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ECN6540 ECN6540 1

Data Provided:

Mathematical, Statistical and Financial Tables for the Social Sciences (Kmietowicz
and Yannoulis).

DEPARTMENT OF ECONOMICS Autumn Semester 2022/23

ECN6540 Econometric Methods

Duration: 2? Hours

Maximum 1500 words excluding equations

The answers to the questions must be type-written. The preference is that
symbols and equations should be inserted into the document using the
equation editor in Word. Alternatively, they can be scanned and inserted as an
image (providing it is clear and readable).

There are two questions, firstly on microeconometrics and secondly on
macroeconometrics. ANSWER ALL QUESTIONS. The marks shown within each
question indicate the weighting given to component sections. Any calculations
must show all workings otherwise full marks will not be awarded.

ECN654540 2
MICROECONOMETRICS

1. The non-mortgage debt behaviour of individuals is modelled using UK
cross sectional data for 2017 from Understanding Society based upon
11,470 employees. The table below describes the variables in the data.

Variable Definitions
-----------------------------------------------------------------------------------------------------
debtor = 1 if has any non-mortgage debt, 0 otherwise
debt_inc = debt to income ratio (outstanding debt ? annual income)
work_fin = 1 if employed in financial sector, 0 otherwise
lincome = natural logarithm of income last month
ghealth = 1 if currently in good or excellent health, 0 otherwise
sex = 1 if male, 0=female
degree = 1 if university degree, 0 = below degree level education
lsavinv_inc = natural logarithm of saving & investment annual income
age = age of individual in years
agesq = age squared
-----------------------------------------------------------------------------------------------------
a. The following Stata output shows an analysis of modelling the probability that
an individual holds non-mortgage debt using a Logit regression.

logit debtor ib(0).work_fin##c.lincome ghealth sex degree age lsavinv_inc

Logistic regression Number of obs = 11,470
LR chi2(8) = 546.50
Prob > chi2 = 0.0000
Log likelihood = -7067.5606 Pseudo R2 = 0.0372

----------------------------------------------------------------------------------
debtor | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------------+--------------------------------------------------------------
1.work_fin | 5.43774 1.271821 4.28 0.000 2.945017 7.930462
lincome | .4584589 .0384631 11.92 0.000 .3830726 .5338451
|
work_fin#c.lincome |
1 | -.6710698 .1587322 -4.23 0.000 -.9821792 -.3599604
|
ghealth | -.0796141 .0413548 -1.93 0.054 -.160668 .0014398
sex | -.0084802 .0433091 -0.20 0.845 -.0933645 .0764041
degree | .0795525 .0462392 1.72 0.085 -.0110748 .1701797
age | -.0316432 .0020753 -15.25 0.000 -.0357106 -.0275757
lsavinv_inc | -.0819022 .0085226 -9.61 0.000 -.0986062 -.0651983
_cons | -2.638081 .2870575 -9.19 0.000 -3.200703 -2.075458
----------------------------------------------------------------------------------

ib(0).work_fin##c.lincome is an interaction effect between a binary
and continuous variable. Summary statistics on variables used in the analysis
are provided below.

sum ib(0).work_fin##c.lincome ghealth sex degree age lsavinv_inc

Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
1.work_fin | 11,767 .0398572 .195632 0 11
lincome | 11,767 7.650333 .6965933 .0861777 9.847781

work_fin#|
c.lincome 1 | 11,767 .3197615 1.574852 0 9.72120
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ghealth | 11,767 .5457636 .4979224 0 1
sex | 11,767 .4812612 .49967 0 1
degree | 11,767 .3192827 .4662186 0 1
age | 11,767 44.43885 10.39257 18 65
lsavinv_inc | 11,767 1.857315 2.600682 0 11.51294
-------------+---------------------------------------------------------

i) What do the coefficients of work_fin, lincome and the interaction
term imply? Explain whether the estimates can be interpreted.
ii) Showing your calculations in full, find the marginal effects evaluated
at the mean from the above output.
iii) Provide an economic interpretation of the marginal effects found in
(a(ii)).
iv) Given the pseudo R-squared what is the value of the constrained
log likelihood function? Show your calculation.

[10 marks]

[25 marks]

[10 marks]

[5 marks]
b. There is also information on the amount of debt held as a proportion of
income. This outcome is modelled using the Heckman sample selection
estimator. The Stata output is shown below.

heckman debt_inc age agesq sex degree lsavinv_inc,
select(debtor = ib(0).work_fin##c.lincome ghealth sex degree age lsavinv_inc)

Heckman selection model Number of obs = 11,470
Wald chi2(5) = 249.22
Log likelihood = -13437.59 Prob > chi2 = 0.0000
------------------------------------------------------------------------------------
| Coefficient Std. err. z P>|z| [95% conf. interval]
-----------------------+------------------------------------------------------------
debt_inc |
age | -.1341474 .0629505 -2.13 0.033 -.2575282 -.0107667
agesq | .0003505 .0001265 2.77 0.006 .0001026 .0005985
sex | .1517503 .0607726 2.50 0.013 .0326382 .2708623
degree | .157981 .0661602 2.39 0.017 .0283095 .2876525
lsavinv_inc | .1130368 .0124696 9.06 0.000 .0885968 .1374767
_cons | 9.727016 .2615992 37.18 0.000 9.214291 10.23974
-----------------------+------------------------------------------------------------
debtor |
1.work_fin | 1.130109 .3719515 3.04 0.002 .4010974 1.85912
lincome | .2965059 .0113274 26.18 0.000 .2743045 .3187072
|
work_fin#c.lincome |
1 | -.1360006 .0461592 -2.95 0.003 -.2264709 -.0455303
|
ghealth | -.0106065 .0106393 -1.00 0.319 -.0314592 .0102462
sex | -.0488734 .0236997 -2.06 0.039 -.095324 -.0024229
degree | -.0369117 .0256652 -1.44 0.150 -.0872146 .0133912
age | -.016944 .0011782 -14.38 0.000 -.0192532 -.0146349
lsavinv_inc | -.0468348 .0047518 -9.86 0.000 -.0561482 -.0375214
_cons | -1.828795 .0961843 -19.01 0.000 -2.017312 -1.640277
-------------------+----------------------------------------------------------------
lambda | -2.579767 .039169 -2.656537 -2.502997
--------------------------------------------------------------------------------

