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# Quantitative Anlysis Essay

1452 words - 6 pages

Question 1
(a)
rc, t is the increase rate of real US private consumption.
re, t is the real return of the share price index of the New York Stock Exchange after CPI adjustment.
rf, t is the real return of the Long-term government bond yield after CPI adjustment.

(b)

The plotting graph indicated that there is no obvious trend of the increase rate of real US private consumption. The rate reached its lowest point in 1974.

There is no clear trend of real return of the share price index of the New York Stock Exchange after CPI adjustment. It is worth to notice that it reached its lowest return in 1974 too, which is about -0.38.

The graph of the real return of the Long-term ...view middle of the document...

1322172 = 0.016982
The variance of the real return of the share price index of the New York Stock Exchange after CPI adjustment is 0.016982 which measures the squared deviation of re, t from its expected mean of 0.026054.

(c)

From the table above we can obtain that δec = 0.000815
The covariance measures how much two variables (re, t and rc, t ) change together.
Question 3
(a)

After estimating the equation, β0 = -0.070244 and β1 = 2.824587
So the estimated equation is:
re, t =-0.070244 +2.824587* rc, t + μt
(b) From the question we know that β1 = ϒ, therefore the coefficient of relative risk aversion ϒ = 2.824587, indicating that the relative risk aversion is not high.
(c) As calculated from question 2:
δ2c = 0.0172322 =0.000288
δ2e = 0.1322172 = 0.016982
δec = 0.000815
The question also tells us that:
β0 = -log δ – 0.5(δ2e + ϒ2 δ2c - 2 ϒ δec)
Since δ is the only unknown number in this equation, we can calculate it after put all known number into the equation.
So, -log δ = -0.02732, thus δ = exp(0.02732) = 1.0649269
Therefore the discount factor parameter δ = 1.0649269. And it is the factor multiplied to discount future cash flow back to present.
Question 4
(a)

H0: ϒ = 0
H1: ϒ ≠ 0
t-statistic = 2.824587/1.241818 = 2.274558
As P-value = 0.0296, which is smaller than 5%, we reject H0 at 5% and concluding that ϒ ≠ 0.
(b) H0: ϒ = 1
H1: ϒ ≠ 1
t-statistic = (2.824587 – 1)/ 1.241818 = 1.4693 which is smaller than 1.96, we do not reject H0 at 5%, concluding that ϒ = 1.
(c)

This is the line graph of the residuals. It is clear that the residual is not random: A string of negative residuals followed by a string of positive residuals. Besides, there is persistent departure from the mean. Therefore we believe autocorrelation exists. And if autocorrelation exists, the residuals are correlated with each other, and then the assumptions for liner regression model are not satisfied. And the hypothesis tests conducted in part (a) and (b) are not reliable.
Question 5
(a)

After estimating the equation, β0 = -0.181332, β1 = 3.005210 and β2 =0.005829
So the estimated equation is:
re, t =-0181332 +3.005210* rc, t + 0.005829*trendt + ѡt
(b) From the question we know that β1 = ϒ, therefore the coefficient of relative risk aversion ϒ = 3.005210 which is also not a high level of risk aversion
(c) As calculated from question 2: δ2c = 0.0172322 =0.000288
δ2e = 0.1322172 = 0.016982
δec = 0.000815
The question also tells us that:
β0 = -log δ – 0.5(δ2e + ϒ2 δ2c - 2 ϒ δec)
Since δ is the only unknown number in this equation, we can calculate it after put all known number into the equation.
So, log δ = 0.075563, thus δ = exp(0.075563) = 1.190043
Therefore the discount factor parameter δ = 1.190043. And it is the factor multiplied to discount future cash flow back to present....

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