TITLE: Applications of Stochastic Control and Statistical Inference in Macroeconomics and High-Dimensional Data

ABTSRACT:

This thesis is focused on the optimality of stochastic control in macroeconomics and the fast algorithm of statistical inference. The first topic involves the proof and the calculation of the optimal drift control policy in foreign exchange reserve management. The second topic involves the fast computing algorithm of partial distance covariance statistics with its application in feature screening in high dimensional data.

In the first part of the dissertation, we study the problem of optimally controlling the level of foreign exchange reserve held by a country. When a reserve authority accumulates foreign exchange reserves to meet changing economic conditions, he faces the challenge of finding the right balance between the holding costs and the operational costs involved in adjusting the reserve size. We consider a foreign exchange reserve whose inventory fluctuation is modeled by a Brownian motion with drift, and at any moment the reserve manager can adjust the inventory level by varying the drift rate at which the reserve accumulates or depletes, but incurs a cost satisfies triangle inequality. When the reserve is accumulating or depleting, it also incurs a maintaining cost related with the current drift rate. The inventory level must be nonnegative at all times and continuously incurs a linear holding cost. The manager's problem is to decide when and how to change the drift rate so that the long run expected discounted cost of maintaining the foreign exchange reserve is minimized. We show that, under certain conditions, the control band policies are optimal for the discounted cost drift control problem and explicitly calculate the parameters of the optimal control band policy. In the two drift case, this form of policy is described by two parameters $\{L, U\}$, $0 < L < U$. When the inventory falls to $L$ (rises to $U$), the controller switch the drift rate to depletion (accumulation). We also extend the result to the multiple drift case and develop an algorithm to calculate the optimal thresholds of the optimal control band policy.

In the second part of the dissertation we study the problem of fast computing algorithm of partial distance covariance. If the computation of partial distance covariance is implemented directly accordingly to its definition then its computational complexity is $O(n^{2})$ which may hinder the application of an algorithm. To illustrate it, if $n$ is equal to $10^{^6}$, an $O(n^{2})$ algorithm will need $10^{12}$ numerical operations, which is impossible even for modern computers. In comparison, an $O(n \log n)$ algorithm will only require around $10^{6}$ numerical operations, which is doable. In this part of the thesis, we show that an $O(n \log n)$ algorithm for a version of the partial distance covariance exits. The derivation of the fast algorithm involves significant reformulation from the original version of partial distance covariance. We also demonstrate its application in feature screening in high dimensional data in the following part of the thesis.

In the final part of the thesis we further study the feature screening problem in high dimensional data. We propose an iterative feature screening procedure based on the partial distance covariance. This procedure can simultaneously address the two issue when using sure independence screening (SIS) procedure. First, an important predictor that is marginally uncorrelated but jointly correlated with the response cannot be picked by SIS and thus will not enter the estimation model. Second, SIS works only for linear models, and performance is very unstable in other nonlinear models. To the best of our knowledge, this is the first time that the ``new metric'' -- partial distance covariance -- is used for feature screening, and the idea of conditional screening is formally developed.