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Introduction  

    Quantitative  Methods  II  (EC)   Academic  Year  2012/2013,  Block  3  

 

 


Uniseminar  –  Quantitative  Methods  II  

Welcome  to  Uniseminar!  

 

 

 

 

                         Introduction  

I n t r o d u c t i o n  

Uniseminar   offers   E x a m   P r e p a r a t i o n   S e m i n a r s ,   S u m m a r y   S c r i p t s   a n d   L e a r n i n g   C a r d s   for  students  of  the  Maastricht  University.  It  is  our  goal  to  op-­‐

timally  prepare  you  for  your  exams  and  to  make  your  own  exam  preparation  as  

efficient  as  possible.  In  order  to  achieve  this  goal,  we  have  developed  a  system  of  

seminars  in  combination  with  an  extensive  summary  script,  which  is  proven  for   several  years  by  now.  

In   university   it   is   often   the   case   that   there   is   a   lot   of   material   available   for   a   course  and  that  the  importance  of  this  material  is  hard  to  evaluate.  Since  we,  as  

students,  have  made  this  experience  as  well,  you  are  provided  with  a  Uniseminar   Summary   Script   of   the   corresponding   course.   This   folder   contains   all   exam-­‐ relevant  material  and  it  gives  you  a  good  summary  of  all  course  topics.  The  con-­‐

tent   of   the   folder   is   created   by   experienced   Master   or   PhD   students,   who   have   taught  this  course  already  several  times.  As  a  consequence,  it  is  possible  for  you   to   concentrate   on   the   actual   exam   preparation,   rather   than   spending   hours   searching  and  printing  the  right  material.  

At   the   end   of   week   6   of   your   block,   normally   during   the   weekend,   our   E x a m  

P r e p a r a t i o n   S e m i n a r s   take   place.   These   seminars   are   taught   by   above-­‐

average   students,   who   have   already   mastered   their   studies   at   the   Maastricht   University  and  have  a  great  deal  of  experience  in  tutoring.  Since  they  have  stud-­‐

ied   and   taught   at   the   Maastricht   University   they   know   exactly   where   potential   problems  may  lie  and  are  therefore  able  to  optimally  teach  you  the  whole  theory  

of  the  course  and  practice  perfectly  tailored  examples  with  you.  Furthermore  you   can  bring  in  your  own  questions  during  the  seminar  and  discuss  individual  prob-­‐ lems  during  the  breaks.  

You   are   able   to   pick   up   your   S u m m a r y   S c r i p t   a n d   L e a r n i n g   C a r d s   in   ad-­‐

vance  of  the  Seminar  in  order  to  already  start  preparing  so  that  you  can  discover  

your  own  difficulties  early  enough.  Later  in  the  Seminar  you  will  then  know  what  


Introduction    

 

                 Uniseminar  –  Quantitative  Methods  II  

your  weaknesses  are  and  be  able  to  pay  special  attention  to  these  sections  or  ask  

questions   about   it.   Our   Summary   Script   and   Learning   Cards   are   updated   every  

year  according  to  the  current  course’s  content  and  we  are  always  trying  to  opti-­‐

mize  the  folder  as  much  as  possible.    

A b o u t    U U s  

 

Uniseminar  was  founded  by  two  students  at  the  University  of  St.  Gallen  in  order   to  make  Exam  Preparation  more  efficient  and  coherent.  Since  2005  we  have  ex-­‐

panded   our   vision   and   are   now   offering   seminars   and   material   for   an   efficient   exam  preparation  in  Switzerland,  the  Netherlands,  Italy  and  Germany.  

Thanks  to  this  longstanding  experience,  we  were  able  to  build  up  a  team  of  highly  

qualified   tutors   and   editors   and   are   therefore   able   to   guarantee   high   quality   of   exam  preparation.  

The  team  of  Uniseminar  is  grown  strongly  over  the  years  and  comprehends  sev-­‐

eral  mathematicians,  statisticians  and  economists,  who  all  bring  a  great  didactical   experience.  All  tutors  of  Uniseminar  have  been  teaching  their  field  for  years  and  

know  exactly  what  is  important  in  order  to  optimally  prepare  and  pass  the  exam.    


Uniseminar  –  Quantitative  Methods  II  

S u m m a r y    SS c r i p t  

 

 

 

 

                         Introduction  

This  aim  of  this  folder  is  to  support  you  with  your  exam  preparation  for  ‘Quanti-­‐

tative   Methods   II’   as   much   as   possible.   Usually   it   consists   of   five   different   sec-­‐ tions.  As  follows,  a  short  overview  of  the  content  of  this  folder:  

 

1 . T h e o r y :   The  Theory  Script  summarizes  the  whole  theory  of  the  course  in  

a  simple  and  understandable  way.  Concepts  are  explained  with  the  help  of   demonstrative  examples.  It  is  structured  according  to  the  seven  weeks  of   the  course  and  is  one  of  the  most  important  parts  of  your  exam  prepara-­‐ tion.      

2. P r a c t i c e :  The  Practice  part  contains  practice  exercises  to  each  week  and   therefore  to  each  chapter  of  the  theory  script.  By  this,  you  can  deepen  your  

theoretical  knowledge  with  practical  exercises  and  you  can  go  through  the   exercises  of  these  topics  again,  which  you  have  not  understood  so  well  un-­‐

til  now.    

3. E x a m s :  In  this  part  you  will  find  old  exams  of  the  Maastricht  University,   as  well  as  one  practice  exams  constructed  by  Uniseminar.  During  the  sem-­‐ inar  you  will  then  receive  a  further  practice  exam.  

4. E x t r a s :   In  the  Extras  part,  you  will  find  a  formula  sheet  as  well  as  an  ex-­‐ planation  of  how  to  read  off  the  statistics  tables.  

5. S e m i n a r :    In  this  part,  we  have  provided  you  with  some  notepaper  so  that  

you   can   take   notes   during   the   seminar.   Furthermore   you   will   receive   a  

fourth  practice  exam  during  the  seminar,  which  you  can  file  in  here.  In  case   you  have  not  subscribed  for  the  Quantitative  Methods  II  seminar  yet,  you   can  do  so  on  our  website  -­‐  www.uniseminar.nl  -­‐  at  any  time.  


Introduction    

Q u a n t i t a t i v e    M M e t h o d s    II I  

 

                 Uniseminar  –  Quantitative  Methods  II  

The  ‘QM2’  course  treats  two  different  main  fields:  Mathematics  and  Statistics.  It   is  the  continuation  of  QM1,  you  have  attended  and  hopefully  passed  in  your  first   block  of  this  academic  year.  The  topics  of  the  course  build  on  QM1.  Although  we   integrated  some  minor  parts  of  the  QM1  course,  it  is  essential  that  you  have  un-­‐ derstood   the   major   theories   and   practices   of   QM1   in   order   to   master   QM2.   De-­‐

pending  on  your  difficulties,  you  should  put  your  focus  on  certain  fields  or  topics,   however,  do  not  forget  that  the  exam  is  equally  distributed  in  terms  of  questions   per  topics.  It  does  not  make  any   sense   to   concentrate   on   Mathematics   only,   be-­‐ cause  this  knowledge  alone  will  not  be  sufficient  to  pass  the  exam.  

The  exam  consists  of  40  multiple  choice  questions  and  you  will  have  3  hours  time   to   calculate   it.   As   mentioned,   the   questions   are   equally   distributed,   i.e.   you   will   have  20  math  questions  and  20  statistics  questions.    

H i n t s    aa n d    T T r i c k s  

Here  are  some  tips  that  may  be  helpful  for  your  exam  preparation.  Many  students   make  typical  errors  when  preparing  for  their  first  exams  at  university.  We  there-­‐

fore  want  to  help  you  to  avoid  these  mistakes,  so  that  you  can  focus  on  the  essen-­‐ tial  stuff,  rather  than  wasting  your  time  with  a  preparation  into  the  wrong  direc-­‐ tion!  

 

H o w    d d o    II    o o p t i m a l l y    p p r e p a r e    ff o r    aa n    ee x a m ?  

