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Definition 8.3.4. Stationary in the Variance.
σ 2 (t) = σ 2 , a constant.
A time series is stationary in the variance if
This is where we need to be careful! With time series we are in a situation where sequential
observations may be correlated. This will have the effect of biasing the estimator, i.e. over- or
under-estimating the true population variance.
This will be particularly problematic in time series where we are short on data and thus only
have a small number of observations. In a high correlation series, such observations will be close
to each other and thus will lead to bias.
In practice, and particularly in high-frequency finance, we are often in a situation of having a
substantial number of observations. The drawback is that we often cannot assume that financial
series are truly stationary in the mean or stationary in the variance.
As we make progress with the section in the book on time series, and develop more sophisticated
models, we will address these issues in order to improve our forecasts and simulations.
We are now in a position to apply our time series definitions of mean and variance to that of
serial correlation.
8.4 Serial Correlation
The essence of serial correlation is that we wish to see how sequential observations in a time
series affect each other. If we can find structure in these observations then it will likely help
us improve our forecasts and simulation accuracy. This will lead to greater profitability in our
trading strategies or better risk management approaches.
Firstly, another definition. If we assume, as above, that we have a time series that is stationary
in the mean and stationary in the variance then we can talk about second order stationarity:
Definition 8.4.1. Second Order Stationary.
A time series is second order stationary if the
correlation between sequential observations is only a function of the lag, that is, the number of
time steps separating each sequential observation.
Finally, we are in a position to define serial covariance and serial correlation!
Definition 8.4.2. Autocovariance of a Time Series.
If a time series model is second order
stationary then the (population) serial covariance or autocovariance, of lag k, C k = E[(x t −
µ)(x t+k − µ)].
The autocovariance C k is not a function of time. This is because it involves an expectation
E(..), which, as before, is taken across the population ensemble of possible time series realisations.
This means it is the same for all times t.
This motivates the definition of serial correlation (autocorrelation) simply by dividing through
by the square of the spread of the series. This is possible because the time series is stationary in
the variance and thus σ 2 (t) = σ 2 .
Definition 8.4.3. Autocorrelation of a Time Series. The serial correlation or autocorrelation
of lag k, ρ k , of a second order stationary time series is given by the autocovariance of the
series normalised by the product of the spread. That is, ρ k = C k
σ 2 .
Note that ρ 0 = C0
σ
= E[(xt−µ)2 ]
2 σ
= σ2
2 σ 2
value of unity.
= 1. That is, the first lag of k = 0 will always give a