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 unknown parameter of interest be $\mu$, and assume we have a statistic $m$ such that the expected value of m is μ: $\mathbb{E}\left[m\right]=\mu$, i.e. m is an unbiased estimator for μ. 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 $\mu$ 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

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

is the correlation coefficient of m and t. The greater the value of $\vert\rho_{m,t}\vert$, the greater the variance reduction achieved.

In the case that $\textrm{Cov}\left(m,t\right)$, $\textrm{Var}\left(t\right)$, and/or $\rho_{m,t}\;$ 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

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

using Monte Carlo integration. This integral is 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

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

Now we introduce $g(x) = 1+x$ as a control variate with a known expected value $\mathbb{E}\left[g\left(U\right)\right]=\int_0^1 (1+x) \, \mathrm{d}x=\frac{3}{2}$ and combine the two into a new estimate

$I \approx \frac{1}{n} \sum_i f(u_i)+c\left(\frac{1}{n}\sum_i g(u_i) -3/2\right).$

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. (The exact result is $I=\ln 2 \approx 0.69314718$.)

See also

• Antithetic variates
• Importance sampling

Notes

1. Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. New York: Springer. ISBN 0-387-00451-3 (p. 185)

References

• Ross, Sheldon M. (2002) Simulation 3rd edition ISBN 978-0-12-598053-1
• 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 978-0-521-88441-9. Downloadable draft (Section 11.4: Control variates and shadow functions)