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A multivariate ordinal time series is a process ${\mathbf{Y}}_{\mathbf{t}}=\left\{{Y}_{k,t}\right\}$ $t=1,...,T$ , $k=1,...,p$ , where ${Y}_{k,t}$ takes a value in the set of ordered categories, $\mathbb{S}=\{{r}_{1},{r}_{2},...,{r}_{J}\}$ . We are interested in identifying the underlying structure of the process: such as serial dependence, trends, and useful explanatory/exogenous variables.
The data is weekly rankings of the following 10 globally traded currency exchange rates.
The period is from Jan 3, 2000 to June 30, 2009. The rankings, $\mathbb{S}=\{1,2,...,10\}$ , are based on spot returns Rankings can also be based on other measures such as a volatility on a position of fixed lot size. Let ${Y}_{k,t}$ denote the week t ranking of exchange rate k . Along with the rankings, the underlying weekly return $\left\{{r}_{k,t}\right\}$ and and weekly historic volatility $\left\{{x}_{k,t}\right\}$ are included. The data is obtained from Bloomberg.
We are interested in the following:
For each ${Y}_{k,t}$ , the historic volatility of all k exchange rates , $X=({x}_{1},{x}_{2},...{x}_{k})$ , are used as exogenous variables. When analyzing ordinal series, one can not rely on the standard techniques for analyzing real-valued time series. For example, consider the following $AR\left(2\right)$ model for Y _{t} taking one of three values 0, 1, or 2.
If we estimate φ _{0} , φ _{1} , and φ _{2} from observed data and predict ${Y}_{t+1}$ by
Thereś no restriction on ${\widehat{y}}_{t+1}$ from taking a value outside the set $\{0,1,2\}$ . Further ${\widehat{\u03f5}}_{t+1}$ can only take one of three values: $-{y}_{t+1}-{\widehat{y}}_{t+1}$ , $1-{y}_{t+1}-{\widehat{y}}_{t+1}$ , or $2-{y}_{t+1}-{\widehat{y}}_{t+1}$ ; it can be shown that the ${\widehat{\sigma}}_{\u03f5}$ is not constant over time, violating the AR model assumptions.
One common approach ([1], Chapter 13 of [2], [3],[5]and [5]) to time domain analysis of an ordinal valued time series, Y _{t} is based on latent state variable models. Assume the existence of underlying state variable ξ _{t} and cut points a _{1} , a _{2} , . . ., ${a}_{J-1}$ such that probability ${Y}_{k,t}$ takes a certain category is determined by ${\xi}_{k,t}$ . In some cases, [3] and [5]for example, Markov assumptions are made on the distribution of ${\xi}_{k,t}$ . The distribution of ${\xi}_{k,t}$ is estimated by a generalized linear model and model parameters are estimated by maximum likelihood.
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