The  exams  of  the  university  are  created  in  such  a  way  that  every  student  can  pass  

them  with  an  average  preparation  time.  Since  there  is  a  lot  of  content  and  time  is  

limited,   planning   is   the   basis   of   your   success.   You   don’t   need   to   be   a   genius   to  

pass  the  exam,  but  you  should  still  take  care  of  a  few  things  and  try  to  develop  a   certain   discipline   in   the   following   weeks.   The   subsequent   hints   may   be   helpful   for  you:  


Seminar

Extras

Exams

Practice

Theory

T


Theory  

    Quantitative  Methods  II  (EC)   Academic  Year  2012/2013,  Block  3  

 

 

 


Theory    

T h e o r y  

 

 

 

 

 

                 Uniseminar  –  Quantitative  Methods  II  

The   Theory   Script   summarizes   the   whole   theory   of   the   course   in   a   simple   and  

understandable   way.   Concepts   are   explained   with   the   help   of   demonstrative   examples.  It  is  structured  according  to  the  seven  weeks  of  the  course  and  is  one   of  the  most  important  parts  of  your  exam  preparation.  Although  practice  is  very  

important,  it  is  even  more  crucial  to  understand  the  basic  concepts  of  the  course  

in  order  to  be  able  to  calculate  and  understand  all  different  kinds  of  exercises  and   exam  questions.    

 

T a b l e    o o f    C C o n t e n t s     M a t h e m a t i c s  

1   Series  of  payments  and  discounting  

2   Matrices,  Determinants  and  Systems  of  Equations  

3   Comparative  Statics  

1  

1  

6  

19  

4   Constrained  Optimization  

25  

S t a t i s t i c s  

3 1  

 

1   Recap  from  QM1  –  Hypothesis  Testing  

31  

3   Statistic  inference  based  on  more  than  two  samples  

42  

2   Statistic  inference  based  on  two  samples  

33  

4   Simple  Regression  

49  

6   Regression  assumptions  

70  

5   Multiple  Regression  

57  


Theory  –  Mathematics      

2        

 

                     Uniseminar  –  Quantitative  Methods  II  

M a t r i c e s ,    D D e t e r m i n a n t s    aa n d    SS y s t e m s    o o f    E E q u a t i o n s  

This  section  introduces  matrix  algebra,  which  is  in  itself  a  branch  of  mathematics.   In   general,   matrices   and   their   determinants   can   be   used   to   solve   or   simplify   a   large  variety  of  problems.  One  of  the  big  applications  is  the  solving  of  systems  of   linear   equations.   However,   there   are   other   fields   when   you   will   encounter  

matrices   that   are   not   part   of   this   course.   One   example   some   of   you   may   encounter   in   later   courses   is   that   Econometricians   use   matrices   to   represent   variables  and  parameters  in  regression  analysis.    

 

2 . 1         M a t r i c e s  

W h a t    ii s    aa    m m a t r i x ?  

In  general  a  m m a t r i x  is  simply  a  rectangular  array  of  numbers.  Thereby  the  order   of  the  numbers  and  the  format  of  the  matrix  are  important  and  cannot  simply  be  

changed.  The  use  of  matrices  is  that  they  facilitate  many  otherwise  more  difficult  

mathematical   operations.   To   make  this   clear   just   consider   the   following   system   of  equations:  

đ?‘Žđ?‘Ž đ?‘Ľđ?‘Ľ + đ?‘Žđ?‘Ž đ?‘Ľđ?‘Ľ = đ?‘?đ?‘?  

đ?‘Žđ?‘Ž đ?‘Ľđ?‘Ľ + đ?‘Žđ?‘Ž đ?‘Ľđ?‘Ľ = đ?‘?đ?‘?  

In  matrix  notation  this  can  be  written  as  follows:   đ?‘Žđ?‘Ž đ??´đ??´đ??´đ??´ = đ?‘?đ?‘?                      đ?‘¤đ?‘¤đ?‘¤đ?‘¤đ?‘¤đ?‘¤â„Ž  đ??´đ??´ = đ?‘Žđ?‘Ž 

đ?‘Žđ?‘Ž đ?‘Ľđ?‘Ľ đ?‘?đ?‘? , đ?‘?đ?‘? =  đ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Žđ?‘Ž  đ?‘Ľđ?‘Ľ = đ?‘Žđ?‘Ž đ?‘Ľđ?‘Ľ   đ?‘?đ?‘?

Thereby,  A  is  a  matrix  of  format   2x2,   while   b   and   x   are   matrices  of   format   2x1.  

Thereby,  the  first  number  always  refers  to  the  row  of  a  matrix,  while  the  second  

refers   to   the   columns.   As   b   and   x   only   have   one   column   they   can   also   be   called   vectors.  

     

6  


Uniseminar  –  Quantitative  Methods  II    

B a s i c    m m a t r i x    o o p e r a t i o n s  

 

 

 

                     Theory  –  Mathematics  

As   with   numbers   it   is   also   possible   to   perform   mathematical   operations   with  

matrices.  However,  the  rules  that  apply  to  matrices  are  different  than  the  rules,  

which   apply   to   numbers.   To   start   easy   let’s   have   a   look   at   how   it   is   possible   to   sum  matrices:   �� ��

đ?‘Žđ?‘Ž đ?‘?đ?‘? + đ?‘Žđ?‘Ž đ?‘?đ?‘?

đ?‘?đ?‘? đ?‘Žđ?‘Ž + đ?‘?đ?‘? =  đ?‘?đ?‘? đ?‘Žđ?‘Ž + đ?‘?đ?‘?

đ?‘Žđ?‘Ž + đ?‘?đ?‘?   đ?‘Žđ?‘Ž + đ?‘?đ?‘?

From   the   formula   you   can   see   that   each   individual   element   is   added   to   the   corresponding  element  of  the  other  matrix.  Moreover,  this  implies  that  it  is  only   possible  to  add  matrices,  which  have  the  same  format  (dimensions),  that  is,  have   an  equal  amount  of  rows  and  columns.  

Additionally   to   adding   matrices   we   can   also   multiply   a   matrix   with   a   constant.  

Doing  so  requires  simply  multiplying  each  individual  element  in  the  matrix  with   the  constant:   đ?‘Žđ?‘Ž đ?‘?đ?‘? ∗ đ?‘Žđ?‘Ž 

đ?‘Žđ?‘Ž đ?‘?đ?‘? ∗ đ?‘Žđ?‘Ž = đ?‘Žđ?‘Ž đ?‘?đ?‘? ∗ đ?‘Žđ?‘Ž

đ?‘?đ?‘? ∗ đ?‘Žđ?‘Ž đ?‘?đ?‘? ∗ đ?‘Žđ?‘Ž  

The  last  step  is  to  take  the  product  of  two  matrices.   �� ��

đ?‘Žđ?‘Ž đ?‘?đ?‘? ∗ đ?‘Žđ?‘Ž đ?‘?đ?‘?

đ?‘?đ?‘? đ?‘Žđ?‘Ž đ?‘?đ?‘? + đ?‘Žđ?‘Ž đ?‘?đ?‘? =   đ?‘?đ?‘? đ?‘Žđ?‘Ž đ?‘?đ?‘? + đ?‘Žđ?‘Ž đ?‘?đ?‘?

đ?‘Žđ?‘Ž đ?‘?đ?‘? + đ?‘Žđ?‘Ž đ?‘?đ?‘?   đ?‘Žđ?‘Ž đ?‘?đ?‘? + đ?‘Žđ?‘Ž đ?‘?đ?‘?

To   avoid   mistakes   it   is   easier   to   write   the   multiplication   as   follows.   In   order   to  

obtain   the   element   in   column   1,   row   1   of   the   resulting   matrix,   you   have   to  

multiply  column  1  of  matrix  A  with  row  1  of  matrix  B.      Following  this  logic,  the   element   in   column   2,   row   1   of   the   resulting   matrix   is   obtained   by   multiplying   column   2   of   matrix   A   with   row   1   of   matrix   B.   This   logic   is   persistent.   Note   that  

not  all  matrices  can  be  multiplied  with  each  other.  Whether  or  not  two  matrices   can  be  multiplied  depends  on  its  dimensions  as  explained  below  

Multiplying   đ??´đ??´ ∗ đ??ľđ??ľ  is   not   equal   to   the   result   of   đ??ľđ??ľ ∗ đ??´đ??´  in   matrix   algebra.  

