Identifying extreme values of exchange market pressure

Abstract

This paper contributes to the existing literature of currency crisis dating in a number of areas. Firstly, we combine the Monte Carlo simulation with a modified version of the Hill’s estimator to obtain robust results and deal with the bias–variance tradeoff in identifying extreme values. Secondly, to avoid sample- specific thresholds, we construct our results upon stationary series with the help of the Hill’s estimator. We also report the whole identified crisis episodes while disregarding the exclusion window technique, which may induce identification problems. To select the reference country when building the exchange market pressure, a statistical search between two exogenously given options is employed. Thirdly, different data frequencies are applied and the results are evaluated. Our findings suggest that higher frequency data are more appropriate when applying extreme value theory (EVT). Our results recommend researchers to be more cautious when applying EVT and interpreting tail incidences that are obtained from lower frequency data.

Appendix

To determine the optimum cutoff value by Monte Carlo simulations, we adopt the simulation steps from Longin and Solnik (2001) and Haile and Pozo (2008) and run them with R software. These steps are as follows:

• 1. $S$ time series containing $n$ observations of EMP are simulated. Each S is derived from a known Student’s $t$distribution with $\alpha$ degrees of freedom, where $\alpha$ ranges from 1 to $K.$The class of the Student’s $t$distributions chosen to consider different degrees of tail fatness. Since the tail index $\gamma$ is inverse of $\alpha$ ($\gamma$=1/$\alpha )$, the lower the degree of freedom is, the fatter the distribution will be. In our simulation, $\alpha$ is allowed to take values from 1 to 10 with increment of 0.1 and number of replications ($S)$ equals 1,000.

• 2. For different numbers of $m$ of the extreme EMPs, a tail index $\gamma _{s}^{h}(m$,$\alpha )$ corresponding to the $s-$th replication from the Student’s $t$ with $\alpha$ degrees of freedom is estimated. Values of $m$ can vary from %1 to %30 of $n,$ where $n$ is the sample size of the actual EMP data.

• 3. For a particular Student’s $t$ distribution with $\alpha$ degrees of freedom and for each value of $m$, MSE of the $S$tail index estimates, which is denoted by MSE ($\gamma _{s}^{h}(m$,$\alpha )_{s=1,2, }$...,$_{S})$, is computed. This computation repeatedly continues for different values of $m$ and particular Student’s $t$ with $\alpha$ degrees of freedom. Then the optimal $m$, denoted by $m^{*}(\alpha )$, which minimizes MSE for the particular Student’s $t$ distribution with $\alpha$ degrees of freedom is selected.$^{25}$ Optimum values of $m$ for different Student’s $t$ distributions are repeatedly selected. A total of $K$ optimal values of $m*, (m*(\alpha ))_{\alpha = 1,2, \ldots ,K},$ are subsequently selected for $K$ possible theoretical distributions.

• 4. Using each of $K$ optimum values of $m$ that are obtained in last step, the Hill index, $\gamma ^{h}(m*(\alpha )$, is estimated from actual EMP series. For all $\alpha$ from 1 to 10, the tail indices, $\gamma ^{h}$, are estimated from actual EMP series.

As the final step, we select one single number(say $m**)$ from the $K$optimum tail indices, $m*,$ for each EMP series such that the estimated tail index from the actual data (step 4) is statistically the closest to the corresponding tail index of the theoretical distribution. The main objective of the whole exercise is to determine the number of extreme observations for each EMP series. This value, $m**$, that is corresponding to the optimum tail index, specifies the number of observations as the largest EMPs or episodes of currency crises.