Marco tectónico

We have provided a formal diffusion-theoretic justification for measuring and analyzing time series of realized volatilities constructed from high-frequency intraday returns. Utilizing a unique data set consisting of ten-years of 5-minute DM/$ and Yen/$ returns, we find that the distributions of realized daily variances, standard deviations and covariances are skewed to the right and leptokurtic, but that the distributions of logarithmic standard deviations and correlations are approximately Gaussian.

Volatility movements, moreover, are highly correlated across the two exchange rates, as would be implied by a factor structure induced by common dependence on U.S. fundamentals. We also find that

the correlation between the exchange rates (as opposed to the correlation between their volatilities) increases with volatility, so that the benefits of portfolio diversification are reduced just when they are needed most. There is, however, no evidence that the correlation is related to the sign of the returns.

Finally, we confirm the wealth of existing evidence of strong volatility clustering effects in daily returns. However, in contrast to earlier work such as Christoffersen and Diebold (1998), which often indicates that volatility persistence decreases fairly quickly with the horizon, we find that even monthly realized volatilities remain highly persistent. But realized volatilities do not have unit roots; instead, they appear fractionally integrated and therefore very slowly mean-reverting. This finding is strengthened by our analysis of temporally aggregated volatility series, which appear governed by remarkably accurate scaling laws, as predicted by the structure of fractional integration.

A key conceptual distinction between this paper and the earlier work on which we build --Andersen and Bollerslev (1998a) in particular -- is the recognition that realized volatility is usefully viewed as the object of intrinsic interest, rather than simply a post-modeling device to be used for evaluating

parametric volatility models such as GARCH. As such, it is of interest to examine and model realized volatility directly. This paper is a first step in that direction, providing a nonparametric characterization of both the unconditional and conditional distributions of bivariate realized exchange rate volatility.

It will be of interest in future work to fit parametric models directly to realized volatility, and to use them for forecasting, risk management, asset allocation, and asset pricing. The results of this paper suggest modeling the joint conditional dependence in the realized logarithmic daily standard deviations by a standard linear Gaussian vector autoregressive (VAR) model, and that doing so may, for example, result in important improvements in the accuracy of long-term volatility forecasts. This idea is pursued in Andersen, Bollerslev, Diebold and Labys (1999).

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Table 1

Statistics Summarizing Unconditional Univariate Distributions Daily Realized DM/$ and Yen/$ Volatilities

______________________________________________________________________________

vard_t vary_t stdd_t stdy_t lstdd_t lstdy_t cov_t corr_t

______________________________________________________________________________

Mean 0.529 0.538 0.679 0.684 -0.450 -0.444 0.243 0.435 Variance 0.238 0.272 0.068 0.071 0.121 0.124 0.075 0.028 Skewness 3.750 5.438 1.696 1.852 0.348 0.283 3.876 -0.203 Kurtosis 24.53 63.61 7.867 10.13 3.265 3.470 26.33 2.716 Quantiles

Min. 0.061 0.029 0.247 0.171 -1.397 -1.767 -0.096 -0.244 0.001 0.068 0.044 0.261 0.209 -1.345 -1.568 -0.050 -0.108 0.005 0.084 0.067 0.290 0.260 -1.238 -1.349 -0.002 -0.009 0.01 0.094 0.084 0.306 0.290 -1.183 -1.238 0.008 0.033 0.05 0.136 0.138 0.368 0.371 -0.999 -0.992 0.034 0.152 0.10 0.169 0.176 0.411 0.420 -0.889 -0.869 0.052 0.214 0.50 0.397 0.399 0.630 0.632 -0.462 -0.459 0.166 0.437 0.90 0.985 1.012 0.992 1.006 -0.008 0.006 0.514 0.649 0.95 1.373 1.365 1.172 1.168 0.158 0.155 0.710 0.697 0.99 2.520 2.570 1.587 1.603 0.462 0.472 1.444 0.767 0.995 3.620 3.189 1.903 1.786 0.643 0.580 1.654 0.801 0.999 4.646 5.301 2.156 2.300 0.768 0.832 2.737 0.851 Max. 5.503 9.965 2.346 3.157 0.853 1.150 3.087 0.918

______________________________________________________________________________

Notes: The daily volatility measures are constructed by summation of the corresponding intraday 5-minute returns, as detailed in the text. The sample period is December 1, 1986 until December 1, 1996.