Additionally,   not   all   matrices   can   be   multiplied   with   each   other.   To   make   the  

multiplication  possible,  the  amount  of  columns  of  the  first  matrix  needs  to  equal   the   amount   of   rows   in   the   second   matrix.   Moreover,   the   dimensions   of   the   7  


Theory  –  Mathematics      

 

                     Uniseminar  –  Quantitative  Methods  II  

resulting  matrix  look  as  follows:  The  number  of  rows  of  matrix  A  and  the  number   of  columns  of  matrix  B.  However,  this  is  only  possible  if  the  number  of  columns  of  

matrix   A   equals   the   number   of   rows   of   matrix   B.   To   make   this   point   clear   consider  the  following  examples:   1.)

2.) 3.)  

đ??´đ??´:  dimension  3  x  2;  đ??ľđ??ľ:  dimension  2  x  3            

→ đ??´đ??´đ??´đ??´: dimension  3  x  3;    đ??ľđ??ľđ??ľđ??ľ: dimension  2  x  2  

đ??´đ??´:  dimension  4  x  2;  đ??ľđ??ľ:  dimension    2  x  2            

→ đ??´đ??´đ??´đ??´: dimension  4  x  2;    đ??ľđ??ľđ??ľđ??ľ: not  defined, as  2 ≠ 4   đ??´đ??´:  dimension  đ?‘šđ?‘š  x  đ?‘›đ?‘›;  đ??ľđ??ľ:  dimension  đ?‘?đ?‘?  x  đ?‘žđ?‘ž          

→ đ??´đ??´đ??´đ??´: dimension  đ?‘šđ?‘š  x  đ?‘žđ?‘ž   only  defined  for = đ?‘?đ?‘?  

         đ??ľđ??ľđ??ľđ??ľ: dimension  đ?‘?đ?‘?  x  đ?‘›đ?‘›  (only  defined  for  đ?‘žđ?‘ž = đ?‘šđ?‘š)  

T h e    tt r a n s p o s e  

Every   matrix   has   a   so-­�called   transpose.   It   is   denoted   that   the   t r a n s p o s e   of  

matrix  A  is  called  A’.  It  can  easily  be  computed  by  switching  rows  and  columns  of   the  original  matrix:   đ?‘Žđ?‘Ž đ??´đ??´ = đ?’‚đ?’‚ đ?&#x;?đ?&#x;?đ?&#x;?đ?&#x;?

đ?’‚đ?’‚đ?&#x;?đ?&#x;?đ?&#x;?đ?&#x;? đ?‘Žđ?‘Ž            → đ??´đ??´ = đ?‘Žđ?‘Ž đ?’‚đ?’‚đ?&#x;?đ?&#x;?đ?&#x;?đ?&#x;?

đ?’‚đ?’‚đ?&#x;?đ?&#x;?đ?&#x;?đ?&#x;? đ?‘Žđ?‘Ž  

As   rows   and   columns   are   simply   switched   this   also   means   that   if   a   matrix   has   format  m  x  n,  then  its  transpose  will  have  format  n  x  m.    

T h e    ii d e n t i t y    m m a t r i x  

The   i d e n t i t y   m a t r i x   is   a   matrix,   which   has   ones   along   its   main   diagonal   and  

zeros   everywhere   else.   Moreover,   it   is   of   square   format,   meaning   that   it   has   as  

many   rows   as   columns.   The   amount   of   rows   and   columns   is,   however,   not   restricted.  In  general  the  identity  matrix  is  denoted  đ??źđ??ź ,  in  which  n  is  the  amount  

of   rows.   The   following   gives   an   identity   matrix   with   three   rows   and   three   columns.   8  


Uniseminar  –  Quantitative  Methods  II    

4        

 

C o n s t r a i n e d    O O p t i m i z a t i o n  

 

 

                     Theory  –  Mathematics  

QM1  has  extensively  dealt  with  the  optimization  of  functions.  QM2  extends  this  

framework   to   the   c o n s t r a i n e d   o p t i m i z a t i o n   of   functions.   The   difference   is  

really  that  we  are  no  longer  maximizing  a  function  on  its  entire  domain,  but  only   consider  a  certain  part  of  the  domain.  Thereby  the  Lagrange  multiplier  method  is   a  useful  tool  in  solving  such  problems.    

 

4 . 1         T h e    L L a g r a n g e    m m u l t i p l i e r    m m e t h o d   S e t t i n g    u u p    tt h e    L L a g r a n g i a n  

The   general   constrained   optimization   problem   consists   of   two   parts.   First   the   function  that  needs  to  be  optimized,  say  maximized  in  this  case:   max    ��(��, ��)  

Second  the  constraint,  which  sets  the  region  on  which,  the  maximum  needs  to  be.  

Thereby  s.t.  stands  for  subject  to  and  b  can  be  any  value.   đ?‘ đ?‘ . đ?‘Ąđ?‘Ą.      đ?‘”đ?‘” đ?‘Ľđ?‘Ľ, đ?‘Śđ?‘Ś ≤ đ?‘?đ?‘?  

The  Lagrangian  can  then  be  written  as:   đ??żđ??ż = đ?‘“đ?‘“ đ?‘Ľđ?‘Ľ, đ?‘Śđ?‘Ś − đ?œ†đ?œ†(đ?‘”đ?‘” đ?‘Ľđ?‘Ľ, đ?‘Śđ?‘Ś − đ?‘?đ?‘?)  

 

F i r s t    O O r d e r    cc o n d i t i o n s  

The   three   f i r s t   o r d e r   c o n d i t i o n s   are   then   the   derivatives   of   the   Lagrangian  

with   respect   to  đ?‘Ľđ?‘Ľ,  đ?‘Śđ?‘Ś  and  đ?œ†đ?œ†.   The   derivative   with   respect   to  đ?œ†đ?œ†  is,   however,   always   equal   to   the   constraint.   As   in   the   case   of   unconstrained   optimization   the   first  

order  constraints  need  to  be  equal  to  zero.  In  a  next  step  it  can  then  be  solved  for  

the  three  unknowns  đ?‘Ľđ?‘Ľ,  đ?‘Śđ?‘Ś  and  đ?œ†đ?œ†  in  these  three  equations.  Please  be  aware  that  any  

solution  only  qualifies  as  a  stationary  point  and  does  not  necessarily  need  to  be  a   maximum  or  minimum.  The  next  section  will  tell  you  how  to  distinguish  between   the  different  stationary  points.  

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Theory  –  Mathematics      

I.

II.

 

III.

 

                     Uniseminar  –  Quantitative  Methods  II  

đ??żđ??ż = 0  

đ??żđ??ż = 0  

đ?‘”đ?‘” đ?‘Ľđ?‘Ľ = đ?‘?đ?‘?  

A n    ee x a m p l e  

Consider  the  following  optimization  problem:   max      đ?‘Ľđ?‘Ľ − 2đ?‘Śđ?‘Ś                        đ?‘ đ?‘ . đ?‘Ąđ?‘Ą. : 2đ?‘Ľđ?‘ĽÂ˛ + đ?‘Śđ?‘ŚÂ˛ = 18   The  Lagrangian  of  this  problem  is.   đ??żđ??ż = đ?‘Ľđ?‘Ľ − 2đ?‘Śđ?‘Ś − đ?œ†đ?œ† 2đ?‘Ľđ?‘Ľ  − đ?‘Śđ?‘Ś  − 18  

This  gives  the  following  first  order  conditions.   I.

II.

III.

1 − 4đ?œ†đ?œ†đ?œ†đ?œ† = 0  

−2 − 2đ?œ†đ?œ†đ?œ†đ?œ† = 0  

2��² + ��² = 18  

The  first  two  equations  can  be  solved  for  đ?œ†đ?œ†  and  then  equated.  

