Introduction:
As an alternative to the traditional Capital Asset Pricing Model (CAPM), Stephen Ross presented
the arbitrage pricing theory (APT) in 1976. The APT model makes the supposition that the
anticipated return on an asset is inversely proportional to that asset's exposure to multiple risk
factors, such as firm-specific components, macroeconomic circumstances, and financial
variables. Investors who are prudent and averse to risk want recompense for the risks they take
on, and they hold the view that keeping riskier assets should provide better returns than holding
less risky ones. The performance of competing assets in the market as well as inflation, interest
rates, and currency rates are important risk variables that might impact asset returns. We can
determine the main risk variables that have an impact on a company's performance, like
Microsoft, using regression analysis, and we may estimate the coefficients for each element.
The APT model also makes the assumption that investors are logical and capable of spotting
market mispricing’s that they may take advantage of through arbitrage. Because of their
susceptibility to various risk variables, assets that are mispriced in comparison to projected
returns will be promptly acquired or sold by investors looking to profit from the mismatch. This
makes it more likely that asset prices will represent their genuine underlying values and the
market will continue to function effectively. In comparison to the CAPM, the APT model
provides a more dynamic and adaptable system of threat evaluation. The APT allows for colorful
sources of threat that are specific to individual means or asset classes, whereas the CAPM
considers that there's only one source of methodical threat in the request, videlicet the request
portfolio. This implies that the APT can help investors produce further different portfolios and
make further accurate threat assessments since it can more represent the complex and constantly
changing character of the investing world.
Question No #1
a)
Solution
The results of our F-test indicate that at least one independent variable has a significant effect on
the dependent variable. This means that there is a 10% probability that our null hypothesis can be
rejected. In this case, we can conclude that at least one independent variable has a significant
effect on the dependent variable.
b) Solution
Restrictions are used in regression analysis to limit or restrict the values that can be used in
the model. Restrictions can be used to test the significance of coefficients, or to test whether
the value of a coefficient is significant.
Four restrictions are commonly used in regression analysis: the restrictions on DPROD,
DCREDIT, DMONEY, and DSPREAD. These restrictions correspond to coefficients of
DPROD, DCREDIT, DMONEY, and DSPREAD being jointly zero.
To express these restrictions in EViews, we can use the following syntax in the command
window:
restrict DPROD=0;
restrict DCREDIT=0
restrict DMONEY=0;
restrict DSPREAD=0
Restrictions are linear in coefficients.
c) Solution
In this case, R-squared is the proportion of variation in the dependent variable that is explained
by the independent variables in the regression model. In other words, it provides an indication of
how well the model fits the data. In this case, the R-squared value is 0.344910, which means that
the independent variables in the model explain 34.49% of the variation in the dependent variable
ERMSOFT.
Adjusted R-squared shows that there are some factors that do not fully explain why people use
ERMSOFT or why they leave it when they leave their job at work place (such as age). It
penalizes the addition of irrelevant variables to the model and rewards the addition of relevant
variables. Adjusted R-squared is always lower than R-squared if more than one independent
variable is used in the model. In this case, adjusted R-squared is slightly lower than R-squared
with a value of 0.334456, which means that there are some factors that do not fully explain why
people use ERMSOFT or why they leave it when they leave their job at work place (such as
age).
d)
Solution
The R-squared value for this particular model is 0.017166, indicating a relatively weak
association between the variables. It is noteworthy that the adjusted R-squared value is also low,
standing at 0.017166. Thus, excluding DPROD from the model does not enhance its accuracy
and should be included in the regression.
.
e)
Solutions
The Durbin-Watson test is a statistical procedure used to test the null hypothesis that there is no
first-order autocorrelation in the residuals of a regression. The null hypothesis is that the
residuals are independently and identically distributed. The test statistic is reported in the output
as 1.998055. The lower and upper critical values for the Durbin-Watson test with 383
observations and 8 regressors (including the constant) at the 5% significance level are
approximately 1.417 and 1.579, respectively.
