Optimality and robustness of combinations of moving averages

Abstract

A combination of moving averages has been shown previously to be more accurate than simple moving averages, under certain conditions, and to be more robust to non-optimal parameter specification. However, the use of the method depends on specification of three parameters: length of greater moving average, length of shorter moving average, and the weighting given to the former. In this paper, expressions are derived for the optimal values of the three parameters, under the conditions of a steady state model. These expressions reduce a three-parameter search to a single-parameter search. An expression is given for the variance of the sampling error of the optimal combination of moving averages and this is shown to be marginally greater than that for exponentially weighted moving averages (EWMA). Similar expressions for optimal parameters and the resultant variance are derived for equally weighted combinations. The sampling variance of the mean of such combinations is shown to be almost identical to the optimal general combination, thus simplifying the use of combinations further. It is demonstrated that equal weight combinations are more robust than EWMA to noise to signal ratios lower than expected, but less robust to noise to signal ratios higher than expected.

Appendix

Johnston et al¹ derived the following expression for the variance of the sampling error of the mean using a combination of moving averages:

Setting ∂C/∂f=0 gives:

Setting ∂C/∂s=0 gives:

Setting ∂C/∂g=0 gives:

Hence the conditions ∂C/∂s=∂C/∂g=∂C/∂f=0 yield the following equations:

Subtracting the equation (A1) from equation (A3)

To ensure that g>s,

Substituting into equation (A2),

From equation (A1),

Note: equation (13) can be confirmed form this result, by substituting N=g−s

Hence, substituting for f,

This gives three solutions, only one of which is greater than zero, namely:

Using the relationship g=s+N,

And using the expression for f derived above,

This solution also satisfies all the second-order minimisation criteria: ∂²C/∂s²>0, ∂²C/∂g²>0, (∂²C/∂s²)(∂²C/∂g²)>(∂²C/∂s∂g)².

In the case of equally weighted combinations, the conditions ∂C/∂s=∂C/∂g=0 yield the following equations:

Rearranging equation (A4) gives:

The roots of this equation are included in the roots of:

This gives six solutions, namely s=±[√(3/10)]N, s=±[√(5/4−√17/4)]N, and s=±[√(5/4+√17/4)]N. By definition, s must be positive and must not be greater than g. After applying these conditions, two roots remain: s=[√(3/10)]N, s=[√(5/4−√17/4)]N. However, the latter root does not satisfy the original simultaneous equations. Hence,

Substitution in the second simultaneous equation gives:

This solution also satisfies all the second-order minimisation criteria: ∂²C/∂s²>0, ∂²C/∂g²>0, (∂²C/∂s²)(∂²C/∂g²)>(∂²C/∂s∂g)².