 |
AEM702S - APPLIED ECONOMETRICS MODELLING - 1ST OP - NOVEMBER 2024 |
|
|
1 Page 1 |
▲back to top |
nAmlBIA UnlVERSITY
OF SCIEnCEAno TECHnOLOGY
FacultyofHealthN, atural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet
Private Bag13388
Windhoek
NAMIBIA
T: +264612072913
E: msas@nust.na
W: www.nust.na
QUALIFICATION: Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BSAM
LEVEL: 7
COURSE CODE: AEM702S
COURSE NAME: Applied Econometrics Modelling
SESSION: NOVEMBER 2024
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 100
EXAMINER
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
Dr D. B. GEMECHU
MODERATOR:
Prof L. PAZVAKAWAMBWA
INSTRUCTIONS
1. There are 7 questions, answer ALL the questions by showing all the necessary steps.
2. Write clearly and neatly.
3. Number the answers clearly.
4. Round your answers to at least four decimal places, if applicable.
PERMISSIBLE MATERIALS
1. Nonprogrammable scientific calculators with no cover.
THIS QUESTION PAPERCONSISTSOF 3 PAGES(Including this front page)
ATTACHMENTS
Four statistical distribution tables (t-, z-, x2- and F-distribution tables)
|
|
2 Page 2 |
▲back to top |
Question 1 (11 Marks]
1.1. Compare and contrast econometrics, economic theory, mathematical economics and economic
statistics
[4]
1.2. Briefly discuss the problem of multicollinearity in multiple linear regression model. Your
discussion should include definition, two consequences, two methods of detections and two
possible remedial measures.
(7J
Question 2 (20 Marks]
/J /J r;~?, 2. Consider a two-variable linear regression model Yi = /31 + (J2Xi + ui for i = 1,2,3, ..., n
= 2.1. Show that the OLS estimator 2 is an unbiased estimator of (32 . Hint 2
= where xi
xi-x
[61
= 2.2. Show that the Var(fA 23)
u2
Ixf
[6]
/J 2.3. Show that 2 is the best estimator of (32 . Hint: Consider an alternative linear unbiased estimator
/Ji= LWi~ of (32 and show that Var(/Ji)2::Var(/2J).
[8]
Question 3 (10 Marks]
3.1. Consider the general (k-variable) linear regression model
y
X p +u
p n x 1 Cl n x k k x 1 n x 1
Show that an OLSestimator = (X' X)- 1X'y is an unbiased estimator of p.
[SJ
3.2. Suppose a researcher collected data on personal saving (S) and personal income {I) for 31-year
period and fitted a linear regression model
Si = /J1+ /J2li + ui
Assume that a graphical inspection suggests that u/s are heteroscedastic so that a researcher
= employed the Goldfeld Quandt test by removing c 9 central observations after arranging the
data based on income. Applying OLSto each subset, a researcher obtained the following results.
For subset I: S1i = -738.84 + 0.008/i
For Subset II: S2i = 1141.07 + 0.029/i
With RSS1 = L ut = 144,771.5
With RSS2 = Iu~i = 769,899.2
Based on the above results, test if there is any evidence of heteroscedasticity at 5% level of
significancy.
[SJ
Question 4 (21 Marks]
4. A marketing department is interested in the effects of changing advertising levels for television and
internet on sales (Y). They vary X2=total expenditure on TV advertisement in $, and X3=total
expenditure on internet advertisement in$. Answer the following questions based on the summary
of sample values:
n = 20; Y = 183.8186; y'y = 686084.6; RSS = 608.6247; TSS = 10299.09
20
416.5343
X'X
=
(
416.5343
46.487
9546.826
8733.245
X'y
=
(
3676.373)
78940.14
77022.41
28.65885
Var-
cov(P) =
(
-0.65588
-0.6498
406.487)
8733.245
9111.308
-0.65588
0.045468
-0.01432
0.800494
(X' X)-
1
=
(
-0.01832
-0.01815
-0.6498)
-0.01432
0.046649
-0.01832
0.00127
-0.0004
-0.01815)
-0.0004
0.001303
Page 1 of2
|
|
3 Page 3 |
▲back to top |
(!:) 4.1. Compute the point estimator of p -
and interpret each partial coefficient
(4]
4.2. Construct ANOVA table and test the hypothesis H0 : /32 = {33 = 0 at 5% level of sign.
[5]
4.3. Compute an unbiased estimate of the residual variance
[2]
4.4. Test the significance of the partial regression coefficient for advertisement expense on TV (X2 ),
{32 . Use 5% level of significancy.
