_{i=1}^nx_i= n {\bar{x}}$)}\\ &= \sum_{i=1}^nx_i^2 - 2 n {\bar{x}}^2 + n {\bar{x}}^2 && \text{(multiply)}\\ &= \sum_{i=1}^nx_i^2 - n {\bar{x}}^2 && \text{(collect the ${\bar{x}}^2$ terms)} \end{aligned} \]

Lemma 2 (Expanded centered sum of squares)

Lemma 2 (Expanded centered sum of squares) The centered sum of squares $S_{xx}$ (Definition 14) equals

\[S_{xx}= \sum_{i=1}^nx_i^2 - n {\bar{x}}^2.\]

Proof

Proof. This is the solution to Exercise 5.

Exercise 6 (Expanding the centered sum of cross-products)

Exercise 6 (Expanding the centered sum of cross-products) Expand the centered sum of cross-products $S_{xy}$ (Definition 15) into an expression in terms of $n$, $\sum_{i=1}^nx_iy_i$, ${\bar{x}}$ and ${\bar{y}}$.

Solution 6

Solution 6. \[ \begin{aligned} S_{xy} &= \sum_{i=1}^n(x_i- {\bar{x}})(y_i- {\bar{y}}) && \text{(definition of $S_{xy}$)}\\ &= \sum_{i=1}^n\mathopen{}\left(x_iy_i- {\bar{y}}x_i- {\bar{x}}y_i+ {\bar{x}}{\bar{y}}\right)\mathclose{} && \text{(expand the product)}\\ &= \sum_{i=1}^nx_iy_i- \sum_{i=1}^n{\bar{y}}x_i- \sum_{i=1}^n{\bar{x}}y_i+ \sum_{i=1}^n{\bar{x}}{\bar{y}} && \text{(distribute the sum over the terms)}\\ &= \sum_{i=1}^nx_iy_i- {\bar{y}}\sum_{i=1}^nx_i- {\bar{x}}\sum_{i=1}^ny_i+ \sum_{i=1}^n{\bar{x}}{\bar{y}} && \text{(constant factors out of the sum)}\\ &= \sum_{i=1}^nx_iy_i- {\bar{y}}\sum_{i=1}^nx_i- {\bar{x}}\sum_{i=1}^ny_i+ n {\bar{x}}{\bar{y}} && \text{(sum of a constant)}\\ &= \sum_{i=1}^nx_iy_i- {\bar{y}}\cdot n {\bar{x}}- {\bar{x}}\cdot n {\bar{y}}+ n {\bar{x}}{\bar{y}} && \text{($\sum_{i=1}^nx_i= n {\bar{x}}$, $\sum_{i=1}^ny_i= n {\bar{y}}$)}\\ &= \sum_{i=1}^nx_iy_i- n {\bar{x}}{\bar{y}}- n {\bar{x}}{\bar{y}}+ n {\bar{x}}{\bar{y}} && \text{(multiply)}\\ &= \sum_{i=1}^nx_iy_i- 2 n {\bar{x}}{\bar{y}}+ n {\bar{x}}{\bar{y}} && \text{(add the two equal terms)}\\ &= \sum_{i=1}^nx_iy_i- n {\bar{x}}{\bar{y}} && \text{(add the remaining term)} \end{aligned} \]

Lemma 3 (Expanded centered sum of cross-products)

Lemma 3 (Expanded centered sum of cross-products) The centered sum of cross-products $S_{xy}$ (Definition 15) equals

\[S_{xy}= \sum_{i=1}^nx_iy_i- n {\bar{x}}{\bar{y}}.\]

Proof

Proof. This is the solution to Exercise 6.

Exercise 7 (Derivative of RSS with respect to the intercept)

Exercise 7 (Derivative of RSS with respect to the intercept) Find the partial derivative of the simple linear regression residual sum of squares $\text{RSS}(\beta_{0}, \beta_{x})$ (Example 3) with respect to the intercept, ${\partial \text{RSS} / \partial \beta_{0}}$.

Solution 7

Solution 7. Differentiate one operation at a time, starting from the definition (Example 3):

\[ \begin{aligned} \frac{\partial \text{RSS}}{\partial \beta_{0}} &= \frac{\partial}{\partial \beta_{0}} \sum_{i=1}^n(y_i- \beta_{0}- \beta_{x} x_i)^2 && \text{(definition of RSS)}\\ &= \sum_{i=1}^n\frac{\partial}{\partial \beta_{0}} (y_i- \beta_{0}- \beta_{x} x_i)^2 && \text{(derivative of a sum)}\\ &= \sum_{i=1}^n2 (y_i- \beta_{0}- \beta_{x} x_i) \frac{\partial}{\partial \beta_{0}} (y_i- \beta_{0}- \beta_{x} x_i) && \text{(chain rule)} \end{aligned} \]

The inner derivative is:

\[ \begin{aligned} \frac{\partial}{\partial \beta_{0}} (y_i- \beta_{0}- \beta_{x} x_i) &= \frac{\partial y_i}{\partial \beta_{0}} - \frac{\partial \beta_{0}}{\partial \beta_{0}} - \frac{\partial (\beta_{x} x_i)}{\partial \beta_{0}} && \text{(derivative of a sum)}\\ &= 0 - 1 - 0 && \text{($y_i$ and $\beta_{x} x_i$ do not depend on $\beta_{0}$)}\\ &= -1 && \text{(subtract)} \end{aligned} \]

Plugging the inner derivative back in gives:

\[ \begin{aligned} \frac{\partial \text{RSS}}{\partial \beta_{0}} &= \sum_{i=1}^n2 (y_i- \beta_{0}- \beta_{x} x_i) (-1) && \text{(inner derivative is $-1$)}\\ &= \sum_{i=1}^n-2 (y_i- \beta_{0}- \beta_{x} x_i) && \text{(multiply $2$ by $-1$)}\\ &= -2 \sum_{i=1}^n(y_i- \beta_{0}- \beta_{x} x_i) && \text{(constant factor out of the sum)} \end{aligned} \]

Exercise 8 (Derivative of RSS in terms of the means)

Exercise 8 (Derivative of RSS in terms of the means) In the setting of Exercise 7, write the partial derivative in terms of ${\bar{x}}$ and ${\bar{y}}$.

Solution 8

Solution 8. Write the derivative from Exercise 7 in terms of ${\bar{x}}$ and ${\bar{y}}$:

\[ \begin{aligned} \frac{\partial \text{RSS}}{\partial \beta_{0}} &= -2 \sum_{i=1}^n(y_i- \beta_{0}- \beta_{x} x_i) && \text{(derivative of the RSS)}\\ &= -2 \mathopen{}\left(\sum_{i=1}^ny_i- \sum_{i=1}^n\beta_{0}- \sum_{i=1}^n\beta_{x} x_i\right)\mathclose{} && \text{(distribute the sum over the terms)}\\ &= -2 \mathopen{}\left(\sum_{i=1}^ny_i- n \beta_{0}- \sum_{i=1}^n\beta_{x} x_i\right)\mathclose{} && \text{(sum of a constant)}\\ &= -2 \mathopen{}\left(\sum_{i=1}^ny_i- n \beta_{0}- \beta_{x} \sum_{i=1}^nx_i\right)\mathclose{} && \text{(constant factor out of the sum)}\\ &= -2 \mathopen{}\left(n {\bar{y}}- n \beta_{0}- \beta_{x} \sum_{i=1}^nx_i\right)\mathclose{} && \text{($\sum_{i=1}^ny_i= n {\bar{y}}$)}\\ &= -2 \mathopen{}\left(n {\bar{y}}- n \beta_{0}- \beta_{x} n {\bar{x}}\right)\mathclose{} && \text{($\sum_{i=1}^nx_i= n {\bar{x}}$)} \end{aligned} \]

Lemma 4 (Derivative of RSS with respect to the intercept)

Lemma 4 (Derivative of RSS with respect to the intercept) The partial derivative of the simple linear regression residual sum of squares $\text{RSS}(\beta_{0}, \beta_{x})$ (Example 3) with respect to the intercept is

\[\frac{\partial \text{RSS}}{\partial \beta_{0}} = -2 \sum_{i=1}^n(y_i- \beta_{0}- \beta_{x} x_i) = -2 \mathopen{}\left(n {\bar{y}}- n \beta_{0}- \beta_{x} n {\bar{x}}\right)\mathclose{}.\]

Proof

Proof. The derivative in sum form is the solution to Exercise 7. The form in terms of means is the solution to Exercise 8.

Exercise 9 (Derivative of RSS with respect to the slope)

Exercise 9 (Derivative of RSS with respect to the slope) Find the partial derivative of the simple linear regression residual sum of squares $\text{RSS}(\beta_{0}, \beta_{x})$ (Example 3) with respect to the slope, ${\partial \text{RSS} / \partial \beta_{x}}$.

Solution 9

Solution 9. Differentiate one operation at a time, starting from the definition (Example 3):

\[ \begin{aligned} \frac{\partial \text{RSS}}{\partial \beta_{x}} &= \frac{\partial}{\partial \beta_{x}} \sum_{i=1}^n(y_i- \beta_{0}- \beta_{x} x_i)^2 && \text{(definition of RSS)}\\ &= \sum_{i=1}^n\frac{\partial}{\partial \beta_{x}} (y_i- \beta_{0}- \beta_{x} x_i)^2 && \text{(derivative of a sum)}\\ &= \sum_{i=1}^n2 (y_i- \beta_{0}- \beta_{x} x_i) \frac{\partial}{\partial \beta_{x}} (y_i- \beta_{0}- \beta_{x} x_i) && \text{(chain rule)} \end{aligned} \]

The inner derivative:

\[ \begin{aligned} \frac{\partial}{\partial \beta_{x}} (y_i- \beta_{0}- \beta_{x} x_i) &= \frac{\partial y_i}{\partial \beta_{x}} - \frac{\partial \beta_{0}}{\partial \beta_{x}} - \frac{\partial (\beta_{x} x_i)}{\partial \beta_{x}} && \text{(derivative of a sum)}\\ &= 0 - 0 - x_i && \text{($y_i$ and $\beta_{0}$ do not depend on $\beta_{x}$)}\\ &= -x_i && \text{(subtract)} \end{aligned} \]

Plugging the inner derivative back in:

\[ \begin{aligned} \frac{\partial \text{RSS}}{\partial \beta_{x}} &= \sum_{i=1}^n2 (y_i- \beta_{0}- \beta_{x} x_i) (-x_i) && \text{(inner derivative is $-x_i$)}\\ &= \sum_{i=1}^n-2 x_i(y_i- \beta_{0}- \beta_{x} x_i) && \text{(rearrange the product)}\\ &= -2 \sum_{i=1}^nx_i(y_i- \beta_{0}- \beta_{x} x_i) && \text{(constant factor out of the sum)} \end{aligned} \]

Lemma 5 (Derivative of RSS with respect to the slope)

Lemma 5 (Derivative of RSS with respect to the slope) The partial derivative of the simple linear regression residual sum of squares $\text{RSS}(\beta_{0}, \beta_{x})$ (Example 3) with respect to the slope is

\[\frac{\partial \text{RSS}}{\partial \beta_{x}} = -2 \sum_{i=1}^nx_i(y_i- \beta_{0}- \beta_{x} x_i).\]

Proof

Proof. This is the solution to Exercise 9.