đ?œ†đ?œ† =

1 1 = − = đ?œ†đ?œ†   4đ?‘Ľđ?‘Ľ đ?‘Śđ?‘Ś

Please   note   that   we   just   divided   by  ��  and   by  ��.   Thereby   we   are   implicitly  

assuming   that   neither   of   the   two   is   equal   to   zero   as   this   is   not   possible.   Unfortunately   this   also   means   that   if  �� = 0  or  �� = 0  was   a   solution   of   to   the   problem  we  would  have  lost  it.  To  be  very  specific,  this  does  not  mean  that  it  is  

necessarily  a  solution,  it  just  means  that  if  it  was  we  would  have  lost  it.  Therefore   we   need   to   check   whether   it   is   a   solution   or   not   by   just   plugging  �� = 0  and  

afterwards  �� = 0  into   the   first   order   conditions.   Already   the   first   condition   shows   that   ��  cannot   be   equal   to   zero   as   that   would   violate   the   equation.  

Moreover,   if  �� = 0  the   second   condition   is   violated.   Hence,   in   this   case   neither  

�� = 0  nor  �� = 0  is   a   solution   and   therefore   dividing   by   it   results   in   no   loss   of  

solution.   If  �� = 0  or  �� = 0  were   to   satisfy   all   three   constraints   it   would   be   a  

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                     Theory  –  Mathematics  

solution   and   could   just   be   added   to   the   solutions   of   the   problem   which   we   are   going  to  find  now.  

Multiplying  the  previous  equation  with  the  two  denominators  and  by  −1  gives.  

−4đ?‘Ľđ?‘Ľ = đ?‘Śđ?‘Ś  

This  can  then  be  plugged  into  the  last  constraint.  

2��² + 16��² = 18  

��  = 1  

�� = ¹1  

Which  we  can  then  plug  back  into  −4đ?‘Ľđ?‘Ľ = đ?‘Śđ?‘Ś.  

If  đ?‘Ľđ?‘Ľ = 1,  then  đ?‘Śđ?‘Ś = −4,  if  đ?‘Ľđ?‘Ľ = −1,  then  đ?‘Śđ?‘Ś = 4.  Hence  we  have  found  two  stationary  

points   which   solve   the   constraint   optimization   problem.   As   always   with   stationary   points   we   cannot   say   yet   whether   those   are   a   maximum   or   a   minimum.  Judging  those  stationary  points  is  the  topic  of  the  next  section.    

4 . 2         I n t e r p r e t i n g    tt h e    rr e s u l t s   T h e    ee x t r e m e    v v a l u e    tt h e o r e m  

The   Lagrange   multiplier   approach   to   solve   a   constraint   optimization   problem  

will   always   give   you   stationary   points.   However,   to   answer   a   problem   it   is   necessary   to   know   whether   this   stationary   point   qualifies   as   a   minimum   or   maximum  solution.  

Every  function  has  at  least  one  maximum  and  one  minimum  if  it  is  optimized  on  a   closed  and  bounded  set.  A  set  is  called  closed  if  the  boundary  of  the  set  is  part  of  

the  set  as  well  and  a  set  is  called  bounded  if  it  is  not  possible  to  “leave�  the  set   27  


Uniseminar  –  Quantitative  Methods  II        

 

 

 

                               Theory  –  Statistics  

S t a t i s t i c s   1  

R e c a p    ff r o m    Q Q M 1    ––    H H y p o t h e s i s    T T e s t i n g  

This   very   first   section   recaps,   when   a   certain   h y p o t h e s i s   is   rejected   at   the  

example  of  the  z-­�table.  The  story  is  exactly  the  same  for  all  other  distributions.  If  

the  critical  value  is  closer  to  the  null  hypothesis  than  you  reject  the  test  statistic,   otherwise  you  do  not.  

To   perform   a   test   we   first   need   to   decide   upon   a   level   of   significance.   As   the  

standard  normal  curve  goes  from  minus  infinity  to  plus  infinity  we  need  to  cut  off  

a   certain   part   of   the   distribution   which   we   think   is   unlikely.   The   standard   significance  level  in  statistics  is  5%.  This  means  that  we  cut  off  an  area  which  is  

2.5%  large   on   the   left   and   on   the   right   respectively.   These   areas   are   then  

considered  as  being  too  unlikely  to  occur.  It  is  now  possible  to  check  the  ��-­�table   for  the  ��-­�value  that  belongs  to  this  area.  Feel  free  to  validate  by  yourself  that  the  

area  to  the  right  of  1.96  is  actually  2.5%  large.  As  the  distribution  is  symmetric,  it  

also  means  that  the  value  of  −1.96  cuts  off  the  2.5%  area  on  the  left.  This  critical   value   of  −1.96  we   can   then   compare   to   the  đ?‘§đ?‘§-­â€?score,   which   follows   from   the   calculation  based  on  the  hypothesis  test.  The  two  graphs  show  that  we  can  reject   whenever  the  đ?‘§đ?‘§-­â€?value  is  more  extreme  than  the  critical  value  and  cannot  reject  if  

it  is  less  extreme.  

 

To  perform  a  đ?‘?đ?‘?-­â€?value  test  we  do  not  need  to  fix  the  significance  level  right  away.   All   we   have   to   do   is   to   find   the   area   to   the   extreme   of   our  đ?‘§đ?‘§-­â€?value.   Then   this  đ?‘?đ?‘?-­â€?

value  can  be  compared  to  any  significance  level  đ?›źđ?›ź  we  like.  Whenever  the  đ?‘?đ?‘?-­â€?value  

is   larger   than  ��  we   cannot   reject   at   that   significance   level  ��,   whenever   it   is   31  


Theory  –  Statistics  

 

                     Uniseminar  –  Quantitative  Methods  II  

smaller  we  can  reject  at  that  đ?›źđ?›ź.  Again  this  is  summarized  by  a  graph.  Note  that  the   fact  that  the  đ?‘?đ?‘?-­â€?value  is  smaller  than  the  significance  level  đ?›źđ?›ź  follows  directly  from  

the   fact   that   the   critical   value   is   smaller   than   the  ��-­�score   as   the   areas   become  

smaller  if  the  ��-­�values  become  larger.  

 

When  using  the  đ?‘?đ?‘?-­â€?value  test  it  is  important  to  understand  the  interpretation  of  a  

đ?‘?đ?‘?-­â€?value.   We   already   defined   it   as   the   area   to   the   extreme   of   the  đ?‘§đ?‘§-­â€?statistic.  

However,   in   words   it   is   the   probability   to   observe   such   a   sample   (or   an   even   more   extreme   one)   GIVEN   that   the   null   hypothesis   is   true.   As   we   center   our  ��-­�

curve  around  the  null  hypothesis  we  first  assume  that  it  is  true.  If  the  đ?‘?đ?‘?-­â€?value  is   then   very   small   we   have   a   small   chance   of   getting   such   a   sample   if   the   null   hypothesis   is   true.   However,   as   we   really   got   this   sample   we   can   only   conclude   that  the  assumption  was  false  and  the  null  hypothesis  cannot  be  true.  

So   far   we   tested   two-­�sided.   The   main   thing   that   changes   if   we   test   one   sided   is   that   we   can   eliminate   one   side   of   the   distribution.   However,   to   be   able   to   do   so  

you   need   a   clear   intuition   that   tells   you   that   one   side   of   the   distribution   is   completely  infeasible.  If  this  is  the  case,  then  this  means  that  we  can  forget  about   one  side  of  the  distribution  and  only  consider  the  other  side.  The  consequence  of  

this   is   that   the   significance   level  ��  does   not   need   to   be   divided   by   two   tails   but  

stays  in  one  tail,  so  you  do  not  need  to  divide  by  two  again.  Moreover,  the  đ?‘?đ?‘?-­â€?value   as  observed  in  one  tail  is  already  the  full  đ?‘?đ?‘?-­â€?value  as  there  is  no  other  tail,  so  you  

do  not  have  to  multiply  it  by  two  when  testing  one  sided.  This  is  also  shown  in  the  

graph  for  the  case  in  which  the  null  hypothesis  would  be  rejected.  The  calculation   of  the  ��-­�value  stays  exactly  the  same  as  in  the  two-­�sided  test.  

