Volume 12

Issue 4

##### Economics

JOURNAL OF

POLISH

AGRICULTURAL

UNIVERSITIES

Available Online: http://www.ejpau.media.pl/volume12/issue4/art-13.html

**
TIME SERIES MODELING WITH SEASONALITY OF NONLINEARLY CHANGEABLE AMPLITUDE ON AN EXAMPLE OF INFLATION
**

Joanna Kisielińska*
Department of Agricultural Economics and International Economic Relations,
Warsaw Agricultural University, Poland*

The paper covers modeling of processes characterized by fluctuations. In the theory of forecasting two kinds of models are used: additive and multiplicative models. The generalization about them is the additive model with nonlinearly changeable amplitude of fluctuations according to any nonlinear time function. If there is the inverse function towards the one amplitude's modeling, dependence on time can be exchanged with dependence on the trend. If there is assumption of constant amplitude, the presented model leads to the simple additive model. Assumption of amplitude directly proportional to the trend leads to the multiplicative model. The model was verified on an example of inflation in Poland. The model with nonlinearly changeable amplitude of fluctuations resulted in smaller errors than the ordinary multiplicative model and the multiplicative models, where seasonality was distinguished with use of centered 12-month moving average and Cenzus Method II X-11.

**Key words:**
inflation forecasting, seasonality, additive and multiplicative models, amplitude of fluctuations.

**INTRODUCTION**

In the theory of forecasting two main types of models are used for time series modeling with fluctuations: additive and multiplicative models. Additive models are used in cases when amplitude of fluctuations around the trend does not change. On the other hand multiplicative models are applied when the amplitude depends linearly on trend. The restrictions of both models can be avoided thanks to a more general model that can be called an additive model with nonlinearly changeable amplitude of fluctuations. In the article the methods of building such a model will be presented. It will also be shown that additive and multiplicative models are just special cases of this model.

The proposed model will be verified on an example of inflation (consumer price index) in Poland. The model will be built based on inflation data covering the period from March 1990 to February 2007. Inflation values covering the period from March to December 2007 will be used for additional models verification and comparison. The year 1989 and the beginning of 1990 were omitted because the economical and political situation in Poland was not stable what was reflected in inflation level and fluctuations over this time.

Inflation periodicity will be considered by the method of indicators under the assumption of the 12 month oscillation cycle. A year lasting oscillation cycle was confirmed by Fourier analysis (see [9]). In the paper models based on harmonic analysis will be omitted because they gave worse results than indicator models. One should however remember that performing Fourier analysis is essential as the length of fluctuation oscillation is in general not obvious. It is known that inflation changes with a year cycle, its graph however on the figure 1 does not allow to find this out directly.

The additive model with nonlinearly changeable amplitude in time around deterministic trend will be compared with tree types of multiplicative model. The first one is the simple multiplicative model with deterministic trend. In the second one the seasonal indicators are calculated by centered 12 month moving averages and the third one is obtained by the Census II X -11. The description of the two last models can be found in e. g. [5].

**SEASONAL MODELS**

Let us take into account particular variable y(t) depending on time, which consists of three elements. First of them is the trend, the second is the consequence of repeating cycles (which especially can be results of seasonality), the third is irregular component. The trend can be deterministic or stochastic. The deterministic trend can be described as any time function y*(t). The stochastic trend is a realization of stochastic process. Trend's type decides on the way of its determination. In the further description the deterministic trend will be used though the elaborations are also true for the stochastic trends.

As it was mentioned in introduction, additive and multiplicative models are used to forecast effects characterized by periodic fluctuations (see [2,3,4,5,6,7,8,10,11] and a lot of others). In the additive model periodic fluctuations are added to the trend, whereas in the multiplicative are multiply by it. Additive models are used when periodic fluctuations are stable in time, on the contrary multiplicative models are used when fluctuations change in time in a way directly proportional to the trend.

The additive model can be denoted in the following way:

y(t) = y*(t) + o(_{a}t) + eps_{t} |
(1) |

whereas the multiplicative as:

y(t) = y*(t) · o(_{m}t) + eps_{t} |
(2) |

where: *eps*_{t} is the irregular component, *o _{a}*(

*t*) is the periodic component in the additive model, while

*o*(

_{m}*t*) in the multiplicative.

If in the model (1) there is an assumption that the periodic component is:

o(t) = y*(t) · w(t) |
(3) |

the following relationship will be the result:

y(t) = y*(t) + y*(t) · w(t) + eps = y*(t) · (1 + _{t}w(t)) + eps _{t} |
(4) |

Taking *o _{m}*(

*t*) = 1 +

*w*(

*t*) into the (4) formula the multiplicative model is obtained. These transformations indicate that the multiplicative model is equivalent to the additive model if there is the assumption of the formula (3) of the periodic component. This assumption can be made because multiplicative models are used if seasonal fluctuations are directly proportional to the trend. The formula (4) is used for the multiplicative model also by [7].

