7.2 Modeling time series of return-based rankings of currencies  (Page 2/4)

To investigate the stability of a ranking and the expected ranking, one approach is to first estimate the the state transition probabilities of $Y_{k,t}$ , $t=1,\cdots ,T$ . For example if T is a stopping time such that

and P is a time homogeneous transition probability function for $Y_{k,t}$ then the expected duration of state j can be determined as follows [7]:

On any week however, the ranking of a given currency, k , in our set is directly dependent on return/ranking of the the rest of the set. We should consider the process ${\mathbf{Y}}_t=(Y_{1,t},...,Y_{k,t})$ that is a permutation of S at time t . There's little information in literature for modeling Y _t directly. Its more common to assume that conditional on some underlying process $Y_{k,t}$ independent of $Y_{j,t}$ , an assumption though convenient may, in the case of return based permutation rankings, engender ranking estimates with ties, an event with low probability of occurrence (two currencies having the same weekly return). Following this approach, I make the simplifying assumption that conditional on X the weekly historic volatility of all currencies in our set, $Y_{k_1,t}$ is independent of $Y_{k_2,t}$ for any pair of currencies k ₁ and k ₂ . The state space model:

• ${\xi }_{k,t}$ The underlying continuous variable. In a practical implementation, ${\xi }_{k,t}$ should be $r_{k,t}$ the corresponding return series, in which case the model would be:
  $$r_{k,t}={\beta }_{k,t}^T{\mathbf{R}}_{t-1,l_k}+{\gamma }_{k,t}{\mathbf{X}}_{t-1,l_k}$$
  where R with $R_{i,k}=r_{k,t-i}$ , $i\le i\le l_k$ , $1\le k\le 10$
  $$Y_{k,t}=j⟺{\alpha }_{j-1}\le {\xi }_{k,t}<{\alpha }_j$$
  But in work done up to this point we assume the more general case (often encountered in non financial settings) that the underlying continuous variable is not observed.
• ${\mathbf{y}}_{k,t-1,l_k}$ Observed past weekly rankings from time $t-1$ to l _k of currency k . Including past rankings of other currencies as predictors for $x{i}_{k,t}$ yields a matrix of predictors that is non-invertable ( ${\mathbf{y}}_{k,t-1,l_k}$ is implemented as a binary matrix with $Y_{k,t,j}$ as entries.
• ${\mathbf{X}}_{t-1,l_k}$ Matrix of weekly historic volatilities of all J instruments ${\mathbf{X}}_{i,k}={\sigma }_{k,t-i}$
• ${ϵ}_{k,t}\sim Logistic(0,1)$ The $Logistic(0,1)$ has thicker tails than the $Normal(0,1)$ distribution; thus the logistic is more appropriate for capturing the probability of extreme rankings 1 and 10, in our case.
• ${\alpha }_{k,t}$ , ${\beta }_{k,t}$ , ${\gamma }_{k,t}$ Parameters to be estimated.

In the study I considered three approaches to modeling the state probabilities, $P(Y_{k,t}\mid {\mathcal{F}}_{t-1})$ .

1. Periodic Markov model Model $P(Y_{k,t}\mid {\mathcal{F}}_{t-1})$ as a time-heterogeneous Markov chain of order l _k , i.e, the transition probability of $Y_{k,t}$ conditional on the past, depends only on the l _k previous states of Y _k . Though heterogeneous, $P(Y_{k,t}\mid {\mathcal{F}}_{t-1})$ is modeled as a periodic function of time with period h _k . This is approach was adopted by Menard et. al [3]. If it can be shown $P(Y_{k,t}\mid {\mathcal{F}}_{t-1})$ has period h _k and if we observed the data over N consecutive periods with $T=h_{k}N$ being total number of observations, then
   $$P(Y_{k,t}=j\mid Y_{k,t-1}=i_{t-1},...,Y_{k,1}=i_1)=f(t;{\mathbf{y}}_{k,t-1,l_k};{\theta }_k)$$
   such that,
   $$f(t+h_k;j,{\mathbf{y}}_{k,t-1,l_k};\theta )=f(t;j,{\mathbf{y}}_{k,t-1,l_k};{\theta }_k)$$
   Given $t=s+h_{k}n$ , and letting $Y_{k,t}=Y_{k,s,n}=Y_{k,s+h_{k}n}$ ( $1\le s\le h_k$ ; $0\le n\le N$ ) be response on week s of period n , then process ${\mathbf{Y}}_{k,n}^{*}=(Y_{k,1,n},...,Y_{k,h_k,n})$ is a time homogeneous Markov chain and under condition that $f(s;j,{\mathbf{y}}_{k,t-1,l_k};{\theta }_k)>0$ for all $s=1,...,h_k$ , $j\in {\mathbb{S}}_t$ and ${\mathbf{y}}_{k,t-1,l_k}\in {\mathbb{S}}_{t-1}\times \cdots \times {\mathbb{S}}_{t-l_k}$ , ${\mathbf{Y}}_k^{*}$ is ergodic. A time homogeneous, recurrent, and ergodic process has a limiting distribution. It follows that the MLEs are consistent and asymptotically normal and the likelihood ratio tests also have the desired asymptotic distribution see [3], [6].
2. Non periodic proportional odds model Model $P(Y_{k,t}\mid {\mathcal{F}}_{t-1})$ as a time heterogeneous Markov chain of order l _k with no periodicity, ergodicity, or recurrence assumptions.
   $$\begin{array}{ccc}\hfill P(Y_{k,t}=j\mid {\mathcal{F}}_{t-1})& =& P({\alpha }_{j-1}\le {\xi }_{k,t}\le {\alpha }_j)\hfill \\ & =& F({\alpha }_j-({\beta }_{k,t}^T{\mathbf{y}}_{k,t-1,l_k}+{\gamma }_k{\mathbf{X}}_{t-1,l_k}))\hfill \\ & & -F({\alpha }_{j-1}-({\beta }_{k,t}^T{\mathbf{y}}_{k,t-1,l_k}+{\gamma }_k{\mathbf{X}}_{t-1,l_k}))\hfill \end{array}$$
   In this case, inference on the parameters α , β , γ , is justified since the maximum partial likelihood estimators of the parameters are consistent and asymptotically normal under reasonable assumptions (see section 3.3.3 of [8]).