(4]
4.5. Compute and interpret the coefficient of multiple determination.
[3]
4.6. If a marketing department decided to invest $10 on TV advertisement and $3 on internet
advertisement, then what will be the predicted sells for such investment based on the fitted
model?
[3]
Question 5 [6 Marks]
5. Consider the following infinite distributed lag model:
Yt = a+ /JoXt + /31Xt-t + /32Xt- 2 + /33Xt_ 3 + ...+ Ut for O < il ::; 1
Show how the Koyck transformation can be used to produce the following type of model
Yt = a(l - il) + /JoXt + ilYt-t + Vt
[6]
Question 6 [20 Marks]
6. Consider the following regression result for expenditure on new plant and equipment (Y) on sales
(X) in billions of dollars and lagged value of Y.
Yt = -15.104 + 0.629Xt + 0.272Yt-1
se = (4.7294) (0.0978) (0.1148)
d = 1.5185,
durbin h = 1.3403
Answer the following questions based on this result.
6.1. If we assume that this model resulted from a Koyck-type transformation, then
6.1.1. What is the estimate for rate of decline or decay this model?
[2]
6.1.2. Compute the median lag
[2]
6.1.3. Compute the mean lag
[2]
6.1.4. What is the short-run or impact multiplier value for this model? Provide interpretation
of the value as well.
[2]
6.1.5. Compute the estimate for the coefficient of the first lag, Xt-t· Hint: Use the Koyck
scheme.
[2]
6.2. Assuming that
Yt°= a + /JoXt + Ut,
where Y* = desired, or long-run, expenditure for new plant and equipment.
Derive the partial adjustment model.
[4)
6.2.1. Compute the coefficient of partial adjustment
[2]
6.2.2. Estimate the parameters of this model
[4)
Question 7 [12 Marks]
7. Consider the following structural equations
Y1t = /310+ /312Yu+ Y11X1t+ Utt
Yu = /320+ /321Y1t + uu
7.1. Derive the reduced form equations expressed in the form of Y1t and Yu,
[10)
7.2. Determine which of the preceding equations are identified (either just or over).
[2]
=== END OF QUESTION PAPER===
Page 2 of2
|
|
4 Page 4 |
▲back to top |
~~l~ll[[M~-l[l"MlMl-~~N~llNll[~Mlllllll~ •
V
\\0
=0r\\-
r-
O'\\
tri
c- i
0- 0
M
00
-
0N
\\o0 o
-
0\\
r-
Vr---
V
V)
0- -
rr--
N
ci
N
V1n
O'\\
O'\\
00
1n
NN
00
00
-
N
=
\\0
O'\\
Vr)-
O'\\
N
0
NN
•
O'\\
-
00
N
O'\\
-
M
M
-
0
00
M
r-
\\0
N
M
0¢ 0
-
C-
M
¢
\\0
V)
-
M
N
00
V)
M
=0
\\0
-
0\\
0
0\\
V
MV
V'l
-
0r\\ -
0-0
VV
= = 00
r-
O'\\
V
V)
N
00
V
V'l
O'\\
N
\\0
0
V V VV
M
V'l
-
0\\
M
C-
0\\
M
\\0
ci
V'l
V'l
V'l
~[[[ll~[[[[~[[[l[[[[~[[~[[~[~[[
~~[~[[[[ll[[l[l[[[[l[[[[~i~[[[[
~lili['"lli'"ll!l_-[Mll~lNl•[•ll~lloll~ll~[N[-Nll_MN [~Ml!_•[•NnU_•[•llM[•[-li ¢
¢
-
-
V
0\\
M
V)
V
r-
0r-
-
0\\
\\0
-
00
0
V'l
O
N
C-
0\\
-
-
-
=
01n
-
O'\\
0
M
r-
\\0
N
0
0 00
-
-
-
-
N
¢
\\0
00
O'\\
0\\
M
\\0
0\\
N
NM
N
N
N
N
MO
00
\\0
V
-
V'l
00
-
V
Or-
¢
N
\\0
0\\
r- -
-
V
V'l
00
-
\\0
00
-
00 0
N
0
00 0
N
N
M
MM
M
MM
M
M
V
M
V'l
r-
M
V)
C-
NM
V
V
V
~~~[~[~[~[l[[~[ll[~[~[[[[~~~[[[
;=
=_ -
~lsi[lilililllillll[[ll[llllll[lllillllllillllllll~llli 0
0\\
V
r- 00 0
V'l
00
r- 00
\\0
V
0\\
0\\
0
00
\\0
00
00
\\0
r- N
M
-
00
V)
V
V
O'\\ C-
0
M
V'l
M
N
\\0
00
8-
00
.