The formula (2) differs from the most common form of the formulas for multiplicative models (see [2,3,4,5,6,8,10,11]) in additive introduction of irregular components. These components *eps _{t}* are usually multiplied by the deterministic part in the multiplicative models:

y(t) = y*(t) · o(_{m}t) · eps_{t} |
(5) |

In this case only taking the logarithm allows to obtain the additive irregular component:

ln(y(t)) = ln(y*(t)) + ln(o(_{m}t) + ln(eps)_{t} |
(6) |

The method of least squares requires that components have the normal distribution. So in the multiplicative model there should be also the assumption, that the irregular components has the lognormal distribution.

In some papers there are recommendations to take
the logarithm of *y*(*t*) in order to calculate the trend (for example
[2,4]). It is not necessary. The trend y*(t) determined by the method of least
squares is the same for *y*(*t*) and ln(*y(t*)).

In further considerations we will use the (2). The additive irregular component in the multiplicative model is used inter alia by authors of [1,3,7].

If the deterministic trend y*(t) is assumed for the models (1) and (2), it can be represented by any function. If it is a linear function, parameters estimation is very easy (linear regression). Functions of other classes are linearizing functions (polynomial, power, exponential or logarithm), which can be transformed into linear problem thanks to simple changes. Different group consists of non-linearizing functions, whose parameters can be found with use of algorithms of non-linear optimization.

After determination of the trend, it should be
eliminated from the series. The way of elimination depends on the model's type.
In the additive model the trend is subtracted from real values of the variable,
whereas in the multiplicative real values are divided by the trend. The function
obtained by the trend's elimination is denoted as *o*(*t*), consists
of the periodic and irregular element. It is calculated in the additive model
as:

o(t) = y(t) – y*(t) |
(7) |

while in the multiplicative as:

(8) |

Transformation (7) or (8) should lead to the
periodic function, which does not include fluctuations in the succeeding periods
or in other words fluctuations amplitude does not change in time (if o(t) is
treated as a time series, the stochastic process, which generates it, should
be at least of weak stationary). It is possible if fluctuations amplitude is
constant in the original function *y*(*t*) and the additive model is
used or if amplitude is directly proportional to the trend and the multiplicative
model is used.

If fluctuations are being changed in time and
are not proportional to the trend, models (1) and (2) do not allow to describe
an effect properly. In this situation the additive model, where amplitude of
fluctuations will change according to any time function *a*(*t*).

In the o(t) function amplitude *a*(*t*)
and fluctuations w(t) with deviations not changeable in time (not depended on
the examined cycle) should be distinguished. The function *o(t*) is then
denoted as:

o (t) = a(t) · w(t) |
(9) |

It can be observed that both *a*(*t*)
and y*(t) are time functions. If there is the inverse function towards y*(t),
amplitude can be presented as the trend function. There is also a possibility
of direct estimation of amplitude as the trend function, not the time function.

In order to distinguish *w*(*t*) component,
it is necessary to determine *a*(*t*) function. The formula (9) resembles
the multiplicative model but it is not the same. The *o*(*t*) function
was obtained by subtracting the trend function from *y*(*t*). As a
result it has both positive as well as negative values and fluctuations are around
zero. That is why the trend function cannot be considered – because it has been
eliminated yet. As far as the *a*(*t*) function is concerned it should
be assumed that it is always positive (fluctuations amplitude cannot be negative).

Elimination of negative values from *o*(*t*)
requires calculation of absolute value |*o*(*t*)|, which according
to the formula (9) can be presented as:

|o(t)| = |a(t) · w(t)| = a(t) · |w(t)| |
(10) |

because a(t) is always positive.

However in the |*o*(*t*)| function
the trend component can be distinguished resulting from increase (or decrease)
in fluctuations amplitude in time. If the equation (10) is treated as the multiplicative
model, *a*(*t*) can be determined as the trend function for

|*o*(*t*)|.

Finally, elimination of variability of fluctuations
amplitude demands dividing *o*(*t*) by the estimated *a*(*t*):

(11) |

Function w(t) obtained in that way represents fluctuations, whose amplitude does not change in time, and which can by modeled with use of seasonal indexes or Fourier series analysis.

The additive model with the amplitude changeable according to any time function (or trend function) can be denoted as:

(12) |

If *a*(*t*) = const = *a*, introducing
denotation *o _{a}*(

*t*) =

*a*·

*w*(

*t*) into the formula (12), the simple additive model is obtained as (1). Moreover, assuming

*a*(

*t*) = const · y*(t) =

*a*· y*(t) and introducing denotation

*o*(

_{m}*t*) =

*a*·

*w*(

*t*) · y*(t) into the formula (12), the multiplicative model is obtained. It is a proof that the additive model with the amplitude changeable according to any time function is a generalization about the simple additive and multiplicative model.

