# Let stated that the series have asymmetry steepness.

Let

be a random sample from a continuous probability distribution

with density

. Recently, a very important problem
is the symmetry of

about some unknown values. The symmetry of

is essential for determining which location
parameters represent the probability distribution the most. The presence of the
mean, median and mode doesn’t function in the asymmetry’s case. Labeling with

the mean or median of

, the null hypothesis for the
symmetry can be formulated as:

Against the alternative hypothesis of asymmetry:

for at least an

. Various procedures to test the asymmetry of the probability
distribution are proposed in the economic literature. These
can be classified according to the use of calculating the coefficient of
skewness. The skewness index is presented from the third standardized moment

, where

and

are respectively the third central moment

and the standard deviation. In an
obvious manner

if the distribution is symmetric. Although

measure of skewness, it nevertheless has its flaws. The coefficient of skewness
is sensitive to outliers and can be vague when it comes to heavy-tailed distribution.
Moreover, even when it is equal to zero for the symmetric distribution, a value
equal to zero doesn’t necessarily signify that the distribution is symmetric (Bacci
& Bartolucci, 2013).

The notion, that the economic time series can evolve in an asymmetric way
during the business cycle, has taken a lot of attention on the economy. A
business cycle is symmetric when the recessions are mirror images with the
expansions (Boldin,
1999)
or is asymmetric when expansions are not the same with the recessions (Sichel,
1993).
Sichel pointed out that the expansions often consist in smaller deviations of
the economic time series from the trend more than the recessions. He called
this quality, asymmetry deepness. If the same quality applies to the growth
rates of the economic time series, then it can be stated that the series have
asymmetry steepness.

This study tests the steepness and deepness asymmetry, using the moments
of the distribution. For this purpose we use times series data for the inflation,
unemployment rates from 1 January 2000 – 31 December 2012, and GDP growth rate from
1 January 2005 – 31 December 2012 in Albania.

The study’s structure is like it follows: The second section starts with a
summary of the asymmetry and the moments of the distribution. The third part
informs about the results and analysis, while the last section presents the
conclusions.

1.
The asymmetry and the moments of distribution

Moments of distribution give us a useful summary of the probability
distribution. For a symmetric probability distribution, the coefficient of
skewness is going to be zero1
and the mean will be equal to the median. A positive coefficient of skewness
shows that the probability distribution inclines to the right, which means that
the right tail of the probability distribution is longer than the tail on the
left. Vice versa, a negative coefficient shows that the probability
distribution inclines to the left. However, if the contractions are shorter and
more severe than the expansions, the probability distribution should be negatively
skewed and it should induce an important coefficient of skewness in frequency
distribution (DeLong & Summers, 1986). DeLong and Summers focused
on the changes’ magnitude and according to them this probability distribution
should have considerably less than half of observations below the mean; just like the average deviation from the observations’ mean below the
mean should be significantly bigger than the average deviation from the
observations’ mean above the mean. The median should surpass the mean by a
considerable amount. According to Sichel, this phenomenon is known as deepness
hypothesis:

“If a
time series exhibits deepness, then it should exhibit negative skewness
relative to mean or trend; that it is should have fewer observations below its
mean or trend than above, but the average deviation of observations below the
mean or trend should exceed the average deviation of observations above…” (Sichel, 1993, f. 227)

Besides this, if the macroeconomic variables in the study fall quickly
from the trend, the negative deviation’s slope should be steepened. This means
that the distribution of the first differences should be skewed negatively as
well. The number of observations beneath the mean should be lower than the
number of the observations above the mean, even though the average deviation
from the observation’s mean beneath the mean should be greater than the average
deviation from the observation’s mean above the mean. This is recognized as the
steepness hypothesis (Sichel,
1993).

“….if
a time series exhibits steepness, then its first differences should exhibit
negative skewness. That is, the sharp decreases in the series should be larger,
but less frequent, than the moderate increases in the series” (Sichel,
1993, f. 228)

2.
Results and analysis

After the
relevant statistics are calculated, the result are shown in Table 3. We test the
following hypothesis:

·
Null hypothesis: The quarterly rate (*)2
in Albania does not have deepness/steepness asymmetry.

·
Alternative hypothesis: The quarterly rate (*) in Albania
does have deepness/steepness asymmetry.

The figure 1 shows the frequency probability distributions for the quarterly
rates of inflation, unemployment, and the GDP growth rate in Albania and the
first differences, given the null hypothesis of the symmetry against the
alternative hypothesis of the asymmetry.

In favor of the above’s characteristics, facts about the asymmetry deepness
in the distribution of the macroeconomic variable are shown in the table 1,
which illustrates the asymmetry deepness hypothesis. When it comes to
inflation’s and GDP’s growth rates the coefficient of skewness is almost zero. When
studying the unemployment rate, presented as variables against the cycle, it
can be seen that the coefficient of skewness is different from zero and the
mean surpasses the median. This only signifies that these variables show
asymmetry deepness, which can be confirmed even if we won’t take the relative
number of the “beneath the mean” observations and the relative
average deviation in the matter. The number of observations beneath the mean is
greater than half of the observations above the mean while the average
deviation is smaller. Inflation and GDP growth rates don’t show signs of the
asymmetry deepness; the mean and median are almost equal and the number of
observations beneath the mean is greater that those above the mean too, while
the average deviation is lesser when it comes to observations beneath the mean.

The hypothesis of steepness are illustrated in the table 2. All the
macroeconomic variables have a positive coefficient of skewness, meaning that
the growth of these macroeconomic variables above the trend can be very quick,
besides the GDP growth rate, which has a negative coefficient of skewness.

For the unemployment rate, taken as variables against the cycle, it can be
noticed that the coefficient of skewness is different from zero and that the
mean exceeds the median. This means that these variable show asymmetry
steepness, which can be confirmed even if the relative number of the
observations below the mean and the average relative deviation is observed.