 

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5  

 

 

 

                               Theory  –  Statistics  

M u l t i p l e    R R e g r e s s i o n  

Once   you   understood   the   idea   of   a   simple   regression   the   step   to   a   m u l t i p l e  

r e g r e s s i o n  is  not  very  hard.  The  main  change  is  that  more  explanatory  variables  

are  added  to  the  regression  model.  By  doing  this  we  need  to  introduce  some  more   tests   to   evaluate   the   performance   of   the   model.   Moreover,   there   are   some   problems  that  can  arise  and  some  new  types  of  variables  that  can  be  introduced.    

5 . 1   T h e    rr e g r e s s i o n    ee q u a t i o n  

R e g r e s s i o n    ee q u a t i o n s    ff o r    tt h e    p p o p u l a t i o n  

As  just  mentioned  we  can  produce  a  multiple  regression  model  by  adding  more   explanatory   variables.   Obviously   the   grade   of   student   does   not   only   depend   on   the   bonus   points,   but   may   also   depend   on   the   subject   he   or   she   is   studying,   the  

gender   or   the   nationality.   Another   impact   might   be   attributed   to   the   amount   of   study   time   that   is   spent   in   group   learning   rather   than   self-­�study.   By   adding   all   these  variables  the  population  regression  model  becomes:  

đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘” = đ?›˝đ?›˝ + đ?›˝đ?›˝ đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘? + đ?›˝đ?›˝ đ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘š + đ?›˝đ?›˝ đ?‘’đ?‘’đ?‘’đ?‘’đ?‘’đ?‘’đ?‘’đ?‘’ + đ?›˝đ?›˝ đ?‘“đ?‘“đ?‘“đ?‘“đ?‘“đ?‘“đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘? + đ?›˝đ?›˝ đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘” + đ?›˝đ?›˝ đ?‘œđ?‘œđ?‘œđ?‘œâ„Žđ?‘’đ?‘’đ?‘’đ?‘’ + đ?›˝đ?›˝ đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘” + đ?›˝đ?›˝ đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”² + đ?›˝đ?›˝ đ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘š ∗ đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘? + đ?œ–đ?œ–  

You  do  not  have  to  worry  if  you  are  unable  to  understand  the  purpose  of  all  these   variables  by  now.  This  whole  section  will  attempt  to  explain  them  one  by  one.      

R e g r e s s i o n    ee q u a t i o n s    ff o r    tt h e    ss a m p l e  

Just  as  we  did  in  the  simple  regression  case  we  need  to  take  a  sample  to  estimate   the  regression  model.  This  results  from  the  fact  that  the  population  is  too  big  to  

be   observed.   Switching   to   the   sample   then   also   means   to   switch   from   Greek   to   normal  letters:    

đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘” = đ?‘?đ?‘? + đ?‘?đ?‘? đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘? + đ?‘?đ?‘? đ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘š + đ?‘?đ?‘? đ?‘’đ?‘’đ?‘’đ?‘’đ?‘’đ?‘’đ?‘’đ?‘’ + đ?‘?đ?‘? đ?‘“đ?‘“đ?‘“đ?‘“đ?‘“đ?‘“đ?‘“đ?‘“đ?‘“đ?‘“đ?‘“đ?‘“ + đ?‘?đ?‘? đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘” + đ?‘?đ?‘? đ?‘œđ?‘œđ?‘œđ?‘œâ„Žđ?‘’đ?‘’đ?‘’đ?‘’ + đ?‘?đ?‘? đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘” + đ?‘?đ?‘? đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”đ?‘”² + đ?‘?đ?‘? đ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘šđ?‘š ∗ đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘?đ?‘? + đ?‘’đ?‘’    

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Again,   all   those   sample   coefficients   are   just   estimates   for   the   population   coefficients  and  another  sample  would  result  in  different  coefficients.  Moreover,   the  last  term  is  still  called  the  residual  and  it  is  on  average  equal  to  zero.    

5 . 2   T h e    E E x c e l    O O u t p u t   R e g r e s s i o n    ss t a t i s t i c s  

First  of  all  note  that  the   r e g r e s s i o n    o o u t p u t  in  this  section  is  for  our  complete  

model  as  it  is  given  above.  We  will  work  with  this  model  throughout  this  chapter.   It   is   not   a   problem   if   you   do   not   understand   some   terms   yet,   as   they   will   be  

explained  soon.  The  purpose  of  this  is  simply  to  give  an  overview  of  the  output,  so   that  you  know  where  to  find  what.    

The  first  row  labeled  m m u l t i p l e    R R  does  not  have  a  very  deep  meaning  in  the  case  

of   a   multiple   regression.   In   a   simple   regression   it   gives   the   correlation   between  

the  dependent  and  the  explanatory  variable.  In  a  multiple  regression  it  is  still  the  

square  root  of  the  R²,  but  does  not  carry  much  intuition.  

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                               Theory  –  Statistics  

The  R R ²  is  the  measure  for  the  goodness  of  fit.  As  already  explained  for  the  simple  

regression  it  tells  you  how  many  percent  of  the  deviations  of  grade  from  its  mean   can   be   explained   by   the   model.   This   holds   still   true   for   a   multiple   regression   model.  

The  aa d j u s t e d    R R ²  does  only  make  sense  for  a  multiple  regression.  The  problem  is   that   the   normal   R²   is   not   a   good   measure   when   different   models   need   to   be  

compared   to   each   other.   The   reason   for   this   is   that   adding   variables   to   a   model  

will  never  decrease  the  R².  If  the  variable  is  extremely  bad,  the  R²  will  stay  almost   the   same.   This   follows   from   the   fact   that   adding   a   variable   will   also   always   add  

some  information.  Even  in  the  worst  case  where  no  explanatory  power  is  added   to   the   model,   the   old   variables   can   explain   as   much   as   before.   Therefore   the  

model   still   explains   at   least   the   same   share   of   the   total   deviation   as   before.   Knowing  that  the  R²  never  decreases  when  a  variable  is  added  to  the  regression  

model  gave  rise  to  the  introduction  of  the  so  called  adjusted  R².  Different  than  the   R²  the  adjusted  R²  increases  only  if  the  variable,  which  is  added  to  the  model,  is  a  

good   variable.   If   the   variable   is   a   bad   variable,   meaning   that   it   has   a   weak   explanatory  power  with  respect  to  the  dependent  variable,  then  the  adjusted  R²   will  decrease.  

The   s t a n d a r d   e r r o r   is   a   measure   for   the   spread   of   the   residuals.   If   it   is   large,  

then  the  residuals  fluctuate  strongly  around  the  regression  line.  In  a  way  it  can  be  

taken  to  measure  the  importance  of  variables,  which  are  not  yet  included  in  the  

model   because   those   variables   will   end   up   in   the   error   term   and   produce   large   and   strongly   fluctuating   residuals.   Do   not   mix   this   term   up   with   the   standard   errors  of  the  coefficients!  In  formulas  it  is  generally  referred  to  as  �� .  

The   o b s e r v a t i o n   row   simply   names   the   total   amount   of   observations   that   are  

included   in   the   sample.   It   is   generally   denoted   with   the   letter  ��  and   can   also   be   referred  to  as  count.       59  


Seminar

Extras

Exams

Practice

P


Practice  Exercises  

    Quantitative  Methods  II  (EC)   Academic  Year  2012/2013,  Block  3  

     

 


Practice    

 

                               Uniseminar  –  Quantitative  Methods  II  

P r a c t i c e    E E x e r c i s e s  

This  part  contains  practice  exercises  to  each  week  and  therefore  to  each  chapter  

of   the   theory   script.   By   this,   you   can   deepen   your   theoretical   knowledge   with  

practical   exercises   and   you   can   go   through   the   exercises   of   these   topics   again,  

which  you  have  not  understood  so  well  until  now.  Although  you  may  think  that   you   already   have   done   enough   exercises   during   the   weeks,   these   exercises   are   tailored  specifically  to  your  needs  and  try  to  teach  you  the  most  important  topics   of  the  exam  in  a  practical  manner.  

