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Analyze OLS After Duplicating Every Observation

SnapchatData Scientist · Technical Screen
MediumData ScientistTechnical ScreenStatistics & Math

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This question tests your understanding of OLS coefficient estimation and standard error calculations under the scenario of duplicate data observations.

Problem Statement

Suppose you have a dataset of n independent observations (X, y) where X is an n x k matrix and y is an n x 1 vector. You estimate the coefficients using Ordinary Least Squares (OLS) as beta_hat = (X'X)^-1 X'y.

Now, imagine you create a new dataset by duplicating every single observation in your original dataset, resulting in a new dataset of size 2n.

  1. How do the estimated coefficients beta_hat_new compare to the original coefficients beta_hat?
  2. How do the estimated standard errors of the coefficients change in the new dataset? Specifically, compare the variance-covariance matrix of the estimators Var(beta_hat) and Var(beta_hat_new).
  3. Explain why the p-values calculated using the new dataset would be misleading and how this relates to the assumption of independent and identically distributed (i.i.d.) errors.

Assume the standard OLS assumptions hold for the original dataset, including homoscedasticity.

Hints

Hint 1

Consider the matrix representation of OLS: adding duplicate rows to X results in a new matrix X_new = [X; X], and similarly for y.

Hint 2

Recall that the variance of the OLS estimator is given by sigma^2 * (X'X)^-1. Calculate how the term (X_new' * X_new) relates to (X'X).

Hint 3

Think about the degrees of freedom used in the estimation of the error variance (sigma^2) and how they change when the number of observations doubles.

Follow-up questions

  • What would happen to the standard errors if, instead of duplicating the data, you generated new, independent observations with the same distribution?
  • If you were analyzing this data and suspected duplication, what diagnostic tests or data preprocessing steps could you perform?
  • How would this impact the calculation of the R-squared value of the regression model?