Question

Carolina Wood Products, Inc., a major manufacturer of household furniture, is interested in predicting expenditures on furniture (FURN) for the entire United States.

Period	FURN ($Billions)	Time
Mar-07	61.1	1
Jun-07	59.8	2
Sep-07	59	3
Dec-07	58	4
Mar-08	56.2	5
Jun-08	58.1	6
Sep-08	59.2	7
Dec-08	61.4	8
Mar-09	63.7	9
Jun-09	67.4	10
Sep-09	71.1	11
Dec-09	74.1	12
Mar-10	77.3	13
Jun-10	80.2	14
Sep-10	82.4	15
Dec-10	85.7	16
Mar-11	88.9	17
Jun-11	92.3	18
Sep-11	95.2	19
Dec-11	99.6	20
Mar-12	100.4	21
Jun-12	104.4	22
Sep-12	108.3	23
Dec-12	110.7	24
Mar-13	111.8	25
Jun-13	113.2	26
Sep-13	116.4	27
Dec-13	117.2	28
Mar-14	122.8	29
Jun-14	127.4	30
Sep-14	129.2	31
Dec-14	132.7	32
Mar-15	136.7	33
Jun-15	138.5	34
Sep-15	138	35
Dec-15	138.7	36
Mar-16	144.4	37
Jun-16	143	38
Sep-16	142.7	39
Dec-16	139.3	40
Mar-17		41
Jun-17		42
Sep-17		43
Dec-17		44

1. Estimate the bivariate linear trend model for the data where TIME=1 for 2007, Q1 through TIME=40 for 2016, Q4.

   FURN = a + b(TIME)

2. Evaluate this model in one paragraph, with particular emphasis on its usefulness in forecasting.
3. Prepare a time-trend forecast of furniture and household equipment expenditures for 2017 based on the model in (b).

   Period	Time	Trend Forecast
   2017 Q1	41
   2017 Q2	42
   2017 Q3	43
   2017 Q4	44

Answer 1

a)

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	820	3966.5	5330	36121.3	13746.35
mean	20.50	99.16	SSxx	SSyy	SSxy

sample size ,   n =   40          
here, x̅ = Σx / n=   20.50   ,     ȳ = Σy/n =   99.16  
                  
SSxx =    Σ(x-x̅)² =    5330.0000          
SSxy=   Σ(x-x̅)(y-ȳ) =   13746.4          
                  
estimated slope , ß1 = SSxy/SSxx =   13746.4   /   5330.000   =   2.5791
                  
intercept,   ß0 = y̅-ß1* x̄ =   46.2919          
                  
so, regression line is   Ŷ =   46.2919   +   2.5791   *x

b)

SSE=   (SSxx * SSyy - SS²xy)/SSxx =    668.715
      
std error ,Se =    √(SSE/(n-2)) =    4.195
      
correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.9907
      
R² =    (Sxy)²/(Sx.Sy) =    0.9815

As we can see R square = 98.15 % means 98.15% variance in FURN can be explained by time period. Model is good.

C)

Predicted Y at X=   41   is                  
Ŷ =   46.29192   +   2.579053   *   41   =   152.033

Predicted Y at X=   42   is                  
Ŷ =   46.29192   +   2.579053   *   42   =   154.612

Predicted Y at X=   43   is                  
Ŷ =   46.29192   +   2.579053   *   43   =   157.191

Predicted Y at X=   44   is                  
Ŷ =   46.29192   +   2.579053   *   44   =   159.770

THANKS

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