i) Interpret the estimates in the outcome equation.
ii) In the context of the above Stata output what does the estimate of
the inverse Mills ratio (lambda) suggest? What does lambda
provide an estimate of in terms of the theory?
[5 marks]

[15 marks]
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c.
iii) What assumption has been made about the covariates
work_fin, lincome and ghealth in the treatment equation?
What are the implications if these assumptions are not met? Are
they individually statistically significant? If these variables are also
included in the outcome equation explain whether the model is
identified or not.

In the context of the above application the following figure shows the
distribution of debt as a proportion of annual income.

Describe a situation in which a Tobit specification would be the preferred
modelling choice rather than a sample selection approach. What
assumptions would the Tobit modelling approach have to make with
regard to the treatment and outcome equations?

ECN6540
ECN6540 5
MACROECONOMETRICS

2. a.

The following Stata output is based upon modelling aggregate
savings as a function of Gross Domestic Product (GDP), both
measured in constant prices, over time () using data for the U.S.
over the period 1960 to 2020. The savings function is a double
logarithmic specification as follows:
log = 0 + 1log +
Where log is the natural logarithm of savings and log is the
natural logarithm of GDP. The Stata output also shows the results
of ADF tests for savings and GDP. Note that in the output L
denotes a lag and D a difference.

regress logS logY

Source | SS df MS Number of obs = 61
-------------+------------------------------ F( 1, 59) = 180.39
Model | 29.3601715 1 29.3601715 Prob > F = 0.0000
Residual | 9.6029125 59 .162761229 R-squared = 0.7535
-------------+------------------------------ Adj R-squared = 0.7494
Total | 38.963084 60 .649384734 Root MSE = .40344
------------------------------------------------------------------------------
logS | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
logY | 1.16096 .0864398 13.43 0.000 .9879948 1.333926
_cons | -4.007335 .6861211 -5.84 0.000 -5.38026 -2.63441
------------------------------------------------------------------------------

Durbin-Watson d-statistic( 2, 61) = .7252386
predict e, resid

i) Interpret the OLS results. Explain whether the analysis is likely
to be spurious?
ii) What do the results of the ADF tests on savings and GDP imply
at the 5 percent level? Show the test statistic used, the null
hypothesis tested and the appropriate critical value.
iii) Explain whether savings and GDP are cointegrated at the 5
percent level. Explicitly state the null hypothesis, show
algebraically the estimated test equation based upon the
output, and provide the appropriate critical value.

dfuller logS, lag(4) regress

Augmented Dickey-Fuller test for unit root Number of obs = 56
------------------------------------------------------------------------------
D.logS | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
logS |
L1. | -.129875 .0534553 -2.43 0.019 -.2372431 -.0225069
LD. | .2335003 .099153 2.35 0.022 .0343457 .4326549
L2D. | .1939032 .0807975 2.40 0.020 .0316167 .3561897
L3D. | -.0834007 .0858594 -0.97 0.336 -.2558545 .089053
L4D. | -.2258198 .0784568 -2.88 0.006 -.3834049 -.0682348
cons | .7246592 .2840536 2.55 0.014 .1541207 1.295198
------------------------------------------------------------------------------

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dfuller logY, lag(4) regress

Augmented Dickey-Fuller test for unit root Number of obs = 56
------------------------------------------------------------------------------
D.logY | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
logY |
L1. | -.0175739 .0092468 -1.90 0.063 -.0361467 .000999
LD. | .4530274 .1290377 3.51 0.001 .1938476 .7122072
L2D. | -.0699222 .1306402 -0.54 0.595 -.3323208 .1924765
L3D. | -.1351664 .1297451 -1.04 0.303 -.3957672 .1254344
L4D. | -.1774947 .1177561 -1.51 0.138 -.4140149 .0590255
_cons | .1720878 .076104 2.26 0.028 .0192285 .3249471
------------------------------------------------------------------------------

dfuller e, lag(4)

Test Statistic
----------------------------
Z(t) -4.042
----------------------------

b. Explain why the Johansen approach to cointegration may be
preferable to the Engle-Granger two step approach, in each of the
following two scenarios:
i) In the above example (part a) when there are variables in the
model, i.e. = 2?
ii) When ?3. In this scenario what is the maximum number of
cointegrating vectors?

c. A researcher has modelled the relationship between personal
consumption expenditure and the money supply as measured by
M2 based upon a double logarithmic specification as follows:
log() = 0 + 1log(2) +
They then build a dynamic forecast of consumption. Two
alternative models are estimated over the period 1969q1 through
to 2008q4: Model 1 an ARIMA(1,1,2) and Model 2 an
ARIMA(1,1,1). Then the researcher forecasts out of sample
through to 2010q3. The results are shown below along with
diagnostic statistics.

i) Based upon the output below for the ARIMA(1,1,1) model draw
both the ACF and PACF for the AR and MA components.
ii) Explain whether the models are stationary and invertible, along
with any potential implications.
iii) Explain in detail which of the above two models is preferred
and why. Outline any further analysis you may want to
undertake giving your reasons.