Table 2 Correlation Matrix

Daily Realized DM/$ and Yen/$ Volatilities

______________________________________________________________________________

vary_t stdd_t stdy_t lstdd_t lstdy_t cov_t corr_t

______________________________________________________________________________

vard_t 0.538 0.961 0.553 0.858 0.513 0.809 0.344

vary_t 1.000 0.547 0.946 0.515 0.828 0.760 0.238

stdd_t - 1.000 0.594 0.965 0.580 0.794 0.385

stdy_t - - 1.000 0.591 0.960 0.761 0.283

lstdd_t - - - 1.000 0.606 0.719 0.391

lstdy_t - - - - 1.000 0.683 0.295

cov_t - - - - - 1.000 0.587

______________________________________________________________________________

Notes: The daily volatility measures are constructed by summation of the corresponding intraday 5-minute returns, as detailed in the text. The sample period is December 1, 1986 until December 1, 1996.

Table 3

Dynamic Dependency Measures Daily Realized DM/$ and Yen/$ Volatilities

______________________________________________________________________________

vard_t vary_t stdd_t stdy_t lstdd_t lstdy_t cov_t corr_t

______________________________________________________________________________

LB 3199.0 3230.2 6314.7 5086.4 9051.1 6706.4 1125.8 12291 ADF -7.495 -9.083 -6.941 -8.008 -6.370 -7.817 -7.565 -5.589 ˆd 0.346 0.353 0.388 0.424 0.421 0.448 0.304 0.423

______________________________________________________________________________

Notes: The realized daily volatilities are constructed by the summation of the DM/$ and Yen/$ 5-minute returns from December 1, 1986 until December 1, 1996, as detailed in the text. The first row reports Ljung-Box tests using sample autocorrelations through displacement 20. The 0.001 critical value is 45.3. The second row reports Augmented Dickey-Fuller tests for unit roots in the autoregressive lag-operator polynomials of univariate ARMA representations, using ten

augmentation lags. The corresponding 0.01 and 0.05 critical values are -2.86 and -3.43. The third row reports the log-periodogram regression estimate for the fractional integration parameter, d. The estimates are based on the m = [T^4/5 ] = 514 lowest-frequency periodogram ordinates.

The asymptotic standard error of the estimator depends only on m; it is B·(24·m)^-½ = 0.028.

Table 4

Statistics Summarizing Unconditional Marginal Distributions

Realized DM/$ and Yen/$ Volatilities at Different Levels of Temporal Aggregation ______________________________________________________________________________

vard_t,h vary_t,h stdd_t,h stdy_t,h lstdd_t,h lstdy_t,h cov_t,h corr_t,h

______________________________________________________________________________

Weekly, h=5

Mean 2.645 2.691 1.555 1.567 0.399 0.406 1.217 0.448 Variance 3.250 3.643 0.227 0.238 0.083 0.083 0.947 0.021 Skewness 2.483 2.815 1.230 1.404 0.242 0.350 2.281 -0.127 Kurtosis 12.40 15.33 5.422 6.204 3.146 3.334 10.30 2.393 Bi-Weekly, h=10

Mean 5.296 5.387 2.216 2.234 0.759 0.767 2.438 0.454 Variance 10.52 11.27 0.387 0.398 0.071 0.070 2.840 0.019 Skewness 2.109 2.260 1.108 1.195 0.262 0.350 1.715 -0.141 Kurtosis 9.378 10.87 4.771 5.050 3.050 3.129 6.533 2.242 Tri-Weekly, h=15

Mean 7.935 8.074 2.718 2.746 0.965 0.978 3.652 0.456 Variance 21.78 23.03 0.548 0.537 0.068 0.061 5.698 0.018 Skewness 2.036 2.289 1.023 1.297 0.216 0.496 1.633 -0.144 Kurtosis 9.636 10.43 4.578 5.341 2.993 3.229 6.019 2.231 Monthly, h=20

Mean 10.59 10.77 3.150 3.179 1.116 1.127 4.875 0.458 Variance 34.35 36.46 0.668 0.668 0.061 0.058 8.992 0.017 Skewness 1.619 1.893 0.959 1.147 0.311 0.472 1.428 -0.188 Kurtosis 5.816 7.457 3.792 4.422 2.818 3.042 4.855 2.130

______________________________________________________________________________

Note: See notes to Table 1. The results for the weekly, bi-weekly, tri-weekly and monthly distributions are based on 489, 244, 163 and 122 observations, respectively.

