Pr1. Time Series.						Written 20%
						Test Grade	0
The data in the table below represent the annual revenues(in billion of dollars)
of McDonald's Corporation over the period from 1975 to 2012.
A) Calculate a five-year moving average to the data (add a column to the table)
Year	Coded Year	Revenues	Forecast
1975	0	1
1976	1	1.2
1977	2	1.4
1978	3	1.7
1979	4	1.9
1980	5	2.2	1.44
1981	6	2.5	1.68
1982	7	2.8	1.94
1983	8	3.1	2.22
1984	9	3.6	2.5
1985	10	3.8	2.84
1986	11	3.9	3.16
1987	12	4.9	3.44
1988	13	5.2	3.86
1989	14	5.9	4.28
1990	15	6.4	4.74
1991	16	6.7	5.26
1992	17	7	5.82
1993	18	7.4	6.24
1994	19	8.3	6.68
1995	20	9.8	7.16
1996	21	10.7	7.84
1997	22	11.4	8.64
1998	23	13.4	9.52
1999	24	14.5	10.72
2000	25	15.6	11.96
2001	26	14.9	13.12
2002	27	15.4	13.96
2003	28	17.1	14.76
2004	29	19	15.5
2005	30	20.5	16.4
2006	31	19.3	17.38
2007	32	22.4	18.26
2008	33	24.5	19.66
2009	34	23.6	21.14
2010	35	24.1	22.06
2011	36	29.5	22.78
2012	37	26.7	24.82
2013			25.68
B) Using a smoothing coefficient of W = 0.45, exponentially smooth the series
(add a column to the table, use data analysis to smooth)
			Exponential Smoothing
Year	Coded Year	Revenues	W=0.45
1975	0	1
1976	1	1.2	1
1977	2	1.4	1.09
1978	3	1.7	1.2295
1979	4	1.9	1.441225
1980	5	2.2	1.64767375
1981	6	2.5	1.896220563
1982	7	2.8	2.167921309
1983	8	3.1	2.45235672
1984	9	3.6	2.743796196
1985	10	3.8	3.129087908
1986	11	3.9	3.430998349
1987	12	4.9	3.642049092
1988	13	5.2	4.208127001
1989	14	5.9	4.65446985
1990	15	6.4	5.214958418
1991	16	6.7	5.74822713
1992	17	7	6.176524921
1993	18	7.4	6.547088707
1994	19	8.3	6.930898789
1995	20	9.8	7.546994334
1996	21	10.7	8.560846884
1997	22	11.4	9.523465786
1998	23	13.4	10.36790618
1999	24	14.5	11.7323484
2000	25	15.6	12.97779162
2001	26	14.9	14.15778539
2002	27	15.4	14.49178197
2003	28	17.1	14.90048008
2004	29	19	15.89026404
2005	30	20.5	17.28964522
2006	31	19.3	18.73430487
2007	32	22.4	18.98886768
2008	33	24.5	20.52387722
2009	34	23.6	22.31313247
2010	35	24.1	22.89222286
2011	36	29.5	23.43572257
2012	37	26.7	26.16464742
2013			26.40555608
c) Plot the results from a) and b) with the time series on a scatter plot.
d) Compute a quadratic trend forecasting equation and plot the predicted result with the data against the coded years.
Y	X
Revenues	Coded Year	X^2	Quadratic
1	0	1	0.09025703
1.2	1	1.44	0.498757525
1.4	2	1.96	0.908705531
1.7	3	2.89	1.326072025
1.9	4	3.61	1.739638805
2.2	5	4.84	2.162433461
2.5	6	6.25	2.588485015
2.8	7	7.84	3.017793467
3.1	8	9.61	3.450358815
3.6	9	12.96	3.911512487
3.8	10	14.44	4.338830612
3.9	11	15.21	4.753302086
4.9	12	24.01	5.313067375
5.2	13	27.04	5.768431006
5.9	14	34.81	6.309559604
6.4	15	40.96	6.821376125
6.7	16	44.89	7.293024244
7	17	49	7.76792926
7.4	18	54.76	8.27268917
8.3	19	68.89	8.928894812
9.8	20	96.04	9.820682706
10.7	21	114.49	10.55505389
11.4	22	129.96	11.23550532
13.4	23	179.56	12.53350071
14.5	24	210.25	13.48934092
15.6	25	243.36	14.48896831
14.9	26	222.01	14.50320327
15.4	27	237.16	15.17786466
17.1	28	292.41	16.57809044
19	29	361	18.21968851
20.5	30	420.25	19.69228979
19.3	31	372.49	19.22866551
22.4	32	501.76	21.96819991
24.5	33	600.25	24.15080484
23.6	34	556.96	23.76806018
24.1	35	580.81	24.60013829
29.5	36	870.25	30.23776863
26.7	37	712.89	27.79105567
SUMMARY OUTPUT
Regression Statistics
Multiple R	0.997512999
R Square	0.995032183
Adjusted R Square	0.994748308
Standard Error	0.611342825
Observations	38
ANOVA
	df	SS	MS	F	Significance F
Regression	2	2620.047782	1310.023891	3505.173962	4.82E-41
Residual	35	13.08090174	0.37374005
Total	37	2633.128684
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	0.072163155	0.240458073	0.300107018	0.765870804	-0.41599	0.560318996	-0.41599	0.560319
Coded Year	0.40053919	0.01915883	20.90624462	2.337E-21	0.361645	0.439433683	0.361645	0.439434
X^2	0.018093875	0.000911953	19.84079847	1.26918E-20	0.016243	0.019945238	0.016243	0.019945
	Y=0.072+0.400X+0.018X^2	quadratic forecasting model
e) Compute an exponential trend forcasting equation and plot the predicted results with the data against the coded years.