 

T a b l e    o o f    C C o n t e n t s    

M a t h e m a t i c s  

1  

  Solutions  

19  

  Exercises    

1  

S t a t i s t i c s  

6 1  

  Solutions  

92  

  Exercises  

 

61  

 


Uniseminar  –  Quantitative  Methods  II    

M a t h e m a t i c s    -­‐-­‐    E E x e r c i s e s  

 

 

 

 

1  

W e e k    1 1  

1.

C a l c u l a t e    tt h e    ff o l l o w i n g    cc o m p o u n d i n g    p p r o b l e m s  

 

paid  annually.  

(b)

How  long  does  it  take  for  the  amount  to  double?  

 

                 Practice  

S e r i e s ,    SS e q u e n c e s ,    II n t e r e s t    R R a t e s    aa n d    A A n n u i t i e s   i.

(a)  

You  put  5000€  into  your  bank  account  at  an  interest  rate  of  5%  per  year  

What  amount  will  you  have  after  10  years?  

Now  assume  that  this  interest  is  paid  biannually.  

(c)

What  amount  will  you  have  after  10  years?  

(e)

What  is  the  effective  interest  rate  per  year?  

ii.

How  much  money  should  you  have  invested  5  years  ago  in  order  to  have  

(d)

How  long  does  it  take  for  the  amount  to  double?    

 

100,000€  today?  The  interest  rate  is  4%.  

iii.

The  annual  interest  rate  is  9%.  What  is  the  effective  interest  rate,  if  money  

 

 

is  compounded  

(g)

biannually?  

(f)

annually?  

(h)

quarterly?  

(j)

continuously?  

(i)

monthly?    

1  


Practice  

i v.

 

Uniseminar  –  Quantitative  Methods  II  

What   is   the   nominal   interest   rate,   if   the   effective   interest   rate   is   12%  

 

compounded  

(b)

quarterly?  

(a) (c)

(d) v.

biannually?  

monthly?  

continuously?    

Your   credit   card   provider   charges   you   1.5%   interest   on   the   outstanding  

 

balance   per   month.   What   is   the   effective   annual   rate,   which   you   pay   to  

 

your  credit  card  provider?  

vi.

Assume   that   the   value   of   your   car   depreciates   at   15%   annual   rate  

 

 

compounded  continuously.    

(f)

How  long  does  it  take  for  the  car  to  be  worth  25%  of  its  original  value?  

(e) vii.

As  compared  to  the  original  value,  what  is  value  of  the  car  after  2  years?    

What  is  the  present  value  of  5000€  which  you  need  to  pay  in  5  years  time  

 

given  that  the  interest  rate  is  5%  

2.

C a l c u l a t e    tt h e    ff o l l o w i n g    ii n f i n i t e    ss u m s :  

i.

 

 

1 1 1 1 1 + + + + + ... 3 9 27 81  

 

ii.

5+

2  

 

5 × 3 5 × 32 5 × 33 5 × 34 + 2 + 3 + 4 ...   8 8 8 8


Uniseminar  –  Quantitative  Methods  II    

iii.

 

 

 

 

 

 

                 Practice  

64 + 32 + 16 + 8 + ...  

 

i v.

 

4 8 16 32 64 128 2− + − + − + ... 5 25 125 525 2625 13125  

 

v.

 

1 1 1 1 1 1 1 + 4 + + 2 + + 1 + + + + ... 3 9 27 2 81 4  

 

3. i.

1+

 

C a l c u l a t e    tt h e    ff o l l o w i n g    ff i n i t e    ss u m s :    

1 1 1 1 1 + 2 + 3 + 4 + ... + 10 4 4 4 4 4  

ii.

 

1 + 2 + 4 + 8 + 16 + 32 + 64 + ...210  

 

iii.

A  grandmother  puts  200€  into  a  savings  account  for  her  grandchild  at  the  

 

beginning  of  every  year  for  the  next  18  years.  Interest  is  paid  annually  and  

(a)

What  is  the  amount  in  the  savings  account  after  18  years?    

 

(b)  

stays  flat  at  5%.    

What   would   be   the   amount   in   the   savings   account   after   18   years   if   payments  were  made  at  the  end  instead  of  the  beginning  of  each  year?  

 

3  


Uniseminar  –  Quantitative  Methods  II    

 

 

 

4  

W e e k    4 4  

1.

F i n d    tt h e    ii n v e r s e    o o f    tt h e    ff o l l o w i n g    m m a t r i c e s  

 

 

                 Practice  

I n v e r s e    M M a t r i c e s   i.

 

⎛ 1 2 ⎞ ⎜ ⎟ ⎝ 3 4 ⎠  

 

ii.

 

⎛ 0 3 ⎞ ⎜ ⎟ ⎝ 2 1 ⎠  

 

iii.

 

⎛ 1 2 4 ⎞ ⎜ ⎟ ⎜ 0 4 1 ⎟ ⎜ 1 −2 1 ⎟ ⎝ ⎠  

 

i v.

 

⎛ 1 2 3 ⎞ ⎜ ⎟ ⎜ 2 1 3 ⎟ ⎜ 3 2 1 ⎟ ⎝ ⎠  

 

v.

 

⎛ 1 3 2 ⎞ ⎜ ⎟ ⎜ 2 0 0 ⎟   ⎜ 3 2 2 ⎟ ⎝ ⎠

13  


Practice  

2. i.

 

Uniseminar  –  Quantitative  Methods  II  

U n d e r    w w h i c h    cc o n d i t i o n s    aa r e    tt h e    ff o l l o w i n g    m m a t r i c e s    ii n v e r t i b l e    

⎛ 3 2 3 ⎞ ⎜ ⎟ ⎜ 2 a 2 ⎟ ⎜ 3 3 a ⎟ ⎝ ⎠  

 

ii.

 

⎛ 1 2 3 ⎞ ⎜ ⎟ ⎜ 4 7 a ⎟ ⎜ 1 a 3 ⎟ ⎝ ⎠  

 

iii.

 

⎛ −1 2 b ⎞ ⎜ ⎟ ⎜ −2 2 2 ⎟ ⎜ 4 a 1 ⎟ ⎝ ⎠  

 

i v.

 

⎛ 3 2 1 ⎞ ⎜ ⎟ ⎜ 2 b 2 ⎟ ⎜ a 2 1 ⎟ ⎝ ⎠  

 

3. i.

C o n s i d e r    tt h e    ff o l l o w i n g        

1 A   and   B   are   square   matrices   with   dimensions   (3 × 3)   A = 4 and   B = .   2

What    is   A × A−1 × B × 4B ' ?    

14  


Uniseminar  –  Quantitative  Methods  II    

ii.    

iii.

 

 

 

 

                 Practice  

A  is  a   (6 × 6)  matrix.   A = 7 .  Matrix  B  arises  from  A,  by  multiplying  the  first  

column   by   1,   the   second   column   by   2,   the   third   column   by   3   (and   so   on)   and  the  6th  column  by  6.  What  is  the  determinant  of  matrix  B?    

A  and  B  are  square  matrices  of  dimensions   (3 × 3) .   A × B = I .  If   A = 7 .  What  

 

is   B ?  

i v.

It  is  given  that   A × A ' = I .  What  is   A ' ?    

 

 

 

 

15  


Uniseminar  –  Quantitative  Methods  II    

 

3  

W e e k    3 3  

1.

T h e    rr e g r e s s i o n    ee q u a t i o n  

 

 

 

 

                 Practice  

S i m p l e    R R e g r e s s i o n  

Before  a  new  movie  comes  out,  a  trailer  is  usually  released  way  in  advance.  While   the   platform   TFMP   does   not   have   the   function   of   rating   movie   trailers,  

Youtube.com  does  provide  this  feature.  There,  you  can  rate  any  video  with  0  –  5   stars.  You  wonder,  whether  the  Youtube  trailer  rating  of  a  movie  is  predicting  the  

final  rating  of  a  movie  on  the  TFMP  platform.  This,  you  want  to  examine  with  the   following  regression  model:  

rat = β0 + β1t_rat + ε   rat

i.

