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

is the traditional

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.