Control variates

The control variates method is a variance reduction technique used in Monte Carlo methods. It exploits information about the errors in estimates of known quantities to reduce the error of an estimate of an unknown quantity.^[1]

Underlying principle

Let the parameter of interest be μ, and assume we have a statistic m such that $\mathbb{E}\left[m\right]=\mu$. Suppose we calculate another statistic t such that $\mathbb{E}\left[t\right]=\tau$ is a known value. Then

$m^\star = m + c\left(t-\tau\right) \,$

is also an unbiased estimator for μ for any choice of the coefficient c. The variance of the resulting estimator $m^{\star}$ is

$\textrm{Var}\left(m^{\star}\right)=\textrm{Var}\left(m\right) + c^2\,\textrm{Var}\left(t\right) + 2c\,\textrm{Cov}\left(m,t\right);$

It can be shown that choosing the optimal coefficient

$c^\star = - \frac{\textrm{Cov}\left(m,t\right)}{\textrm{Var}\left(t\right)};$

minimizes the variance of $m^{\star}$, and that with this choice,

$\begin{align} \textrm{Var}\left(m^{\star}\right) & =\textrm{Var}\left(m\right) - \frac{\left[\textrm{Cov}\left(m,t\right)\right]^2}{\textrm{Var}\left(t\right)} \\ & = \left(1-\rho_{m,t}^2\right)\textrm{Var}\left(m\right); \end{align}$

where

$\rho_{m,t}=\textrm{Corr}\left(m,t\right); \,$

hence, the term variance reduction. The greater the value of $\vert\rho_{mt}\vert$, the greater the variance reduction achieved.

In the case that $\textrm{Cov}\left(m,t\right)$, $\textrm{Var}\left(t\right)$, and/or ρ_mt are unknown, they can be estimated across the Monte Carlo replicates. This is equivalent to solving a certain least squares system; therefore this technique is also known as regression sampling.

Example

We would like to estimate

$I = \int_0^1 \frac{1}{1+x} \, \mathrm{d}x.$

The exact result is $I=\ln 2 \approx 0.69314718$. Using Monte Carlo integration, this integral can be seen as the expected value of f(U), where

$f(x) = \frac{1}{1+x}$

and U follows a uniform distribution [0, 1]. Using a sample of size n denote the points in the sample as $u_1, \cdots, u_n$. Then the estimate is given by

$I \approx \frac{1}{n} \sum_i f(u_i);$

If we introduce $T=\int_0^1 1+x \, \mathrm{d}x.$ as a control variate with a known expected value $\textrm{E}\left[T\left(U\right)\right]=\frac{3}{2}$

Using n = 1500 realizations and an estimated optimal coefficient $c^\star \approx 0.4773$ we obtain the following results

	Estimate	Variance
Classical estimate	0.69475	0.01947
Control variates	0.69295	0.00060

The variance was significantly reduced after using the control variates technique.

See also

• Antithetic variates
• Importance sampling

Notes

1. ^ Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. New York: Springer. ISBN 0387004513 (p. 185)

References

• Ross, Sheldon M. (2002) Simulation 3rd edition ISBN 978-0125980531
• Averill M. Law & W. David Kelton (2000), Simulation Modeling and Analysis, 3rd edition. ISBN 0-07-116537-1
• S. P. Meyn (2007) Control Techniques for Complex Networks, Cambridge University Press. ISBN 9780521884419. Downloadable draft (Section 11.4: Control variates and shadow functions)