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ECN6540 Econometric Methods
Duration: 2½ 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.
MICROECONOMETRICS
1. Using a cross section from the Chinese Health and Nutrition Survey for 2015 the following Stata output shows an analysis of modelling the hours worked per week of females. This is estimated by a Tobit regression with censoring on zero hours.
Variable Definitions
----------------------------------------------------------------------------------------------------
hours =
number of weekly hours worked
age =
age of individual in years
agesq =
age squared
linc =
natural logarithm of weekly income (mean=4.783)
degree =
1 if university or college degree, 0 = below degree
rural = 1 if household registration is rural, 0 = urban
----------------------------------------------------------------------------------------------------
tobit hours age agesq linc ib(0).degree rural ib(0).degree#c.linc, ll(0)
Tobit regression Number of obs = 4,239
Uncensored = 2,205
Limits: Lower = 0 Left-censored = 2,034
LR chi2(6) = 3146.41
Prob > chi2 = 0.0000
Log likelihood = -10708.96 Pseudo R2 = 0.1281
-------------------------------------------------------------------------------
hours | Coefficient Std. err. t P>|t| [95% conf. interval]
--------------+----------------------------------------------------------------
age | 1.945303 .2477056 7.85 0.000 1.45967 2.430936
agesq | -.0292011 .0028844 -10.12 0.000 -.034856 -.0235461
linc | 8.769653 .2446168 35.85 0.000 8.290075 9.24923
1.degree | 8.271809 7.227195 1.14 0.252 -5.897285 22.4409
rural | 4.187504 .8555 4.89 0.000 2.510275 5.864733
|
degree#c.linc |
1 | -.6570594 .9609595 -0.68 0.494 -2.541044 1.226925
|
_cons | -67.88748 5.388033 -12.60 0.000 -78.45085 -57.32411
--------------+----------------------------------------------------------------
\sigma| 21.07271 3.80836 416.5156 473.4239
-------------------------------------------------------------------------------
Average marginal effects Number of obs = 4,239
Expression: E(hours|hours>0), predict(e(0,.))
dy/dx wrt: age agesq linc 1.degree rural
------------------------------------------------------------------------------
| Delta-method
| dy/dx std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
age | .9096171 .1159253 7.85 0.000 .6824077 1.136827
agesq | -.0136543 .0013506 -10.11 0.000 -.0163015 -.0110071
linc | 4.072079 .1068943 38.09 0.000 3.86257 4.281588
1.degree | 2.002049 .9401151 2.13 0.033 .1594573 3.844641
rural | 1.958063 .3986838 4.91 0.000 1.176657 2.739469
------------------------------------------------------------------------------
(*) dy/dx is for discrete change of dummy variable from 0 to 1 ECN6540
ECN6540 3 TURN OVER
a.The value of the log likelihood from a restricted model (constant only) is equal to -12282.166. Based upon this information calculate the missing pseudo R-squared in the above output.[5 marks]
b.With reference to the coefficients what does the interaction term show in the tobit model? Is it statistically significant at the 5% level?[5 marks]
c.The above output also shows the marginal effects on the expected value of the dependent variable for uncensored observations for a unit change in the kth covariate, xk .
i) Provide an interpretation of these marginal effects.[10 marks]
ii) Calculate the income elasticity from the marginal effect for income using the scaling factor.[10 marks]
d. Calculate the predicted hours worked for the following individual: a female aged 30, who has a degree, has income equal to the sample mean, and has an urban registration. [5 marks]
e. For the individual described in (d) what is the probability that they work between 10 and 20 hours per week?[15 marks]
f. Can you identify any problems with the empirical specification of the Tobit model in terms of the independent variables included? Explain.[10 marks]
g. An alternative approach to modelling the hours worked by females is to use the Heckman sample selection estimator. This model is applied to the same data where additional variables are defined as follows:
Variable Definitions
--------------------------------------------------------------------------------------------------------
participate = 1 if hours worked per week>0; 0 if hours worked=0
depkid_u6 = 1 if has child under 6 years of age; 0 otherwise
partnerwork = 1 if their spouse or partner works; 0 otherwise
ghealth = 1 if currently in good or excellent health; 0 otherwise
bmi = body mass index
--------------------------------------------------------------------------------------------------------
