Bayesian Dynamic Factor Models - Department of Statistical Science ...

els by implementing sequential portfolio allocations based on forecast means and variances of FX
series.
5.1 Portfolio allocation rule
We base the following forecasting strategy and portfolio allocation rules. Suppose that we
initially have the first n = 880 observations of our FX data analyzed above. That is, in our FX data
set, we have the remaining time intervals, a total of 100 business days, as periods of investment
study. After observing the closing rates on the business day t − 1 (≥ n), one-step-ahead forecasts
for means and variances, denoted by (g t, Q t), of y t are computed via the MCMC based on draws
from one-step-ahead predictive posterior distribution using the available data, y 1:t−1. Resources
are allocated according to a vector of portfolio weights wt optimized by a specific allocation rule
described below. The realized portfolio return at time t is rt = w ′ ty t. The portfolio is reallocated
on each business day based on one-step-ahead forecasting computed via the MCMC given updated
data. We fix the total sum invested on each business day by restricting w ′ t1 = 1.
We use the traditional optimization technique for portfolio allocation originally developed by
Markowitz (1959). Given a fixed scalar return target m, we optimize the portfolio weights wt,
by minimizing the one-step ahead variance of returns among the restricted portfolios whose one-
step-ahead expectation is equal to m. Specifically, at time t, we minimize an ex-ante portfolio
variance w ′ tQtwt, subject to w ′ tgt = m, and w ′ t1 = 1. The solution of this quadratic minimization
problem is derived via Lagrange multipliers as w (m)
t = Q−1 t (atgt + bt1), where at = 1 ′ Q −1
t e, and
bt = −g ′ tQ −1
t e, where e = (1m − gt)/d, and d = (1 ′ Q −1
t 1)(g′ tQ −1
t gt) − (1 ′ Q −1
t gt) 2 . This optimal
portfolio is called as efficient frontier. In our analysis, we also consider a target-free minimum-
variance portfolio, whose solution is given by w∗ t = Q −1
t 1/(1′ Q −1
t 1). We implicitly assume that we
can freely reallocate the resources to arbitrary long or short positions across the currencies without
any transaction cost.
In addition to one-step-ahead prediction, we also examine the analysis from five-step-ahead
forecasts. Every five business days, the posterior predictive distribution of five-step horizons,
(y t, y t+1, . . . , y t+4), is computed via MCMC based on the available data y 1:t−1. This experiment
assumes a possible situation that investors allocate their resource every business day based on
weekly-updated forecasts.
5.2 Model comparisons
We consider the following three models from the class of LTDFM:
• LM-AF: Local means, and autoregressive factors.
• LM-IF: Local means, and time-independent factors (Φ = O).
16