Multiple linear regression: quadratic regression model

We have one predictor \(x\) but the plot of \(y\) vs \(x\) exhibits a quadratic pattern. See simulated data below:

x1 <- runif(100)
y <- 0.2 + 2 * x1 + 5 * x1^2 + rnorm(100, 0, 0.3)
plot(x1, y)

Then we can fit a multiple regression model:

\[\mathbb{E}[y] = \beta_0 + \beta_1 x + \beta_2 x^2. \]

This is also called a quadratic regression model or, more generally, a polynomial regression model. To fit in R:

fit <- lm(y ~x1 + I(x1^2))
coef(fit)
## (Intercept)          x1     I(x1^2) 
##   0.2315906   1.5892734   5.3925940

Visualize this fit:

#make a grid
xnew <- seq(0,1, by = 0.1)
xnewsq <- xnew^2

f <- function(x,xsq) { r <- coef(fit)[1] + coef(fit)[2] * x + coef(fit)[3] * xsq }
z <- outer(xnew, xnewsq, f)
persp3d(xnew, xnewsq, z,  col = "lightblue", xlab = "X1", ylab = "X1^2", zlab = "Y")
points3d(x1,x1^2,y,col="red")

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