Department:
Economics
Title of Exam:
Econometrics 1 & 2
Time Allowed:
24 Hours (PLEASE NOTE: Late papers will not be marked)
Time Recommended:
We would expect this to last two hours.
Word limit:
There are no word limits.
Allocation of Marks:
Section A is worth 40 marks (40%).
Section A contains SEVEN short questions. The first TWO questions are worth 5
marks each and the last FIVE questions are worth 6 marks each.
Section B is worth 60 marks (60%).
Section B contains THREE questions, each worth 20 marks.
Instructions for Candidates:
You should attempt both sections of the paper.
Section B – To get good marks your answer should contain clear statements that
explain the logical structure of your argument.
Materials Required:
Statistics Tables – these will be provided on the VLE.

## A note on Academic Integrity

We are treating this online examination as a time-limited open assessment, and you are
therefore permitted to refer to written and online materials to aid you in your answers.
However, you must ensure that the work you submit is entirely your own, and for the whole
time the assessment is live you must not:
● communicate with departmental staff on the topic of the assessment
● communicate with other students on the topic of this assessment.
● seek assistance with the assignment from the academic and/or disability support
services, such as the Writing and Language Skills Centre, Maths Skills Centre and/or
Disability Services. (The only exception to this will be for those students who have
been recommended an exam support worker in a Student Support Plan. If this
applies to you, you are advised to contact Disability Services as soon as possible to
discuss the necessary arrangements.)
● seek advice or contribution from any third party, including proofreaders, friends, or
family members.
We expect, and trust, that all our students will seek to maintain the integrity of the
assessment, and of their award, through ensuring that these instructions are strictly followed.
Failure to adhere to these requirements will be considered a breach of the Academic
Misconduct regulations, where the offences of plagiarism, breach/cheating, collusion and
commissioning are relevant – see AM.1.2.1” (Note this supersedes section 7.3 of the Guide
to Assessment).

## Section A

Answer ALL SEVEN questions in section A.

1. Each part of the following is either a true or false statement. If you think Part $n$ is True, your answer should be: “(n) $\mathrm{T}$ ” and if you think it is false, “(n) F”. Consider the linear model with $k$ regressors, including a constant
$$\boldsymbol{y}=\mathbf{X}^{\prime} \boldsymbol{\beta}+\varepsilon$$
where it is known that $p \lim \frac{1}{n} \boldsymbol{X}^{\prime} \varepsilon \neq 0$ ( $n$ denotes sample size). Suppose the columns of the matrix $\mathrm{Z}$ contain $m>k$ instruments, including a constant, such that $\operatorname{plim} \frac{1}{n} \mathrm{Z}^{\prime} \varepsilon=0$, plim $\frac{1}{n} \mathrm{Z}^{\prime} \mathrm{X}=Q_{\mathrm{zx}}$, a matrix of rank $k$, and plim $\frac{1}{n} \mathrm{Z}^{\prime} \mathrm{Z}=Q_{\mathrm{zz}}$, a matrix of rank $m$. Let $\hat{X}$ and $\hat{y}$ denote the fitted values from regressing the columns of $X$ and $y$ respectively on $Z$. Which of the following will give consistent estimates of $\beta ?$
(a) $\left(\hat{\mathrm{X}}^{\prime} \hat{\mathrm{X}}\right)^{-1} \hat{\mathrm{X}}^{\prime} y . \quad(1$ mark $)$
(b) $\left(\mathrm{X}^{\prime} \hat{\mathrm{X}}\right)^{-1} \mathrm{X}^{\prime} \hat{y} . \quad(1$ mark $)$
(c) $\left(\mathrm{X}^{\prime} \mathrm{X}\right)^{-1} \mathrm{X}^{\prime} y . \quad(1$ mark $)$
(d) $\left(\mathrm{Z}^{\prime} \mathrm{X}\right)^{-1} \mathrm{Z}^{\prime} \boldsymbol{y} . \quad(1$ mark $)$
(e) $\left(\mathbf{Z}^{\prime} \mathbf{X}\right)^{-1} \mathbf{X}^{\prime} \boldsymbol{y} . \quad(1$ mark $)$
2. Each part of the following is either a true or false statement. If you think Part $n$ is True, your answer should be: “(n) $\mathrm{T}$ ” and if you think it is false, “(n) F”.
A researcher estimates the following dynamic regression using annual data, where $y_{t}$ is the log demand for a certain good and $x_{t}$ is its log price, which we take to be exogenous.
$$\begin{array}{rcccc} y_{t}= & 117.7 & +0.373 y_{t-1}-0.156 x_{t}-0.062 x_{t-1}+e_{t} \ & (50.284) & (0.177) & (0.019) & (0.029) \ n= & 46 & & & \end{array}$$
(Standard errors in parentheses.)
(a) A t-test that the variable $x_{t}$ should be omitted from the regression would be rejected at the $5 \%$ level. ( 1 mark)
(b) If $x_{t}$ were to fall by one unit we would expect $y_{t}$ to fall by $0.156$. ( 1 mark)
(c) A small coefficient for $y_{t-1}$ means that it takes longer for the system to adjust to a new equilibrium value. (1 mark)

