# 7.2 Modeling time series of return-based rankings of currencies

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This report summarizes work done as part of the Computational Finance: Ordinal Time Series PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed round the study of a common problem. In this module the author presents an ordinal time series model for return based weekly rankings of a set of currency instruments. A logistic proportional odds model is used to model periodic and non periodic transition probability functions for each currency. The models are evaluated by examining one step predictions based on estimated probabilities.

## 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, $\mathbb{S}=\left\{{r}_{1},{r}_{2},...,{r}_{J}\right\}$ . 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)
• 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, $\mathbb{S}=\left\{1,2,...,10\right\}$ , 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 , $X=\left({x}_{1},{x}_{2},...{x}_{k}\right)$ , 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.

${Y}_{t}={\phi }_{0}+{\phi }_{1}{Y}_{t-1}+{\phi }_{2}{Y}_{t-2}+{ϵ}_{t}$

If we estimate φ 0 , φ 1 , and φ 2 from observed data and predict ${Y}_{t+1}$ by

${\stackrel{^}{y}}_{t+1}=E\left({Y}_{t+1}\mid {\mathcal{F}}_{t}\right)={\phi }_{0}+{\phi }_{1}{y}_{t}+{\phi }_{2}{y}_{t-1}$
${\stackrel{^}{ϵ}}_{t+1}={y}_{t+1}-{\stackrel{^}{y}}_{t+1}$

Thereś no restriction on ${\stackrel{^}{y}}_{t+1}$ from taking a value outside the set $\left\{0,1,2\right\}$ . Further ${\stackrel{^}{ϵ}}_{t+1}$ can only take one of three values: $-{y}_{t+1}-{\stackrel{^}{y}}_{t+1}$ , $1-{y}_{t+1}-{\stackrel{^}{y}}_{t+1}$ , or $2-{y}_{t+1}-{\stackrel{^}{y}}_{t+1}$ ; it can be shown that the ${\stackrel{^}{\sigma }}_{ϵ}$ 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.

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