Transcript Analysis of Sales of Food Services & Drinking Places

Analysis of Sales of Food
Services & Drinking Places
Julianne Shan
Ho-Jung Hsiao
Christian Treubig
Lindsey Aspel
Brooks Allen
Edmund Becdach
Outline
Introduction
Original and Differenced data
Modeling
Model validation
Forecasting
Summary
Introduction
 We chose to analyze the food services and drinking
industry, looking at total sales over the last 18 years or.
Our data thus includes sales from many restaurants and
bars across the US
 As young adults, we work
and spend time in these places,
so we were interested to see
what the trend in sales looks like
Original Data
 We gathered our data from the US Census
Bureau, at http://www.census.gov/marts/www/timeseries.html
 The data appears to be evolutionary at first, with
a clear upwards trend in sales.
 Looking at the histogram of the data, we also
see that the data is not Normal at the 5%
significance level since the Jarque-Bera statistic
has a probability of 0.00047 < 0.05; the data is
slightly skewed and almost looks uniform.
Original Data
Original Data
Original Data
 We next looked at the ACF and PACF, and found
Augmented Dickey Fuller statistic of 1.54,indicating that there is a unit root at the 5%
level
 We found no apparent trend in the variance, but
we did apply the log transformation of the data
(for completeness) and there were no apparent
improvements. Therefore, we chose not to
apply the log transformation in our final model
Seasonal Difference
 Next, we applied a seasonal difference to the
data, with a seasonality of 12. The trace still
looks evolutionary; there is a positive trend
from 1992-2006, and a negative trend from
2006-present.
 The data is now normal at the 5% level, with a
Jarque-Bera statistic of probability 0.1683 >
0.05; the data is slightly skewed.
 The ACF decays slowly; the PACF has
significant spikes at lags 1, 2, 12, 13, 24.
 The Dickey-Fuller stat shows that there is still
a unit root at the 5% level
Seasonal Difference
First Difference
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We apply a first difference to the seasonally adjusted
data, as its trace looked evolutionary; once again, the
trace appears evolutionary, with a positive trend from
1992-2006, and a negative trend from 2006-present.
A look at the histogram shows that the data is not
normal at the 5% significance level, since the JarqueBera statistic has a probability of 0.002990 < 0.05; the
histogram is slightly skewed and kurtotic, but has one
main peak.
The correlogram shows that the ACF has significant
spikes at lags 1, 2, 11, 12, 13; the PACF has
significant spikes at lags 1, 11, 12, 24, 36
The Dickey-Fuller stat indicates that there is now NO
unit root at the 5% significance level
First Difference
First Difference
 The correlogram shows that the ACF has significant spikes at lags 1, 2, 11,
12, 13; the PACF has significant spikes at lags 1, 11, 12, 24, 36
 The Dickey-Fuller stat indicates that there is now NO unit root at the 5%
significance level
Modeling!
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Since there is no unit root, the data is now stationary;
we chose ten appropriate models to try to fit the data,
and eventually chose the model
ARIMA(1,1,1)x(1,1,1)12
This model had the lowest AIC (Akaike Information
Criterion) among the other models, at 2740
Our last three models do not have a seasonal
component, but were used to confirm that the seasonal
component is necessary, as evidenced by their
diagnostic plots.
The correlogram of the residuals also appears to lie
within the confidence interval
Modeling!
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We used the following ten models to try to estimate the trend in
food service and drinking place sales:
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ARIMA(1,1,1)x(0,1,0)12
ARIMA(1,1,1)x(1,1,0)12
ARIMA(1,1,1)x(0,1,1)12
ARIMA(1,1,1)x(1,1,1)12
ARIMA(0,1,2)x(1,1,1)12
ARIMA(1,1,2)x(1,1,1)12
ARIMA(1,1,1)x(0,0,0)12
ARIMA(1,1,1)x(1,0,0)12
ARIMA(1,1,1)x(0,0,1)12
ARIMA(1,1,1)x(1,0,1)12
Modeling!
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ARIMA(1,1,1)x(1,1,1)12
AIC = 2740.571
Coefficients:
s.e.
ar1
ma1
-0.1516 -0.2635
0.1813 0.1791
sar1
sma1
-0.1571 -0.9609
s.e. 0.0845 0.1794
Model Validation
 From the diagnostic plots we notice the
following:
 The Durbin Watson Statistic = 1.98  2, indicating no
serial correlation
 The ACF looks like white noise: it is one at lag 1, and
approximately zero at all other lags
 The p-values for the Ljung-Box Statistic are all
sufficiently large
 The residuals are not normal since the probability of
the Jarque-Bera statistic is approx. 0 < 0.05; they are
slightly skewed and highly kurtotic, but there is one
main peak
Model Validation
Model Validation
Next, we re-estimated Model 4 excluding
the last 12 values (i.e. we used the data
from 01/1992 – 04/2008). Then we
forecasted the withheld values.
From the graph we can see that all twelve
of the actual values (05/12008 – 04/2009)
fall within the 95% confidence band, so
our model has good predictive powers.
This is our final model.
Model Validation
 A plot of the actual values, forecasted values, and a 95% confidence interval:
Forecasting:
 We used our final model to predict the sales of Food Services &
Drinking Places (in Millions of Dollars) for the next twelve months.
Our model clearly indicates that sales will continue to rise
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5/2009:
6/2009:
7/2009:
8/2009:
9/2009:
10/2009:
11/2009:
12/2009:
1/2010:
2/2010:
3/2010:
4/2010:
$ 38,275.66
$ 38,358.69
$ 38,481.70
$ 38,610.12
$ 38,710.74
$ 38,891.78
$ 38,977.95
$ 39,116.53
$ 39,258.38
$ 39,284.23
$ 39,466.58
$ 39,557.42
Forecasting
Summary
Our final model, ARIMA(1,1,1)x(1,1,1)12,
predicts an increase in Food Services &
Drinking Places over the next twelve
months, although from the trace of the
forecast it appears to be slowing down a
little bit in the current recession.
This is no surprise, as our country still has
a growing young population that likes to
eat out and party
Questions?