Question

1. This question is to give you a feel for the actual calculations involved with OLS regressions. In this case, the independent variable is time (t). We will talk more about adding time as a variable later. For now, just treat it like any other independent variable x. You should do the calculations manually without a computer. It is very important that you show your work as answers will vary a bit due to rounding errors so I need to know that you followed the correct methodology. I recommend not trying to type up your answers as typing this many numbers will likely involve at least some typos. Consider the following data (continued on next page):

ti	yi
1	104
2	250
3	310
4	410
5	510
6	610
7	680
8	818
9	943

(a) Find the values of b0 and b1.

(b) Find the Coefficient of Determination.

(c) Find the sample correlation coefficient.

(d) Find the estimated standard deviation of b1 and the corresponding t-statistic. At the 1% level of significance, can you reject the null hypothesis? Make sure you state the null and alternative hypotheses.

Please try to include the entire calculations, I need to understand how to solve the question. Also, we can't use Excel to solve this question it must be manual. Thank you SO much in advance.

Answer 1

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
1	104	16.00	168921.00	1644.00
2	250	9.00	70225.00	795.00
3	310	4.00	42025.00	410.00
4	410	1.00	11025.00	105.00
5	510	0.00	25.00	0.00
6	610	1.00	9025.00	95.00
7	680	4.00	27225.00000	330.0000
8	818	9.00	91809.00000	909.000
9	943	16.00	183184.00	1712.00

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	45.00	4635.00	60.00	603464.00	6000.0
mean	5.00	515.00	SSxx	SSyy	SSxy

a)

sample size ,   n =   9          
here, x̅ = Σx / n=   5.000   ,     ȳ = Σy/n =   515.000  
                  
SSxx =    Σ(x-x̅)² =    60.0000          
SSxy=   Σ(x-x̅)(y-ȳ) =   6000.0          
                  
estimated slope , ß1 = SSxy/SSxx =   6000.0   /   60.000   =   100.00000
                  
intercept,   ß0 = y̅-ß1* x̄ =   15.00000          

b) R² =    (Sxy)²/(Sx.Sy) =    0.9943

c) correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.9971

d)  

Ho:   ß1=   0
H1:   ß1╪   0

SSE=   (SSxx * SSyy - SS²xy)/SSxx =    3464.0000
      
std error ,Se =    √(SSE/(n-2)) =    22.2454

estimated std error of slope =Se(ß1) = Se/√Sxx =    22.245   /√   60.00   =   2.8719

t stat = estimated slope/std error =ß1 /Se(ß1) =    100.0000   /   2.8719   =   34.8206
Degree of freedom ,df = n-2=   7  
p-value =    0.000000  
decison :    p-value<α , reject Ho