(d) The estimated long run elasticity of demand is approximately $0.218 .$ ( 1 mark)
(e) OLS estimates of this regression are inconsistent if the disturbance term is a first order moving average process. ( 1 mark)

1. Suppose we are trying to estimate the linear relationship $y_{i}=\mathrm{x}{i}^{\prime} \boldsymbol{\beta}+\sigma \epsilon{i}$, where $\epsilon_{i} \sim N(0,1) .$ Our sample has been truncated from below, so that it contains only individuals who report a positive value for the dependent variable, $y_{i}>0 .$ Explain, using diagrams or otherwise, why ordinary least squares estimates of $\beta$ will not be consistent. Outline a method that would produce consistent results. (6 marks)
2. In order to begin to model a quarterly univariate time series, the sample autocorrelation function (SACF) and the sample partial autocorrelation function (SPACF) are calculated. The following results are obtained.
$\begin{array}{rll}\text { Lag } & \text { SACF } & \text { SPACF } \ 1 & 0.832 & 0.832 \ 2 & 0.694 & -0.066 \ 3 & 0.519 & -0.019 \ 4 & 0.372 & -0.083 \ 5 & 0.236 & 0.017 \ 6 & 0.176 & -0.134 \ 7 & 0.112 & -0.104 \ 8 & 0.095 & -0.030\end{array}$
Briefly explain how the sample partial autocorrelation function (SPACF) is calculated and discuss what type of time series process would fit the table. (6 marks)
3. A researcher is interested in the relationship between the elements of an $m \times 1$ vector, $\mathrm{Y}{t}$, using the equivalent expressions \begin{aligned} \mathbf{Y}{t} &=\boldsymbol{\alpha}+\Phi_{1} \mathrm{Y}{t-1}+\Phi{2} \mathrm{Y}{t-2}+\epsilon{t} \ \Delta \mathbf{Y}{t} &=\Pi\left(\mathrm{Y}{t-1}-\boldsymbol{\mu}\right)+\Gamma_{1} \Delta \mathrm{Y}{t-1}+\epsilon{t}, t=3, \ldots, n \end{aligned}
where $\Delta \mathrm{Y}{t}=\mathrm{Y}{t}-\mathrm{Y}{t-1}$ and $\epsilon{t} \sim I I D(0, \Omega) .$ Show how the parameters $\mu, \Pi$ and $\Gamma_{1}$ relate to the parameters $\alpha, \Phi_{1}$ and $\Phi_{2} .$ (6 marks)
4. Define what it means for a time series process, $y_{t}$, to be stationary. Under what conditions is an autoregressive moving average (ARMA) process stationary? Show that the following ARMA $(2,1)$ process is stationary: $y_{t}=y_{t-1}-0.21 y_{t-2}+\varepsilon_{t}+0.7 \varepsilon_{t-1}$, $t \in(-\infty,+\infty)$ where $\varepsilon_{t}$ is a white noise process with variance $\sigma^{2}$. (6 marks)
1. Suppose that an econometrician is interested in the relationship between two variables, $y_{1 t}$ and $y_{2 t}$ using a sample $t=1, \ldots, n$. They want to estimate the following simultaneous regressions model
\begin{aligned} &y_{1 t}=\gamma_{12} y_{2 t}+\beta_{1} z_{1 t}+\epsilon_{1 t} \ &y_{2 t}=\gamma_{21} y_{1 t}+\beta_{2} z_{2 t}+\epsilon_{2 t} \end{aligned}
where $z_{1 t}$ and $z_{2 t}$ are exogenous explanatory variables, $\gamma_{12}, \gamma_{21} \neq 0$ and $\epsilon=\left(\epsilon_{1 t}, \epsilon_{2 t}\right)^{\prime}$ contains unobserved disturbances with $\mathrm{E}(\epsilon)=0, \operatorname{Var}(\epsilon)=\Omega$. Explain why ordinary least squares estimates of Equation (1) and Equation (2) will not be consistent.
Propose a method to obtain consistent estimates of $\gamma_{12}$ and $\beta_{1}$ in Equation (1), taking care to state any conditions and assumptions required. (6 marks)

# Section B

Answer ALL THREE questions in section B. To get good marks your answer should contain clear statements that explain the logical structure of your argument.

