Exchange rate volatility and the mixture of distribution hypothesis

Abstract

This study sheds new light on the mixture of distribution hypothesis by means of a study of the weekly exchange rate volatility of the Norwegian krone. In line with other studies we find that the impact of information arrival on exchange rate volatility is positive and statistically significant, and that the hypothesis that an increase in the number of traders reduces exchange rate volatility is not supported. The novelties of our study consist in documenting that the positive impact of information arrival on volatility is relatively stable across three different exchange rate regimes, and in that the impact is relatively similar for both weekly volatility and weekly realised volatility. It is not given that the former should be the case since exchange rate stabilisation was actively pursued by the central bank in parts of the study period. We also report a case in which undesirable residual properties attained within traditional frameworks are easily removed by applying the log-transformation on volatilities.

Data sources and transformations

The data transformations were undertaken in Ox 3.4 and EViews 5.1.

S n(t)	85 n(t)=1(t), 2(t),.., N(t), where S 1(t) is the first BID NOK/1EUR opening exchange rate of week t, S 2(t) is the first closing rate, S 3(t) is the second opening rate, and so on, with S N(t) denoting the last closing rate of week t. Before 1.1.1999 the BID NOK/1EUR rate is obtained by the formula BID NOK/100DEM×0.0195583, where 0.0195583 is the official DEM/1EUR conversion rate 1.95583 DEM=1 EUR divided by 100. The first untransformed observation is the opening value of BID NOK/100DEM on Wednesday 6.1.1993 and the last is the BID NOK/1EUR closing value on Friday 26.12.2003. The source of the BID NOK/100DEM series is Olsen and the source of the BID NOK/1EUR series is Reuters.
S t	SN(t), the last closing value of week t
r t	log S t −log S t−1
V t w	{{log[S t +I(S t =S t−1)×0.0009]−log(S t−1)} ×100}2. I(S t =S t−1) is an indicator function equal to 1 if S t =S t−1 and 0 otherwise, and S t =S t−1 occurs for t=10/6/1994, t=19/8/1994 and t=17/2/2000.
v t w	log V t w
V t r	Σ n [log(S n /S n−1)×100]2, where n=1(t), 2(t),..., N(t) and 1(t)−1≔N(t−1)
v t r	log V t r
M n(t)	n(t)=1(t), 2(t),.., N(t), where M1(t) is the first BID USD/EUR opening exchange rate of week t, M2(t) is the first closing rate, M3(t) is the second opening rate, and so on, with MN(t) denoting the last closing rate of week t. Before 1.1.1999 the BID USD/EUR rate is obtained with the formula 1.95583/(BID DEM/USD). The first untransformed observation is the opening value of BID DEM/USD on Wednesday 6.1.1993 and the last is the closing value on Friday 30.12.2003. The source of the BID DEM/USD and BID USD/EUR series is Reuters.
M t	MN(t), the last closing value of week t
m t	log M t
M t w	{{log[M t +I(M t =M t-1)×k t ]−log(M t-1)}×100}2. I(M t =M t−1) is an indicator function equal to 1 if M t =M t−1 and 0 otherwise, and k t is a positive number that ensures the log-transformation is not performed on a zero-value. M t =M t−1 occurs for t=23/2/1996, t=19/12/1997 and t=20/2/1998, and the value of k t was set on a case to case basis depending on the number of decimals in the original, untransformed data series. Specifically the values of k t were set to 0.00009, 0.0009 and 0.00009, respectively.
m t w	log M t w
M t r	Σ n [log(M n /M n−1)×100]2, where n =1(t), 2(t),.., N(t) and 1(t)−1≔N(t−1)
m t r	log M t r
Q t	Weekly number of NOK/EUR quotes (NOK/100DEM before 1.1.1999). The underlying data is a daily series from Olsen Financial Technologies, and the weekly values are obtained by summing the values of the week.
q t	log Q t . Note that this series is “synthetic” in that it has been adjusted for changes in the underlying quote-collection methodology at Olsen Financial Technologies. More precisely q t has been generated under the assumption that Δq t was equal to zero in the weeks containing Friday 17 August 2001 and Friday 5 September 2003, respectively. In the first week the underlying feed was changed from Reuters to Tenfore, and on the second a feed from Oanda was added.
O n(t)	n(t)=2(t), 4(t),.., N(t), where O2(t) is the first closing value of the Brent Blend spot oilprice in USD per barrel in week t, O4(t) is the second closing value of week t, and so on, with On(t) denoting the last closing value of week t. The untransformed series is Bank of Norway database series D2001712, which is based on Telerate page 8891 at 16.00.
O t	ON(t), the last closing value in week t
o t	log O t
O t w	{log[O t +I(O t =O t−1)×0.009]−log(O t−1) }2. I(O t =O t−1) is an indicator function equal to 1 if O t =O t−1 and 0 otherwise, and O t =O t−1 occurs three times, for t=1/7/1994, t=13/10/1995 and t=25/7/1997.
o t w	log O t w
O t r	Σ n [log(O n /O n−2)]2, where n=2(t), 4(t),.., N(t) and 2(t)−2≔N(t−1)
o t r	log O t r
X n(t)	n(t)=2(t), 4(t),.., N(t), where X2(t) is the first closing value of the main index of the Norwegian Stock Exchange (TOTX) in week t, X4(t) is the second closing value, and so on, with XN(t) denoting the last closing value of week t. The source of the daily untransformed series is EcoWin series ew:nor15565.
X t	XN(t), the last closing value in week t
x t	log X t
X t w	[log (X t /X t−1)]2. X t =X t−1 does not occur for this series.
x t w	log X t w
X t r	Σ n [log(X n /X n−2)]2, where n=2(t), 4(t),.., N(t) and 2(t)−2≔N(t−1)
x t r	log X t r
U n(t)	n(t)=2(t), 4(t),.., N(t), where U2(t) is the first closing value in USD of the composite index of the New York Stock Exchange (the NYSE index) in week t, U4(t) is the second closing value, and so on, with UN(t) denoting the last closing value of week t. The source of the daily untransformed series is EcoWin series ew:usa15540.
U t	UN(t), the last closing value in week t
U t w	[log (U t /U t-1)]2. U t =U t-1 does not occur for this series.
u t w	log U t w
U t r	Σ n [log(U n /U n−2)]2, where n=2(t), 4(t),.., N(t) and 2(t)−2≔N(t−1)
u t r	log U t r
F t	The Norwegian central bank's main policy interest-rate, the so-called “folio”, at the end of the last trading day of week t. The source of the untransformed daily series is Bank of Norway's web-pages.
f t a	|Δ F t |×I a , where I a is an indicator function equal to 1 in the period 1 January 1999–Friday 30 March 2001 and 0 elsewhere
f t b	|Δ F t |×I b , where I b is an indicator function equal to 1 after Friday 30 March 2001 and 0 before
id t	Russian moratorium impulse dummy, equal to 1 in the week containing Friday 28 August 1998 and 0 elsewhere.
sd t	Step dummy, equal to 0 before 1997 and 1 thereafter.