In that task we need to calculate the variance ratios for q=2 using overlapping qth differences. The variance ratio test is straightforward and often a powerful for detecting departures from randomness. Variance ratio tests examine the ratio between return variances for time intervals of different lengths. Just as implied by all versions of the random walk hypothesis, the increment variance is linear in the observation interval. The variance ratio for a q-period variance is given by:

VR(q) = var(rt(q))q*var(rt)

This is the sample variance ratio

Vr(q)=σc2(q)σa2

To find VR(q) it is necessary to obtain σa2 - variance of one period return and σc2-1/q*variance of overlapping q period returns. σa2 and σc2 can be calculated

σc2 = 1mt=qT(rtq-q*μ)2 ,

where μ is a mean of rt

m=q*(T-q+1)(1- qT)

The results got in Excel are following

Column1 | Column2 |

q | 2 |

m | 1384 |

σa2 | 0,000710204 |

σc2 | 0,000723739 |

overlapping VR | 1,019058043 |

The plausibility of a random walk model may be checked by comparing the variance of rt+rt-1 to twice the variance of rt. In practice they will not be numerically identical but their ratio should be statistically indistinguishable from one. The first random walk hypothesis is the strongest version, which states that price changes are independently identically distributed:

Pt=µ+pt-1+εt, εt~iid (0,σ2)

The µ in the equation is the drift term of the returns. The random walks first hypothesis is restrictive in that its returns have to be both independent and uncorrelated. When the RWH1 is true the returns process is uncorrelated and hence the best linear prediction of a future return is its unconditional mean, which RWH1 assumes is a constant. RWH1 implies that the mean squared forecast error is minimised by the constant predictor.

To provide some intuition for the test, initially suppose that the stochastic process generating returns is stationary, with V(1)=Var(rt). As q=2 we looked at the...