The additive model with any fluctuations amplitude described by the equation (12) includes the deterministic trend. It can be also used for stochastic trends. It is essential to determine the trend at first and after that to model seasonality, not conversely – seasonality at first and the trend after that.

**MODELS OF INFLATION IN POLAND**

Graph 1 represents changes in inflation in Poland from March 1990 to February 2007. It clearly shows, that inflation fluctuations are not constant in time so it would be aimless to use the simple additive model to modeling of these phenomena. That is why the additive model with fluctuations amplitude changeable in time (ADZ), the ordinary multiplicative model (MU) and for comparison two multiplicative models, where seasonality is distinguished with use of centered 12-month moving average (SR12) and Cenzus Method II X-11 (CX-11), will be constructed. In the two first models the deterministic trend is determined at the beginning and seasonality after its elimination. In the last two ones the seasonality is modeled at first whereas the trend after its elimination. In all models we use seasonal indexes.

Graph 1. Consumer Price Index from March 1990 to February 2007 |

Source: Own elaboration on the base of data from Central Statistical Office of Poland |

Data used for graph 1 |

Year |
I |
II |
III |
IV |
V |
VI |
VII |
VIII |
IX |
X |
XI |
XII |

1990 |
104.3 |
107.5 |
104.6 |
103.4 |
103.6 |
101.8 |
104.6 |
105.7 |
104.9 |
105.9 |
||

1991 |
112.7 |
106.7 |
104.5 |
102.7 |
102.7 |
104.9 |
100.1 |
100.6 |
104.3 |
103.2 |
103.2 |
103.1 |

1992 |
107.5 |
101.8 |
102 |
103.7 |
104 |
101.6 |
101.4 |
102.7 |
105.3 |
103 |
102.3 |
102.2 |

1993 |
104.1 |
103.4 |
102.1 |
102.3 |
101.8 |
101.4 |
101.1 |
102.3 |
102.5 |
101.9 |
104 |
105.6 |

1994 |
101.8 |
101.1 |
102 |
102.9 |
101.7 |
102.3 |
101.5 |
101.7 |
104.5 |
102.9 |
101.8 |
101.9 |

1995 |
104.1 |
102.1 |
101.7 |
102.3 |
101.8 |
101 |
99.1 |
100.4 |
103 |
101.8 |
101.3 |
101.5 |

1996 |
103.4 |
101.5 |
101.5 |
102.2 |
101.4 |
101 |
99.9 |
100.5 |
101.9 |
101.4 |
101.3 |
101.3 |

1997 |
102.9 |
101.1 |
100.8 |
101 |
100.6 |
101.5 |
99.8 |
100.1 |
101.4 |
101.1 |
101.2 |
101 |

1998 |
103.1 |
101.7 |
100.6 |
100.7 |
100.4 |
100.4 |
99.6 |
99.4 |
100.8 |
100.6 |
100.5 |
100.4 |

1999 |
101.5 |
100.6 |
101 |
100.8 |
100.7 |
100.2 |
99.7 |
100.6 |
101.4 |
101.1 |
100.9 |
100.9 |

2000 |
101.8 |
100.9 |
100.9 |
100.4 |
100.7 |
100.8 |
100.7 |
99.7 |
101 |
100.8 |
100.4 |
100.2 |

2001 |
100.8 |
100.1 |
100.5 |
100.8 |
101.1 |
99.9 |
99.7 |
99.7 |
100.3 |
100.4 |
100.1 |
100.2 |

2002 |
100.8 |
100.1 |
100.2 |
100.5 |
99.8 |
99.6 |
99.5 |
99.6 |
100.3 |
100.3 |
99.9 |
100.1 |

2003 |
100.4 |
100.1 |
100.3 |
100.2 |
100 |
99.9 |
99.6 |
99.6 |
100.5 |
100.6 |
100.3 |
100.2 |

2004 |
100.4 |
100.1 |
100.3 |
100.8 |
101 |
100.9 |
99.9 |
99.6 |
100.3 |
100.6 |
100.3 |
100.1 |

2005 |
100.1 |
99.9 |
100.1 |
100.4 |
100.3 |
99.8 |
99.8 |
99.9 |
100.4 |
100.4 |
99.8 |
99.8 |

2006 |
100.2 |
100 |
99.9 |
100.7 |
100.5 |
99.7 |
100 |
100.3 |
100.2 |
100.1 |
100 |
99.8 |

2007 |
100.4 |
100.3 |
100.5 |
100.5 |
100.5 |
100 |
99.7 |
99.6 |
100.8 |
100.6 |
100.7 |
100.3 |

For ADZ and MU inflation models the best fitted was the power trend:

(13) |

After elimination of the trend in the additive model, the fluctuations amplitude should be determined, which requires determination of the absolute value as well as the trend fitting. The logarithm function occurred to be the best modeling function for the amplitude.