Y		X
Revenues	log(Y)	Coded Year	Exponential
1	0	0	1.476038177
1.2	0.079181246	1	1.61028543
1.4	0.146128036	2	1.756742615
1.7	0.230448921	3	1.916520238
1.9	0.278753601	4	2.090829806
2.2	0.342422681	5	2.280993017
2.5	0.397940009	6	2.488451775
2.8	0.447158031	7	2.71477913
3.1	0.491361694	8	2.961691199
3.6	0.556302501	9	3.231060186
3.8	0.579783597	10	3.524928571
3.9	0.591064607	11	3.845524601
4.9	0.69019608	12	4.195279182
5.2	0.716003344	13	4.576844317
5.9	0.770852012	14	4.993113211
6.4	0.806179974	15	5.447242206
6.7	0.826074803	16	5.942674719
7	0.84509804	17	6.48316735
7.4	0.86923172	18	7.072818365
8.3	0.919078092	19	7.716098772
9.8	0.991226076	20	8.41788622
10.7	1.029383778	21	9.183501989
11.4	1.056904851	22	10.01875133
13.4	1.127104798	23	10.9299675
14.5	1.161368002	24	11.92405975
15.6	1.193124598	25	13.00856575
14.9	1.173186268	26	14.19170874
15.4	1.187520721	27	15.48245986
17.1	1.23299611	28	16.89060616
19	1.278753601	29	18.42682489
20.5	1.311753861	30	20.10276435
19.3	1.285557309	31	21.9311323
22.4	1.350248018	32	23.92579227
24.5	1.389166084	33	26.10186871
23.6	1.372912003	34	28.47586163
24.1	1.382017043	35	31.06577174
29.5	1.469822016	36	33.89123696
26.7	1.426511261	37	36.97368127
	SUMMARY OUTPUT
	Regression Statistics
	Multiple R	0.987441402
	R Square	0.975040523
	Adjusted R Square	0.974347204
	Standard Error	0.068146237
	Observations	38
	ANOVA
		df	SS	MS	F	Significance F
	Regression	1	6.530906101	6.530906101	1406.338	1.8889E-30
	Residual	36	0.167180744	0.00464391
	Total	37	6.698086845
		Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
log(b0))	Intercept	0.169097591	0.021680206	7.799630256	3.02E-09	0.125128095	0.213067	0.125128	0.213067
log(b1)	Coded Year	0.037805273	0.001008109	37.50117225	1.89E-30	0.035760733	0.03985	0.035761	0.03985
	b0=10^0.16=	1.476038177
	b1=10^0.037=	1.090951071
	Y= 1.476*1.090^X	Exponential Forecasting Model
f) Compute a second -order autoregressive model, test for the significance of the second-order
	X	Y
Year	Coded Year	Revenues	Lag1	Lag2	Lag3
1975	0	1	#N/A	#N/A	#N/A
1976	1	1.2	1	#N/A	#N/A
1977	2	1.4	1.2	1	#N/A
1978	3	1.7	1.4	1.2	1
1979	4	1.9	1.7	1.4	1.2
1980	5	2.2	1.9	1.7	1.4
1981	6	2.5	2.2	1.9	1.7
1982	7	2.8	2.5	2.2	1.9
1983	8	3.1	2.8	2.5	2.2
1984	9	3.6	3.1	2.8	2.5
1985	10	3.8	3.6	3.1	2.8
1986	11	3.9	3.8	3.6	3.1
1987	12	4.9	3.9	3.8	3.6
1988	13	5.2	4.9	3.9	3.8
1989	14	5.9	5.2	4.9	3.9
1990	15	6.4	5.9	5.2	4.9
1991	16	6.7	6.4	5.9	5.2
1992	17	7	6.7	6.4	5.9
1993	18	7.4	7	6.7	6.4
1994	19	8.3	7.4	7	6.7
1995	20	9.8	8.3	7.4	7
1996	21	10.7	9.8	8.3	7.4
1997	22	11.4	10.7	9.8	8.3
1998	23	13.4	11.4	10.7	9.8
1999	24	14.5	13.4	11.4	10.7
2000	25	15.6	14.5	13.4	11.4
2001	26	14.9	15.6	14.5	13.4
2002	27	15.4	14.9	15.6	14.5
2003	28	17.1	15.4	14.9	15.6
2004	29	19	17.1	15.4	14.9
2005	30	20.5	19	17.1	15.4
2006	31	19.3	20.5	19	17.1
2007	32	22.4	19.3	20.5	19
2008	33	24.5	22.4	19.3	20.5
2009	34	23.6	24.5	22.4	19.3
2010	35	24.1	23.6	24.5	22.4
2011	36	29.5	24.1	23.6	24.5
2012	37	26.7	29.5	24.1	23.6
	SUMMARY OUTPUT
	Regression Statistics
	Multiple R	0.969101336
	R Square	0.9391574
	Adjusted R Square	0.90968955
	Standard Error	2.109544412