Exercise 10 (Solving the normal equations)

Exercise 10 (Solving the normal equations) Using Exercise 7 and Exercise 9, solve the normal equations of Example 6 for $\beta_{0}$ and $\beta_{x}$, assuming $S_{xx}> 0$ (Definition 14).

Solution 10

Solution 10. First normal equation. Set the derivative from Exercise 7 to zero, then isolate $\beta_{0}$:

\[ \begin{aligned} 0 &= -2 \mathopen{}\left(n {\bar{y}}- n \beta_{0}- \beta_{x} n {\bar{x}}\right)\mathclose{} && \text{(first normal equation)}\\ 0 &= n {\bar{y}}- n \beta_{0}- \beta_{x} n {\bar{x}} && \text{(divide both sides by $-2$)}\\ 0 &= {\bar{y}}- \beta_{0}- \beta_{x} {\bar{x}} && \text{(divide both sides by $n$)}\\ \beta_{0}&= {\bar{y}}- \beta_{x} {\bar{x}} && \text{(add $\beta_{0}$ to both sides)} \end{aligned} \]

Second normal equation. Substitute that intercept into the derivative from Exercise 9:

\[ \begin{aligned} \frac{\partial \text{RSS}}{\partial \beta_{x}} &= -2 \sum_{i=1}^nx_i(y_i- \beta_{0}- \beta_{x} x_i) && \text{(derivative with respect to $\beta_{x}$)}\\ &= -2 \sum_{i=1}^nx_i\mathopen{}\left(y_i- ({\bar{y}}- \beta_{x} {\bar{x}}) - \beta_{x} x_i\right)\mathclose{} && \text{(substitute $\beta_{0}$)}\\ &= -2 \sum_{i=1}^nx_i\mathopen{}\left(y_i- {\bar{y}}+ \beta_{x} {\bar{x}}- \beta_{x} x_i\right)\mathclose{} && \text{(remove the inner parentheses)}\\ &= -2 \sum_{i=1}^nx_i\mathopen{}\left((y_i- {\bar{y}}) - (\beta_{x} x_i- \beta_{x} {\bar{x}})\right)\mathclose{} && \text{(group terms)}\\ &= -2 \sum_{i=1}^nx_i\mathopen{}\left((y_i- {\bar{y}}) - \beta_{x} (x_i- {\bar{x}})\right)\mathclose{} && \text{(factor out $\beta_{x}$)} \end{aligned} \]

To shorten the next steps, write $d_i \stackrel{\text{def}}{=}(y_i- {\bar{y}}) - \beta_{x} (x_i- {\bar{x}})$, so that ${\partial \text{RSS} / \partial \beta_{x}} = -2 \sum_{i=1}^nx_id_i$. The $d_i$ sum to zero, because deviations from a sample mean sum to zero (Exercise 4):

\[ \begin{aligned} \sum_{i=1}^nd_i &= \sum_{i=1}^n\mathopen{}\left((y_i- {\bar{y}}) - \beta_{x} (x_i- {\bar{x}})\right)\mathclose{} && \text{(definition of $d_i$)}\\ &= \sum_{i=1}^n(y_i- {\bar{y}}) - \sum_{i=1}^n\beta_{x} (x_i- {\bar{x}}) && \text{(distribute the sum over the terms)}\\ &= \sum_{i=1}^n(y_i- {\bar{y}}) - \beta_{x} \sum_{i=1}^n(x_i- {\bar{x}}) && \text{(constant factor out of the sum)}\\ &= 0 - \beta_{x} \cdot 0 && \text{(deviations sum to zero)}\\ &= 0 - 0 && \text{(multiply)}\\ &= 0 && \text{(subtract)} \end{aligned} \]

So subtracting ${\bar{x}}\sum_{i=1}^nd_i$ from $\sum_{i=1}^nx_id_i$ does not change it:

\[ \begin{aligned} \sum_{i=1}^nx_id_i &= \sum_{i=1}^nx_id_i - {\bar{x}}\sum_{i=1}^nd_i && \text{($\sum_{i=1}^nd_i = 0$)}\\ &= \sum_{i=1}^nx_id_i - \sum_{i=1}^n{\bar{x}}d_i && \text{(constant factor into the sum)}\\ &= \sum_{i=1}^n(x_i- {\bar{x}}) d_i && \text{(combine the sums)}\\ &= \sum_{i=1}^n(x_i- {\bar{x}}) \mathopen{}\left((y_i- {\bar{y}}) - \beta_{x} (x_i- {\bar{x}})\right)\mathclose{} && \text{(definition of $d_i$)}\\ &= \sum_{i=1}^n\mathopen{}\left((x_i- {\bar{x}})(y_i- {\bar{y}}) - \beta_{x} (x_i- {\bar{x}})^2\right)\mathclose{} && \text{(distribute)}\\ &= \sum_{i=1}^n(x_i- {\bar{x}})(y_i- {\bar{y}}) - \sum_{i=1}^n\beta_{x} (x_i- {\bar{x}})^2 && \text{(distribute the sum over the terms)}\\ &= \sum_{i=1}^n(x_i- {\bar{x}})(y_i- {\bar{y}}) - \beta_{x} \sum_{i=1}^n(x_i- {\bar{x}})^2 && \text{(constant factor out of the sum)}\\ &= S_{xy}- \beta_{x} \sum_{i=1}^n(x_i- {\bar{x}})^2 && \text{(definition of $S_{xy}$)}\\ &= S_{xy}- \beta_{x} S_{xx} && \text{(definition of $S_{xx}$)} \end{aligned} \]

So ${\partial \text{RSS} / \partial \beta_{x}} = -2 \mathopen{}\left(S_{xy}- \beta_{x} S_{xx}\right)\mathclose{}$. Set it to zero and isolate $\beta_{x}$:

\[ \begin{aligned} 0 &= -2 \mathopen{}\left(S_{xy}- \beta_{x} S_{xx}\right)\mathclose{} && \text{(second normal equation)}\\ 0 &= S_{xy}- \beta_{x} S_{xx} && \text{(divide both sides by $-2$)}\\ \beta_{x} S_{xx}&= S_{xy} && \text{(add $\beta_{x} S_{xx}$ to both sides)}\\ \beta_{x} &= \frac{S_{xy}}{S_{xx}} && \text{(divide both sides by $S_{xx}> 0$)} \end{aligned} \]

So the normal equations have exactly one solution: $\beta_{x} = S_{xy}/ S_{xx}$ and $\beta_{0}= {\bar{y}}- \beta_{x} {\bar{x}}$.

Lemma 6 (Solution of the normal equations)

Lemma 6 (Solution of the normal equations) If $S_{xx}> 0$ (Definition 14), the normal equations of Example 6 have exactly one solution:

\[\beta_{x} = \frac{S_{xy}}{S_{xx}}, \qquad \beta_{0}= {\bar{y}}- \beta_{x} {\bar{x}}.\]

Proof

Proof. This is the solution to Exercise 10.

Exercise 11 (Second derivatives of RSS)

Exercise 11 (Second derivatives of RSS) Find the matrix of second derivatives of $\text{RSS}(\beta_{0}, \beta_{x})$ (Example 3) with respect to $\beta_{0}$ and $\beta_{x}$.

Solution 11

Solution 11. Differentiate each first derivative again.

Derivatives of ${\partial \text{RSS} / \partial \beta_{0}}$. By Exercise 7, ${\partial \text{RSS} / \partial \beta_{0}} = -2 \mathopen{}\left(n {\bar{y}}- n \beta_{0}- \beta_{x} n {\bar{x}}\right)\mathclose{}$. Differentiating with respect to $\beta_{0}$:

\[ \begin{aligned} \frac{\partial^2 \text{RSS}}{\partial \beta_{0}^2} &= -2 \frac{\partial}{\partial \beta_{0}} \mathopen{}\left(n {\bar{y}}- n \beta_{0}- \beta_{x} n {\bar{x}}\right)\mathclose{} && \text{(constant factor)}\\ &= -2 \mathopen{}\left( \frac{\partial (n {\bar{y}})}{\partial \beta_{0}} - \frac{\partial (n \beta_{0})}{\partial \beta_{0}} - \frac{\partial (\beta_{x} n {\bar{x}})}{\partial \beta_{0}}\right)\mathclose{} && \text{(derivative of a sum)}\\ &= -2 (0 - n - 0) && \text{(differentiate each term)}\\ &= -2 (-n) && \text{(drop the zeros)}\\ &= 2n && \text{(multiply)} \end{aligned} \]

Differentiating with respect to $\beta_{x}$:

\[ \begin{aligned} \frac{\partial^2 \text{RSS}}{\partial \beta_{x} \partial \beta_{0}} &= -2 \frac{\partial}{\partial \beta_{x}} \mathopen{}\left(n {\bar{y}}- n \beta_{0}- \beta_{x} n {\bar{x}}\right)\mathclose{} && \text{(constant factor)}\\ &= -2 \mathopen{}\left( \frac{\partial (n {\bar{y}})}{\partial \beta_{x}} - \frac{\partial (n \beta_{0})}{\partial \beta_{x}} - \frac{\partial (\beta_{x} n {\bar{x}})}{\partial \beta_{x}}\right)\mathclose{} && \text{(derivative of a sum)}\\ &= -2 (0 - 0 - n {\bar{x}}) && \text{(differentiate each term)}\\ &= -2 (-n {\bar{x}}) && \text{(drop the zeros)}\\ &= 2 n {\bar{x}} && \text{(multiply)} \end{aligned} \]

