← Two Sigma Interview Insights
You need the joint covariance matrix to be positive semi-definite, so the determinant constraint gives you p >= -1/2.
Recognize that the correlation matrix must be positive semidefinite. Construct the 3x3 matrix with 1s on the diagonal and p off-diagonal, then compute its eigenvalues and require them to be nonnegative to find the valid range of p.
Pro tip: Mention that the lower bound -1/2 is a classic result and that the matrix is a special case of an equicorrelation matrix; also note that if p = -1/2, the variables are linearly dependent, which is allowed.
Write the 3x3 correlation matrix with 1s on the diagonal and p on all off-diagonal entries.
Recall that any valid correlation matrix must be positive semidefinite, meaning all eigenvalues must be nonnegative.
Find the eigenvalues of the matrix: they are 1+2p (multiplicity 1) and 1-p (multiplicity 2).
Require 1+2p >= 0 and 1-p >= 0, which gives p >= -1/2 and p <= 1.
Conclude that the valid range is -1/2 <= p <= 1.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Factor model construction: write each variable as sqrt(p)*F + sqrt(1-p)*e_i where F and the e_i's are independent standard normals.
First, clarify the problem context: identify the three variables and the range of p (likely a probability). Then, construct a simple probabilistic model (e.g., independent Bernoulli trials) where the probability of an event can be tuned to any p in the range by adjusting a parameter. Finally, provide an explicit example with specific values that achieves the desired p, and verify the calculation.
Pro tip: Choose the simplest possible model (like independent coin flips) to minimize complexity and reduce the chance of errors. Explicitly state any assumptions you make about the variables and the range of p.
Restate the question in your own words to ensure you understand what the three variables are and what range of p is valid. Ask clarifying questions if needed.
Select a model with three variables that can produce a range of probabilities, such as three independent Bernoulli trials with success probability q. The probability of at least one success is 1 - (1-q)^3, which can achieve any p in (0,1) by solving for q.
Express p in terms of the model parameters. For the Bernoulli example, p = 1 - (1-q)^3, so q = 1 - (1-p)^(1/3). This shows that for any p in (0,1), there exists a q in (0,1).
Pick a specific p (e.g., p = 0.5) and compute the corresponding q. Then define the three variables explicitly (e.g., X1, X2, X3 ~ Bernoulli(q)) and show that the probability of at least one success is p.
Check the calculation and mention edge cases like p=0 or p=1 (which may require degenerate distributions). Also, note that the construction works for any p in the valid range.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Lower bound becomes -1/(n-1), which makes sense because as n grows you can't have everything negatively correlated with everything else.
First, clarify that the equal-correlation setup means all pairwise correlations are equal to ρ. Then, derive the admissible range by considering the positive semidefiniteness of the n×n correlation matrix with 1 on the diagonal and ρ off-diagonal. The range is -1/(n-1) ≤ ρ ≤ 1.
Pro tip: Mention that the lower bound approaches 0 as n grows, which is a key insight for high-dimensional data. Also, relate this to the concept of equicorrelation and its applications in finance and statistics.
Write the n×n matrix with 1 on the diagonal and ρ on all off-diagonal entries. This is the equicorrelation matrix.
A valid correlation matrix must be positive semidefinite. Compute its eigenvalues or use the fact that for such a matrix, eigenvalues are 1 + (n-1)ρ (multiplicity 1) and 1 - ρ (multiplicity n-1).
Require all eigenvalues to be nonnegative: 1 + (n-1)ρ ≥ 0 and 1 - ρ ≥ 0. Solve these to get ρ ≥ -1/(n-1) and ρ ≤ 1.
Combine the inequalities to get the final range: -1/(n-1) ≤ ρ ≤ 1. Note that for n=2, this reduces to -1 ≤ ρ ≤ 1, as expected.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
This is the part I was least prepared for.
Set up the true model in matrix form and derive the OLS estimator from the misspecified regression of y on X alone. Then substitute the true model into the estimator to express the bias as a function of the omitted variable Z and the correlation between X and Z.
Pro tip: Emphasize that the bias depends on both the effect of Z on y and the linear projection of Z on X; if X and Z are orthogonal, the bias vanishes. This shows you understand the conditions under which omitted variable bias is a concern.
Write the true linear regression model in matrix form: y = Xβ + Zγ + ε, where X and Z are matrices of regressors and ε is the error term with E[ε|X,Z] = 0.
Regress y on X only, obtaining the OLS estimator β_hat = (X'X)^{-1}X'y.
Plug y = Xβ + Zγ + ε into β_hat to get β_hat = β + (X'X)^{-1}X'Zγ + (X'X)^{-1}X'ε.
Take the expectation (conditional on X and Z) to find E[β_hat|X,Z] = β + (X'X)^{-1}X'Zγ. The bias term is (X'X)^{-1}X'Zγ, which depends on the regression of Z on X and the true coefficient γ.
Explain that the bias is zero if X'Z = 0 (orthogonality) or if γ = 0 (Z has no effect). Otherwise, the bias is the product of the coefficient of Z in the true model and the coefficient from regressing Z on X.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.