Ratio Transformation for Stationary Time Series with Special

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Transcript Ratio Transformation for Stationary Time Series with Special

Ratio Transformation for Stationary
Time Series with
Special Application in Consumer Price Index
in Qatar
By:
Adil Yousif, Hind Alrkeb, Doha Alhashmi
Department of Mathematics and Statistics
Qatar University
Doha, Qatar
Abstract
•This article was intended to perform a comprehensive time series
analysis for Consumer Price Index (CPI) in Qatar. The data was
obtained from the Statistics Authority in Qatar. For the period
between: (2002-2009) a quarterly data was analyzed using several
time series techniques such as Exponential Trend method, Holts’
Trend method, and ARIMA. These methods were used to examine
trends and built a forecast model. Ratio transformation technique
was used to obtain a stationary time series and found to be
efficient with small size of data.
Despite the small size of the data the analysis indicated that
ARIMA model is more adequate forecast one.
Key words: Time series, index, CPI, ARIMA, Ratio Transformation
Introduction
• Qatar witnessed a tremendous increase in the
economy in the last decade.
• The economic policy in Qatar is moving towards
economic diversifications through investing the
returns of oil and gas in profitable projects.
• In addition to that, the state adopts new
methodologies in economic and trade
liberalization to attract foreign investments and
strengthen the private sector.
• The GDP has increased rapidly during the last period. It
reached 268 billion Qatari Riyals (73.4 billion US Dollar)
in 2009 as per the estimation of Statistics Authority.
• The growth rates of the GDP during last three years
were 34%, 13%, and 15% in the years 2006, 2007 and
2008 respectively.
• As Qatar’s economy is in the booming phase, the prices
level increased rapidly during last years as a result of
expanding in the Building & Construction sector. This
expansion caused inflation in rent rates, and hence
affected the prices level in Qatar
Index Number
• The index number is defined as a statistical
indicator expresses the proportional change in
value of specific phenomenon or group of
phenomena compared to specific base.
• When the change is over the time the previous
period is called the base period, where as the
measured period is called the comparison period.
Several methods are used to calculate the index
number and some of them are discussed below.
Laspere’s index number
• It is assumed within this index that individuals
will consume, in the new period and in the
shadow of change in prices, the same original
group of goods, weighted by base year
quantities, as shown in the following
equation:
Paasche’s index number
• This index is based on an assumption differs
from Laspere’s assumption, which is,
individuals will consume in the new period
and in the shadow of change in prices, a new
group or quantities of goods for each year
separately, provided that prices are weighted
by comparison year quantities, as shown in
the following equation:
Fisher’s optimum index number
• It is the index which comprises the Laspere’s
index and the Paasche’s index by finding the
geometric mean of both numbers, as shown
in the following equation:
Collecting Price Data
•
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•
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The Statistics Authority in Qatar is the only official body responsible of collecting
data in the country.
Price data were collected through personal interview by the trained data
collectors.
Data collection is carried out by registering prices from shelves of commercial
stores( trade malls, etc).
Or through personal interview with establishment managers education centers,
clinics, real estate bureaus, government establishments,…etc).
Each data collector has been provided with questionnaire for each group of
prices and sources of each questionnaire.
The data were collected quarterly for the period of 2002-2009 with the base year
considered to be 2001.
After price data sets are collected, it will be audited at the office and entered to
the computer, and then the average price and consumer prices index number are
calculated. Laspere’s formula was used to calculate the index number weighted
by 2001 prices.
Time series plot of HOUSEHOLD data
Fig[1]
Time Series Plot of HOUSEHOLD and GDP
Fig[2]
Methodology
• The original data was used to fit the exponential trend
and the Holts Winter model.
• Since ARIMA requires a stationary set of data and the
original data is violating this condition transformation
was used.
• The traditional difference transformation was used and
the third difference yield in a stationary set.
• Since the data is relatively small a new approach of
transformation in which ratios
was considered to
avoid the loss in the sample size as a result of several
differences. Results of the two transformation methods
are almost identical see table
ARIMA
• When using this transformation with first
order autoregressive model
• It becomes:
• Which eliminate the constant terms and the
random shock term will be a multiple of the
response variable. The final model will be
Ratio for ARIMA
• When the ratio transformation (
) to the
first order moving average model is used
• Substituting the ratio form we get:
• Which involve the response variable directly
unlike the original model, however the
constant term is again eliminated
Double Exponential Smoothing
• Double exponential smoothing technique was
introduced by Holt (and Brown as a special
case) smoothes the data.
• Double exponential smoothing provides shortterm forecasts.
• This procedure can work well when a trend is
present but it can also serve as a general
smoothing method.
• Both exponential trend method and Holts’
Double Exponential Smoothing were used and
the last outperformed the first when their MADs
are compared.