Derivatives of ${\partial \text{RSS} / \partial \beta_{x}}$. By Exercise 9, ${\partial \text{RSS} / \partial \beta_{x}} = -2 \sum_{i=1}^nx_i(y_i- \beta_{0}- \beta_{x} x_i)$. Differentiating with respect to $\beta_{0}$, and reusing the inner derivative $-1$ from Exercise 7:

\[ \begin{aligned} \frac{\partial^2 \text{RSS}}{\partial \beta_{0} \partial \beta_{x}} &= -2 \frac{\partial}{\partial \beta_{0}} \sum_{i=1}^nx_i(y_i- \beta_{0}- \beta_{x} x_i) && \text{(constant factor)}\\ &= -2 \sum_{i=1}^n\frac{\partial}{\partial \beta_{0}} \mathopen{}\left(x_i(y_i- \beta_{0}- \beta_{x} x_i)\right)\mathclose{} && \text{(derivative of a sum)}\\ &= -2 \sum_{i=1}^nx_i\frac{\partial}{\partial \beta_{0}} (y_i- \beta_{0}- \beta_{x} x_i) && \text{($x_i$ does not depend on $\beta_{0}$)}\\ &= -2 \sum_{i=1}^nx_i(-1) && \text{(inner derivative is $-1$)}\\ &= -2 \sum_{i=1}^n(-x_i) && \text{(multiply)}\\ &= -2 \mathopen{}\left(-\sum_{i=1}^nx_i\right)\mathclose{} && \text{(constant factor out of the sum)}\\ &= 2 \sum_{i=1}^nx_i && \text{(multiply)}\\ &= 2 n {\bar{x}} && \text{($\sum_{i=1}^nx_i= n {\bar{x}}$)} \end{aligned} \]

Differentiating with respect to $\beta_{x}$, and reusing the inner derivative $-x_i$ from Exercise 9:

\[ \begin{aligned} \frac{\partial^2 \text{RSS}}{\partial \beta_{x}^2} &= -2 \frac{\partial}{\partial \beta_{x}} \sum_{i=1}^nx_i(y_i- \beta_{0}- \beta_{x} x_i) && \text{(constant factor)}\\ &= -2 \sum_{i=1}^n\frac{\partial}{\partial \beta_{x}} \mathopen{}\left(x_i(y_i- \beta_{0}- \beta_{x} x_i)\right)\mathclose{} && \text{(derivative of a sum)}\\ &= -2 \sum_{i=1}^nx_i\frac{\partial}{\partial \beta_{x}} (y_i- \beta_{0}- \beta_{x} x_i) && \text{($x_i$ does not depend on $\beta_{x}$)}\\ &= -2 \sum_{i=1}^nx_i(-x_i) && \text{(inner derivative is $-x_i$)}\\ &= -2 \sum_{i=1}^n\mathopen{}\left(-x_i^2\right)\mathclose{} && \text{(multiply)}\\ &= -2 \mathopen{}\left(-\sum_{i=1}^nx_i^2\right)\mathclose{} && \text{(constant factor out of the sum)}\\ &= 2 \sum_{i=1}^nx_i^2 && \text{(multiply)} \end{aligned} \]

The matrix. Collecting the four second derivatives gives the matrix of second derivatives:

\[ \begin{pmatrix} 2n & 2 n {\bar{x}}\\ 2 n {\bar{x}}& 2 \sum_{i=1}^nx_i^2 \end{pmatrix}. \]

Exercise 12 (Positive definiteness of the second-derivative matrix of RSS)

Exercise 12 (Positive definiteness of the second-derivative matrix of RSS) In the setting of Exercise 11, show that the matrix of second derivatives is positive definite when $S_{xx}> 0$ (Definition 14).

Solution 12

Solution 12. The top-left entry of the matrix of second derivatives from Exercise 11, $2n$, is positive. The determinant of this matrix, using the expanded form of $S_{xx}$ from Exercise 5, is

\[ \begin{aligned} \det \begin{pmatrix} 2n & 2 n {\bar{x}}\\ 2 n {\bar{x}}& 2 \sum_{i=1}^nx_i^2 \end{pmatrix} &= (2n) \mathopen{}\left(2 \sum_{i=1}^nx_i^2\right)\mathclose{} - (2 n {\bar{x}}) (2 n {\bar{x}}) && \text{($2 \times 2$ determinant)}\\ &= 4 n \sum_{i=1}^nx_i^2 - 4 n^2 {\bar{x}}^2 && \text{(multiply)}\\ &= 4 n \mathopen{}\left(\sum_{i=1}^nx_i^2 - n {\bar{x}}^2\right)\mathclose{} && \text{(factor out $4n$)}\\ &= 4 n S_{xx} && \text{(expanded form of $S_{xx}$)} \end{aligned} \]

which is positive when $S_{xx}> 0$. A symmetric $2 \times 2$ matrix with a positive top-left entry and a positive determinant is positive definite.

Lemma 7 (Second derivatives of RSS)

Lemma 7 (Second derivatives of RSS) The matrix of second derivatives of $\text{RSS}(\beta_{0}, \beta_{x})$ (Example 3) with respect to $\beta_{0}$ and $\beta_{x}$ is positive definite at every $(\beta_{0}, \beta_{x})$ when $S_{xx}> 0$ (Definition 14), and equals

\[\begin{pmatrix} 2n & 2 n {\bar{x}}\\ 2 n {\bar{x}}& 2 \sum_{i=1}^nx_i^2 \end{pmatrix}.\]

Proof

Proof. The matrix of second derivatives is the solution to Exercise 11. The positive definiteness is the solution to Exercise 12.

Theorem 5 (Closed-form OLS estimates)

Theorem 5 (Closed-form OLS estimates) Suppose $S_{xx}> 0$ (Definition 14), that is, not all $x_i$ are equal. Then the OLS estimates (Example 5) are unique, and

\[\hat{\beta}_{x} = \frac{S_{xy}}{S_{xx}}, \qquad \hat{\beta}_{0}= {\bar{y}}- \hat{\beta}_{x} {\bar{x}}.\]

Proof

Proof. $\text{RSS}$ is differentiable, so any point that minimizes it solves the normal equations. By Lemma 6, the normal equations have exactly one solution, $\beta_{x} = S_{xy}/ S_{xx}$ and $\beta_{0}= {\bar{y}}- \beta_{x} {\bar{x}}$. By Lemma 7, the matrix of second derivatives of $\text{RSS}$ is positive definite at every point, so $\text{RSS}$ is strictly convex and its only stationary point is its unique global minimum. So the OLS estimates exist, are unique, and equal that solution.

The same estimates follow from a derivation in vector notation, which treats $(\beta_{0}, \beta_{x})$ as a single vector instead of differentiating with respect to each component separately.

Definition 16 (Covariate vector)

Definition 16 (Covariate vector) In simple linear regression (Definition 8), the covariate vector $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i$ of observation $i$ is its covariate value $x_i$, preceded by a 1 for the intercept:

\[ \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\stackrel{\text{def}}{=}\begin{pmatrix} 1 \\ x_i\end{pmatrix}. \]

Exercise 13 (The conditional mean and the covariate vector)

Exercise 13 (The conditional mean and the covariate vector) In simple linear regression (Definition 8), express the conditional mean $\mu(x_i)$ as a dot product of the covariate vector $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i$ (Definition 16) and the coefficient vector $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}= {\mathopen{}\left(\beta_{0}, \beta_{x}\right)\mathclose{}}^{\top}$ (Example 6).

Solution 13

Solution 13. Start from the dot product:

\[ \begin{aligned} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}} &= \begin{pmatrix} 1 \\ x_i\end{pmatrix} \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}} && \text{(definition of $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i$)}\\ &= \begin{pmatrix} 1 \\ x_i\end{pmatrix} \cdot \begin{pmatrix} \beta_{0}\\ \beta_{x} \end{pmatrix} && \text{(definition of $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$, written as a column)}\\ &= 1 \cdot \beta_{0}+ x_i\beta_{x} && \text{(definition of the dot product)}\\ &= \beta_{0}+ x_i\beta_{x} && \text{($1 \cdot a = a$)}\\ &= \beta_{0}+ \beta_{x} x_i && \text{(reorder the factors)}\\ &= \mu(x_i) && \text{(definition of $\mu(x)$)} \end{aligned} \]

So $\mu(x_i) = \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$.

Lemma 8 (The conditional mean is a dot product)

Lemma 8 (The conditional mean is a dot product) In simple linear regression (Definition 8), the conditional mean of observation $i$ is the dot product of its covariate vector (Definition 16) and the coefficient vector $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$ (Example 6):

\[\mu(x_i) = \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}.\]

Proof

Proof. This is the solution to Exercise 13.

Example 7 (Residual sum of squares in vector notation)

Example 7 (Residual sum of squares in vector notation) In simple linear regression, take $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$ and substitute $\mu(x_i) = \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$ (Lemma 8) into each residual of the residual sum of squares (Definition 11). The residual sum of squares is then a function of the coefficient vector $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$, the same function as $\text{RSS}(\beta_{0}, \beta_{x})$ in Example 3:

\[ \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}) = \sum_{i=1}^n\mathopen{}\left(y_i- \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{}^2. \]

Exercise 14 (Gradient of RSS in vector notation)

Exercise 14 (Gradient of RSS in vector notation) Find the gradient of the residual sum of squares in vector notation (Example 7), $\frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}})$, by differentiating with respect to the vector $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$ directly, rather than one component at a time.

