Statistical Analysis and Forecasting of Beer Volume in New Zealand

This assignment analyzes the time series data of beer volume in New Zealand from 2010 to 2018, employing various forecasting techniques. The solution begins with an exploration of Exponential Smoothing (ETS) methods, including Simple Exponential Smoothing, Holt Linear Trend, and Holt-Winter's seasonal methods, to determine the most appropriate model for the data. The analysis then delves into stationarity tests using ACF and PACF plots, Box-Cox transformation, and Augmented Dickey-Fuller tests to identify the need for differencing. An ARIMA model is then developed, with a focus on identifying the order of the model based on ACF and PACF plots. The assignment details the model's components, including parameter estimates, and residual diagnostics to validate the model's assumptions of constant variance, normality, independence, and autocorrelation. Finally, the solution provides forecast values and compares the ARIMA model's performance with an autoarima model, concluding that ARIMA(4, 1, 0) is the most suitable model for forecasting beer volume, considering the data's trend and seasonality.

Running head: FORECASTING VOLUME OF BEER IN NEW ZEALAND 1
Forecasting Volume of Beer in New Zealand
Name
Institution

FORECASTING VOLUME OF BEER IN NEW ZEALAND 2
Forecasting Volume of Beer in New Zealand
Question 1: Exponential Smoothing (ETS)
(a) Figure 1 shows the time plot of the total beer available (million litres) in New Zealand
between the first quarter of 2010 to last quarter of 2018.
The plot shows that the series is stationary in trend since there is no observable trend
in the plot. However, seasonality is present since the volume of beer available for
consumption decrease in the second and third quarter but rises to a peak value in the
last quarter. The change in volume is observed consistently over time.
(b) The figures 2, 3 and 4 shows the plots for the forecast
(1) Simple Exponetial Smoothing forecast

FORECASTING VOLUME OF BEER IN NEW ZEALAND 3
(2) Holt Linear Trend plot

FORECASTING VOLUME OF BEER IN NEW ZEALAND 4
(3) Holt’s Damped Plot
The plots does not provide a distinct seasonality observed in the series. There
forecasted values appear to be constant over the period (Angadi & Kulkarni,
2015). Therefore, these method of forecasting does not fit the data appropriately.
(c) The figure 5 shows the plot of Holt-Winter’ seasonal methods

FORECASTING VOLUME OF BEER IN NEW ZEALAND 5
The multicative seasonality is necessary since it replicate the actual series more
closelyy than additive seasonality method.
(d) The table 1 shows the estimated MSE and MAE.
Table 1: MSE and MAE for the models
Model MSE MAE
Exponetial Smoothing NA NA
Holt’s linear trend 0.4048 0.2255
Holt’s dampedtrend 0.4002 0.2267
One-step-ahead 0.1011 0.0212
Four-step-ahead 0.1011 0.0212
Source: author (2019)
The one-step-ahead forecast appear to forecast the data more accurate since it has the
samllest MAE and MSE. The selection does not depend on the number of pre-
specified (steps-ahead) forecast since both one-step and four-step have the same MSE
and MAE.
Question 2: Stationarity
(a) The figure 6 shows the ACF and PACF plot of the data

FORECASTING VOLUME OF BEER IN NEW ZEALAND 6
The ACF shows a slowlyy decaying insignificnat values an indication that the series
is non-stationanry. Further, the value of the lags at the fourth, eighth, and twelveth
lags are large and positive an indication of seasonality with a period of four. Also,
there are large negative values at the second, sixth, tenth, and fourteenth lags
indicating that the series is cyclical. The seasonality is further, confirmed by the
PACF which shows large values for the first four lags while the rest are not different
from zero. The results conform with those in question 1 (a). The series does not have
trend but is having seasonality and cyclic behaviour therefore, differencing is
appropriate to make the series stationary.
(b) The figure 7 shows plot of the Box-Cox transformed data.