.
.
.
,
.
.0
00
-
00
N
0
V)
MV
M
\\0
V)
00
V'l
r-
00
0\\
00
0
V
-
C- C- V
0\\
N
MV
\\0
-
-
00
V
r- 00 0\\
0
00
0
-
N
0
rO'\\
MV
11
-=~
I
l
~V:oOll:-li~N~M;l¢l:~:ll:lt;li:ll:6- li-::JlL-;:J;[- ;l-;lL- ;-0[;l-;[;- [;-[~6N[;:N[! ;lNl;N[;l:N;l;lNl;N[;lNl;l:Nl;NlliU_~
-
o
•
·~~lill[li[ll[lLlililillllli-lL~[N[~liNlloli~[M[~[MlNllMli•[•[-lL~[- 0
-
M
N
N
V
V
0V 0
O
0000
N
V-
0V 0
O
M
-N
-
N-
VV
M
MM
M
VM
V
1n
O
-
N
M
00
M
-
0
V
M
N
-
0
00
V
.
6
NM
00
0
NM
t_
~o.lt-:lL-0 :0Vl0 i:MlLO- :l0Vl:lMMi:~00 :l00i:~:li:M0 ~:V- ll0;0 lLV ;lN- t:000l0 i~000 li:llVVN;lNMMl~OVll~0V0 [~[V;llN ;l--l;lM0;0 [:[;0ll;ll;lL;ll~
~[[[[[[[~[[~~~~~[li~[li[[~~[[[~[[
~[~[[[[[[[[~[[~[~[~[~[~[[[[[~~[
~llllllLll~ll~~lllll~ll~~ll~l~~ ~ll0;U0_;0ll~0 l0t~0li;0lt~- lL- ~NllN;[;Mll;Mlt~ll~lL;lt;U_;[;ll~00 li;li;l0-l;l-l;l- l~-NllM-ill-M:ll~lt~ll~ll:li=[:
|
|
5 Page 5 |
▲back to top |
l!lililt~lilll!_li[l!_[lill nlllilllinlilllilililililililllillllll -
N
M 00 N M -
-
MO
-
-
M
00 00
M
-
00 N
00 -
M
00 0
-
M
00 00 00
99
0 0 0 0 0 0 0 0 0 0 0 OOOOOOOOOOOOOOOOOOO0
l!li-M9 [-9[M- llool00i~-li~ll-0l0l~-[oM[~l!_0~0 l!_--l!_M~[NllOllN[~li0N0 l0i0~l00!_-li-l!_~li~li-l!_M[~l!_~li~lloollooll~
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1l:~_1l~llL~li~Jlil.ltL~.Ll~~~lJ;_l~ll-~~l~Jl.tli~Ll~~l~Jli~lli~~l~lll~lil~ltl~~l~~ll~lll;il~LIE~l.~Jll~tli~lll~Ll~~I~J_;_llll~L;_ll_li
l!li[[[[[[li[~li[[[lilllilllllilllllllllilllllilillll
C
lc0!i n9- li~9 li9~lMl~0ll00~l0 lliN' l0irll0 NNlioo0 lt!,_~l!-_~Nll-lM!_~[~lloll~l!_~l!_~l!_~l!_~l!_~l!_ollNll~ll~li~l!_~l!_ool!_oo
0
N
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO0
1.