Solution 14

Solution 14. Differentiate one operation at a time, starting from $\text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}})$ in vector notation (Example 7):

\[ \begin{aligned} \frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}) &= \frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \sum_{i=1}^n\mathopen{}\left(y_i- \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{}^2 && \text{(RSS in vector form)}\\ &= \sum_{i=1}^n\frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \mathopen{}\left(y_i- \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{}^2 && \text{(derivative of a sum)}\\ &= \sum_{i=1}^n2 \mathopen{}\left(y_i- \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} \frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \mathopen{}\left(y_i- \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} && \text{(chain rule)} \end{aligned} \]

For the inner derivative, we need the gradient of a linear function. For any constant vector $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{a}}\vphantom{a}}= {(a_0, a_x)}^{\top}$:

\[ \begin{aligned} \frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{a}}\vphantom{a}} \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}} &= \frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \mathopen{}\left(a_0 \beta_{0}+ a_x \beta_{x}\right)\mathclose{} && \text{(write out the dot product)}\\ &= \begin{pmatrix} \frac{\partial}{\partial \beta_{0}} \mathopen{}\left(a_0 \beta_{0}+ a_x \beta_{x}\right)\mathclose{} \\ \frac{\partial}{\partial \beta_{x}} \mathopen{}\left(a_0 \beta_{0}+ a_x \beta_{x}\right)\mathclose{} \end{pmatrix} && \text{(definition of the gradient)}\\ &= \begin{pmatrix} a_0 + 0 \\ 0 + a_x \end{pmatrix} && \text{(derivative of each term)}\\ &= \begin{pmatrix} a_0 \\ a_x \end{pmatrix} && \text{(add zero)}\\ &= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{a}}\vphantom{a}} && \text{(definition of $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{a}}\vphantom{a}}$)} \end{aligned} \]

With $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{a}}\vphantom{a}}= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i$:

\[ \begin{aligned} \frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \mathopen{}\left(y_i- \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} &= \frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} y_i- \frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}} && \text{(derivative of a difference)}\\ &= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{0}}\vphantom{0}}- \frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}} && \text{($y_i$ does not depend on $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$)}\\ &= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{0}}\vphantom{0}}- \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i && \text{(gradient of a linear function)}\\ &= -\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i && \text{(subtract from zero)} \end{aligned} \]

Plugging the inner derivative back in:

\[ \begin{aligned} \frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}) &= \sum_{i=1}^n2 \mathopen{}\left(y_i- \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} \mathopen{}\left(-\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\right)\mathclose{} && \text{(inner derivative is $-\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i$)}\\ &= \sum_{i=1}^n2 \mathopen{}\left(-\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\right)\mathclose{} \mathopen{}\left(y_i- \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} && \text{(reorder the scalar and vector factors)}\\ &= \sum_{i=1}^n\mathopen{}\left(-2\right)\mathclose{} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\mathopen{}\left(y_i- \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} && \text{($2 \mathopen{}\left(-\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\right)\mathclose{} = -2 \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i$)}\\ &= -2 \sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\mathopen{}\left(y_i- \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} && \text{(constant factor out of the sum)} \end{aligned} \]

Lemma 9 (Gradient of RSS in vector notation)

Lemma 9 (Gradient of RSS in vector notation) The gradient of the residual sum of squares in vector notation (Example 7) is

\[\frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}) = -2 \sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\mathopen{}\left(y_i- \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{}.\]

Proof

Proof. This is the solution to Exercise 14.

Exercise 15 (Normal equations in vector notation)

Exercise 15 (Normal equations in vector notation) Define the matrix and vector

\[ A \stackrel{\text{def}}{=}\sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i}^{\top}, \qquad \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} \stackrel{\text{def}}{=}\sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_iy_i. \]

Using Exercise 14, show that the normal equations can be written as $A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$.

Solution 15

Solution 15. Distribute the sum in the gradient from Exercise 14:

\[ \begin{aligned} \frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}) &= -2 \sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\mathopen{}\left(y_i- \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} && \text{(gradient of RSS)}\\ &= -2 \sum_{i=1}^n\mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_iy_i- \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{}\right)\mathclose{} && \text{(distribute $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i$)}\\ &= -2 \mathopen{}\left(\sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_iy_i- \sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{}\right)\mathclose{} && \text{(sum of differences)} \end{aligned} \]

Each term of the second sum regroups as a matrix times $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$:

\[ \begin{aligned} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} &= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\mathopen{}\left({\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i}^{\top} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} && \text{(dot product as a transpose product)}\\ &= \mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i}^{\top}\right)\mathclose{} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}} && \text{(matrix multiplication is associative)} \end{aligned} \]

So the gradient is

\[ \begin{aligned} \frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}) &= -2 \mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} - \sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{}\right)\mathclose{} && \text{(definition of $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$)}\\ &= -2 \mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} - \sum_{i=1}^n\mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i}^{\top}\right)\mathclose{} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} && \text{(regroup each term)}\\ &= -2 \mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} - \mathopen{}\left(\sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i}^{\top}\right)\mathclose{} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} && \text{(factor $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$ out of the sum)}\\ &= -2 \mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} - A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} && \text{(definition of $A$)} \end{aligned} \]

The normal equations set this gradient equal to $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{0}}\vphantom{0}}$. Solving for $A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$ one operation at a time:

\[ \begin{aligned} -2 \mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} - A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} &= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{0}}\vphantom{0}} && \text{(normal equations)}\\ \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} - A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}&= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{0}}\vphantom{0}} && \text{(divide both sides by $-2$)}\\ \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} &= A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}} && \text{(add $A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$ to both sides)}\\ A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}&= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} && \text{(swap the sides)} \end{aligned} \]

Lemma 10 (Normal equations in vector notation)

Lemma 10 (Normal equations in vector notation) Let

The normal equations of a simple linear regression are equivalent to

\[A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}.\]

Proof

Proof. This is the solution to Exercise 15.

Exercise 16 (Solving the normal equations in vector notation)

Exercise 16 (Solving the normal equations in vector notation) Assuming $S_{xx}> 0$ (Definition 14), show that $A$ from Exercise 15 is invertible, solve $A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$ for $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$, and compare the result with Exercise 10.

Solution 16

Solution 16. Write out the entries of each term of $A$ and $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$, then add them up:

\[ \begin{aligned} A &= \sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i}^{\top} && \text{(definition of $A$)}\\ &= \sum_{i=1}^n\begin{pmatrix} 1 & x_i\\ x_i& x_i^2 \end{pmatrix} && \text{(outer product of $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i$)}\\ &= \begin{pmatrix} \sum_{i=1}^n1 & \sum_{i=1}^nx_i\\ \sum_{i=1}^nx_i& \sum_{i=1}^nx_i^2 \end{pmatrix} && \text{(matrices add entrywise)}\\ &= \begin{pmatrix} n & \sum_{i=1}^nx_i\\ \sum_{i=1}^nx_i& \sum_{i=1}^nx_i^2 \end{pmatrix} && \text{(sum of a constant)}\\ &= \begin{pmatrix} n & n{\bar{x}}\\ n{\bar{x}}& \sum_{i=1}^nx_i^2 \end{pmatrix} && \text{($\sum_{i=1}^nx_i= n {\bar{x}}$)} \end{aligned} \]

\[ \begin{aligned} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} &= \sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_iy_i && \text{(definition of $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$)}\\ &= \sum_{i=1}^n\begin{pmatrix} y_i\\ x_iy_i\end{pmatrix} && \text{(scalar times a vector)}\\ &= \begin{pmatrix} \sum_{i=1}^ny_i\\ \sum_{i=1}^nx_iy_i\end{pmatrix} && \text{(vectors add entrywise)}\\ &= \begin{pmatrix} n{\bar{y}}\\ \sum_{i=1}^nx_iy_i\end{pmatrix} && \text{($\sum_{i=1}^ny_i= n {\bar{y}}$)} \end{aligned} \]

The determinant of $A$ is

\[ \begin{aligned} \det A &= n \sum_{i=1}^nx_i^2 - \mathopen{}\left(n{\bar{x}}\right)\mathclose{} \mathopen{}\left(n{\bar{x}}\right)\mathclose{} && \text{(determinant of a $2 \times 2$ matrix)}\\ &= n \sum_{i=1}^nx_i^2 - n^2 {\bar{x}}^2 && \text{(multiply)}\\ &= n \mathopen{}\left(\sum_{i=1}^nx_i^2 - n {\bar{x}}^2\right)\mathclose{} && \text{(factor out $n$)}\\ &= n S_{xx} && \text{(expansion of $S_{xx}$)} \end{aligned} \]

where the last step uses Exercise 5. Since $n > 0$ and $S_{xx}> 0$, $\det A > 0$, so $A$ is invertible, and $A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$ has exactly one solution, $A^{-1} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$.

To simplify $A^{-1} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$, we need two expansions of uncentered sums. Adding $n {\bar{x}}^2$ to both sides of Exercise 5 gives $\sum_{i=1}^nx_i^2 = S_{xx}+ n {\bar{x}}^2$. The cross-product sum expands the same way:

\[ \begin{aligned} S_{xy} &= \sum_{i=1}^n(x_i- {\bar{x}})(y_i- {\bar{y}}) && \text{(definition of $S_{xy}$)}\\ &= \sum_{i=1}^n\mathopen{}\left(x_iy_i- x_i{\bar{y}}- {\bar{x}}y_i+ {\bar{x}}{\bar{y}}\right)\mathclose{} && \text{(expand the product)}\\ &= \sum_{i=1}^nx_iy_i- \sum_{i=1}^nx_i{\bar{y}}- \sum_{i=1}^n{\bar{x}}y_i+ \sum_{i=1}^n{\bar{x}}{\bar{y}} && \text{(sum of a sum)}\\ &= \sum_{i=1}^nx_iy_i- {\bar{y}}\sum_{i=1}^nx_i- \sum_{i=1}^n{\bar{x}}y_i+ \sum_{i=1}^n{\bar{x}}{\bar{y}} && \text{(constant factor out of the sum)}\\ &= \sum_{i=1}^nx_iy_i- {\bar{y}}\sum_{i=1}^nx_i- {\bar{x}}\sum_{i=1}^ny_i+ \sum_{i=1}^n{\bar{x}}{\bar{y}} && \text{(constant factor out of the sum)}\\ &= \sum_{i=1}^nx_iy_i- {\bar{y}}\sum_{i=1}^nx_i- {\bar{x}}\sum_{i=1}^ny_i+ n {\bar{x}}{\bar{y}} && \text{(sum of a constant)}\\ &= \sum_{i=1}^nx_iy_i- {\bar{y}}\mathopen{}\left(n {\bar{x}}\right)\mathclose{} - {\bar{x}}\sum_{i=1}^ny_i+ n {\bar{x}}{\bar{y}} && \text{($\sum_{i=1}^nx_i= n {\bar{x}}$)}\\ &= \sum_{i=1}^nx_iy_i- {\bar{y}}\mathopen{}\left(n {\bar{x}}\right)\mathclose{} - {\bar{x}}\mathopen{}\left(n {\bar{y}}\right)\mathclose{} + n {\bar{x}}{\bar{y}} && \text{($\sum_{i=1}^ny_i= n {\bar{y}}$)}\\ &= \sum_{i=1}^nx_iy_i- n {\bar{x}}{\bar{y}}- n {\bar{x}}{\bar{y}}+ n {\bar{x}}{\bar{y}} && \text{(reorder the factors)}\\ &= \sum_{i=1}^nx_iy_i- n {\bar{x}}{\bar{y}} && \text{(combine like terms)} \end{aligned} \]

