Question

Date	Sales (000,000)
Sep-12	206
Oct-12	245
Nov-12	185
Dec-12	169
Jan-13	162
Feb-13	177
Mar-13	207
Apr-13	216
May-13	193
Jun-13	230
Jul-13	212
Aug-13	192
Sep-13	162
Oct-13	189
Nov-13	244
Dec-13	209
Jan-14	207
Feb-14	211
Mar-14	210
Apr-14	173
May-14	194
Jun-14	234
Jul-14	156
Aug-14	206
Sep-14	188
Oct-14	162
Nov-14	172
Dec-14	210
Jan-15	205
Feb-15	244
Mar-15	218
Apr-15	182
May-15	206
Jun-15	211
Jul-15	273
Aug-15	248
Sep-15	262
Oct-15	258
Nov-15	233
Dec-15	255
Jan-16	303
Feb-16	282
Mar-16	291
Apr-16	280
May-16	255
Jun-16	312
Jul-16	296
Aug-16	307
Sep-16	281
Oct-16	308
Nov-16	280
Dec-16	345

1. Calculate both the three-month and the five-month averages for these data.
2. Plot the data to examine the possible existence of trend and seasonality.
3. Prepare the following two (2) separate forecasting models to examine the restaurant sales data using monthly data:
   1. An exponential smoothing model (α=0.38, smoothing constant for the level)
   2. Holt’s model (α=0.04, smoothing constant for the level; β=0.07, smoothing constant for the trend)

d. Examine the accuracy of the forecast given by each model ( four models: MA 3, MA 5, Simple Exponential smoothing and Holt's Method) by calculating the root mean square error (RMSE) for each during the historical period.

e. Which model does minimize the RMSE? Carefully explain which characteristics of the original data caused one of these models to minimize the RMSE.

f. Using Holt's method forecast 12 months of restaurant sales for 2017.

Answer 1

(a)

Moving averages - 3 period moving average
Forecast
212
199.6667
172
169.3333
182
200
205.3333
213
211.6667
211.3333
188.6667
181
198.3333
214
220
209
209.3333
198
192.3333
200.3333
194.6667
198.6667
183.3333
185.3333
174
181.3333
195.6667
219.6667
222.3333
214.6667
202
199.6667
230
244
261
256
251
248.6667
263.6667
280
292
284.3333
275.3333
282.3333
287.6667
305
294.6667
298.6667
289.6667

Moving averages - 5 period moving average
Forecast
193.4
187.6
180
186.2
191
204.6
211.6
208.6
197.8
197
199.8
199.2
202.2
212
216.2
202
199
204.4
193.4
192.6
195.6
189.2
176.8
187.6
187.4
198.6
209.8
211.8
211
212.2
218
224
240
250.4
254.8
251.2
262.2
266.2
272.8
282.2
282.2
284
286.8
290
290.2
300.8
294.4

(b) The plot is:

(c) (i)

Forecast
78.28
141.6336
158.1128
162.25
162.155
167.7961
182.6936
195.35
194.457
207.9633
209.4973
202.8483
187.326
187.9621
209.2565
209.159
208.3386
209.3499
209.597
195.6901
195.0479
209.8497
189.3868
195.6998
192.7739
181.0798
177.6295
189.9303
195.6568
214.0272
215.5369
202.7929
204.0116
206.6672
231.8736
238.0017
247.121
251.255
244.3181
248.3772
269.1339
274.023
280.4743
280.294
270.6823
286.383
290.0375
296.4832
290.5996
297.2118
290.6713

(ii)

Smoothed Forecast, Ft
206
206
207.56
206.6576
205.1513
203.4252
202.3682
202.5535
203.0914
202.6877
203.7802
204.109
203.6246
201.9596
201.4413
203.1436
203.3779
203.5228
203.8218
204.069
202.8262
202.4732
203.7342
201.8249
201.9919
201.4322
199.8549
198.7407
199.1911
199.4234
201.2065
201.8782
201.0831
201.2798
201.6686
204.5219
206.261
208.4905
210.4709
211.3721
213.1172
216.7125
219.324
222.191
224.5034
225.7233
229.1743
231.8474
234.8535
236.6993
239.5514
241.1693

(d)

	Trend adjusted exponential smoothing	Exponential smoothing	Moving averages - 5 period moving average	Moving averages - 3 period moving average
MSE	1823.36	1309.198	835.0165	867.6236