7.2 Modeling time series of return-based rankings of currencies

Introduction

      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.

Data

      The data is weekly rankings of the following 10 globally traded currency exchange rates.

• AUD: Australian Dollar (AUD/USD)
• CAD: Canadian Dollar(USD/CAD)
• CHF: Swiss Franc (USD/CHF)
• DKK: Danish Krone (USD/DKK)
• EUR: Euro (EUR/USD)
• GBP: British Pound (GPB/USD)
• NOK: Norwegian Krone (USD/NOK)
• NZD: New Zealand Dollar (NZD/USD)
• JPY: Japanese Yen (USD/JPY)
• SEK: Swedish Krona (USD/SEK)

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.

Objectives

      We are interested in the following:

• the stability of a ranking Y _k over time. The stability of Y _k will be measured by determining the expected duration Y _k remains in each of the J ordinal categories.
• the lagged dependence of $Y_{k,t}$ on $Y_{k,t-h}$ and expected ranking of $Y_{k,t+h}$ given $Y_{k,t}$

Model

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 ${\widehat{ϵ}}_{t+1}$ 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 ${\widehat{\sigma }}_{ϵ}$ 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.