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

Answer three written questions from a quantitative online assessment. Show your working for each one. ### Clarifying Questions - Part 1: Does the model include an intercept? If it does, is $x_2$ also orthogonal to the constant, meaning it is uncorrelated with $x_1$ in the sample? - Part 1: Is $t_1$ computed with the usual homoskedastic standard error? - Part 2: Is the Sharpe ratio of 8 annualized, and how many trading days per year should be assumed? - Part 2: What confidence level should trigger suspicion? - Part 2: May daily returns be treated as independent with a stable mean and volatility? - Part 3: May you assume that $E[|X|]$ is finite? ### Part 1 — t-Statistic After Adding an Orthogonal Regressor You fit ordinary least squares $y = a_1 x_1 + \varepsilon$ on $n$ observations. Let $t_1$ be the t-statistic of $\hat a_1$. You then add a new regressor $x_2$ that is orthogonal to $x_1$ and refit $y = a_1 x_1 + a_2 x_2 + \varepsilon$ by OLS. How does $t_1$ change? ```hint Split the statistic Write $t_1$ as the estimated coefficient divided by its standard error, and ask separately what orthogonality does to each piece. ``` #### What This Part Should Cover - What happens to the coefficient estimate $\hat a_1$. - What happens to the residual variance estimate and the degrees of freedom, and the exact condition that decides whether $t_1$ rises or falls. - How an intercept changes the orthogonality condition. ### Part 2 — When to Doubt a High-Sharpe Backtest A strategy's backtest shows a Sharpe ratio of 8. After it has run live for $k$ days, its cumulative return is 0. At what value of $k$ should you start to suspect the strategy? ```hint Match the horizons Convert the backtest figure to the frequency of the live observations, then consider how the mean and the standard deviation of a $k$-day cumulative return each grow with $k$. ``` #### What This Part Should Cover - Converting a Sharpe ratio between horizons, and the assumptions that conversion needs. - A test statistic for a zero realized return, and the resulting $k$ for a stated threshold. - Reasons a backtest Sharpe ratio this high may be unreliable in the first place. ### Part 3 — Estimator That Minimizes Expected Absolute Error A random variable $X$ has an unknown distribution, and you observe an iid sample $x_1, \dots, x_n$. Find the estimator $c$ that minimizes $E[|X - c|]$. ```hint Move c slightly Consider how $E[|X - c|]$ changes when $c$ increases by a small amount, in terms of how much probability lies on each side of $c$. ``` #### What This Part Should Cover - A derivation of the minimizer at the population level. - The sample estimator, including what happens when $n$ is even. - How the answer differs from the minimizer of squared error. ### What a Strong Answer Covers - Derivations that state their assumptions (intercept, annualization, independence) instead of quoting a result. - Exact conditions where the answer depends on the data, not a one-word answer. - A numerical answer to Part 2 as a function of the chosen threshold, plus a concrete value. - Awareness of the limits of each result: finite samples, fat-tailed or autocorrelated returns, and overfitting in backtests. ### Follow-up Questions 1. In Part 1, what happens to $t_1$ if $x_2$ is strongly correlated with $x_1$ instead of orthogonal to it? 2. How would you change the Part 2 threshold if the strategy were the best of many backtested candidates? 3. Which value of $c$ minimizes the expected loss if under-predictions cost twice as much per unit as over-predictions?