In pinbasic: Fast and Stable Estimation of the Probability of Informed Trading (PIN)

HAC Algorithm

Another methodology which utilizes hierarchical agglomerative clustering (HAC) to generate starting values is proposed by @Cluster. The daily order imbalance $\orderimb, \daysym = 1, \dots, \totaldays$, serves as criterion to assign the trading days to three clusters representing no-news, good-news and bad-news trading days. HAC is a bottom-up clustering technique in which at the beginning of the algorithm all order imbalances $\orderimb$ illustrate a cluster of their own, e.g. if trading data for one quarter of a year is used to estimate the probability of informed trading roughly 60 clusters exist when the algorithm is initialized.

@Cluster use the complete-linkage clustering to sequentially merge the small clusters to bigger ones. Two clusters with the shortest distance are combined in each step. The definition of shortest distance distinguishes between several available agglomerative clustering methods.^[ @Cluster mention that they tested different agglomerative clustering methods but that the complete-linkage method performed marginally better than the others. For a description of the other available methods, e.g. single-linkage or centroid-linkage, see @ClusterAna.] In complete-linkage clustering or farthest-neighbour clustering, the distance between clusters is calculated as the distance between those two elements, whereat the elements are in separated clusters, that are farthest away from each other. The minimal computed distance in each step causes the merging of both clusters involved.

To be precise, in the complete-linkage clustering, the distance $\operatorname{D}(X, Y)$ between two clusters $X$ and $Y$ can be written as $$ \begin{align} \operatorname{D}(X,Y) = \underset{x \in X, y \in Y}{\max} d(x, y), \end{align} $$ where $d(x, y)$ is the distance between the cluster elements $x \in X$ and $y \in Y$. @Cluster use the euclidean norm as measure for $d(x,y)$.

The following is a step-by-step instruction how to use the clustering algorithm to generate initial values for the parameters in the EHO model [see @Cluster, p. 1809].

1. Calculate a series of daily order imbalances, $\orderimb = B_{\daysym} - S_{\daysym}$ with $\daysym = 1, \dots \totaldays$, and use the daily order imbalances, buys and sells as inputs for the following steps.
2. Perform HAC on the daily order imbalances using the complete-linkage clustering.^[ According to @Cluster we use the R function hclust to perform this task [see @fastcluster].] Stop the algorithm when there are three clusters left.
3. The mean of the clusters serve as a criterion to assign them to no-news, good-news and bad-news. The cluster with the highest mean is assumed to consists of trading days with positive private information. Likewise the cluster with the lowest mean adheres trading days with negative private information. The remaining cluster is then defined as the no-news cluster.
4. Compute the average daily buys $\bar{B}_c$ and sells $\bar{S}_c$ for $c \in \statesset$. Then, assign each cluster a weight $w_c$ which is calculated as the proportion this cluster occupies of the total number of trading days $\totaldays$. Hence, the cluster weights sum up to 1.
5. With the help of the classification of trading days and the cluster weights from the third and fourth step, we are able to compute initial values for the intensities of uninformed buys and sells as weighted sums of average buys $B_c$ and sells $S_c$, respectively. $$ \begin{align} \intensuninfbuys^0 &= \dfrac{w_{\badnews}}{w_{\badnews} + w_{\nonews}} \bar{B}{\badnews} + \dfrac{w{\nonews}}{w_{\badnews} + w_{\nonews}} \bar{B}{\nonews} \notag \ \intensuninfsells^0 &= \dfrac{w{\goodnews}}{w_{\goodnews} + w_{\nonews}} \bar{S}{\badnews} + \dfrac{w{\nonews}}{w_{\goodnews} + w_{\nonews}} \bar{S}_{\nonews} \notag \end{align} $$
6. The intensity of informed trading is then calculated as the weighted sum of the intensities of informed buys $\intensinf_b^0$ and sells $\intensinf_s^0$.^[ A splitting of the intensity of informed trading is not present in the EHO model. However, one could extend the existing model with this feature and already had a suitable technique for generating initial values.] ^[ It is not ensured that $\intensinf_b^0$ and $\intensinf_s^0$ are positive. Hence, if $\intensinf_b^0$ or $\intensinf_s^0$ are negative we set them to 0.] $$ \begin{align} \intensinf_b^0 &= \bar{B}{\goodnews} - \intensuninfbuys^0 \notag \ \intensinf_s^0 &= \bar{S}{\badnews} - \intensuninfsells^0 \notag \ \intensinf^0 &= \dfrac{w_{\goodnews}}{w_{\goodnews} + w_{\badnews}} \intensinf_b^0 + \dfrac{w_{\badnews}}{w_{\goodnews} + w_{\badnews}} \intensinf_s^0 \notag \end{align} $$
7. Cluster sizes are utilized to compute starting values for the probability of an information event $\probinfevent$ and the probability of bad news given that private information enter the market $\probbadnews$. $$ \begin{align} \probinfevent^0 &= w_{\goodnews} + w_{\badnews} \notag \notag \ \probbadnews^0 &= \dfrac{w_{\badnews}}{\probinfevent^0} \notag \end{align} $$
8. An initial estimate of the probability of informed trading is given by $$ \begin{align} \pintext = \dfrac{\probinfevent^0 \intensinf^0}{% \intensuninfbuys^0 + \intensuninfsells^0 + \probinfevent^0 \intensinf^0} \notag \end{align} $$ In contrast to the brute force grid search method discussed in the previous section, the HAC algorithm returns only a single vector of initial values. Furthermore, no computation time is spent in generating infeasible sets of values which are then immediately discarded.

pinbasic documentation built on May 2, 2019, 2:07 a.m.