Adding $n {\bar{x}}{\bar{y}}$ to both sides gives $\sum_{i=1}^nx_iy_i= S_{xy}+ n {\bar{x}}{\bar{y}}$. Now compute the solution:

\[ \begin{aligned} A^{-1} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} &= \frac{1}{\det A} \begin{pmatrix} \sum_{i=1}^nx_i^2 & -n{\bar{x}}\\ -n{\bar{x}}& n \end{pmatrix} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} && \text{(inverse of a $2 \times 2$ matrix)}\\ &= \frac{1}{n S_{xx}} \begin{pmatrix} \sum_{i=1}^nx_i^2 & -n{\bar{x}}\\ -n{\bar{x}}& n \end{pmatrix} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} && \text{($\det A = n S_{xx}$)}\\ &= \frac{1}{n S_{xx}} \begin{pmatrix} \sum_{i=1}^nx_i^2 & -n{\bar{x}}\\ -n{\bar{x}}& n \end{pmatrix} \begin{pmatrix} n{\bar{y}}\\ \sum_{i=1}^nx_iy_i\end{pmatrix} && \text{(substitute $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$)}\\ &= \frac{1}{n S_{xx}} \begin{pmatrix} \sum_{i=1}^nx_i^2 \cdot n{\bar{y}}+ \mathopen{}\left(-n{\bar{x}}\right)\mathclose{} \sum_{i=1}^nx_iy_i\\ \mathopen{}\left(-n{\bar{x}}\right)\mathclose{} \cdot n{\bar{y}}+ n \sum_{i=1}^nx_iy_i \end{pmatrix} && \text{(matrix times a vector)}\\ &= \frac{1}{n S_{xx}} \begin{pmatrix} n {\bar{y}}\sum_{i=1}^nx_i^2 - n {\bar{x}}\sum_{i=1}^nx_iy_i\\ -n^2 {\bar{x}}{\bar{y}}+ n \sum_{i=1}^nx_iy_i \end{pmatrix} && \text{(multiply out the products)}\\ &= \frac{1}{n S_{xx}} \, n \begin{pmatrix} {\bar{y}}\sum_{i=1}^nx_i^2 - {\bar{x}}\sum_{i=1}^nx_iy_i\\ -n {\bar{x}}{\bar{y}}+ \sum_{i=1}^nx_iy_i \end{pmatrix} && \text{(factor $n$ out of the vector)}\\ &= \frac{1}{S_{xx}} \begin{pmatrix} {\bar{y}}\sum_{i=1}^nx_i^2 - {\bar{x}}\sum_{i=1}^nx_iy_i\\ -n {\bar{x}}{\bar{y}}+ \sum_{i=1}^nx_iy_i \end{pmatrix} && \text{(cancel $n$)}\\ &= \frac{1}{S_{xx}} \begin{pmatrix} {\bar{y}}\sum_{i=1}^nx_i^2 - {\bar{x}}\sum_{i=1}^nx_iy_i\\ \sum_{i=1}^nx_iy_i- n {\bar{x}}{\bar{y}} \end{pmatrix} && \text{(reorder the terms of the second entry)}\\ &= \frac{1}{S_{xx}} \begin{pmatrix} {\bar{y}}\mathopen{}\left(S_{xx}+ n{\bar{x}}^2\right)\mathclose{} - {\bar{x}}\sum_{i=1}^nx_iy_i\\ \sum_{i=1}^nx_iy_i- n {\bar{x}}{\bar{y}} \end{pmatrix} && \text{($\sum_{i=1}^nx_i^2 = S_{xx}+ n{\bar{x}}^2$)}\\ &= \frac{1}{S_{xx}} \begin{pmatrix} {\bar{y}}\mathopen{}\left(S_{xx}+ n{\bar{x}}^2\right)\mathclose{} - {\bar{x}}\mathopen{}\left(S_{xy}+ n{\bar{x}}{\bar{y}}\right)\mathclose{} \\ \mathopen{}\left(S_{xy}+ n{\bar{x}}{\bar{y}}\right)\mathclose{} - n {\bar{x}}{\bar{y}} \end{pmatrix} && \text{($\sum_{i=1}^nx_iy_i= S_{xy}+ n{\bar{x}}{\bar{y}}$)}\\ &= \frac{1}{S_{xx}} \begin{pmatrix} {\bar{y}}S_{xx}+ {\bar{y}}n{\bar{x}}^2 - {\bar{x}}S_{xy}- {\bar{x}}n{\bar{x}}{\bar{y}}\\ S_{xy}+ n{\bar{x}}{\bar{y}}- n {\bar{x}}{\bar{y}} \end{pmatrix} && \text{(distribute)}\\ &= \frac{1}{S_{xx}} \begin{pmatrix} {\bar{y}}S_{xx}+ n{\bar{x}}^2{\bar{y}}- {\bar{x}}S_{xy}- n{\bar{x}}{\bar{x}}{\bar{y}}\\ S_{xy}+ n{\bar{x}}{\bar{y}}- n {\bar{x}}{\bar{y}} \end{pmatrix} && \text{(reorder the factors)}\\ &= \frac{1}{S_{xx}} \begin{pmatrix} {\bar{y}}S_{xx}+ n{\bar{x}}^2{\bar{y}}- {\bar{x}}S_{xy}- n{\bar{x}}^2{\bar{y}}\\ S_{xy}+ n{\bar{x}}{\bar{y}}- n {\bar{x}}{\bar{y}} \end{pmatrix} && \text{(${\bar{x}}{\bar{x}}= {\bar{x}}^2$)}\\ &= \frac{1}{S_{xx}} \begin{pmatrix} {\bar{y}}S_{xx}- {\bar{x}}S_{xy}\\ S_{xy}+ n{\bar{x}}{\bar{y}}- n {\bar{x}}{\bar{y}} \end{pmatrix} && \text{(first entry: cancel $\pm n{\bar{x}}^2{\bar{y}}$)}\\ &= \frac{1}{S_{xx}} \begin{pmatrix} {\bar{y}}S_{xx}- {\bar{x}}S_{xy}\\ S_{xy} \end{pmatrix} && \text{(second entry: cancel $\pm n{\bar{x}}{\bar{y}}$)}\\ &= \begin{pmatrix} {\bar{y}}S_{xx}/ S_{xx}- {\bar{x}}\, S_{xy}/ S_{xx}\\ S_{xy}/ S_{xx} \end{pmatrix} && \text{(multiply each entry by $1 / S_{xx}$)}\\ &= \begin{pmatrix} {\bar{y}}- {\bar{x}}\, S_{xy}/ S_{xx}\\ S_{xy}/ S_{xx} \end{pmatrix} && \text{(cancel $S_{xx}$)} \end{aligned} \]

Its entries are $\beta_{0}= {\bar{y}}- \beta_{x} {\bar{x}}$ and $\beta_{x} = S_{xy}/ S_{xx}$, the same solution as Exercise 10.

Lemma 11 (Solution of the normal equations in vector notation)

Lemma 11 (Solution of the normal equations in vector notation) If $S_{xx}> 0$ (Definition 14), the matrix $A = \sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i}^{\top}$ is invertible, and the normal equations $A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$ with $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} = \sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_iy_i$ (Lemma 10) have exactly one solution,

\[ \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}= A^{-1} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} = \begin{pmatrix} {\bar{y}}- \beta_{x} {\bar{x}}\\ S_{xy}/ S_{xx}\end{pmatrix} \quad \text{with} \quad \beta_{x} = \frac{S_{xy}}{S_{xx}}, \]

the same solution as Lemma 6.

Proof

Proof. This is the solution to Exercise 16.

Exercise 17 (Hessian of RSS in vector notation)

Exercise 17 (Hessian of RSS in vector notation) Find the Hessian $\frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \frac{\partial}{\partial {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}}^{\top}} \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}})$.

Solution 17

Solution 17. Differentiate the gradient $-2 \mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} - A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{}$ from Exercise 15 with respect to ${\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}}^{\top}$. $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$ does not depend on $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$. Each entry of $A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$ is the dot product of a row of $A$ with $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$, so by the gradient of a linear function (Exercise 14), the derivative of $A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$ with respect to ${\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}}^{\top}$ is $A$. The Hessian is then:

\[ \begin{aligned} \frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \frac{\partial}{\partial {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}}^{\top}} \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}) &= \frac{\partial}{\partial {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}}^{\top}} \mathopen{}\left(-2 \mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} - A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{}\right)\mathclose{} && \text{(differentiate the gradient)}\\ &= -2 \frac{\partial}{\partial {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}}^{\top}} \mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} - A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} && \text{(constant factor)}\\ &= -2 \mathopen{}\left(\frac{\partial}{\partial {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}}^{\top}} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} - \frac{\partial}{\partial {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}}^{\top}} A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} && \text{(derivative of a difference)}\\ &= -2 \mathopen{}\left(\mathbf{0} - \frac{\partial}{\partial {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}}^{\top}} A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}\right)\mathclose{} && \text{($\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$ does not depend on $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$)}\\ &= -2 \mathopen{}\left(\mathbf{0} - A\right)\mathclose{} && \text{(derivative of a linear function)}\\ &= -2 \mathopen{}\left(-A\right)\mathclose{} && \text{(subtract from zero)}\\ &= 2 A && \text{(multiply)}\\ &= 2 \sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i}^{\top} && \text{(definition of $A$)} \end{aligned} \]

Exercise 18 (Positive definiteness of the Hessian of RSS)

Exercise 18 (Positive definiteness of the Hessian of RSS) In the setting of Exercise 17, show that the Hessian is positive definite when $S_{xx}> 0$ (Definition 14).

