(a) The simple linear regression equation
= β0 + β1 Xi, β0 = -74.5926, β1 =0.0954
= -74.5926+ 0.0954Xi
Where is personal consumption expenditure in billion of dollars Xi is disposable personal income (PCE) in billions of dollars (*PCE is used as a short form of disposable personal income in the following report) (b) Interpret the coefficient
The slope of disposable income (β1= 0.0953) tell us for each additional billion of dollars in disposable personal income, the PCE increase by 0.0953 billion dollars
(c) Comment on the significance of model (α = 0.05) Hypotheses:
H0: β1 =0 H1 : β1 ≠ 0
Decision rule: reject H0, if |tcalc|> |t(α/2, n-k-1)|
Where tcrit = t (0.025, 98) =1.9845
Test statistic: t = = = 48.368
Decision: Reject H0 because t calc > t crit
Conclusion: There is sufficient evidence to conclude that there is significant relationship between disposable personal income and PCE at 5% level of significance. (d) The coefficient of determination
It implies 95.98% of the variation in PCE can be explained by the variation in disposable personal income. The rest 4.02% of the variation in PCE is due to factors other than disposable income. (e) Test the assumptions
1. Linear test
We can find the assumption of linearity is not violated because this plot is approximately a straight line in this diagram.
2. Independent test
H0:=0 H1: ≠ 0
Decision rule: Reject H0 if D< dL (α,k,n)
Do not reject H0 if D > du
No decision if dL < D < du
Where dL (0.05,1,100)=1.65 and du(0.05,1,100)=1.69
Test statistic: Dcalc=6192.4602748/21774.3187301= 0.2844
Decision: Reject H0 because D < dL
Conclusion: There is sufficient evidence to conclude that there is positive autocorrelation at 5% level of significance. We can evaluate the assumption of independence is violated.
3. Normality test
We can evaluate the assumption of normality is not violated because the graph is normally distributed with bell-shaped and a little slight skew. 4. Equal variance test
We can point out the assumptions of equal variance is violated as the wide spread onwards which have a fan shape as a whole. In summary, the assumption test is failed as independent & equal variance test is violated, this regression model is not considered as an appropriate model to estimate PCE.
(a) the reason of proper explanatory variables
Besides disposable income, bank prime loan rate and population are considered as contributing factors which affect personal consumption expenditure (PCE).
In terms of bank prime loan rate which has a strong relationship with our daily expenditure. It would probably imply the negative relationship between this variable with personal consumption expenditure. For instance, if bank loan rate has increased, it would eliminate the intensives of people to borrow money from bank for their spending. Hence, personal consumption expenditure would decrease. When it comes to population factors, it is clearly implied that the increase of population would bring the personal consumption expenditure significantly by the large amount of basic demands (e.g., furniture for new families). While the population declines, it would result in a reduction in PCE. (b) Report
1. Multiple regression equation
=182.6974 + 0.1523- 4.3897- 0.0018
: Disposable income in billions of dollars
: Bank loan rate in percentage
: Population in thousands of people
2. Interpret the coefficient:
The intercept (b0=182.6974), when X1, X2 and X3 equal to zero(though it is impossible all the variables equal zero, we use extrapolation to estimate), the personal consumption expenditure is 182.6974 The slope of disposable income (= 0.1523) indicates that when disposable income increases in 1 billion dollar, the personal consumption...
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