Question

1. Suppose we have the following annual sales data for an automobile dealership:

Year                Sales                Trend

2009                121                     1

2010                187                     2

2011                165                     3

2012                134                     4

2013                155                     5

2014                167                     6

2015                200                     7

2016                206                     8

2017                221                    9

2018                231                 10

We want to forecast sales for 2019 and 2020 using either a simple trend model or a quadratic trend model. Use a within sample forecasting technique to determine the best model using the RMSE measure discussed in lecture. Once this model has been determined, provide actual forecasts for 2019 and 2020.

Answer 1

Since RMSE of quadratic trend model is less than simple linear trend model so quadratic model is better and using this model the forecast for year 2019 and 2020 are given as

using estimated quadratic model, y^=146.18-1.17*t+1.01*t^2

for year 2019, t=11, and y^=146.18-1.17*11+1.01*11*11=255.52

for year 2020, t=12, and y^=146.18-1.17*12+1.01*12*12=277.58

the simple linear trend the model is given as y^=123.93+9.96*t

RMSE=19.42

regression analysis

Observation	y	y^	y-y^=Residuals(e)	e2
1	121	133.89	-12.89	166.18
2	187	143.85	43.15	1862.05
3	165	153.81	11.19	125.30
4	134	163.76	-29.76	885.87
5	155	173.72	-18.72	350.48
6	167	183.68	-16.68	278.18
7	200	193.64	6.36	40.50
8	206	203.59	2.41	5.79
9	221	213.55	7.45	55.48
10	231	223.51	7.49	56.11
sum=	1556	1563.490909	-7.490909091	3769.838
			RMSE=sqrt(e2/n)=	19.42

ANOVA
	df	SS	MS	F	Significance F
Regression	1	8180.148	8180.148	17.10455	0.003272
Residual	8	3825.952	478.2439
Total	9	12006.1
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	123.9333	14.93923	8.295833	3.36E-05	89.48341	158.3833
t	9.957576	2.407674	4.135765	0.003272	4.405469	15.50968

using estimated quadratic model, y^=146.18-1.17*t+1.01*t^2

RMSE=18.07

Observation	y	y^	y-y^=Residuals(e)	e2
1	121	146.03	-25.03	626.36
2	187	147.89	39.11	1529.28
3	165	151.78	13.22	174.68
4	134	157.70	-23.70	561.47
5	155	165.63	-10.63	113.00
6	167	175.59	-8.59	73.75
7	200	187.57	12.43	154.55
8	206	201.57	4.43	19.61
9	221	217.60	3.40	11.58
10	231	235.65	-4.65	21.58
sum=	1556	1551.35	4.65	3264.303
			RMSE=sqrt(e2/n)=	18.07

ANOVA
	df	SS	MS	F	Significance F
Regression	2	8720.217	4360.108	9.288449	0.010724
Residual	7	3285.883	469.4119
Total	9	12006.1
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	146.1833	25.48241	5.736637	0.000708	85.92701	206.4397
t	-1.16742	10.64253	-0.10969	0.91573	-26.333	23.99816
t^2	1.011364	0.942888	1.072623	0.31902	-1.21821	3.240939