Solution 18

Solution 18. By Exercise 17, the Hessian is $2A$ with $A = \sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i}^{\top}$. To show that this Hessian is positive definite, take any vector $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}} = {(v_0, v_x)}^{\top} \neq \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{0}}\vphantom{0}}$:

\[ \begin{aligned} {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}}^{\top} \mathopen{}\left(2 A\right)\mathclose{} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}} &= 2 \, {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}}^{\top} A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}} && \text{(constant factor)}\\ &= 2 \, {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}}^{\top} \mathopen{}\left(\sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i}^{\top}\right)\mathclose{} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}} && \text{(definition of $A$)}\\ &= 2 \sum_{i=1}^n{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}}^{\top} \mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i}^{\top}\right)\mathclose{} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}} && \text{(matrix multiplication distributes over the sum)}\\ &= 2 \sum_{i=1}^n\mathopen{}\left({\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}}^{\top} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\right)\mathclose{} \mathopen{}\left({\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i}^{\top} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}\right)\mathclose{} && \text{(matrix multiplication is associative)}\\ &= 2 \sum_{i=1}^n\mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}} \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\right)\mathclose{} \mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}\right)\mathclose{} && \text{(transpose products as dot products)}\\ &= 2 \sum_{i=1}^n\mathopen{}\left(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}} \cdot \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i\right)\mathclose{}^2 && \text{(dot product is symmetric)}\\ &= 2 \sum_{i=1}^n\mathopen{}\left(v_0 + v_x x_i\right)\mathclose{}^2 && \text{(write out the dot product)}\\ &\ge 0 && \text{(sum of squares)} \end{aligned} \]

Equality would need the condition $v_0 + v_x x_i= 0$ for every $i$. If $v_x \neq 0$, the condition would force every $x_i= -v_0 / v_x$, contradicting $S_{xx}> 0$; if $v_x = 0$, the condition would force $v_0 = 0$, contradicting $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}} \neq \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{0}}\vphantom{0}}$. So the inequality is strict, and the Hessian is positive definite at every $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$.

Lemma 12 (Hessian of RSS in vector notation)

Lemma 12 (Hessian of RSS in vector notation) The Hessian of the residual sum of squares is

\[ \frac{\partial}{\partial \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}} \frac{\partial}{\partial {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}}^{\top}} \text{RSS}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}) = 2 \sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i}^{\top}, \]

and it is positive definite at every $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$ when $S_{xx}> 0$ (Definition 14).

Proof

Proof. The Hessian is the solution to Exercise 17. The positive definiteness is the solution to Exercise 18.

Theorem 6 (OLS estimates in vector notation)

Theorem 6 (OLS estimates in vector notation) Suppose $S_{xx}> 0$ (Definition 14). Then the matrix $A = \sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i {\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_i}^{\top}$ is invertible, and the vector of OLS estimates (Definition 12) is

\[ \hat{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}}= A^{-1} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}, \qquad \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}} = \sum_{i=1}^n\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}_iy_i. \]

Its entries are the estimates of Theorem 5.

Proof

Proof. $\text{RSS}$ is differentiable, so any point that minimizes it solves the normal equations. By Lemma 10, the normal equations are $A \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$. By Lemma 11, $A$ is invertible, so their only solution is $A^{-1} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$, whose entries are those of Theorem 5. By Lemma 12, the Hessian of $\text{RSS}$ is positive definite at every $\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\beta}}\vphantom{\beta}}$, so $\text{RSS}$ is strictly convex and $A^{-1} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{c}}\vphantom{c}}$ is its unique global minimum.

Exercise 19 (OLS slope and the correlation)

Exercise 19 (OLS slope and the correlation) Assuming $S_{xx}> 0$ and $S_{yy}> 0$ (Definition 14), write the OLS slope $\hat{\beta}_{x}$ from Theorem 5 in terms of the Pearson correlation coefficient $r$ and the sample standard deviations $s_x$ and $s_y$ of the $x_i$ and the $y_i$.

Solution 19

Solution 19. In terms of the centered sums (Definition 14, Definition 15),

\[ r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}, \qquad s_x = \sqrt{\frac{S_{xx}}{n-1}}, \qquad s_y = \sqrt{\frac{S_{yy}}{n-1}}. \]

So:

\[ \begin{aligned} r \, \frac{s_y}{s_x} &= \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} \cdot \frac{\sqrt{S_{yy}/ (n-1)}}{\sqrt{S_{xx}/ (n-1)}} && \text{(substitute $r$, $s_x$, $s_y$)}\\ &= \frac{S_{xy}}{\sqrt{S_{xx}} \sqrt{S_{yy}}} \cdot \frac{\sqrt{S_{yy}/ (n-1)}}{\sqrt{S_{xx}/ (n-1)}} && \text{(square root of a product)}\\ &= \frac{S_{xy}}{\sqrt{S_{xx}} \sqrt{S_{yy}}} \cdot \frac{\sqrt{S_{yy}} / \sqrt{n-1}}{\sqrt{S_{xx}} / \sqrt{n-1}} && \text{(square root of a quotient)}\\ &= \frac{S_{xy}}{\sqrt{S_{xx}} \sqrt{S_{yy}}} \cdot \frac{\sqrt{S_{yy}}}{\sqrt{S_{xx}}} && \text{(cancel the factors of $\sqrt{n - 1}$)}\\ &= \frac{S_{xy}\sqrt{S_{yy}}}{\sqrt{S_{xx}} \sqrt{S_{yy}} \sqrt{S_{xx}}} && \text{(multiply the fractions)}\\ &= \frac{S_{xy}}{\sqrt{S_{xx}} \sqrt{S_{xx}}} && \text{(cancel $\sqrt{S_{yy}}$, which is positive)}\\ &= \frac{S_{xy}}{S_{xx}} && \text{($\sqrt{S_{xx}} \sqrt{S_{xx}} = S_{xx}$)}\\ &= \hat{\beta}_{x} && \text{(closed-form OLS slope)} \end{aligned} \]

Corollary 2 (OLS slope in terms of the correlation)

Corollary 2 (OLS slope in terms of the correlation) If $S_{xx}> 0$ and $S_{yy}> 0$ (Definition 14), then

\[\hat{\beta}_{x} = r \, \frac{s_y}{s_x},\]

where $r$ is the Pearson correlation coefficient and $s_x$ and $s_y$ are the sample standard deviations of the $x_i$ and the $y_i$.

Proof

Proof. This is the solution to Exercise 19.

Video lecture

Hutchinson’s Linear Regression (pt1) (17 min) covers fitting a linear regression by least squares, framed as machine learning [11]. The login for the video site is posted on Canvas.

Read

Hogg et al. [3, Sec. 6.5], pp. 277–284 covers a simple regression problem.
Casella and Berger [12, Sec. 11.3], pp. 539–562 covers simple linear regression.
Rice [4, Sec. 14.2.1], pp. 547–549 covers slope and intercept properties.

3.3 Fitting a simple linear regression in R

Example 8 (Regression of fasting glucose on BMI in HERS)

Example 8 (Regression of fasting glucose on BMI in HERS) The OLS estimates from Theorem 5, and the slope from Corollary 2, for the participants with a BMI measurement:

hers_bmi <- hers |> dplyr::filter(!is.na(BMI))
x <- hers_bmi$BMI
y <- hers_bmi$glucose
b1 <- sum((x - mean(x)) * (y - mean(y))) / sum((x - mean(x))^2)
c(
  intercept = mean(y) - b1 * mean(x),
  slope = b1,
  slope_from_r = cor(x, y) * sd(y) / sd(x)
)
#>    intercept        slope slope_from_r 
#>     60.07405      1.82218      1.82218

lm() gives the same estimates:

slr_fit <- lm(glucose ~ BMI, data = hers)
parameters::parameters(slr_fit) |>
  parameters::print_md(digits = 3)

Parameter	Coefficient	SE	95% CI	t(2756)	p
(Intercept)	60.074	3.565	(53.084, 67.064)	16.852	< .001
BMI	1.822	0.122	(1.582, 2.062)	14.878	< .001

performance::model_performance(slr_fit) |>
  insight::print_md(digits = 3)

Indices of model performance
AIC	AICc	BIC	R2	R2 (adj.)	RMSE	Sigma
27517.9	27517.9	27535.7	0.074	0.074	35.471	35.484

Figure 5 shows the data with the fitted line.

Show code

ggplot2::ggplot(hers_bmi, ggplot2::aes(x = BMI, y = glucose)) +
  ggplot2::geom_point(alpha = 0.2) +
  ggplot2::geom_abline(
    intercept = coef(slr_fit)[[1]],
    slope = coef(slr_fit)[["BMI"]],
    color = "firebrick"
  ) +
  ggplot2::labs(x = "BMI (kg/m^2)", y = "Fasting glucose (mg/dL)")

Figure 5: Fasting glucose against BMI in HERS, with the OLS line

The estimated slope is $\hat{\beta}_{\text{BMI}} = 1.82$ mg/dL per kg/m²: mean fasting glucose is about 1.8 mg/dL higher among participants whose BMI is 1 kg/m² higher. The t statistic for the slope equals the correlation test statistic of Example 1.

3.4 The coefficient of determination

Definition 17 (Total sum of squares)

Definition 17 (Total sum of squares) The total sum of squares of $y_1, \ldots, y_n$ is

\[\text{TSS} \stackrel{\text{def}}{=}\sum_{i=1}^n(y_i- {\bar{y}})^2.\]

Source: Wikipedia contributors [13].

Example 9 (Total sum of squares of three values)

Example 9 (Total sum of squares of three values) For $y = 1, 2, 2$, $\bar y = 5/3$, so

\[ \begin{aligned} \text{TSS} &= \mathopen{}\left(1 - \tfrac{5}{3}\right)\mathclose{}^2 + 2\mathopen{}\left(2 - \tfrac{5}{3}\right)\mathclose{}^2 && \text{(definition)}\\ &= \tfrac{4}{9} + \tfrac{2}{9} && \text{(square the deviations)}\\ &= \tfrac{2}{3} && \text{(arithmetic)} \end{aligned} \]

Definition 18 (Coefficient of determination)

Definition 18 (Coefficient of determination) The coefficient of determination of a model fitted with parameter estimate $\hat{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}}$ is one minus the ratio of its residual sum of squares at $\hat{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}}$ to the total sum of squares of its outcomes:

\[R^2 \stackrel{\text{def}}{=}1 - \frac{\text{RSS}(\hat{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}})}{\text{TSS}}.\]

Source: Wikipedia contributors [14].