FORECASTING VOLUME OF BEER IN NEW ZEALAND 7
The plot does not show any change in the series therefore, Box-Cox transformation does help
in making the series stationary. The Aurgmented dickey fuller tests shows that the tranformed
series is not stationary since the p-value for the ADF test is 0.6645 which is larger than 0.05.
The figure 8 shows the plot of the 1st diffrenced series.
The first difference makes the series stationary as shown in figure 8. The ADF test on the
differenced series shows that the differenced series is stationary p-value = 0.01 whish is less
than 0.05. Therefore, the transformation needed for this series is differencing of order 1.

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Question 3: ARIMA model
(a) First differencing makes the series stationary thus d = 1, for p and q we examine the
acf and pacf plot of the differenced series. Figure 9 shows the ACF and PACF plot of
the differenced series.
The acf decays exponential an indication that the series is an autoregressive series.
The order of the AR is determined by the PACF (Shumway & Stoffer, 2017) . Figure
9 (b) shows that the PACF cuts off at lag 4 and confirming that the series is an AR(4).
Therefore, the approriate ARIMA with p = 4, d = 1, and q = 0. That is ARIMA(4, 1,
0).
(b) Constant term should not be included in the model since d = 1 >0.
(c) The proposed model is of the form:
∑
i=0
4
αi Bi ( 1−B ) Xt =ε t
Where; α 0=1, B- backshift operator, Xt – series (volume of beer availablein litres),
and ε t – white noise.
(d) Table 2 shows the estimates of the model

FORECASTING VOLUME OF BEER IN NEW ZEALAND 9
Table 2: Results of the ARIMA(4,1,0) model
Statistic ar1(α1) ar2(α2) ar3(α 3) ar4(α 4)
Coefficient -1.1191 -0.8658 -0.4751 -0.4208
Std. Error 0.2282 0.3548 0.3419 0.2266
Sigma2
Log likelihood
AIC
0.03101
5.8200
-1.6400
Source: author (2019)
The estimated model is of the form:
Xt =−1.1191 Xt −1−0.8658 Xt −2−0.4751 Xt−3−0.4208 Xt−4 (1)
The figure 10 shows the plot of the residual diagnostics. The plots help in verifying the
following assumptions: (1) Constant variance (homoskedasticity), (2) normality of the
residuals, (3) independence of residuals, and (4) autocoreelations (Chatfield, 2016).

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From (a), the plot of residuals are not showning any outright pattern and they are close to the
zero line and indication that the variances are constant. The assumption of constant variance
is valid. Next, plot (b) shows that majority of the points lie on or close to the straight red line
implying the residuals are assumed to be normally distributed. The plot (c) reemphasize the
normality the residuals. Plot (d) shows that acf of the lags greater or equal to 1 are all not
significantly different from zero implying the assumption of independence of residuals and
autocorrelation is valid. The four assumptiosn are met therefore, the model fits the data
adequately and can be used for forecasting.
(e) The forcast values are shown in table 3.
Table 3: Forecast values for model
Time Forecast Lower
80%CI
Upper
80%CI
Lower
95%CI
Upper
95%CI
2019 Q1 3.1916 2.9659 3.4173 2.8465 3.5367
2019 Q2 3.2404 3.0131 3.4678 2.8928 3.5880
2019 Q3 3.3848 3.1497 3.6200 3.0252 3.7444
2019 Q4 3.1263 2.8797 3.3729 2.7492 3.5034
Source: author (2019)
(f) The figure 11 shows the plots of the forecast and actual values

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(g) The autoarima model output is presetented in the screenshot below
The fitted model does not have any MA or AR component hence to approriate for this data.
Therefore, the ARIMA(4, 1, 0) is the most appropriate model for the data.
(h) ARIMA is best sinec the forecasts are not constant but are having a similar trend as
the original series.

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References
Angadi, M. C., & Kulkarni, A. P. (2015). Time Series Data Analysis for Stock Market
Prediction using Data Mining Techniques with R. International Journal of Advanced
Research in Computer Science, 6(6).
Chatfield, C. (2016). The analysis of time series: an introduction. Chapman and Hall/CRC.
Shumway, R. H., & Stoffer, D. S. (2017). Time series analysis and its applications: with R
examples. Springer.