	Observations	38
	ANOVA
		df	SS	MS	F	Significance F
	Regression	2	2472.92229	1236.461145	555.6907	3.0404E-27
	Residual	36	160.2063946	4.450177627
	Total	38	2633.128684
		Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
a0	Intercept	-2.733198381	0.671135483	-4.072498697	0.000244	-4.09432423	-1.37207	-4.09432	-1.37207
a1	X Variable 1	0	0	65535	#NUM!	0	0	0	0
a2	X Variable 2	0.735649415	0.031207166	23.57309256	#NUM!	0.672358348	0.79894	0.672358	0.79894
p=2	Yi= -2.733+ 0-1+ 0.735Yi-2
tscore=a2/Sa2	23.57309256
t critical for 38-2*2-1=36
	tcritical is 2.042
t-diagram
		0
Since t score of the a2 23.57 is more than the critical value 2.042 the initial assumption is
denied. The second-order autoregressive parameter is significantly large
and the contribution of its term to the model is significant
g) If necessary, compute a first-order autoregressive model, test for the significance of the first-oder
autoregressive parameter, and plot the predicted results with the data against the coded years.
	SUMMARY OUTPUT
	Regression Statistics
	Multiple R	0.994005399
	R Square	0.988046733
	Adjusted R Square	0.987705211
	Standard Error	0.89518193
	Observations	37
	ANOVA
		df	SS	MS	F	Significance F
	Regression	1	2318.363537	2318.363537	2893.07	3.05696E-35
	Residual	35	28.04727406	0.801350687
	Total	36	2346.410811
		Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
a0	Intercept	0.434552578	0.244600786	1.776578827	0.084331	-0.06201342	0.931119	-0.06201	0.931119
a1	X Variable 1	1.021870052	0.018998364	53.78726503	3.06E-35	0.983301322	1.060439	0.983301	1.060439
p = 1	Yi = 0.434 + 1.021Yi-1
tscore = a1/Sa1 =	53.78726503
t critical for 32-2*1-1=29
	tcritical is 2.045
t-diagram
		0
Since t score of the a1 53.78 is more than the critical value 2.045 the initial assumption is
rejected. The first-order autoregressive parameter is significantly large
and the contribution of its term to the model is significant
h) predict the values for years 2013 and 2014 using the best model out of d and e.
And the apropriate autoregressive model of f) or g).
For year 2013 as year 38	0.434552578
For year 2014 as year 39	0.878608843
X	Y	Yhat
Coded Year	Revenue	Yi = 0.434 + 1.021Yi-1
0	1
1	1.2	1.456422629
2	1.4	1.66079664
3	1.7	1.86517065
4	1.9	2.171731666
5	2.2	2.376105676
6	2.5	2.682666691
7	2.8	2.989227707
8	3.1	3.295788722
9	3.6	3.602349738
10	3.8	4.113284764
11	3.9	4.317658774
12	4.9	4.419845779
13	5.2	5.441715831
14	5.9	5.748276846
15	6.4	6.463585882
16	6.7	6.974520908
17	7	7.281081923
18	7.4	7.587642939
19	8.3	7.996390959
20	9.8	8.916074006
21	10.7	10.44887908
22	11.4	11.36856213
23	13.4	12.08387117
24	14.5	14.12761127
25	15.6	15.25166833
26	14.9	16.37572538
27	15.4	15.66041635
28	17.1	16.17135137
29	19	17.90853046
30	20.5	19.85008356
31	19.3	21.38288863
32	22.4	20.15664457
33	24.5	23.32444173
34	23.6	25.47036884
35	24.1	24.55068579
36	29.5	25.06162082
37	26.7	30.5797191
The owner of a chain of ice cream store would like to study
the effect of atmosperic temperature on sales during the summer season.
A sample of 24 consecutive days is selected, with the results stored in the table below