Remark 6 (Interpreting the coefficient of determination)

Remark 6 (Interpreting the coefficient of determination). $R^2$ is often described as the proportion of the variation in $Y$ explained by the regression on $X$.

Exercise 20 (Residual sum of squares of the OLS fit)

Exercise 20 (Residual sum of squares of the OLS fit) Assuming $S_{xx}> 0$ (Definition 14), write the residual sum of squares at the OLS estimates (Theorem 5), $\text{RSS}(\hat{\beta}_{0}, \hat{\beta}_{x}) = \sum_{i=1}^n\hat{\varepsilon}_i^2$, in terms of $S_{xx}$, $S_{yy}$ and $S_{xy}$.

Solution 20

Solution 20. The fitted values of the OLS fit are $\hat y_i= \hat{\beta}_{0}+ \hat{\beta}_{x} x_i$, so each residual is

\[ \begin{aligned} \hat{\varepsilon}_i &= y_i- \hat y_i && \text{(definition of a residual)}\\ &= y_i- \mathopen{}\left(\hat{\beta}_{0}+ \hat{\beta}_{x} x_i\right)\mathclose{} && \text{(substitute $\hat y_i$)}\\ &= y_i- \mathopen{}\left({\bar{y}}- \hat{\beta}_{x} {\bar{x}}+ \hat{\beta}_{x} x_i\right)\mathclose{} && \text{(substitute $\hat{\beta}_{0}$)}\\ &= y_i- {\bar{y}}+ \hat{\beta}_{x} {\bar{x}}- \hat{\beta}_{x} x_i && \text{(distribute the minus sign)}\\ &= (y_i- {\bar{y}}) - \mathopen{}\left(\hat{\beta}_{x} x_i- \hat{\beta}_{x} {\bar{x}}\right)\mathclose{} && \text{(group terms)}\\ &= (y_i- {\bar{y}}) - \hat{\beta}_{x} (x_i- {\bar{x}}) && \text{(factor out $\hat{\beta}_{x}$)} \end{aligned} \]

Squaring and summing:

\[ \begin{aligned} \sum_{i=1}^n\hat{\varepsilon}_i^2 &= \sum_{i=1}^n\mathopen{}\left((y_i- {\bar{y}}) - \hat{\beta}_{x} (x_i- {\bar{x}})\right)\mathclose{}^2 && \text{(substitute $\hat{\varepsilon}_i$)}\\ &= \sum_{i=1}^n\mathopen{}\left((y_i- {\bar{y}})^2 - 2 \hat{\beta}_{x} (x_i- {\bar{x}})(y_i- {\bar{y}}) + \hat{\beta}_{x}^2 (x_i- {\bar{x}})^2\right)\mathclose{} && \text{(expand the square)}\\ &= \sum_{i=1}^n(y_i- {\bar{y}})^2 - \sum_{i=1}^n2 \hat{\beta}_{x} (x_i- {\bar{x}})(y_i- {\bar{y}}) + \sum_{i=1}^n\hat{\beta}_{x}^2 (x_i- {\bar{x}})^2 && \text{(sum of a sum)}\\ &= \sum_{i=1}^n(y_i- {\bar{y}})^2 - 2 \hat{\beta}_{x} \sum_{i=1}^n(x_i- {\bar{x}})(y_i- {\bar{y}}) + \sum_{i=1}^n\hat{\beta}_{x}^2 (x_i- {\bar{x}})^2 && \text{(constant factor out of the sum)}\\ &= \sum_{i=1}^n(y_i- {\bar{y}})^2 - 2 \hat{\beta}_{x} \sum_{i=1}^n(x_i- {\bar{x}})(y_i- {\bar{y}}) + \hat{\beta}_{x}^2 \sum_{i=1}^n(x_i- {\bar{x}})^2 && \text{(constant factor out of the sum)}\\ &= S_{yy}- 2 \hat{\beta}_{x} \sum_{i=1}^n(x_i- {\bar{x}})(y_i- {\bar{y}}) + \hat{\beta}_{x}^2 \sum_{i=1}^n(x_i- {\bar{x}})^2 && \text{(definition of $S_{yy}$)}\\ &= S_{yy}- 2 \hat{\beta}_{x} S_{xy}+ \hat{\beta}_{x}^2 \sum_{i=1}^n(x_i- {\bar{x}})^2 && \text{(definition of $S_{xy}$)}\\ &= S_{yy}- 2 \hat{\beta}_{x} S_{xy}+ \hat{\beta}_{x}^2 S_{xx} && \text{(definition of $S_{xx}$)}\\ &= S_{yy}- 2 \frac{S_{xy}}{S_{xx}} S_{xy}+ \mathopen{}\left(\frac{S_{xy}}{S_{xx}}\right)\mathclose{}^2 S_{xx} && \text{(substitute $\hat{\beta}_{x} = S_{xy}/ S_{xx}$)}\\ &= S_{yy}- 2 \frac{S_{xy}^2}{S_{xx}} + \mathopen{}\left(\frac{S_{xy}}{S_{xx}}\right)\mathclose{}^2 S_{xx} && \text{(multiply)}\\ &= S_{yy}- 2 \frac{S_{xy}^2}{S_{xx}} + \frac{S_{xy}^2}{S_{xx}^2} S_{xx} && \text{(square of a quotient)}\\ &= S_{yy}- 2 \frac{S_{xy}^2}{S_{xx}} + \frac{S_{xy}^2}{S_{xx}} && \text{(cancel one factor of $S_{xx}$)}\\ &= S_{yy}- \frac{S_{xy}^2}{S_{xx}} && \text{(combine like terms)} \end{aligned} \]

Lemma 13 (Residual sum of squares of the OLS fit)

Lemma 13 (Residual sum of squares of the OLS fit) If $S_{xx}> 0$ (Definition 14), the residual sum of squares at the OLS estimates (Theorem 5) is

\[\text{RSS}(\hat{\beta}_{0}, \hat{\beta}_{x}) = \sum_{i=1}^n\hat{\varepsilon}_i^2 = S_{yy}- \frac{S_{xy}^2}{S_{xx}}.\]

Proof

Proof. This is the solution to Exercise 20.

Exercise 21 ($R^2$ and the correlation)

Exercise 21 ($R^2$ and the correlation) Assuming $S_{xx}> 0$ and $S_{yy}> 0$ (Definition 14), use Exercise 20 to write the coefficient of determination $R^2$ of the OLS fit in terms of the Pearson correlation coefficient $r$, and show that $0 \le R^2 \le 1$.

Solution 21

Solution 21. The total sum of squares is $S_{yy}$ (Definition 14), and Exercise 20 gives the residual sum of squares, so:

\[ \begin{aligned} R^2 &= 1 - \frac{\text{RSS}(\hat{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{\theta}}\vphantom{\theta}}})}{\text{TSS}} && \text{(definition of $R^2$)}\\ &= 1 - \frac{\text{RSS}(\hat{\beta}_{0}, \hat{\beta}_{x})}{\text{TSS}} && \text{(the OLS estimate is $(\hat{\beta}_{0}, \hat{\beta}_{x})$)}\\ &= 1 - \frac{\text{RSS}(\hat{\beta}_{0}, \hat{\beta}_{x})}{S_{yy}} && \text{(TSS is $S_{yy}$)}\\ &= 1 - \frac{S_{yy}- S_{xy}^2 / S_{xx}}{S_{yy}} && \text{(residual sum of squares of the OLS fit)}\\ &= 1 - \mathopen{}\left(\frac{S_{yy}}{S_{yy}} - \frac{S_{xy}^2 / S_{xx}}{S_{yy}}\right)\mathclose{} && \text{(distribute the division over the difference)}\\ &= 1 - \mathopen{}\left(1 - \frac{S_{xy}^2 / S_{xx}}{S_{yy}}\right)\mathclose{} && \text{($S_{yy}/ S_{yy}= 1$)}\\ &= 1 - 1 + \frac{S_{xy}^2 / S_{xx}}{S_{yy}} && \text{(distribute the minus sign)}\\ &= \frac{S_{xy}^2 / S_{xx}}{S_{yy}} && \text{($1 - 1 = 0$)}\\ &= \frac{S_{xy}^2}{S_{xx}S_{yy}} && \text{(divide a fraction by $S_{yy}$)}\\ &= \mathopen{}\left(\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}\right)\mathclose{}^2 && \text{(write as a square)}\\ &= r^2 && \text{($r = S_{xy}/ \sqrt{S_{xx}S_{yy}}$)} \end{aligned} \]

Since $-1 \le r \le 1$ (range of the correlation coefficient), $0 \le r^2 \le 1$.

Theorem 7 ($R^2$ of a simple linear regression)

Theorem 7 ($R^2$ of a simple linear regression) For the OLS fit of a simple linear regression, with $S_{xx}> 0$ and $S_{yy}> 0$ (Definition 14), $R^2$ is the square of the Pearson correlation coefficient $r$ of the $x_i$ and $y_i$, and so lies between 0 and 1:

\[ \begin{aligned} R^2 &= r^2, & 0 \le R^2 &\le 1. \end{aligned} \]

Proof

Proof. This is the solution to Exercise 21, which uses the residual sum of squares from Lemma 13.

Example 10 ($R^2$ for the regression of glucose on BMI in HERS)

Example 10 ($R^2$ for the regression of glucose on BMI in HERS) For the fit in Example 8, $R^2$ computed from Definition 18, the square of the Pearson correlation, and lm()’s value agree:

fit <- lm(glucose ~ BMI, data = hers)
y <- fit$model$glucose
c(
  r_squared = 1 - sum(residuals(fit)^2) / sum((y - mean(y))^2),
  cor_squared = cor(fit$model$BMI, y)^2,
  lm = summary(fit)$r.squared
)
#>   r_squared cor_squared          lm 
#>   0.0743455   0.0743455   0.0743455

BMI accounts for only about 7% of the variation in baseline fasting glucose.

Read

Hogg et al. [3, Sec. 7.6], pp. 344–352 covers more regression.
Rice [4, Sec. 14.2.2], pp. 550–559 covers assessing the fit of a regression line.

