Linear Predictability Through A Linear Relationship

ECONOMETRICS
Problem 1
P(Z≥0.3)=1-P(Z≤ 0.3 ¿=1-0.3821=0.6179
P(Z≤0.7)=1-P(Z ≥ 0.7 ¿=1−0.2420=0.758
P(0.3≥Z≥0.7)= P(Z ≤0.7- P(Z≥0.3)
=0.758-0.6179
=0.1401
Problem 2
Assume that the return is Y.
Let Y=ex
InY=Inex =x
Hence InY=x
But P(Y<x)= 1
x . √2 ⫪σ2 exp{(-Inx-u)/√ 2 σ2}
Given that;
σ =4 and μ =0
P(Y<0.7) = 1
0.7 √2 ⫪.16 e-{In(0.7)-0}/
√32
= 1
7.0185 e-2.0895 = 1
7.0185*0.1237=0.01763
Problem 3
Skewness is the measure of symmetry of a distribution of data that is shifted either to the left or
right of the central mean data.
Investors may be more interested in skewness of data. The investors would say that the log
returns are asymmetric when the collected data are skewed from the normal mean distribution
and shifted more either to the left or right.

Problem 4.
Kurtosis is the measure of concentration of data, that is,the lightness or heaviness of data as its
shift towards the tails or outliers. High tailed distributions are characterized by outliers. Low tail
distribution are characterized by the absence of outliers.
The investors would conclude on leptokurtic when they realize that more data is concentrated at
the tails that around the mean. This case is more preferred by the investors as the return is about
three times the mean return.
Problem 5
$r_t$ is normally distributed. Its sample is obtained from two means of equal distribution and
equal variances. Hence its return is normally distributed around its mean.
Consider two hypothesis:
a)The null hypothesis
b)The alternative hypothesis.
H0:There is a significant difference of returns in the stock market.
H1:There is no significant difference of return in the stock market.
From the two hypotheses, you can calculate the t-statististic test with alpha value =0.05
When the p-value of return is less than the alpha value α=0.05, reject the null hypothesis i.e.
there is insignificant difference of return. Otherwise, accept the alternative hypothesis
Problem 6
The standard normal distribution tests includes:
i)Kolmogorov-Smirnov (K-S) test
ii)Lilliefors Corrected K-S test
iii)Cramer-Von Mises Test.
iv)Anderson-Darling Test

Hypotheses:
Null hypothesis:
H0:The mean of the two sample distributions are equal.
H1:The mean of the two sample distribution are not equal
Accept the null hypothesis. The means of standard normal distributions are always equal.
Problem 7
Reject the null hypothesis. The p-value 0.005 <0.05=α alpha
The difference in mean of the two returns is insignificant. Hence there is no enough evidence to
support the null hypothesis that the difference is significant.
Problem 8
quartile is a part of the total distribution whose frequency of distribution is categorized into equal
segments whereby each segment is a representative of the whole population.
Given that α is the probability is the standard normal distribution and Zα the quartile values.
If α =0.1,
Zα=Z0.1 =0.257 from a standard normal z table.
-Zα=Z0.1=-0.257
A two-tailed distribution with α=0.1
Problem 9
qq-plot is a set of two quantiles against each other whereby ,the line of best fit is determined.
QQ plot is used to investigate any set of data that may had been drawn from the same the
theoretical distribution.
QQ-plot is mainly used to detect shift in symmetry and outliers which the normal distribution
cannot help realize.
During return analysis ,it is realized to be more efficient than other test when used to find out the
degree of normal distribution of data. It also highlights major discrepancy on the tail distribution.