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ECS3706 ECONOMETRICS
ASSIGNMENT TWO SOLUTIONS
S1&S2 YEAR 2021
QUESTION A1
(a) For each of the statements provided below, write down the assumptions essential for the
statements to hold. In each of the cases, state why the assumptions are necessary.
(i) The assumptions needed in order to estimate the coefficients β0 and β1 using the OLS
technique. (5)
According to Studenmund, A.H., 2014, the regression model is linear in the coefficients and the
error term. This assumption addresses the functional form of the model. In statistics, a regression
model is linear when all terms in the model are either the constant or a parameter multiplied by an
independent variable. You build the model equation only by adding the terms together.
In the equation, the betas (βs) are the parameters that OLS estimates. Epsilon (ε) is the random
error.
In fact, the defining characteristic of linear regression is this functional form of the parameters
rather than the ability to model curvature. Linear models can model curvature by including
nonlinear variables such as polynomials and transforming exponential functions. To satisfy this
assumption, the correctly specified model must fit the linear pattern.
(ii) The assumptions required to ensure that the OLS estimates are unbiased, consistent and
the most efficient (5)
The error term has a population mean of zero. The error term accounts for the variation in the
dependent variable that the independent variables do not explain. Random chance should determine
the values of the error term. For the model to be unbiased, the average value of the error term must
equal zero.
(iii) The assumptions required for one to be able to carry out t-test and F-tests (5)
T-Test Assumptions
a. The first assumption made regarding t-tests concerns the scale of measurement. The assumption
for a t-test is that the scale of measurement applied to the data collected follows a continuous or
ordinal scale, such as the scores for an IQ test.
, 2
b. The second assumption made is that of a simple random sample, that the data is collected from
a representative, randomly selected portion of the total population.
c. The third assumption is the data, when plotted, results in a normal distribution, bellshaped
distribution curve. When a normal distribution is assumed, one can specify a level of probability
(alpha level, level of significance, p) as a criterion for acceptance. In most cases, a 5% value can
be assumed.
d. The fourth assumption is a reasonably large sample size is used. A larger sample size means the
distribution of results should approach a normal bell-shaped curve.
e. The final assumption is homogeneity of variance. Homogeneous, or equal, variance exists when
the standard deviations of samples are approximately equal.
F-Test Assumptions
a. Each group sample is drawn from a normally distributed population.
b. All populations have a common variance.
c. All samples are drawn independently of each other.
d. Within each sample, the observations are sampled randomly and independently of each
other
e. Factor effects are additive.
Question A2 (15 marks)
(a)Explain briefly the meaning of the following terms.
(I)Level of significance and p-value (3)
The level of significance is the measure of strength for given evidence against or for a given
variable in a regression model that suffices the rejection of the null hypothesis. It is denoted by
alpha and the lower the value the more statistically significant. In research 1% and up to 5% level
of significance is generally accepted in selected cases 10% may also be seen to be statistically
significant, however any greater value is not considered. The P- value is the lowest level of
significance at which the null hypothesis starts to be rejected the P-value presents a specific value
or actual probability that corresponds with a T test.
(ii)The consequences of an omitted variable versus the inclusion of an irrelevant variable (3)
When legitimate variables are omitted from a model, the consequences can be very serious: The
OLS estimators of the variables retained in the model not only are biased but are inconsistent as
well. Additionally, the variances and standard errors of these coefficients are incorrectly estimated,
thereby vitiating the usual hypothesis-testing procedures.
ECS3706 ECONOMETRICS
ASSIGNMENT TWO SOLUTIONS
S1&S2 YEAR 2021
QUESTION A1
(a) For each of the statements provided below, write down the assumptions essential for the
statements to hold. In each of the cases, state why the assumptions are necessary.
(i) The assumptions needed in order to estimate the coefficients β0 and β1 using the OLS
technique. (5)
According to Studenmund, A.H., 2014, the regression model is linear in the coefficients and the
error term. This assumption addresses the functional form of the model. In statistics, a regression
model is linear when all terms in the model are either the constant or a parameter multiplied by an
independent variable. You build the model equation only by adding the terms together.
In the equation, the betas (βs) are the parameters that OLS estimates. Epsilon (ε) is the random
error.
In fact, the defining characteristic of linear regression is this functional form of the parameters
rather than the ability to model curvature. Linear models can model curvature by including
nonlinear variables such as polynomials and transforming exponential functions. To satisfy this
assumption, the correctly specified model must fit the linear pattern.
(ii) The assumptions required to ensure that the OLS estimates are unbiased, consistent and
the most efficient (5)
The error term has a population mean of zero. The error term accounts for the variation in the
dependent variable that the independent variables do not explain. Random chance should determine
the values of the error term. For the model to be unbiased, the average value of the error term must
equal zero.
(iii) The assumptions required for one to be able to carry out t-test and F-tests (5)
T-Test Assumptions
a. The first assumption made regarding t-tests concerns the scale of measurement. The assumption
for a t-test is that the scale of measurement applied to the data collected follows a continuous or
ordinal scale, such as the scores for an IQ test.
, 2
b. The second assumption made is that of a simple random sample, that the data is collected from
a representative, randomly selected portion of the total population.
c. The third assumption is the data, when plotted, results in a normal distribution, bellshaped
distribution curve. When a normal distribution is assumed, one can specify a level of probability
(alpha level, level of significance, p) as a criterion for acceptance. In most cases, a 5% value can
be assumed.
d. The fourth assumption is a reasonably large sample size is used. A larger sample size means the
distribution of results should approach a normal bell-shaped curve.
e. The final assumption is homogeneity of variance. Homogeneous, or equal, variance exists when
the standard deviations of samples are approximately equal.
F-Test Assumptions
a. Each group sample is drawn from a normally distributed population.
b. All populations have a common variance.
c. All samples are drawn independently of each other.
d. Within each sample, the observations are sampled randomly and independently of each
other
e. Factor effects are additive.
Question A2 (15 marks)
(a)Explain briefly the meaning of the following terms.
(I)Level of significance and p-value (3)
The level of significance is the measure of strength for given evidence against or for a given
variable in a regression model that suffices the rejection of the null hypothesis. It is denoted by
alpha and the lower the value the more statistically significant. In research 1% and up to 5% level
of significance is generally accepted in selected cases 10% may also be seen to be statistically
significant, however any greater value is not considered. The P- value is the lowest level of
significance at which the null hypothesis starts to be rejected the P-value presents a specific value
or actual probability that corresponds with a T test.
(ii)The consequences of an omitted variable versus the inclusion of an irrelevant variable (3)
When legitimate variables are omitted from a model, the consequences can be very serious: The
OLS estimators of the variables retained in the model not only are biased but are inconsistent as
well. Additionally, the variances and standard errors of these coefficients are incorrectly estimated,
thereby vitiating the usual hypothesis-testing procedures.
