Questions

• MC1: Recall that the coefficient estimate $\hat{\beta }$ for the linear regression of $Y$ onto $X$ without an intercept ($Y=\beta X$)is given by

  $$\hat{\beta }=\left(\sum _{i=1}^n{x}_i{y}_i\right)/\left(\sum _{j=1}^n{x}_j^2\right)$$

  Under what circumstance is the coefficient estimate for the regression of $X$ onto $Y$ the same as the coefficient estimate for the regression of $Y$ onto $X$?

  1. When $\sum _i{x}_i{y}_i=\sum _j{y}_j^2$
  2. When $\sum _i{x}_i{y}_i=\sum _j{x}_j^2$
  3. When $\sum _j{x}_j^2=\sum _j{y}_j^2$
  4. When $\sum _j{x}_j^2=\frac{1}{\sum _j{y}_j^2}$

• MC2: Using the knowledge of the question MC1, which of these statements is true based on the code given below.

  1. The coefficient estimate for the regression of $X$ onto $Y$ is different from the coefficient estimate for the regression of $Y$ onto $X$.
  2. The coefficient estimate for the regression of $X$ onto $Y$ is equal to the coefficient estimate for the regression of $Y$ onto $X$.

                
            set.seed(1)
x <- 1:100
y <- 2 * x + rnorm(100, sd = 0.1)
fit.Y <- lm(y ~ x + 0)
fit.X <- lm(x ~ y + 0)

• MC3: Using the knowledge of the question MC1, which of these statements is true based on the code given below.
  1. The coefficient estimate for the regression of $X$ onto $Y$ is different from the coefficient estimate for the regression of $Y$ onto $X$.
  2. The coefficient estimate for the regression of $X$ onto $Y$ is equal to the coefficient estimate for the regression of $Y$ onto $X$.

                
            x <- 1:100
y <- 100:1
fit.Y <- lm(y ~ x + 0)
fit.X <- lm(x ~ y + 0)