C
=lci !90o09li09~0[~0 [0 ~0 ~0 ~0 lg0 [0 ~0 !0~0[g0 !0 :!0 ~0 10 ~0 ~0 ~0 l0l~0 l0lg0 ~0 ~l0lll0i!l_l0 l~l0 ll~li~li~li~lt~
0
0M
M
00
00
N
0
N
N 00
-
-
M
00
M 00
l!lilililllilllililllilll!_lllllililllilllililililllllililllillll M N -
-
-
M 00 0 0 00 M
00 00
M 00 M
0N
00 00
d
0000000000000000000000000000000
l!l!_ o09 o9ll~090 liN~[~llNM [~liNN ll~l!_00M0 [-N[-Mlloollooll~l!000_lN0l0~l0li0 l~~l[i~Nl[i~li!~_l~iNl!_[M~lil~i'llooll~
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 OOOOOOOO0
0~llM00liMN [-[~[~N ll-li-l!-_-[0 olol~ll00,l~0ll~Nl!_M oli~llM[~l!_0~0 l!00_-00l!_~l!_Nl!_~l!_~l!_Nll~l!_~l!_~l!_~lloo
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
ll!!lgl!l[_~ll~[~[~[i~~~~~8[[~~[l~i~[l~[[~~l[!:_l:il~
l
Table for a=.05
df2~dfl
l
2
3
4
1 I2 I3
I I 161.448 199.500 215.707
I 18.513 ,1 19.ooo 19.164
I 10.128 9.5521 9.277
7.7091 6.944 6.591
4 I5
I 224.583 230.162
19.2471 19.296
I 9.111 9.014
6.3881 6.256
6 I 7 I8
I I 233.986 236.768 238.883
I 19.329 1 19.353 19.371
8.941 1 8.887
I 6.163 6.0942
8.845
6.041
I 9
10
I 240.543 241.882
I 19.384 19.396
I 8.812 8.786
5.998 I 5.964
12
243.906
19.413
8.745
5.912
5
6.6081 5.786 5.409 5.1921 5.050 4.950
4.876
4.818
4.7721 4.735
4.678
6
5.9871 5.143
4.757
4.533 I 4.387
4.284
4.207 4.147 4.0991 4.060 3.999
7
I 5.591 4.737
4.347
I 4.120 3.972
3.866
3.787
3.726
3.676 I 3.637
3.575
8
5.318 I 4.459
4.066
3.838
3.688
3.581
3.501 3.438 3.388 1 3.347 3.284
9
5.1171 4.256 3.863 3.633 3.482 3.374
3.293 3.229 3.1781 3.137 3.073
10
4.9651 4.103 3.708 3.478 3.326 3.217
3.136
3.072
3.020 I 2.978
2.913
11
4.844 3.982 3.587 3.358 3.204 3.095
3.012 2.948 2.8961 2.854 2.788
12
4.747 3.885 3.490 3.259 3.106 2.996
2.913 2.849 2.796 2.753 2.687
13
I 14
I 15
I 16
I 17
I 18
I 19
I 20
4.667 3.806 3.411 1 3.179
4.600
4.543
3.739 3.3441
I 3.682 3.2871
3.112
3.056
I 4.494
3.6341 3.2391 3.007
I 4.451
I 3.591 3.197 1 2.965
I 4.414
I 3.555 3.160 1 2.928
I I 4.381 3.5221 3.1271 2.895
I 4.351 1 3.493 I 3.098 I 2.866
3.025 2.915
2.958 2.848
2.901 2.791
2.852 2.741
2.810 2.699
2.773
I 2.740
I 2.111
2.661
2.628
2.599
2.832
2.764
2.707
2.657 1
2.6141
2.5771
2.5441
2.5141
2.767
2.699
I 2.641
2.591 I
2.548 1
2.510 I
2.4771
2.441 I
2.714
2.645
2.587
2.537
2.494
2.456
2.423 1
I 2.393
2.611 1
2.6021
2.5441
2.4941
2.450 I
2.412 I
2.378 I
2.348 I
2.604
2.534
2.475
2.425
2.381
2.342
2.308
2.278