The null hypothesis of the Durbin-Watson test is that there is no first-order autocorrelation in the
residuals (i.e., the errors are independently and identically distributed). The Durbin-Watson test
statistic is close to 2, which suggests that there is no strong evidence of first-order
autocorrelation in the residuals. Therefore, we cannot reject the null hypothesis at the 5%
significance level but we also cannot accept it at this time because we have not ruled out any
other possible explanations for our findings.
However, note that the Breusch-Godfrey test for serial correlation also suggests that there is no
serial
f) Solutions
In step a), we applied the Ordinary Least Squares (OLS) method to estimate the regression
coefficients. However, the OLS method assumes that the errors are homoscedastic, which means
that the error variance is constant for all observations. If this assumption is violated, OLS
estimates can be biased and inefficient. To detect heteroscedasticity, we can use the BreuschPagan-Godfrey test, which tests the null hypothesis of homoscedasticity against the alternative
hypothesis of heteroscedasticity. In our case, the test statistic is 0.445770 with a p-value of
0.8729, indicating that we cannot reject the null hypothesis of homoscedasticity at the 5%
significance level. Therefore, we do not have sufficient evidence to conclude that the errors are
heteroscedastic. However, even if we fail to reject the null hypothesis of homoscedasticity, we
can still use a more robust estimator, such as the Heteroscedasticity and Autocorrelation
Consistent (HAC) estimator. The HAC estimator adjusts for both heteroscedasticity and
autocorrelation in the errors, making it more appropriate for our data. The HAC estimator results
are presented in the second table. Although the coefficients of the variables are different from
those obtained in step a), the signs and statistical significance of the coefficients remain the
same. However, the standard errors are larger in the HAC estimator, indicating that the OLS
standard errors may have underestimated the true standard errors due to heteroscedasticity.
Therefore, we can conclude that the stock market index (ERSANDP) and the long-term interest
rate (RTERM) have a positive and statistically significant effect on the real estate prices, while
the other variables do not have a statistically significant effect. While the Breusch-PaganGodfrey test does not provide enough evidence to reject the null hypothesis of homoscedasticity,
it is still advisable to use the HAC estimator to obtain more reliable standard errors.
g)
Solutions
The regression model used in this study contains some variables that are highly correlated with
each other, with a correlation coefficient of 0.80 used as a threshold. For example, ERMSOFT
and ERSANDP have a correlation coefficient of 0.568, which is above the threshold. Likewise,
DCREDIT and DMONEY have a correlation coefficient of 0.150, which is relatively high but
still below the threshold. These correlations suggest that including both ERMSOFT and
ERSANDP, or both DCREDIT and DMONEY in the same model may lead to issues of
multicollinearity. Multicollinearity can affect the estimation and interpretation of coefficients,
and hence needs to be addressed. This can be done by dropping one of the highly correlated
variables, or by employing methods such as principal component analysis or ridge regression.
h)
Solutions
70
Series: Residuals
Sample 1986M05 2018M03
Observations 383
60
50
40
30
20
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-4.24e-16
-0.403413
24.47989
-36.07535
7.772638
-0.005613
4.994826
Jarque-Bera
Probability
63.50547
0.000000
10
0
-30
-20
-10
0
10
20
Simply examining the shape of the histogram of residuals may not be enough to
determine if they follow a normal distribution. Therefore, other methods, such as
examining skewness and kurtosis values or conducting a formal normality test, may be
required.
70
Series: Residuals
Sample 1986M05 2018M03
Observations 383
60
50
40
30
20
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-3.15e-16
-0.367238
24.59304
-36.03588
7.704458
-0.129649
5.124355
Jarque-Bera
Probability
73.09109
0.000000
10
0
-30
-20
-10
0
10
20
Based on the shape of the histogram, it appears that the residuals may not be normally
distributed. There seems to be a slight skewness to the left and a thicker tail on the left
side. However, further statistical tests, such as the Shapiro-Wilk or Anderson-Darling
tests, are needed to make a definitive conclusion.
i)
Solution
The p-value being larger than the significance level of 0.05 indicates that the null hypothesis
cannot be rejected, as suggested by the F-statistic.
