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 \(\sum_i x_i y_i = \sum_jy_j^2\)
  2. When \(\sum_i x_i y_i = \sum_jx_j^2\)
  3. When \(\sum_jx_j^2 = \sum_jy_j^2\)
  4. When \(\sum_jx_j^2 = \frac{1}{\sum_jy_j^2}\)

• 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)