i) In the context of the Stata output below what does the estimate of the inverse Mills ratio (lambda) suggest? [10 marks]
ii) Is the model identified? Explain your answer.[5 marks]
iii) What does the Wald statistic imply, what hypothesis is tested?[5 marks]
heckman hours age agesq linc ib(0).degree rural ib(0).degree#c.linc,
select(participate = depkid_u6 relcare ghealth bmi)
Heckman selection model Number of obs = 4,239
(regression model with sample selection) Selected = 2,205
Nonselected = 2,034
Wald chi2(6) = 343.72
Log likelihood = -10645.37 Prob > chi2 = 0.0000
-------------------------------------------------------------------------------
| Coefficient Std. err. z P>|z| [95% conf. interval]
--------------+----------------------------------------------------------------
hours |
age | .4468108 .1577823 2.83 0.005 .1375633 .7560584
agesq | -.0074671 .0018653 -4.00 0.000 -.011123 -.0038112
linc | -.19395 .2681249 -0.72 0.469 -.7194651 .3315651
1.degree | 2.679251 6.507854 0.41 0.681 -10.07591 15.43441
rural | -7.024816 .5478824 -12.82 0.000 -8.098646 -5.950986
|
degree#c.linc |
1 | -.156791 .847851 -0.18 0.853 -1.818549 1.504967
|
_cons | 33.58792 4.261296 7.88 0.000 25.23593 41.9399
--------------+----------------------------------------------------------------
participate |
depkid_u6 | -.0896882 .053792 -1.67 0.095 -.1951185 .0157422
relcare | .3037178 .0667023 4.55 0.000 .1729836 .434452
ghealth | .2660577 .0405846 6.56 0.000 .1865133 .345602
bmi | -.0127371 .0028127 -4.53 0.000 -.0182498 -.0072243
_cons | .1748344 .0745191 2.35 0.019 .0287797 .3208892
--------------+----------------------------------------------------------------
lambda | -.4360539 2.443039 -5.224322 4.352214
-------------------------------------------------------------------------------
h.With reference to the figure below showing the distribution of weekly hours worked by females, explain why a Tobit specification might 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?[20 marks]
MACROECONOMETRICS
2.You are working as an analyst for the UK Monetary Policy Committee. Using quarterly seasonally adjusted data from 1997q1 to 2020q4 your assistant has modelled the demand for money (MD) as a function of GDP (Y). They have provided you with tests of the time series properties of the data. There is also information provided on a forecast of the demand for money, where two alternative models are estimated over the period 2010q1 through to 2020q4: an ARIMA(1,0,1) and an ARIMA(2,0,1). Data are in natural logarithmic format.
Write a short detailed report for the Monetary Policy Committee based upon your assessment of the Stata output below, where ‘L’ denotes a lag operator and ‘D’ a difference operator. The report should be based upon a detailed discussion of the following areas:
a.An assessment of the OLS regression and the time series properties of the data from the ADF tests provided, including whether a long-run relationship exists between the demand for money (MD) and GDP (Y).[35 marks]
b.An examination of the ARIMA models and a detailed explanation of which specification is preferred.[35 marks]
c.Provide an explanation of any further analysis you may want to undertake giving your reasons.[30 marks]
*1. Regression of money demand on GDP
reg logMD logY
Source | SS df MS Number of obs = 96
-------------+---------------------------------- F(1, 94) = 3.74
Model | 3.30703199 1 3.30703199 Prob > F = 0.0562
Residual | 83.1412774 94 .884481674 R-squared = 0.0383
-------------+---------------------------------- Adj R-squared = 0.0280
Total | 86.4483094 95 .909982204 Root MSE = .94047
------------------------------------------------------------------------------
logMD | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
logY | 1.589514 .8220337 1.93 0.056 -.042653 3.221681
_cons | -10.55294 10.71159 -0.99 0.327 -31.82105 10.71517
------------------------------------------------------------------------------
estat dwatson
Durbin-Watson d-statistic( 2, 96) = .022292
predict e, resid
*2. ADF tests for stationarity
dfuller logY, regress lags(4)
Augmented Dickey-Fuller test for unit root Number of obs = 91
------------------------------------------------------------------------------
D.logY | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
logY |
L1. | -.0600237 .0269995 -2.22 0.029 -.1137059 -.0063415
LD. | -.3995889 .1065209 -3.75 0.000 -.611381 -.1877968
L2D. | -.3259163 .1729871 -1.88 0.063 -.6698611 .0180284
L3D. | .0163639 .7094485 0.02 0.982 -1.39421 1.426937
L4D. | .2489451 .6479647 0.38 0.702 -1.039382 1.537272
_cons | .7870643 .3526486 2.23 0.028 .0859045 1.488224
