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,

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

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 ${\displaystyle \mathbb {E} \left[g\left(U\right)\right]=\int _{0}^{1}(1+x)\,\mathrm {d} x={\tfrac {3}{2}}}$ 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)