Question No: 2
a)
Solution
To compute the logarithms of share prices in EViews, you can navigate to the "Generate
Series" tab and use the "LOG ()" function. For instance, if you want to calculate the
logarithm of the "CLOSE01" series for SSE Plc and save it in a new series named
"LSSE," you can use the equation: LSSE = LOG(CLOSE).
b)
Solution
LSSE
8.6
8.4
8.2
8.0
7.8
7.6
7.4
13
14
15
16
17
18
19
20
21
22 23
19
20
21
22 23
LUKXX
9.0
8.9
8.8
8.7
8.6
8.5
13
14
15
16
17
18
c)
Solution
According to the Engle-Granger cointegration test, the coefficient on LUKXX is -0.756,
with a corresponding p-value of 0.45. Since the p-value is greater than 0.05, we cannot
reject the null hypothesis that there is no cointegration between the two-time series, LSSE
and LUKXX. Therefore, we cannot conclude that there is a long-run relationship between
the two based on this test.
d)
Solution
The ADF test is a statistical tool used to determine whether a time series dataset has a unit root.
In this instance, the dependent variable is D(ECT), which is the first difference of the ECT
variable. The ADF test equation utilized in this case involves the lagged value of the ECT
variable, a constant term, and an error term.
The coefficient for ECT (-1) is negative, indicating an inverse relationship between the current
value of D(ECT) and its lagged value. However, this coefficient is not statistically significant at
the 5% level (p-value = 0.0675), suggesting that we cannot dismiss the null hypothesis of the
presence of a unit root in the dataset.
Furthermore, the constant term (C) is positive, but it is also not statistically significant (p-value =
0.8832). Taken together, the ADF test results suggest that the D(ECT) series might have a unit
root, indicating that it is not stationary and may exhibit a trend or be non-stationary. Additional
analysis may be necessary to confirm this and identify the appropriate modeling approach for the
data.
Conclusion:
Retrogression analysis is a statistical system constantly used in numerous disciplines to assay the
connection between a dependent variable and one or further independent variables. This
assignment gives a thorough review of retrogression analysis. The retrogression equation,
measure estimates, and statistical tests are only a many of the core ideas and language covered in
this assignment. The composition starts by introducing the retrogression equation and explaining
how it's used to estimate the relationship between the dependent variable and the independent
variables. The composition also describes how the measure estimates are calculated using the
Ordinary Least Places (OLS) system, which is the most generally used system for estimating the
parameters of the retrogression equation. The composition also explains how to interpret the
measure estimates and the statistical significance of the variables. The assignment also includes a
number of statistical tests, including the F- test, Durbin- Watson test, Breusch- Godfrey test, and
Breusch- Pagan- Godfrey test, that are used to gauge the retrogression model's quality of fit. The
composition offers a step- by- step tutorial on how to carry out these tests using the well- known
econometric software programmed EViews. The problem of multicollinearity, which occurs
when two or further independent variables are nearly linked, is one of the biggest obstacles in
retrogression analysis. The assignment goes on the goods of multicollinearity as well as ways
like top element analysis and crest retrogression that are used to are nearly task also emphasizes
how pivotal it's to check for heteroscedasticity, which happens when the friction of the error
element isn’t harmonious across all data. The Breusch- Pagan- Godfrey test and the
Heteroscedasticity and Autocorrelation harmonious (HAC) estimator are used in the assignment
to demonstrate how to test for heteroscedasticity’s significance of precisely interpreting the
retrogression findings and the necessity of taking the retrogression hypotheticals into
consideration are emphasized in the composition's conclusion. The study also makes several
recommendations for implicit unborn exploration areas, similar to expanding retrogression
analysis to time series data or probing the use of further sophisticated retrogression algorithms.