• Figure [3] shows the fitted values versus actual
values and one year ahead of forecast together
with their 95% prediction interval for the Holt’s
exponential model.
• It appears that there is an increasing trend in the
Household variable with a MAD (1.21962).
• The one year a head forecasts of this model are
displayed in table [1]
Holt’s Trend Exponential Smoothing for Household
Fig[3]
Auto Regressive Integrated Moving
Avareg (ARIMA)
• The data shows an exponential trend in the time series
plot, without seasonal variation , so we can apply nonseasonal ARIMA model.
• The analysis of the time series of the CPI is conducted
by using MINITAB version 16.0. After the examination
of the behavior of the sample autocorrelation (SAC)
function and partial autocorrelation (SPAC) function for
the regular difference transformation data.
• Referring to figures [4] and [5], it can be concluded that
the data became stationary after the third difference. It
can also be observed that both of the SAC and SPAC cut
off after lag 1.
ARIMA
• However, since the SPAC cut off more fairly
quickly, we tentatively identify the non-seasonal
autoregressive model of order 1.
• The ARIMA for this data shows that the two
parameters in the AR model of order 1 have p value less than 0.05. This implies that we have
very strong evidence that each term in this is
important since can be rejected at the
significance level α equals to 0.05.
• Also from the table [1] it can be noticed that Ljung-Box test
has a p –value associated with for lags K = 12 and K = 24 are
both greater than 0.05. Hence, it can be concluded that
model is adequate. Moreover since =0.4095 < 1 then the
third difference data is stationary.
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Sq.
DF
P-Value
Lag
12
18.3
10
0.051
12
24
28.7 *
22 *
0.0154 *
24
36
48
*
*
*
36
48
Ratio Transformation
• For the ratio transformation the sample
autocorrelation function dies cuts off after lag 1,
whereas the partial autocorrelation dies in a dammed
fashion and a suggested model is a moving average of
order 1.
• The p-values for the two parameters are 0.000 which
implies they are significantly different from zero. From
table [2] below the Ljumg-Box test has p-values at lag
K =12 and K=24 equal to 0.000 and therefore we can
confidently say the model MA of order 1 is adequately
fit the ratio transformation data. The invertibility
condition is also satisfied since
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Sq.
DF
P-Value
Lag
12
80.6
10
0.000
12
24
220.9
22
0.000
24
36
*
*
*
36
48
*
*
*
48
Forecast
• The forecasts for one year ahead, (2010) for the three
models along with actual values (obtained from Qatar
Statistical Authority) are then compared and listed in
the table [3] below.
• The ARIMA model for the ratio transformation data
has the best estimates followed by the ARIMA model
for the regular transformation data.
• Although the Holt’s exponential model is less accurate
than ARIMA still its error forecast is very small and its
predicted values are not far from the actual ones.
One Year Ahead Forecast (2010)
ARIMA1
107.361
109.493
103.661
105.845
ARIMA2
106.012
108.955
104.985
107.864
Holts
101.553
104.675
107.872
101.144
Actual
106.7
107.1
106.7
NA
Conclusion
• From time series analysis it appeared Double Exponential
Smoothing model is better than exponential trend model since
the values of MAD and MSD in Double Exponential Smoothing are
smaller.
• On the other hand ARIMA was used for two different sets of data
and both of them provided forecasts more accurate than the
Double Exponential Smoothing model.
• For the ARIMA in the first set the regular difference
transformation was used and a third difference yield into a
stationary time series, while in the second set the ratio difference
was used for the purpose of preventing the decrease the sample
size due to more than one differences.
• The ARIMA model for the ratio transformation produced more
accurate forecast.
References
• [1] Aka, B. F. and Pieretti, P., Consumer Price Index Dynamics in a Small
Open Economy: Structural Time Series Model for Luxembourg,
International Journal of Applied Economics, 5910, p 1-13, 2008.
• [2] Bascand G., Product Price Index: June Quarter Highlights, Statistics
New Zealand, ISSN 1178-0622, 2009
• [3]Bulletin of Prices Index Numbers 2006 -5th Issue July 2007
• [4] Bower B., O’Conell R. and Koehler A., Forecasting Time Series and
Regression: An Applied Approach
• [5] Bowles, T. J. and Cris Lewis, W., A Time Series Analysis of the Medical
Care Price Index: Implication for Appraising Economic Losses, Journal of
Forensic Economics 13(3), pp 245-254, 2000.
• [6] Box G., Jenkins G. and Reinsel G., Time Series Analysis: Forecasting and
Control. Prentice Hall.
• [7] US Bureau of Labor Statistics: Response Rate for the Consumer Price
Index: Consumer Price index Program, Report 2004