# Control variate

In Monte Carlo methods, one or more control variates may be employed to achieve variance reduction by exploiting the correlation between statistics.

## Example

Let the parameter of interest be ${\displaystyle \mu }$, and assume we have a statistic ${\displaystyle m}$ such that ${\displaystyle \mathbb {E} \left[m\right]=\mu }$. If we are able to find another statistic ${\displaystyle t}$ such that ${\displaystyle \mathbb {E} \left[t\right]=\tau }$ and ${\displaystyle \rho _{mt}={\textrm {corr}}\left[m,t\right]}$ are known values, then

${\displaystyle m^{\star }=m-c\left(t-\tau \right)}$

is also unbiased for ${\displaystyle \mu }$ for any choice of the constant ${\displaystyle c}$. It can be shown that choosing

${\displaystyle c={\frac {\sigma _{m}}{\sigma _{t}}}\rho _{mt}}$

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

${\displaystyle {\textrm {var}}\left[m^{\star }\right]=\left(1-\rho _{mt}^{2}\right){\textrm {var}}\left[m\right]}$;

hence, the term variance reduction. The greater the value of ${\displaystyle \vert \rho _{tm}\vert }$, the greater the variance reduction achieved.

In the case that ${\displaystyle \sigma _{m}}$, ${\displaystyle \sigma _{t}}$, and/or ${\displaystyle \rho _{mt}}$ 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.

## References

• Averill M. Law & W. David Kelton, Simulation Modeling and Analysis, 3rd edition, 2000, ISBN 0-07-116537-1