1. Consider the time series process
$$y_{t}=\alpha+\phi y_{t-1}+\varepsilon_{t}$$
where $|\phi|<1$ and $\varepsilon_{t}$ follows an $\operatorname{ARCH}(1)$ process with $\varepsilon_{t} \mid Y_{t-1} \sim N\left(0, \sigma_{t}^{2}\right)$, where $Y_{t-1}=\left{y_{t-1}, y_{t-2}, \ldots\right}$ denotes the past history of the process and $$\sigma_{t}^{2}=\alpha_{0}+\alpha_{1} \varepsilon_{t-1}^{2}$$ with $\alpha_{0}>0$ and $\alpha_{1} \geq 0$
(a) Using the fact that we may write $\varepsilon_{t}=\eta_{t} \sigma_{t}$, where $\eta_{t} \sim N(0,1)$ is a sequence of independent standard normal variables, show that $\varepsilon_{t}$ is a white noise process with mean zero and (unconditional) variance $\alpha_{0} /\left(1-\alpha_{1}\right)$ as long as $\varepsilon_{t}$ is stationary. (7 marks)
(b) Show that $\varepsilon_{t}^{2}$ follows an $\mathrm{AR}(1)$ process. Hence or otherwise discuss the conditions under which $\varepsilon_{t}$ is stationary. (7 marks)
(c) Suppose the parameters $\alpha, \phi, \alpha_{0}, \alpha_{1}$ are known. Write down the one-step ahead forecast, $\hat{y}{n+1}=\mathrm{E}\left[y{n+1} \mid y_{n}, y_{n-1}, \ldots\right] .$ Derive the variance of the forecast error. ( 6 marks)
1. Consider the salary data of a sample of 258 male bank employees. The dependent variable MANAGERIAL takes the value 1 if worker $i$ is in a managerial role, and 0 otherwise. The regressors EDUC measures the number of completed years of education, MINORITY is a dummy taking the value 1 if the employee is a member of an ethnic minority and PREVEXP measures previous work experience, in months. A researcher employs a logit model, $P\left(y_{i}=1\right)=\Lambda\left(\mathbf{x}{i}^{\prime} \boldsymbol{\beta}\right)$, where $\mathrm{x}{i}$ contains the regressors for worker $i$ and $\Lambda(t)=\frac{1}{1+e^{-t}}$ the logistic function.
(a) As a first attempt, the researcher estimates the following model.
\begin{tabular}{l}
\hline \hline Table 9.1. Dependent Variable: MANAGERIAL \
Method: ML – Binary Logit & \
\hline Variable & Coefficient & std error & Prob. \
C & $-26.95253$ & $4.400895$ & $0.0000$ \
EDUC & $1.674803$ & $0.280049$ & $0.0000$ \
MINORITY & $-2.395242$ & $0.847981$ & $0.0047$ \
PREVEXP & $0.003865$ & $0.003078$ & $0.2092$ \
\hline Mean dependent var & $0.286822$ & Akaike info criterion & $0.522420$ \
Log likelihood & $-63.39219$ & Avg. log likelihood & $-0.245706$ \
\hline
\end{tabular}
Comment briefly on the signs and significance of the coefficients in this model. Are they in line with your expectations? (4 marks)
(b) The researcher re-estimates the model, omitting the variable PREVEXP.
\begin{tabular}{l}
$\overline{\hline \text { Table 9.2. Dependent Variable: MANAGERIAL }}$ \
Method: ML – Binary Logit & \
\hline Variable & Coefficient & std error & Prob. \
C & $-26.21472$ & $4.311652$ & $0.0000$ \
EDUC & $1.644798$ & $0.276714$ & $0.0000$ \
MINORITY & $-2.119683$ & $0.793999$ & $0.0076$ \
\hline Mean dependent var & $0.286822$ & Akaike info criterion & $0.520543$ \
Log likelihood & $-64.15011$ & Avg. log likelihood & $-0.248644$ \
\hline
\end{tabular}
Using all relevant information from the tables, discuss which of the two models you prefer. Use the second model to estimate the probability that a male employee not from an ethinic minority with 15 years of education is in a managerial position. What might explain the increase in the estimated coefficient MINORITY when the variable PREVEXP is excluded from the regression?
(10 marks)
(c) According to the regression in part $9 b$, the probability that a male employee has a management job depends on his time in education. Explain carefully why that probability does not increase by $1.645$ for every additional year of education. Calculate an estimate of the marginal effect of an extra year of education for an ethnic minority male with 17 years of education. (6 marks)

# Econometrics I & II – ECO00047M

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• Department: Economics and Related Studies
• Module co-ordinator: Dr. Michael Thornton
• Credit value: 20 credits
• Credit level: M
• Academic year of delivery: 2019-20

## Module aims

To equip students with intermediate level knowledge of the core techniques employed in modern econometric analysis so that they are able:

to follow the techniques and arguments used in a range of empirical papers in Economics and Finance; and,

to undertake a successful empirical dissertation.

## Module learning outcomes

On completing the module a student should be able:

To recognise and interpret various mathematical objects that arise in the theory of least ssquares estimation and testing.

To extend these skills to the estimation and testing of models under conditions that commonly arise in economic and financial data, including:

non-linear models
disturbances that are heteroskedastic and/orserially correlated
depedent variables that are qualitative (can only take one of a finite number of values) or limited to the range of values they can take
regressors that are endogeneous, through instrumental variable estimation and the generalised method of moments
and
variables that are driven by the long-run trends.

To present and derive key statistical results discussed during the module at an appropriate mathematical level

and

To interpret correctly the results of empirical statistical analysis as performed using contemporary econometric software.

None

## Module feedback

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