Question on Quantitative Method Concept

CFA Level II Quant
#1

Could someone elaborate on the following concepts?

  1. As n increases, the value of the sample correlation, r, required to reject the null hypothesis decreases.

  2. Correlation coefficients computed from sample data are valid as long as “the means and variances of the two variables and the covariance of the two variables are finite and constant.”

  3. Why is the width of a prediction interval negatively related to the standard deviation of the independent variable?

#2

Please provide reference pages of 1) and 2).

As for 3), if the standard deviation is larger, then a hypothesized value of a variable will stray from its mean more frequently, thus, it is much more difficult to predict with precision that it will be within a prescribed interval.

#3

^that is wrong hahah sorry

my best guess is that if the variance of the the independent variable is large, then it explains more of the variance of the prediction error…just guess tho

#4

  1. Look at the formula for t-stat for correlation. Remember that you need a high (absolute) value of t to reject the null.

  2. If the mean/variance of the variables is not constant, the computed correlation is simply a snapshot and not representative of things to come.

  3. The width of prediction interval depends on standard error of forecast. Standard error of forecast is higher when standard deviation of the independent variable is high (if you keep standard error of estimate constant). However, if you don’t keep standard error of estimate constant (which is indeed uncommon in reality), then NO SUCH GENERALIZATION can be made.

#5

  1. If you look at the formula for the t-test for sample correlation: t-value = r*sqrt(n-2)/sqrt(1-r^2). Then for a given t-critical value, as n increases, the value of the r needed to reject the null will decrease.

  2. IF “the means and variances of the two variables and the covariance of the two variables are NOT finite and constant.” => the regression is not covariance stationary and thus, the coefficient is not valid.

  3. I’m not so sure about this. IMO, if the standard deviation of the independent variable is high, then the standard error should be high, which will lead to higher width of the confidence interval.