3.5 Comparing nested regression models

Definition 19 (Nested linear regression models)

Definition 19 (Nested linear regression models) Let a linear regression model have $p$ features and an intercept. A second linear regression model is nested in it if it uses the same outcome and the same observations and is the special case in which $q$ of the $p$ coefficients equal $0$. The first model is the larger model, and the second is the smaller model; the smaller model has $p - q$ features.

Source: Vittinghoff et al. [1, Sec. 4.3].

Example 11 (Nested models for fasting glucose in HERS)

Example 11 (Nested models for fasting glucose in HERS) In the HERS data, take the linear regression of glucose on BMI as the smaller model. The larger model adds age and exercise, so $q = 2$:

\[ \begin{aligned} \text{smaller:}\quad \mu(\text{glucose}) &= \beta_{0}+ \beta_{\text{BMI}} \cdot \text{BMI}, \\ \text{larger:}\quad \mu(\text{glucose}) &= \beta_{0}+ \beta_{\text{BMI}} \cdot \text{BMI} + \beta_{\text{age}} \cdot \text{age} + \beta_{\text{exercise}} \cdot \text{exercise}. \end{aligned} \]

Both models must be fitted to the same HERS participants, so keep the participants with all four variables measured:

hers_nested <- hers |>
  tidyr::drop_na(glucose, BMI, age, exercise)
fit_smaller <- lm(glucose ~ BMI, data = hers_nested)
fit_larger <- lm(glucose ~ BMI + age + exercise, data = hers_nested)
nrow(hers_nested)
#> [1] 2758

The smaller model sets $\beta_{\text{age}} = \beta_{\text{exercise}} = 0$ in the larger one.

Definition 20 (Partial F-test)

Definition 20 (Partial F-test) Let a smaller linear regression model be nested in a larger model with $p$ features, so that they differ by $q$ coefficients, and let both be fitted by ordinary least squares to $n$ observations. Write $\text{RSS}_0$ and $\text{RSS}_1$ for the residual sums of squares of the smaller and the larger fit. The partial F-test of

\[ \begin{aligned} H_0: \text{the } q \text{ coefficients are all } 0 \end{aligned} \]

uses the statistic

\[ F \stackrel{\text{def}}{=}\frac{(\text{RSS}_0 - \text{RSS}_1) / q}{\text{RSS}_1 / (n - p - 1)}. \]

Its p-value is $\Pr(F^* \ge F)$, where $F^*$ has the $F_{q,\, n-p-1}$ distribution (F-distribution).

Source: Vittinghoff et al. [1, Sec. 4.3] and Dobson and Barnett [15, Sec. 6.3].

Theorem 8 (Null distribution of the partial F statistic)

Theorem 8 (Null distribution of the partial F statistic) Let the errors of the larger model be independent and Gaussian with mean $0$ and the same variance. If $H_0$ of Definition 20 holds, then $F$ has the $F_{q,\, n-p-1}$ distribution for every number of observations $n > p + 1$.

Source: Vittinghoff et al. [1, Sec. 4.3].

Proof

Proof. Under $H_0$, $(\text{RSS}_0 - \text{RSS}_1) / \sigma^2$ has the $\chi^2_q$ distribution and $\text{RSS}_1 / \sigma^2$ has the $\chi^2_{n-p-1}$ distribution, where $\sigma^2$ is the error variance. The two are independent, because $\text{RSS}_1$ is the squared length of the part of the outcome vector orthogonal to the larger model’s fitted values, while $\text{RSS}_0 - \text{RSS}_1$ is the squared length of the part in the larger model but orthogonal to the smaller one. Dividing each by its degrees of freedom, $\sigma^2$ cancels from the quotient, which is therefore $F_{q,\, n-p-1}$ by the definition of the F-distribution.

Example 12 (Partial F-test of age and exercise in HERS)

Example 12 (Partial F-test of age and exercise in HERS) For the models of Example 11, anova() gives the partial F-test:

anova(fit_smaller, fit_larger) |>
  parameters::parameters() |>
  parameters::print_md(digits = 3, footer = "")

Parameter	RSS	Sum_Squares	df	df (error)	Mean_Square	F	p
Model 1: glucose ~ BMI	3.470e+06			2756
Model 2: glucose ~ BMI + age + exercise	3.461e+06	9188.413	2	2754	4594.207	3.656	0.026

The definition gives the same statistic:

rss0 <- sum(residuals(fit_smaller)^2)
rss1 <- sum(residuals(fit_larger)^2)
n <- nobs(fit_larger)
p <- length(coef(fit_larger)) - 1
q <- p - (length(coef(fit_smaller)) - 1)
f_stat <- ((rss0 - rss1) / q) / (rss1 / (n - p - 1))
c(
  f = f_stat,
  p_value = pf(f_stat, q, n - p - 1, lower.tail = FALSE)
)
#>         f   p_value 
#> 3.6558211 0.0259658

With $q = 2$ and $n - p - 1 = 2754$, $F = 3.66$, so the data reject $H_0$ at the 5% level.

Remark 7 (The partial F-test and the likelihood ratio test)

Remark 7 (The partial F-test and the likelihood ratio test). Let the errors be Gaussian. The maximum likelihood estimate of the error variance of a model with residual sum of squares $\text{RSS}$ is $\text{RSS} / n$, so its maximized log-likelihood is $-\tfrac{n}{2} \log(2\pi\, \text{RSS} / n) - \tfrac{n}{2}$. The likelihood ratio statistic of the null hypothesis $H_0$ in the partial F-test is then

\[ \begin{aligned} \Lambda &= 2\mathopen{}\left(\ell(\hat\theta_{\text{ML}}) - \ell(\hat{\theta}_0)\right)\mathclose{} \\ &= n \log \frac{\text{RSS}_0}{\text{RSS}_1} \\ &= n \log\mathopen{}\left(1 + \frac{\text{RSS}_0 - \text{RSS}_1}{\text{RSS}_1}\right)\mathclose{}. \end{aligned} \]

For large $n$ under $H_0$, the quantity $(\text{RSS}_0 - \text{RSS}_1) / \text{RSS}_1$ is near $0$, so $\log(1 + u) \approx u$ gives $\Lambda\approx n (\text{RSS}_0 - \text{RSS}_1) / \text{RSS}_1$. Also $q F = (n - p - 1) (\text{RSS}_0 - \text{RSS}_1) / \text{RSS}_1$, which is close to the same value when $n$ is much larger than $p$. Under $H_0$, $\Lambda$ and $q F$ therefore have approximately the $\chi^2_q$ distribution, but the F-test is exact for Gaussian errors at every $n$, while the likelihood ratio test holds only approximately.

3.6 Further reading

Linear Models Overview covers linear regression in depth, including inference for the coefficients and multiple predictors. Vittinghoff et al. [1] cover linear regression in Chapter 4.

Definitions on this page

Definition 1 (Population correlation)
Definition 2 (Test of zero correlation)
Definition 3 (Spearman rank correlation)
Definition 4 (Conditional Gaussian model)
Definition 5 (Deviation from the conditional mean)
Definition 6 (Homoskedastic model)
Definition 7 (Linear regression model)
Definition 8 (Simple linear regression model)
Definition 9 (Intercept)
Definition 10 (Slope)
Definition 11 (Residual sum of squares)
Definition 12 (Ordinary least squares)
Definition 13 (Normal equations)
Definition 14 (Centered sums of squares)
Definition 15 (Centered sum of cross-products)
Definition 16 (Covariate vector)
Definition 17 (Total sum of squares)
Definition 18 (Coefficient of determination)
Definition 19 (Nested linear regression models)
Definition 20 (Partial F-test)

References

[1]

E. Vittinghoff, D. V. Glidden, S. C. Shiboski, and C. E. McCulloch, Regression methods in biostatistics: Linear, logistic, survival, and repeated measures models, 2nd ed. Springer, 2012. doi: 10.1007/978-1-4614-1353-0.

[2]

Wikipedia contributors, “Pearson correlation coefficient — Wikipedia, the free encyclopedia.” 2026. Available: https://en.wikipedia.org/wiki/Pearson_correlation_coefficient

[3]

R. V. Hogg, E. A. Tanis, and D. L. Zimmerman, Probability and statistical inference, Tenth edition. Boston: Pearson, 2019.

[4]

J. A. Rice, Mathematical statistics and data analysis, 3rd ed. Belmont, CA: Cengage Learning, 2007.

[5]

Wikipedia contributors, “Spearman’s rank correlation coefficient — Wikipedia, the free encyclopedia.” 2026. Available: https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient

[6]

Wikipedia contributors, “Homoscedasticity and heteroscedasticity — Wikipedia, the free encyclopedia.” 2026. Available: https://en.wikipedia.org/wiki/Homoscedasticity_and_heteroscedasticity

[7]

Wikipedia contributors, “Linear regression — Wikipedia, the free encyclopedia.” 2026. Available: https://en.wikipedia.org/wiki/Linear_regression

[8]

Wikipedia contributors, “Simple linear regression — Wikipedia, the free encyclopedia.” 2026. Available: https://en.wikipedia.org/wiki/Simple_linear_regression

[9]

Wikipedia contributors, “Residual sum of squares — Wikipedia, the free encyclopedia.” 2026. Available: https://en.wikipedia.org/wiki/Residual_sum_of_squares

[10]

Wikipedia contributors, “Ordinary least squares — Wikipedia, the free encyclopedia.” 2026. Available: https://en.wikipedia.org/wiki/Ordinary_least_squares

[11]

B. Hutchinson, “DATA 471/571 (machine learning) and CSCI 481/581 (deep learning) video lectures.” Western Washington University. Accessed: Sep. 28, 2026. [Online]. Available: https://facultyweb.cs.wwu.edu/~hutchib2/video_lectures/data371/

[12]

G. Casella and R. Berger, Statistical inference, 2nd ed. Cengage Learning, 2002. Available: https://www.cengage.com/c/statistical-inference-2e-casella-berger/9780534243128/

[13]

Wikipedia contributors, “Total sum of squares — Wikipedia, the free encyclopedia.” 2026. Available: https://en.wikipedia.org/wiki/Total_sum_of_squares

[14]

Wikipedia contributors, “Coefficient of determination — Wikipedia, the free encyclopedia.” 2026. Available: https://en.wikipedia.org/wiki/Coefficient_of_determination

[15]

A. J. Dobson and A. G. Barnett, An introduction to generalized linear models, 4th ed. CRC press, 2018. doi: 10.1201/9781315182780.