Winsorization Impact on OLS

Worked Solution

How to Think About It: The mixture model here is designed to produce fat tails -- 99% of the time returns are well-behaved $N(0,1)$, but 1% of the time you get draws from $N(0,25)$, which has standard deviation $5$. Winsorization at $c = 3$ clips extreme values, replacing anything beyond $\pm 3$ with $\pm 3$. For the $N(0,1)$ component, $|R| > 3$ is a roughly $0.3\%$ event -- rare, barely affected. For the $N(0,25)$ component, $|R| > 3$ happens about $45\%$ of the time -- winsorization is doing heavy lifting on the fat tail. The key economic insight: the mixture is symmetric around zero, so winsorization does not change the mean (it stays zero), but it dramatically reduces the variance by chopping off the heavy tails.

Quick Estimate: The unconditional variance of $R$ is $0.99 \times 1 + 0.01 \times 25 = 1.24$. After winsorization, the $N(0,1)$ component barely changes (its contribution stays near $1$). The $N(0,25)$ component, however, gets truncated hard at $\pm 3$. The truncated second moment of $N(0,25)$ inside $[-3,3]$ is roughly $E[R^2 \mid |R| \le 3] \approx 3.5$ (since $3/5 = 0.6$ in standardized units), and the mass beyond $\pm 3$ contributes $9 \cdot P(|R|>3) \approx 9 \times 0.45 = 4.05$. So the second moment from the heavy-tail component is roughly $3.5 \times 0.55 + 4.05 \approx 5.98$, contributing $0.01 \times 5.98 \approx 0.06$ to the total. Winsorized variance is approximately $0.99 \times 1.0 + 0.01 \times 5.98 \approx 1.05$, down from $1.24$ -- a meaningful reduction driven entirely by clipping the heavy-tail component.

Approach: We compute moments by conditioning on each mixture component and splitting the integral at $\pm c$.

Formal Solution:

Let $\sigma_1 = 1$, $\sigma_2 = 5$, $w_1 = 0.99$, $w_2 = 0.01$, and $c = 3$. Write the density of $R$ as

$$f_R(r) = w_1 \, \phi_1(r) + w_2 \, \phi_2(r)$$

where $\phi_j(r) = \frac{1}{\sigma_j} \phi\left(\frac{r}{\sigma_j}\right)$ and $\phi(\cdot)$ is the standard normal density.

Part (a): Mean and variance of $R^{(c)}$.

By symmetry of the mixture (both components have mean zero), $R$ is symmetric about zero. Winsorization preserves this symmetry: $R^{(c)}$ and $-R^{(c)}$ have the same distribution. Therefore

$$E[R^{(c)}] = 0$$

For the variance, since the mean is zero, $\text{Var}(R^{(c)}) = E[(R^{(c)})^2]$. We compute the second moment by conditioning on each component. For component $j$:

$$E[(R^{(c)})^2 \mid \text{component } j] = \int_{-c}^{c} r^2 \, \phi_j(r) \, dr \;+\; c^2 \, P_j(|R| > c)$$

where $P_j(|R| > c) = 2\left(1 - \Phi\left(\frac{c}{\sigma_j}\right)\right)$.

For the integral, substituting $z = r/\sigma_j$:

$$\int_{-c}^{c} r^2 \, \phi_j(r) \, dr = \sigma_j^2 \int_{-c/\sigma_j}^{c/\sigma_j} z^2 \, \phi(z) \, dz$$

Using the identity $\int_{-a}^{a} z^2 \phi(z) \, dz = 2\Phi(a) - 1 - 2a\phi(a)$ (from integration by parts), with $a_j = c/\sigma_j$:

$$\int_{-c}^{c} r^2 \, \phi_j(r) \, dr = \sigma_j^2 \left[2\Phi(a_j) - 1 - 2a_j \phi(a_j)\right]$$

Putting it together:

$$E[(R^{(c)})^2 \mid \text{comp } j] = \sigma_j^2 \left[2\Phi(a_j) - 1 - 2a_j \phi(a_j)\right] + c^2 \cdot 2\left[1 - \Phi(a_j)\right]$$

where $a_j = c / \sigma_j$. For $j=1$: $a_1 = 3$; for $j=2$: $a_2 = 3/5 = 0.6$.

The total variance is:

$$\text{Var}(R^{(c)}) = \sum_{j=1}^{2} w_j \left\{ \sigma_j^2 \left[2\Phi(a_j) - 1 - 2a_j\phi(a_j)\right] + 2c^2\left[1 - \Phi(a_j)\right] \right\}$$

Part (b): Qualitative effects.

Effect on estimated Sharpe ratio: The Sharpe ratio is $\hat{\mu}/\hat{\sigma}$. Winsorization does not change the population mean (it is zero here by symmetry), so $\hat{\mu}$ remains centered at $0$ -- and therefore the estimated Sharpe stays essentially zero and does not meaningfully rise. Shrinking the denominator cannot systematically lift a ratio whose numerator is centered at zero; it merely rescales its sampling noise (for $\mu = 0$ the sampling variance of $\hat{\mu}/\hat{\sigma}$ is $\approx 1/n$ with or without winsorization). The real gain is elsewhere: the variance estimate itself becomes far more stable across subsamples -- without winsorization, a single $5\sigma$ draw from the heavy-tail component can blow up $\hat{\sigma}^2$ (in simulation, winsorizing cuts the sampling standard deviation of $\hat{\sigma}^2$ by roughly a factor of $3$), making the volatility estimate -- and hence any Sharpe built on it -- noisy and unreliable.

Effect on OLS stability: When $R$ is the dependent variable, OLS estimates are $\hat{\beta} = (X^T X)^{-1} X^T R$. Extreme values of $R$ have high leverage on the residuals and inflate the residual variance, which makes $\hat{\beta}$ noisy. The heavy-tail component generates rare but extreme $R$ values that dominate the sum of squared residuals, producing large variance in $\hat{\beta}$ across samples. Winsorization clips these extremes, which: (1) reduces the variance of $\hat{\beta}$ (more stable coefficient estimates); (2) reduces the influence of outlying observations (robust-ifies the fit); and (3) shrinks confidence intervals for $\hat{\beta}$. The trade-off: winsorization introduces bias in $\hat{\beta}$ because it distorts the conditional mean $E[R^{(c)} \mid X] \neq E[R \mid X]$ when the true relationship generates extreme fitted values. In practice, for fat-tailed returns, this bias-variance trade-off almost always favors winsorization -- you lose a tiny bit of accuracy in modeling the tails but gain much more in estimation stability.

Answer:

(a) $E[R^{(c)}] = 0$ by symmetry. The variance is:

$$\text{Var}(R^{(c)}) = \sum_{j=1}^{2} w_j \left\{ \sigma_j^2 \left[2\Phi(a_j) - 1 - 2a_j\phi(a_j)\right] + 2c^2\left[1 - \Phi(a_j)\right] \right\}$$

with $a_1 = 3$, $a_2 = 0.6$, $w_1 = 0.99$, $w_2 = 0.01$, $\sigma_1 = 1$, $\sigma_2 = 5$, $c = 3$.

(b) Winsorization leaves the estimated Sharpe ratio essentially at zero -- by symmetry $\hat{\mu}$ stays centered at $0$, so $\hat{\mu}/\hat{\sigma}$ does not meaningfully rise; its real benefit is a much smaller and far more stable variance estimate (and hence a more reliable Sharpe estimate). It also stabilizes OLS coefficient estimates by clipping extreme residuals, at the cost of introducing mild bias from distorting the conditional mean in the tails.