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.

Problem 10
Efficient market hypothesis is a theory that states that the stock prices is dependent on the
information available in the market.
You can assume that the stock price is Pt.
Pt=Et[Mt+1(Pt+1+Dt+1)]
Let the stochastic discount factor be a constant.
Pt=MEt[Pt+1]
LogPt=LogM +Et[logPt+1]
This implies a random walk on the log of the stock prices.
Problem 11
Random walk is a mathematical model that describes a path of succession of random steps.
The means and variances of a random walk are increasing adding up .It is assumed that the mean
of random walk is zero.However,when zero adds up,it goes back to zero.
Assume Yt to be the drift.
Yt-Yt-1=μ +Ƹt
Problem 12.
From the past observation of returns, log-prices are predictable. Active investors would use the
past stock prices to reveal some patterns of returns in the market. The information obtain is
analyzed and forecasted to predict the future outcome.
Problem 13
Changes in trends and patterns of stock is an indicator of the changes in stock prices from the
random walk trajectory process. Increases in stock prices relatively leads to increase in retain and
increase ion returns relatively leads to increases in stock prices. The investment advisors cannot
effectively contribute to the value of change of returns of any portfolio due to random walk
process.
Problem 14.

Efficient Market Hypothesis can be used to predict the future returns using available information
which correlate to the stock prices and returns. This is an evidence to show that r_t model is
predictable with an efficient market hypothesis
Problem 15.
The asset returns are also predictable. Useful data is available to construct the patterns and draw
relevant information. Investors would use the analysis of return in decision making for future
preparation and adjustments.
Problem 16
Consider Yt=b0 +b1*Y(t-1) +Ƹt
When b1=1,it implies that the model has got a unit root hence the level of mean-reverting is not
defined.
When b1 and b0=0,it applies that the model is a random walk.
When b1 =1 and b0≠0,it implies that the model is a random walk but with a drift.
Hence,any model with a unit root:
Is a random walk when b0=0.
Is a random walk with drift when b0≠ 0
Problem 17
LogP3=logP2+logP2 + Ƹ3
Problem 18
A white noise process is a random process of random variables which are not correlated and with
mean equal zero and variance of a finite value.
You can consider two processes in question:

E[X(t)] =0 for E[X(t))=S1
E[X(t)xh=0 for t ≠ h
Then X(t) is a white noise.
Problem 19
Assume that Yt is weakly stationary.
Yt is a weakly stationary if and only if:
i)μt(t) is independent of t
ii)Yx(t+h,t) is independent of t for each h
iii)Mean of Xt is constant
iv)Variance of Xt is also constant.
Problem 20
The log of stock prices are not always stationary.Stationarity depends on the previous data and
would only be gotten from the test of stationary of returns. Therefore, you need to assess and
ensure that the stock prices are stationary in order to avoid getting spurious result from the
analysis. On-stationary series can be transformed by either DE trending or differencing to help
remove deterministic and variance trend.
Problem 21
Autocorrelation of short periods of returns e.g. the daily returns are relatively small for
accountability. Their autocorrelation tends towards zero for an efficient market hypothesis. As a
result, independence is rejected for daily returns but accepted for long-term returns of weekly,
monthly and yearly. You can conclude that decreasing returns are least consistent with an
efficient market hypothesis while increasing returns are most efficient with an increasing market
hypothesis.
Problem 22
Yt=β0 +β1Yt-1 +Ƹt
Where by;
Yt is the return,β1 =the coefficient of correlation,Ƹt =Errors.

But β1=0.1
Given r_5 and r_8,
ρ (0)=1
ρ (1)= 5
15=0.3333
ρ (5)=a1ρ(1) -a1ρ(0)≈0.1(0.3333)-0.1(1)≈-0.06667
ρ (6)=a1ρ(6) -a1ρ(1)≈0.1(-0.06667)-0.1(0.3333)≈-0.03997
ρ (7)=a1ρ(7) -a1ρ(1)≈0.1(-0.03997)-0.1(0.3333)≈-0.03733
ρ (8)=a1ρ(8) -a1ρ(1)≈0.1(-0.03733)-0.1(0.3333)≈-0.04066
Problem 23
Yt=β0 + β1Yt-1 +Ƹt
ρ (0)=1
ρ (1)= 0.3333.
ρ (2)=a1ρ(1) –a1ρ(0)≈0.5(0.3333)-0.5(1)≈-0.33335
Problem 24
Consider Xt=σ +ϕXt-1 +wt
Where;
wt N(0,σ) and x1,x2,….. are weakly stationary. And |ϕ|<1
Mean=E(Xt)= σ
1−ϕ = 0.8
1−0.4 = 1.3333
Problem 25
Var(X2)= σ2 w
1−ϕ2 1 = 42
1− ( 0.4 ) 2 =19.04765

Autocovariance
γ = σ 2
1−ϕ2 ϕ = 42
1− ( 0.4 ) 2 =7.619
Problem 26
The correlation decreases significantly with increase to the number of lags. The pattern will often
appear random because the lags are serially correlated. This provides a linear predictability of
the variables through a linear relationship.
Problem 27
The process would be stationary if:
i)E(Yt)=Constant
ii)Var(Yt)=constant
iii)Cov(Yt1,Yt2)=E[(Yt1-ut2)(Yt2-ut2)]=r(t1-t2)=f(t1-t2)
Problem 28
A random walk is a special autogressive process whose gradient parameter is equal to
1.Similarly,its auto covariance function progressively decays to zero, thus, a long lasting impacts
on the current values.
Problem 29
Random walk is not stationary. The auto covariance function changes on different lags. If the lag
prices follow a random walk, it is not consistent with the efficient market hypothesis. The market
would be unfavorable as those who access the right information would take the advantage of
controlling the monopoly.
Problem 30
Returns are mainly serially correlated .The returns of different periods responds to their
respective factors. The stock returns are correlated to their return period of t-1,whereby there is
the prediction of high returns within the stocks and a prediction of low returns away from the
stock. The current stock prices depends on the previous returns.
Problem 31

Consider Xt =σ +ϕXt-1 +wt
Where wt is a white noise i.e wt N(0,1)
E(Xt)= σ
1−ϕ = 1
1−0.6 =2.5
E(Xt+1)= 1
1−0.6 =2.5
E(Xt+2)= 1
1−0.6 =2.5
There is a dependence on daily returns. i.e. Xt is stationary throughout the week hence E(Xt) is
the same for all period t.