Table 5

Correlation Matrices

Realized DM/$ and Yen/$ Volatilities at Different Levels of Temporal Aggregation ______________________________________________________________________________

vary_t,h stdd_t,h stdy_t,h lstdd_t,h lstdy_t,h cov_t,h corr_t,h

______________________________________________________________________________

Weekly, h=5

vard_t,h 0.496 0.977 0.507 0.911 0.494 0.780 0.314 vary_t,h 1.000 0.522 0.975 0.518 0.905 0.771 0.207

stdd_t,h - 1.000 0.547 0.978 0.545 0.786 0.338

vard_t,h 0.469 0.982 0.480 0.929 0.475 0.768 0.279 vary_t,h 1.000 0.499 0.981 0.507 0.929 0.749 0.165

vard_t,h 0.501 0.982 0.505 0.933 0.495 0.770 0.245 vary_t,h 1.000 0.518 0.983 0.517 0.937 0.764 0.134

vard_t,h 0.477 0.988 0.481 0.950 0.474 0.767 0.233 vary_t,h 1.000 0.499 0.987 0.507 0.949 0.745 0.107

stdd_t,h - 1.000 0.510 0.987 0.508 0.777 0.248

Note: See notes to Table 2. The weekly, bi-weekly, tri-weekly and monthly correlations are based on 489, 244, 163 and 122 observations, respectively.

Table 6

Dynamic Dependency Measures

Realized DM/$ and Yen/$ Volatilities Under Temporal Aggregation

______________________________________________________________________________

vard_t,h vary_t,h stdd_t,h stdy_t,h lstdd_t,h lstdy_t,h cov_t,h corr_t,h

______________________________________________________________________________

Weekly, h=5

LB 621.5 503.3 799.1 612.0 938.2 627.4 447.7 2857 ˆ

d 0.485 0.455 0.494 0.511 0.528 0.533 0.425 0.488 Bi-weekly, h=10

LB 220.3 200.9 270.3 219.0 311.7 207.5 176.5 1246 ˆd 0.510 0.501 0.501 0.500 0.479 0.485 0.480 0.499 Tri-weekly, h=15

LB 107.6 103.7 130.1 117.9 147.8 116.6 113.6 672.0 ˆd 0.430 0.416 0.432 0.430 0.424 0.429 0.357 0.551 Monthly, h=20

LB 68.7 67.2 81.2 71.4 95.0 68.7 77.9 412.5 ˆd 0.433 0.474 0.426 0.485 0.463 0.482 0.435 0.628

______________________________________________________________________________

Notes: See notes to Table 3. The Ljung-Box tests, LB, using sample autocorrelations through displacement 20, are based on 489, 244, 163, and, 122 observations for the weekly, bi-weekly, tri-weekly and monthly estimates, respectively. The 0.001 critical value is 45.3. The reported log periodogram estimates of d are calculated from the first m = [(T/h)^4/5 ], or 141, 81, 58, and 46, periodogram ordinates, respectively. The corresponding asymptotic standard errors are 0.054, 0.071, 0.084 and 0.095, respectively.

0.0

Distributions of Daily Realized Exchange Rate Volatilities:

Variances, Standard Deviations, and Log Standard Deviations

Notes: The figure shows estimates of the unconditional densities of daily realized DM/$ and Yen/$

volatility, measured in three ways. The top panel shows estimated distributions for the DM/$ variance (solid line, vard in the notation of the text) and Yen/$ variance (dashed line, vary in the notation of the text). The middle panel shows estimated distributions for the DM/$ standard deviation (solid line, stdd) and Yen/$ standard deviation (dashed line, stdy). The bottom panel shows estimated distributions for the DM/$ and Yen/$ logarithmic standard deviations (lstdd, solid line, and lstdy, dashed line). The sample period runs from December 1, 1986, until December 1, 1996.

