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VEE Time Series Student Project
Session: Winter 2013
Introduction
Olive oil is made from the crushing and then subsequent pressing of olives. The fact that olives
are rich in oil is reflected in the botanical name of the olive tree—Olea europea—since the word
"oleum" means oil in Latin. Olive oil is available in a variety of grades, which reflect the degree
to which it has been processed. Extra virgin olive oil is derived from the first pressing of the
olives and has the most delicate flavor and strongest overall health benefits. Therefore, I decided
to examine the price of extra virgin olive oil over time for my Time Series project.
Data
I obtained my data from the Index Mundi website at
http://www.indexmundi.com/commodities/?commodity=olive-oil&months=120 and the data that
I collected was from February 2004 to January 2014. This data contains the average monthly
price in US dollars per Metric Ton.
Olive Oil Monthly Prices
7,000.00
6,000.00
5,000.00
4,000.00
3,000.00
Price
2,000.00
1,000.00
8/1/2013
2/1/2013
8/1/2012
2/1/2012
8/1/2011
2/1/2011
8/1/2010
2/1/2010
8/1/2009
2/1/2009
8/1/2008
2/1/2008
8/1/2007
2/1/2007
8/1/2006
2/1/2006
8/1/2005
2/1/2005
8/1/2004
2/1/2004
-
The prices range from a low of $2780.67 and high of $5853.98. From the graph, you see the
price has been dropping overall until the middle of year 2012 and is quite stable in year 2013.
Seasonality
To check further for seasonality, I graphed each of the years by month to observe any monthly
trends in the graph below. There do not appear to be any seasonal trends that will require
adjustments to the data.
Monthly Olive Oil Prices from 2004 to 2014
6,500.00
2004
6,000.00
2005
5,500.00
2006
5,000.00
2007
4,500.00
2008
4,000.00
2009
2010
3,500.00
2011
3,000.00
2012
2,500.00
2013
2,000.00
2014
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Analysis
Firstly, it must be determined if the data is stationary. In order to do so, we look at the sample
autocorrelations by lag period.
Correlogram of Olive Oil Prive Time Series
1.0000
auto-corr
-
1
6
11
16
21
26
31
36
41
46
51
56
61
66
71
76
81
86
91
96
101
106
111
116
121
Sample Autocorrelation
0.5000
(0.5000)
We look for a correlogram that declines to zero after several lags to demonstrate stationarity. The
correlogram above does not fall to zero quickly and when it does drop to zero at lag 36, it does
not stay zero at subsequent lags with minimum fluctuations. It decreases until negative, then rises
back up. Thus the series could be represented by an AR(1) or AR(2) process.
First Difference
Below is the graph of the first difference with lags. It does not show any particular trend.
First Diff
800.00
600.00
400.00
200.00
First Diff
(200.00)
1
6
11
16
21
26
31
36
41
46
51
56
61
66
71
76
81
86
91
96
101
106
111
116
121
-
(400.00)
(600.00)
The regression results are as follows:
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.98236591
R Square
0.96504278
Adjusted R Square
0.964744
Standard Error173.580015
Observations
119
ANOVA
Regression
Residual
Total
Significan
df
SS
MS
F
ce F
1 97318406 9.7E+07 3229.9481 4.69E-87
117 3525212.5
30130
118 100843618
Coefficient Standard
Lower
Lower
Upper
s
Error
t Stat
P-value
95%
Upper 95% 95.0%
95.0%
Intercept
65.6729286 72.663853 0.90379 0.3679634 -78.23402 209.57988 -78.23402 209.5799
X Variable 1 0.98186651 0.0172765 56.8326 4.693E-87 0.947651 1.0160816 0.947651 1.016082
The result of the equation is Yt = 65.6729+0.9819Yt-1. The R2 and adjusted R2 values are fairly
high, so this is a good model. Significance value is close to zero and it indicates that regression is
significant.
Below is the predicted AR(1) prices and the original prices.
Olive Oil Prices from February 2004 to January
2014
7,000.00
6,000.00
5,000.00
4,000.00
3,000.00
AR(1)
2,000.00
Actual Prices
1,000.00
8/1/2013
2/1/2013
8/1/2012
2/1/2012
8/1/2011
2/1/2011
8/1/2010
2/1/2010
8/1/2009
2/1/2009
8/1/2008
2/1/2008
8/1/2007
2/1/2007
8/1/2006
2/1/2006
8/1/2005
2/1/2005
8/1/2004
2/1/2004
-
The AR(1) is almost a perfect fit. Second Difference will be examined to see if it provides better
fit.
Second Difference
Second Diff
1000
800
600
400
200
Second Diff
-200
1
6
11
16
21
26
31
36
41
46
51
56
61
66
71
76
81
86
91
96
101
106
111
116
121
0
-400
-600
-800
Again, the graph above, the second difference with lags, does not show any particular trend.
The AR(2) regression is as follows:
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.983053471
R Square
0.966394127
Adjusted R Square
0.965809677
Standard Error
171.3779826
Observations
118
ANOVA
df
Significance
F
F
1653.51 1.85726E-85
Regression
Residual
Total
SS
MS
2 97128568 48564284
115 3377597 29370.41
117 1.01E+08
Intercept
X Variable 1
X Variable 2
Standard
Coefficients
Error
t Stat
P-value
82.3107064 72.15429 1.14076 0.25634
1.182375074 0.09128 12.95325 3.26E-24
-0.20417039 0.091244 -2.23763 0.027172
Upper
Lower
Upper
Lower 95%
95%
95.0%
95.0%
-60.6130531 225.2345 -60.6131 225.2345
1.001566649 1.363183 1.001567 1.363183
-0.38490739 -0.02343 -0.38491 -0.02343
The equation is Yt = 82.3107+1.1824Yt-1-0.2042Yt-2. The R^2 and adjusted R^2 values are a
little bit higher than AR(1) model. The coefficients follow the rules of AR92) stationarity:
X1+X2 = 1.1824-0.2042 <1
Abs(X2) = 0.2042<1
X2-X1 = -0.2042-1.1824 <1
Olive Oil Prices from February 2004 to January
2014
7000
6000
5000
4000
AR(2)
3000
Actual Prices
2000
1000
8/1/2013
2/1/2013
8/1/2012
2/1/2012
8/1/2011
2/1/2011
8/1/2010
2/1/2010
8/1/2009
2/1/2009
8/1/2008
2/1/2008
8/1/2007
2/1/2007
8/1/2006
2/1/2006
8/1/2005
2/1/2005
8/1/2004
2/1/2004
0
The graph above shows the predicted prices vs. the actual prices for the AR(2) model. It is also a
very good fit, but it is not as good as AR(1).
Conclusion
Based on the analysis, AR(1) model, Yt = 65.6729+0.9819Yt-1 provides a great prediction of the
olive oil price. The model has proven to provide a very close result when compared to the actual
prices.