(a)

(b)

8 7 9.5 4.5 5.5

t_rat

3.5 4 4 1 2.5  

Use  the  five  observations  to  estimate   β 0 and   β1  

Plot  the  values  in  a  plain  

Draw  the  best  fitting  line  through  the  plain  

 

111  


Practice  

(c)  

 

Uniseminar  –  Quantitative  Methods  II  

Find   the   equation   of   the   best   fitting   line   (remember,   b0   is   your   best   esti-­‐ mate  of   β 0 and  b1  is  your  best  estimate  of   β1 )  

Remember,  we  are  trying  to  find  the  coefficients  of  the  “best  fitting  line”.  As  you  

will  remember  from  your  QM1  lecture,  you  need  to   minimize  the  sum  of  squared   residuals.   The   residual   is   defined   as:   ε i = yi − yˆi .   This   reads:   the   difference   be-­‐ tween  the  observed  value,   yi ,  and  the  predicted  value,   yˆi .  While  it  would  be  bur-­‐

densome  to  write  out  the  entire  sum  of  the  squared  residuals,  you  might  remem-­‐ ber  that  there  is  a  shortcut  to  estimate  the  coefficients:   n

∑ SS b1 = XY = i =1 SS XX

( xi − x )( yi − y ) n

∑(x − x ) i =1

i

2

 and   b0 = y − b1 x  

Thus,  to  use  both  equations,  we  first  need  to  determine   x  and   y :   Simple  math  will  give  you  

x = 3.0 .   y = 6.9

Next,  you  can  determine  b1  and  b0.   n

∑ SS b1 = XY = i =1 SS XX

( xi − x )( yi − y ) n

∑(x − x ) i =1

i

2

(3.5 − 3)(8 − 6.9) + (4 − 3)(7 − 6.9) + (4 − 3)(9.5 − 6.9) + (1 − 3)(4.5 − 6.9) + (2.5 − 3)(5.5 − 6.9)   (3.5 − 3) 2 + (4 − 3) 2 + (4 − 3) 2 + (1 − 3) 2 + (2.5 − 3) 2 ≈ 1.3462 =

and    

b0 = 6.9 − 1.3462 × 3 = 2.8615  

ii.

 

Determine  the  R²  

R²   is   the   ratio   between   the   explained   variation   and   the   total   variation.   The   ex-­‐

plained   variation   is   the   difference   between   the   sum   of   square   differences   be-­‐

tween   the   predicted   value,   yˆi ,   and   the   observed   value,   yi .   The   total   variation   is   112  


Uniseminar  –  Quantitative  Methods  II    

 

 

 

 

 

                 Practice  

the  sum  of  square  differences  between  the  average  value  of  the  dependent  varia-­‐ ble,   y ,  and  each  individual  observation,   yi .     Thus,  the  explained  variation  is   n

SST = ∑ ( yˆi − y ) 2 = i =1

(2.8615 + 1.3462*3.5 − 6.9) 2 + (2.8615 + 1.3462* 4 − 6.9) 2 + (2.8615 + 1.3462* 4 − 6.9) 2 +   (2.8615 + 1.3462*1 − 6.9) 2 + (2.8615 + 1.3462* 2.5 − 6.9) 2 = 11.7788

and  the  total  variation:   n

SSTO = ∑ ( yi − y ) 2 = (8 − 6.9) 2 + (7 − 6.9) 2 + (9.5 − 6.9) 2 + (4.5 − 6.9) 2 + (5.5 − 6.9) 2 = 15.70   i =1

Thus,     R2 =

SST 11.7788 = = 0.750   SSTO 15.70

According   to   the   5   observations,   the   explanatory   variable   t_rat   explains   75%   of  

the  change  in  the  dependent  variable  rat.   iii.  

 

Based   on   your   model,   which   rating   would   you   expect   for   a   movie   whose  

trailer  has  a  rating  of  3.5?    

∂ = 2.862 + 1.346t_rat   rat

Plugging  in  3.5  for  t_rat  would  yield:   ∂ = 2.862 + 1.346 × 3.5 = 7.573   rat

 

 

113  


Uniseminar  –  Quantitative  Methods  II    

 

 

 

 

6  

W e e k    6 6  

1.

E x t e n d i n g    tt h e    m m o d e l    w w i t h    d d u m m y    v v a r i a b l e s  

 

                 Practice  

D u m m y    aa n d    II n t e r a c t i o n    V V a r i a b l e s  

Whereas  we  have  progressed  from  a  simple  linear  regression  model  to  a  multiple  

regression   model,   we   will   extend   the   multiple   model   with   dummy   variables   in  

this  section.  The  first  dummy  variable  we  add  to  the  model  is  the  gender  of  the   lead  actor/actress.  Thus,  model  4)  looks  as  follows:   i.

(a)

rat = β 0 + β1t_rat + β 2 ln(# star ) + β3 age + β 4 age 2 + β5 male + ε  

Interpreting  the  dummy  variable  male  

What  is  the  base  case  of  the  dummy  variable  male?  

The   base   case   of   this   dummy   variable   would   be   that   the   lead   actor   is   female   (a   lead  actress).  You  are  always  free  to  choose  on  how  to  code  the  dummy  variable.  

It   is   only   important   to   remember   that   a   dummy   variable   needs   to   have   a   base   case,  and  this  base  case  is  not  put  as  an  explicit  variable  into  the  model.  Rather,   the  base  case  reflected  in  the  model,  if  the  dummy  variable  takes  on  the  value  0.   (b)

 

 

If  the  dummy  variable  turns  out  to  be  significant,  how  do  you  interpret  the  

coefficient?  

In   the   case   at   hand,   the   dummy   variable   would   be   interpreted   in   the   following  

manner:   If   positive,   than   movies   with   a   lead   actor   would   be   better,   on   average,  

than  movies  with  a  lead  actress;  if  negative,  than  the  interpretation  would  be  the   other  way  round.             131  


Practice  

 

Uniseminar  –  Quantitative  Methods  II  

Running  model  4)  in  Excel  yields  the  following  summary  output:   SUMMARY OUTPUT

Regression Statistics Multiple R 0.764529 R Square 0.584504 Adjusted R Square 0.577031 Standard Error 1.417063 Observations 284 ANOVA df

SS 785.3160 558.2429 1343.5589

MS 157.0632 2.0081

Coefficients Standard Error 0.56903 0.42573 1.04352 0.09161 0.86111 0.12268 0.13270 0.02676 -0.00319 0.00062 -0.48241 0.17656

t Stat 1.33659 11.39108 7.01895 4.95875 -5.13279 -2.73231

Regression Residual Total

Intercept t_rat ln(#stars) Age Age_sq male

(c)

5 278 283

F Significance F 78.2161 0.0000

P-value 0.18245 0.00000 0.00000 0.00000 0.00000 0.00669

Interpret  the  coefficient  of  the  dummy  variable  male  

Lower 95% -0.26904 0.86318 0.61960 0.08002 -0.00442 -0.82997

Upper 95% 1.40710 1.22385 1.10262 0.18538 -0.00197 -0.13485  

The   coefficient   of   male   is   negative   and   statistically   significant   at   the   1%   level.  

Thus,  we  can  conclude  that  movies  with  a  lead  actor  are,  on  average,  rated  0.48   grades  lower  than  movies  with  a  lead  actress.    