0 1 2 3 4

-0.5 0.0 0.5 1.0 1.5

0 1 2 3 4

-0.5 0.0 0.5 1.0 1.5

Covariance

Correlation

Figure 2

Distributions of Daily Realized Exchange Rate Comovement:

Covariances and Correlations

Notes: The figure shows estimates of the unconditional densities of daily realized DM/$ and Yen/$

comovement, measured in two ways: covariance and correlation (cov and corr, in the notation of the text).

The sample period runs from December 1, 1986, until December 1, 1996.

-2 -1 0 1 2

-2 -1 0 1 2

DM/$ Volatility

Yen/$ Volatility Figure 3

Daily Realized Exchange Rate Volatilities: DM/$ vs. Yen/$

Notes: The figure shows a scatterplot of daily realized DM/$ and Yen/$ volatilities, measured as

logarithmic standard deviations (lstdd vs. lstdy, in the notation of the text). The sample period runs from December 1, 1986, until December 1, 1996.

-2 -1 0 1 2

-1.0 -0.5 0.0 0.5 1.0

Correlation Between DM/$ and Yen/$

DM/$ Volatility

-2 -1 0 1 2

-1.0 -0.5 0.0 0.5 1.0

Correlation Between DM/$ and Yen/$

Yen/$ Volatility

Figure 4

Daily Realized Exchange Rate Volatilities vs. Correlation

Notes: The top panel shows a scatterplot of daily realized DM/$ volatility measured as logarithmic standard deviation (lstdd, in the notation of the text) vs. daily realized (DM/$, Yen/$) correlation (corr, in the notation of the text). The bottom panel shows a scatterplot of daily realized Yen/$ volatility measured as logarithmic standard deviation (lstdy) vs. daily realized (DM/$, Yen/$) correlation (corr). The sample period runs from December 1, 1986, until December 1, 1996.

0.0

Distributions of Daily Realized Exchange Rate Correlation:

Low Volatility vs. High Volatility Days

Notes: The figure shows estimates of the unconditional densities of daily realized (DM/$, Yen/$)

correlation (corr, in the notation of the text) in low-volatility and high-volatility periods, where volatility is measured as logarithmic standard deviation (lstdd and lstdy). The top panel shows the estimated

distribution of corr when both lstdd and lstdy are less than their median (“low volatility”) and when both are greater than their median (“high volatility”). The bottom panel shows the estimated distribution of corr when both lstdd and lstdy are less than their tenth percentile (“very low volatility”) and when both are greater than their ninetieth percentile (“very high volatility”). The sample period runs from December 1, 1986, until December 1, 1996.

0.0

Distribution of Realized Volatilities and Correlation:

Positive- vs. Negative-Return Days

Notes: The figure shows estimates of the unconditional densities of daily realized DM/$ and Yen/$

volatilities (lstdd and lstdy, in the notation of the text) and correlation (corr) on positive and negative return days. The top panel shows the distribution of lstdd on positive-return days (solid line) and negative-return days (dashed line). The middle panel shows the distribution of lstdy on positive-return days (solid line) and negative-return days (dashed line). The bottom panel shows the distribution of corr on positive-return days (solid line) and negative-return days (dashed line). The sample period runs from December 1, 1986, until December 1, 1996.

-1.5

Times Series of Daily Realized Volatilities and Correlation

Notes: The figure shows the time series of daily realized DM/$ and Yen/$ volatilities (lstdd and lstdy, in the notation of the text) and correlation (corr). The top panel shows lstdd, the middle panel shows lstdy,

Notes: The figure shows the time series of daily realized DM/$ and Yen/$ volatilities (lstdd and lstdy, in the notation of the text) and correlation (corr). The top panel shows lstdd, the middle panel shows lstdy,

In document Estudio de fábricas magnéticas y su relación con la deformación en el sector centro- occidental del Pirineo Central (Aragón y Navarra) (página 82-88)