1
ECS3706 ECONOMETRICS ASSIGNMENT TWO
S1&S2 YEAR 2021
TL 001
(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)
The model with k explanatory variables:
The regression model is linear in the coefficients. This assumption addresses the functional form
of the model. 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 itself cannot be observed. But let’s suppose the mean of εi = 10
So we could just add and subtract 10 to the model to force the error term to have a mean of zero:
The assumptions required for one to be able to carry out t-test and F-tests (5)
T-Test Assumptions
The common assumptions made when doing a t-test include those regarding the scale of
measurement, random sampling, normality of data distribution, adequacy of sample size, and
equality of variance in standard deviation.
F-Test Assumptions
, 2
An F-test assumes that data are normally distributed and that samples are independent from one
another. Data that differs from the normal distribution could be due to a few reasons. The data
could be skewed or the sample size could be too small to reach a normal distribution.
(a) Explain briefly the meaning of the following terms.
(i) Level of significance and p-value
The significance level, also denoted as alpha or α, is the probability of rejecting the null
hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of
concluding that a difference exists when there is no actual difference. In null hypothesis
significance testing, the p-value is the probability of obtaining test results at least as extreme as
the results actually observed, under the assumption that the null hypothesis is correct (Gujarati
2012).
(ii) The consequences of an omitted variable versus the inclusion of an irrelevant variable.
Consequences of an omitted variable consequences of an inclusion of an
irrelevant variable
Model underfitting Model overfitting
The usual confidence interval and The usual confidence interval and
hypothesis-testing procedures are likely to hypothesis-testing procedures remain
give misleading conclusions about the valid.
statistical significance of the estimated
parameters.
The disturbance variance σ2 is incorrectly The error variance σ2 is correctly
estimated. estimated.
(iii) Linear in variables versus linear in coefficients
Linearity in variables
A regression function such as E(Y | Xi) = β1 + β2Xi2 is not a linear function because the variable X
appears with a power or index of 2. It is Linear if the regression function is as E(Y | Xi) = β1 +
β2Xi.
Linearity in the Coefficients
E(Y | Xi) = β1 + β2Xi2 is a linear (in the coefficients) regression model. However the regression
model E(Y | Xi) = β1 + 3β22, which is nonlinear in the coefficient β2.
(b) Provide a practical example where you can apply the dummy variable technique. Make
sure that you show the difference between a slope dummy variable and an intercept
dummy.(6)
ECS3706 ECONOMETRICS ASSIGNMENT TWO
S1&S2 YEAR 2021
TL 001
(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)
The model with k explanatory variables:
The regression model is linear in the coefficients. This assumption addresses the functional form
of the model. 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 itself cannot be observed. But let’s suppose the mean of εi = 10
So we could just add and subtract 10 to the model to force the error term to have a mean of zero:
The assumptions required for one to be able to carry out t-test and F-tests (5)
T-Test Assumptions
The common assumptions made when doing a t-test include those regarding the scale of
measurement, random sampling, normality of data distribution, adequacy of sample size, and
equality of variance in standard deviation.
F-Test Assumptions
, 2
An F-test assumes that data are normally distributed and that samples are independent from one
another. Data that differs from the normal distribution could be due to a few reasons. The data
could be skewed or the sample size could be too small to reach a normal distribution.
(a) Explain briefly the meaning of the following terms.
(i) Level of significance and p-value
The significance level, also denoted as alpha or α, is the probability of rejecting the null
hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of
concluding that a difference exists when there is no actual difference. In null hypothesis
significance testing, the p-value is the probability of obtaining test results at least as extreme as
the results actually observed, under the assumption that the null hypothesis is correct (Gujarati
2012).
(ii) The consequences of an omitted variable versus the inclusion of an irrelevant variable.
Consequences of an omitted variable consequences of an inclusion of an
irrelevant variable
Model underfitting Model overfitting
The usual confidence interval and The usual confidence interval and
hypothesis-testing procedures are likely to hypothesis-testing procedures remain
give misleading conclusions about the valid.
statistical significance of the estimated
parameters.
The disturbance variance σ2 is incorrectly The error variance σ2 is correctly
estimated. estimated.
(iii) Linear in variables versus linear in coefficients
Linearity in variables
A regression function such as E(Y | Xi) = β1 + β2Xi2 is not a linear function because the variable X
appears with a power or index of 2. It is Linear if the regression function is as E(Y | Xi) = β1 +
β2Xi.
Linearity in the Coefficients
E(Y | Xi) = β1 + β2Xi2 is a linear (in the coefficients) regression model. However the regression
model E(Y | Xi) = β1 + 3β22, which is nonlinear in the coefficient β2.
(b) Provide a practical example where you can apply the dummy variable technique. Make
sure that you show the difference between a slope dummy variable and an intercept
dummy.(6)