(14) |

In the SR12 and CX-11 models also the power trend with coefficients slightly different from (13) was the best:

(15) |

Seasonal indexes were denoted as *c _{j}*(

*i*), where

*i*=1, ..., 12 is cycle phase number (month), whereas j = ADZ, MU, SR12, CX-11 is a denotation of the model. The formulas representing particular models are as the following ones:

(16) |

In the formula (16) t is equal 1 for March 1990.
The index describing the month (cycle phase) has the value of one for January.
It is the residual from dividing *t*+2 by 12 (*i *= *t*+2 mod 12).
Seasonal indexes for the determined models are included in the Table 1.

Table 1. Seasonal indexes for the determined inflation models |

Month |
ADZ |
MU |
SR12 |
CX-11 |

1 |
1.8324 |
1.0144 |
0.9999 |
1.0004 |

2 |
-0.023 |
1.0006 |
1.0014 |
1.0008 |

3 |
-0.1963 |
0.9972 |
1.0145 |
1.0103 |

4 |
0.5349 |
1.0021 |
0.9998 |
1.0008 |

5 |
0.1133 |
0.9988 |
0.9983 |
0.9986 |

6 |
-0.5627 |
0.9964 |
1.001 |
1.0023 |

7 |
-1.6364 |
0.9891 |
0.9989 |
0.9990 |

8 |
-1.2944 |
0.9915 |
0.9969 |
0.9966 |

9 |
0.8944 |
1.0058 |
0.9891 |
0.9903 |

10 |
0.4596 |
1.0021 |
0.9928 |
0.9926 |

11 |
-0.0853 |
1.0003 |
1.0057 |
1.0060 |

12 |
-0.0365 |
1.0017 |
1.0018 |
1.0023 |

Source: Own researches. Note: ^{1) } One represents January. |

Root mean square errors were determined ex post expired forecasts were determined (Table 2), separately in two intervals. The first consists of the period from March 1990 to February 2007 and the second from March 2007 to December 2007. The first interval was used for the models building (so called the base period), the second one for its testing (so called the test period).

Table 2. Root mean square error (RMSE) ex post for the determined models |

Model |
RMSE
for the period from |
RMSE
for the period from |

Additive with nonlinearly changeable amplitude of fluctuations (ADZ) |
0.832 |
0.469 |

Multiplicative (MU) |
0.945 |
0.537 |

Multiplicative with seasonality distinguished with use of centered 12-month moving average (SR12) |
1.348 |
0.848 |

Model estimated with use of Cenzus Method II X-11 (CX-11) |
1.294 |
0.778 |

Source: Own researches |

The additive model with nonlinearly changeable
amplitude of fluctuations (ADZ) occurred to be the best for the base period as
well as the test period. The ordinary multiplicative model (MU) was slightly
worse. The multiplicative models, where seasonality is distinguished with use
of centered 12-month moving average (SR12) and Cenzus Method II X-11 (CX-11),
were far worse. This result could be expected because inflation has the clear
deterministic trend in Poland (see graph 1), whose elimination allows to properly
determine seasonal effects. Stationarity of inflation around the deterministic
trend was confirmed by the Dickey-Fuller test.

**CONCLUSIONS**

The idea and method of estimation of model allowing to analyze phenomena characterized by seasonality were presented in the paper. It is the additive model with nonlinearly changeable amplitude of fluctuations according to any time function (ADZ). The dependence on time can be exchanged with the trend function (if there is the inverse function to function which models amplitude or if estimated amplitude is estimated as the trend function, not the time function). It was presented that this model is the generalization about common used additive and multiplicative models. If the amplitude is constant we obtain additive model. Taking into consideration amplitude directly proportional to the trend leads to the multiplicative model.

Proposed model was used for inflation forecasting in Poland. It allowed to obtain smaller errors than the multiplicative model, which means that inflation fluctuations are suppressed more quickly than in the way directly proportional to the trend.

Two multiplicative models with seasonality distinguished with use of 12-month moving average (centered and Cenzus Method II X-11) occurred to be worse than both models, where the deterministic trend was estimated at first (additive with changeable amplitude as well as multiplicative). It confirms the deterministic character of the inflation trend in Poland.

This result does not surprise. It provides consequences
and efficiency of monetary policy in Poland.

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Accepted for print: 7.10.2009

Joanna Kisielińska

Department of Agricultural Economics and International Economic Relations,

Warsaw Agricultural University, Poland

166 Nowoursynowska Street, 02-787 Warsaw, Poland

email: joanna_kisielinska@sggw.pl

Responses to this article, comments are invited and should be submitted within three months of the publication of the article. If accepted for publication, they will be published in the chapter headed 'Discussions' and hyperlinked to the article.