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t-Stat With an Added Orthogonal Regressor, Live Sharpe Check, and Absolute-Loss Estimator

HeadlandsData Scientist · Online Assessment
MediumData ScientistOnline AssessmentStatistics & Math

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Solve three core quantitative problems covering OLS regression mechanics under orthogonal regressors, statistical significance of backtested Sharpe ratios in live deployment, and loss-function minimization for L1 error.

Answer the following three independent quantitative questions commonly encountered in quantitative trading and research assessments.

Part 1: OLS with an Added Orthogonal Regressor

Consider a simple linear regression model without an intercept:

y = beta_1 * x_1 + epsilon

Suppose you estimate beta_1 via Ordinary Least Squares (OLS) on sample data of size n, obtaining the coefficient estimate hat(beta)_1, its standard error SE(hat(beta)_1), and the t-statistic t_1 = hat(beta)_1 / SE(hat(beta)_1).

You now add a second regressor x_2 to form the multiple regression model:

y = beta_1 * x_1 + beta_2 * x_2 + u

Assume that x_1 and x_2 are strictly orthogonal in the sample (x_1^T x_2 = 0), and x_2 has non-zero explanatory power (i.e., its estimated coefficient hat(beta)_2 != 0).

  1. Does the point estimate hat(beta)_1 change in the new model? Explain mathematically.
  2. How does SE(hat(beta)_1) change? Consequently, what happens to the magnitude of t_1? Specify the condition regarding degrees of freedom and residual sum of squares.

Part 2: Evaluating a High-Sharpe Strategy in Live Trading

A quantitative researcher presents a backtest with an annualized Sharpe ratio of SR = 8.0. Assume 252 trading days per year, and assume daily returns are independently and identically distributed normal random variables: r_t ~ N(mu, sigma^2).

Upon deployment, the strategy produces flat performance: over N consecutive trading days, the realized cumulative return is approximately zero, while daily volatility matches the backtested sigma.

How many consecutive flat days N must elapse before you can reject the backtested mean return at a 2-standard-deviation threshold (i.e., Z <= -2.0)?

Part 3: Optimal Estimator Under Absolute Loss

Let Y be a continuous real-valued random variable with probability density function f(y) and cumulative distribution function F(y).

You wish to find a single constant prediction c that minimizes the expected absolute loss:

L(c) = E[|Y - c|]

Identify the optimal estimator c* and show that it minimizes L(c) by differentiating L(c) with respect to c.

Hints

Hint 1

For Part 1, use the Frisch-Waugh-Lovell theorem or the explicit formula for beta_1 in terms of (X^T X)^(-1) X^T y to check the point estimate, then consider what happens to the residual sum of squares (RSS) versus degrees of freedom (n - p).

Hint 2

For Part 2, convert the annualized Sharpe ratio to a daily Sharpe ratio using the square root of 252, and compute the z-score of observing a sum of zero returns over N days under the null hypothesis.

Hint 3

For Part 3, split the integral defining E[|Y - c|] into (-infinity, c) and (c, infinity), apply Leibniz's rule to take the derivative with respect to c, and set it to zero.

Follow-up questions

  • In Part 1, under what exact F-statistic condition on x_2 would the t-statistic of x_1 decrease instead of increase when adding x_2?
  • In Part 2, how would serial autocorrelation (e.g., AR(1) with positive phi) or non-zero realized volatility during the flat period affect the required sample size N?
  • In Part 3, what is the optimal point estimator if the loss function is asymmetric, such as pinball loss L_tau(y, c) = (tau - I(y < c)) * (y - c)?