------------------------------------------------------------------------------
dfuller logMD, regress lags(4)
Augmented Dickey-Fuller test for unit root Number of obs = 91
------------------------------------------------------------------------------
D.logMD | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
logMD |
L1. | -.283871 .1351519 -2.10 0.039 -.5525891 -.0151529
LD. | -.2439251 .1416164 -1.72 0.089 -.5254964 .0376462
L2D. | .0491399 .1319961 0.37 0.711 -.2133037 .3115835
L3D. | -.1819375 .1271534 -1.43 0.156 -.4347525 .0708775
L4D. | -.2349869 .1096274 -2.14 0.035 -.4529556 -.0170182
_cons | 2.916696 1.368559 2.13 0.036 .1956342 5.637759
------------------------------------------------------------------------------
dfuller D.logY, regress lags(4)
Augmented Dickey-Fuller test for unit root Number of obs = 90
------------------------------------------------------------------------------
D2.logY | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
logY |
LD. | -1.428392 .6056263 -2.36 0.021 -2.632746 -.2240378
LD2. | .027961 .5974807 0.05 0.963 -1.160195 1.216117
L2D2. | -.317543 .6469494 -0.49 0.625 -1.604073 .9689868
L3D2. | -.1259343 .7178216 -0.18 0.861 -1.553401 1.301533
L4D2. | .6701141 .672076 1.00 0.322 -.6663827 2.006611
_cons | .004176 .0040985 1.02 0.311 -.0039743 .0123264
------------------------------------------------------------------------------
dfuller D.logMD, regress lags(4)
Augmented Dickey-Fuller test for unit root Number of obs = 90
------------------------------------------------------------------------------
D2.logMD | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
logMD |
LD. | -1.616965 .3611211 -4.48 0.000 -2.335094 -.898836
LD2. | .2498867 .3029198 0.82 0.412 -.3525026 .852276
L2D2. | .2199463 .2413287 0.91 0.365 -.2599623 .6998549
L3D2. | -.0767043 .1869149 -0.41 0.683 -.4484052 .2949966
L4D2. | -.2724451 .1068518 -2.55 0.013 -.4849316 -.0599586
_cons | .0380046 .0835392 0.45 0.650 -.1281223 .2041315
------------------------------------------------------------------------------
dfuller e, regress lags(4)
Augmented Dickey-Fuller test for unit root Number of obs = 91
------------------------------------------------------------------------------
D.e | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
e |
L1. | -.3330585 .1404399 -2.37 0.020 -.6122906 -.0538265
LD. | -.2005587 .1426445 -1.41 0.163 -.4841742 .0830567
L2D. | .0900581 .1325314 0.68 0.499 -.1734499 .353566
L3D. | -.1445777 .1279506 -1.13 0.262 -.3989778 .1098223
L4D. | -.23066 .110092 -2.10 0.039 -.4495524 -.0117675
_cons | .0154316 .0837729 0.18 0.854 -.1511314 .1819946
------------------------------------------------------------------------------
*3. Forecast of money demand 2010Q1 onwards – ARIMA(1,0,1)
arima D.logMD if date>=2010q1, arima(1,0,1)
ARIMA regression
Sample: 2010q1 - 2020q4 Number of obs = 44
Wald chi2(2) = 20.58
Log likelihood = -510.4454 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
D.logMD | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
logMD |
_cons | .223469 .5121901 0.44 0.663 -.7804049 .1227343
-------------+----------------------------------------------------------------
ARMA |
ar |
L1. | -.5924344 .2967502 -2.00 0.046 -1.174054 -.0108148
|
ma |
L1. | .873606 .2405563 3.63 0.000 .4021243 1.345088
-------------+----------------------------------------------------------------
-------------------
AIC BIC
-------------------
1028.891 1036.028
-------------------
Portmanteau test for white noise
--------------------------------------------
Portmanteau (Q) chi2(40) statistic = 23.7341
.9808
RMSE = 0.6951
*4. Forecast of money demand 2010Q1 onwards – ARIMA(2,0,1)
arima D.logMD if date>=2010q1, arima(2,0,1)
ARIMA regression
Sample: 2010q1 - 2020q4 Number of obs = 44
Wald chi2(3) = 29.29
Log likelihood = -509.654 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
D.logMD | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
logMD |
_cons | .2468865 .5177976 0.48 0.634 -.7679782 .1261751
-------------+----------------------------------------------------------------
ARMA |
ar |
L1. | -.5427889 .3851356 -1.41 0.159 -1.297641 .212063
L2. | -.2453184 .1670213 -1.47 0.142 -.5726742 .0820373
|
ma |
L1. | .7542615 .3798303 1.99 0.047 .0098078 1.498715
-------------+----------------------------------------------------------------
-------------------
AIC BIC
-------------------
1029.308 1038.229
-------------------
Portmanteau test for white noise
--------------------------------------------
Portmanteau (Q) chi2(40) statistic = 26.8514
RMSE = 0.7117
END OF QUESTION PAPER