The additional arguments for the mean and median methods are:
For the kernel mean estimator, the arguments meanBandwidth can be used to control the bandwidth of the
drift estimator and the preAverage argument, which can be used to control the pre-averaging horizon.
These arguments default to 300 and 5 respectively.
The following estimation methods can be specified in method:
Rolling window mean ("mean")
Estimates the spot drift by applying a rolling mean over returns.
$$
\hat{\mu_{t}} = \sum_{t = k}^{T} \textrm{mean} \left(r_{t-k : t} \right),
$$
where \(k\) is the argument periods.
Parameters:
periods how big the window for the estimation should be. The estimator will have periods NAs at the beginning of each trading day.
align alignment method for returns. Defaults to "left", which includes only past data, but other choices, "center" and "right" are available.
Warning: These values includes future data.
Outputs:
Rolling window median ("median")
Estimates the spot drift by applying a rolling mean over returns.
$$
\hat{\mu_{t}} = \sum_{t = k}^{T} \textrm{median} \left(r_{t-k : t} \right),
$$
where \(k\) is the argument periods.
Parameters:
periods How big the window for the estimation should be. The estimator will have periods NAs at the beginning of each trading day.
align Alignment method for returns. Defaults to "left", which includes only past data, but other choices, "center" and "right" are available.
These values includes FUTURE DATA, so beware!
Outputs:
kernel spot drift estimator ("kernel")
$$
dX_{t} = \mu_{t}dt + \sigma_{t}dW_{t} + dJ_{t},
$$
where \(\mu_{t}\), \(\sigma_{t}\), and \(J_{t}\) are the spot drift, the spot volatility, and a jump process respectively.
However, due to microstructure noise, the observed log-price is
$$
Y_{t} = X_{t} + \varepsilon_{t}
$$
In order robustify the results to the presence of market microstructure noise, the pre-averaged returns are used:
$$
\Delta_{i}^{n}\overline{Y} = \sum_{j=1}^{k_{n}-1}g_{j}^{n}\Delta_{i+j}^{n}Y,
$$
where \(g(\cdot)\) is a weighting function, \(min(x, 1-x)\), and \(k_{n}\) is the pre-averaging horizon.
The spot drift estimator is then:
$$
\hat{\bar{\mu}}_{t}^{n} = \sum_{i=1}^{n-k_{n}+2}K\left(\frac{t_{i-1}-t}{h_{n}}\right)\Delta_{i-1}^{n}\overline{Y},
$$
The kernel estimation method has the following parameters:
preAverage a positive integer denoting the length of pre-averaging window for the log-prices. Default is 5
meanBandwidth an integer denoting the bandwidth for the left-sided exponential kernel for the mean. Default is 300L
Outputs: