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

• ${\sum }_{j=1}^{10}\widehat{P}(Y_{k,t}=j|{\mathcal{F}}_{t-1})=1$
• $\widehat{P}(Y_{k,t}=j|{\mathcal{F}}_{t-1})>0$ .

are not met for all instruments and for all t . Figure 7 compares the periodic and nonperiodic transition probabilities for the YEN. For to simplify the plot only the first 6 rankings are shown over a period of 50 weeks.

To evaluate the accuracy of the estimated onestep transition probabilities, define the following rank predictors

• UL $G_{k,t}=\underset{j}{argmax}\left({\widehat{P}}_{k,j,t}\right)$
• Per $G_{k,t}=\underset{j}{argmax}\left({\widehat{\mathbf{P}}}_{h_k},(j,t)\right)$
• ML Let $C_{jt}=\left\{{\widehat{P}}_{k,j,t}\right\}$ and $C_{jt}-k=C_{jt}\setminus {\widehat{P}}_{k,j,t}$ . for $j=1$ to 10 $k_j=\underset{k}{argmax}\left(C_{jt}\right)$ $if($ currency k corresponds to k _j ) THEN $G_{k,t}=j$ $C_{j+1,t}=C_{j+1,t}\setminus \{k_1,\cdots ,k_{j-1}\}$ End

UL classifies the week t ranking of currency k to the ranking category with the highest week t odds based on the non periodic estimated probabilities, ${\widehat{P}}_{k,j,t}$ . Per assigns currency k's week t ranking to the category with the highest week t odds but based on the periodic estimated probabilities, ${\widehat{\mathbf{P}}}_{h_k}(j,t)$ . ML predicts currency k to ranking j at week t if k is unranked at week t and has highest week t probability of j amongst all currencies unranked at week t . One step predictions were compared against true rankings from week 27 to week 126; results from week 27 to 56 (first 30 weeks) are shown in Figure 8. Absolute error estimates for the ML and UL estimators were obtained by non parametric bootstrap for currency k, as follows:

1. Sample 100 ranking and volatility observations (with replacement) from k's ranking and volatility series from week 27 to 126, treating each resulting sample as a time series with respect to the ordering observations are sampled.
2. Estimate onestep transition probability ${\tilde{P}}_{k,j,t}$ based on sampled rankings.
3. Obtain predictions (UL or ML) based on ${\tilde{P}}_{k,j,t}$ 's.
4. Calculate absolute error of each prediction made over the period.

Due to the computation time of each iteration, I was only able to repeated 50 iterations. Averaging the absolute errors accross the period for both estimators (UL and ML). Refer to Figure 5.

In summary, the UL estimator tends to have lower absolute prediction error than the ML estimator. For the period considered, prediction error varies by currency, with the CAD being most predictable (consistent across the 3 estimators) than the other currencies. The results also highlight 6 to 8 week periods in which Per and UL estimators fail to adjust with the true rankings. This results suggests model parameters for estimating one-step probabilities should be estimated from more recent history (instead of six month ranking history). Additionally, spline based regression functions should be considered to improve accuracy of the one step transition probability estimates.

Acknowledgements

The author would like to thank Dr. Kathy Ensor, Department of Statistics, Rice University for her valuable insights. This Connexions module describes work conducted as part of Rice University's VIGRE program, supported by National Science Foundation grant 0739420.

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