(d)

Draw  the  dummy  effect  in  a  plain  (Hint:  take  the  dependent  variable  rat  on  

 

on  the  x-­‐axis)  

 

 the   y-­‐axis,   and   an   unspecified   vector   of   independent   variables   (call   it   X  

 

132  


Seminar

Extras

Exams

E


Exams  

    Quantitative  Methods  II  (EC)   Academic  Year  2012/2013,  Block  3    

   

 


Exams  

E x a m s  

 

   

                 Uniseminar  –  Quantitative  Methods  II  

You   should   start   early   with   the   calculation   of   exams,   because   you   need   to   get   a   general   feeling   of   how   the   exams   are   built   up.   You   will   soon   discover   how   the  

exams  are  constructed  and  that  there  are  general  tendencies,  which  repeat  from  

exam  to  exam.  In  this  part  you  will  find  old  exams  of  the  Maastricht  University,  as  

well   as   one   practice   exams   constructed   by   Uniseminar.   During   the   seminar   you   will  then  receive  a  further  practice  exam.  

 

T a b l e    o o f    C C o n t e n t s    

P r a c t i c e    E E x a m    1 1    (( i n c l .    ss o l u t i o n s )  

1  

O l d    E E x a m s  

3 3  

 

  11/12  Resit  

 

  11/12  First  Sit     10/11  Resit  

  10/11  First  Sit  

  09/10  Resit  

  09/10  First  Sit    

 


Uniseminar  –  Quantitative  Methods  II    

M a t h e m a t i c s    

 

1.)

 What  is  the  value  of  the  following  infinite  series?  

a.)

2.50  

c.)

3.00  

b.) d.)  

2.)

2+

2.75  

Practice  Exam  1  

3 3 3 3 3 − + − + + ⋯   2 4 8 16 32

3.25   Consider  the  following  series  of  payments.  Today  you  receive  200€,  in  one  

year  you  receive  400€,  in  two  years  again  200€,  in  three  years  again  400€,  

in  four  years  again  200€  and  so  on.  The  project  ends  after  thirteen  years  

and  the  yearly  interest  rate  is  5%.  What  is  the  present  value  of  this  series   a.)

b.)

c.)

d.)

 

3.)

of  payments?  

between  3000  and  3100  

between  3100  and  3200    

between  3200  and  3300   between  3300  and  3400  

Consider  the  following  investment  opportunity:   You  are  offered  a  payment  

of  200€  right  now  and  a  second  payment  of  200€  next  year.  However,  to  

get  those  payments  you  have  to  pay  410€  in  two  years  from  now.  What  is  

a.)

b.) c.)

d.)

 

the  internal  rate  of  return  of  this  project?   2.8  %  

1.6  %.   9.4  %  

11.2  %  

1  


Practice  Exam  1      

4.) a.)

                                     Uniseminar  –  Quantitative  Methods  II  

Consider  the  following  three  matrixes:  A  is  of  format  m  x  n,  B  is  format  k  x  l  

and  C  is  of  format  p  x  q.  When  is  the  following  defined:  đ??´đ??´ ∗ đ??ľđ??ľ + đ??śđ??ś?  

đ?‘˜đ?‘˜ = đ?‘™đ?‘™ = đ?‘šđ?‘š = đ?‘žđ?‘ž  and  đ?‘›đ?‘› = đ?‘?đ?‘?  

b.)

đ?‘˜đ?‘˜ = đ?‘™đ?‘™  and  đ?‘šđ?‘š = đ?‘›đ?‘› = đ?‘?đ?‘? = đ?‘žđ?‘ž  

d.)

đ?‘˜đ?‘˜ = đ?‘™đ?‘™ = đ?‘?đ?‘? = đ?‘žđ?‘ž  and  đ?‘šđ?‘š = đ?‘›đ?‘›  

c.)

     

5.)

đ?‘˜đ?‘˜ = đ?‘™đ?‘™ = đ?‘šđ?‘š = đ?‘?đ?‘?  and  đ?‘›đ?‘› = đ?‘žđ?‘ž  

 Consider  the  following  invertible  matrix:   1  đ??´đ??´ = 1 2

2 1 2

1 0   1

2 The  ss e c o n d  column  of  the  inverse  đ??´đ??´  is  given  as   đ?‘Ľđ?‘Ľ   đ?‘Ľđ?‘Ľ 

What  are  the  values  of  ��  and  �� ?   a.)

�� = 1, �� = 0  

b.)

đ?‘Ľđ?‘Ľ = −1, đ?‘Ľđ?‘Ľ = 0  

d.)

đ?‘Ľđ?‘Ľ = 0, đ?‘Ľđ?‘Ľ = −1  

c.)

   

�� = 0, �� = 1  

6.)

Consider  the  following  matrix  đ??´đ??´ =

b.)

đ?‘Žđ?‘Ž = −đ?‘?đ?‘?  

a.) c.)

d.)  

2  

đ?‘Žđ?‘Ž ≠ đ?‘?đ?‘?  

đ?‘Žđ?‘Ž = đ?‘?đ?‘?  and  đ?‘Žđ?‘Ž = −đ?‘?đ?‘?   đ?‘Žđ?‘Ž ≠ đ?‘?đ?‘?  and  đ?‘Žđ?‘Ž ≠ −đ?‘?đ?‘?  

 

1 đ?‘?đ?‘? 

1 .  It  only  has  an  inverse  if‌   ��²


Uniseminar  –  Quantitative  Methods  II    

7.)

 

Practice  Exam  1  

When  can  we  be  absolutely  sure  that  a  certain  matrix  đ??´đ??´  is  invertible?  

I.)  There  exists  a  matrix  đ??ľđ??ľ  such  that   đ??´đ??´đ??´đ??´ = 0  

II.)  The  system  of  equations  đ??´đ??´đ??´đ??´ = đ?‘?đ?‘?  has  a  unique  solution    

a.)

b.)

c.)

d.)    

8.)

In  I.)  we  can  be  sure,  in  II.)  not  

In  II.)  we  can  be  sure,  in  I.)  not  

In  both  cases  we  can  be  sure  

Neither  in  I.)  nor  in  II.)  can  we  be  sure  

The   determinant   of   the   3x3   matrix   A   is   equal   to   4.   First,   we   multiply   the  

first  row  of  A  with  4.  Second,  we  multiply  the  second  column  of  A  with  3.   Finally,  we  multiply  all  elements  of  A  with  2.  What  will  the  determinant  of  

a.)

this  new  matrix  be  equal  to?   384  

b.)

96  

d.)

48  

c.)

   

192  

9.)

 The   system   đ??´đ??´đ??´đ??´ = đ?‘?đ?‘?   has   a   unique   solution.   What   can   you   say   about   the  

a.)

The  system  will  have  no  solution  

b.)

c.)

d.)

 

solutions  of  the  system  đ??´đ??´đ??´đ??´ = đ?‘?đ?‘??  

The  system  will  have  infinitely  many  solutions  

The  system  will  have  a  unique  solution  

Nothing  can  be  said  on  the  basis  of  the  given  information  

  3  


Practice  Exam  1      

                                     Uniseminar  –  Quantitative  Methods  II  

S t a t i s t i c s  

This   trial   exam   is   based   on   a   dataset   that   contains   315   students.   Of   these   students   multiple   characteristics   like   the   QM1   grade,   the   gender   or   the   nationality,  etc.  are  collected.  The  variables  are  named  as  follows:   g r a d e  

QM1  grade  of  a  student  on  a  scale  from  0  to  10,  where  10  is  best  

 

 

to  8  

I B  

 

dummy:  1  for  IB  students,  0  otherwise  

b o n u s   m a l e       E c o n      

F i s c a l    

The  bonus  points  a  student  got  for  the  QM1  exam  on  a  scale  from  0  

A  gender  dummy,  1  for  male  and  0  for  female  students   dummy:  1  for  Economics  students,  0  otherwise  

dummy:  1  for  Fiscal  Economics  students,  0  otherwise  

Some  people  claim  that  the  performance  in  a  QM1  exam  does  solely  depend  on   the  level  of  mathematical  and  statistical  talent.  If  this  claim  was  true  this  would  

mean   that   the   QM1   score   of   a   student   must   be   equal   to   the   QM2   score   of   a   student.    

2 1 . ) Which  test  can  be  used  to  evaluate  this  claim?  

 

a.)

A  Chi-­‐Square  test  for  Independence  to  check  for  the  relationship  between  

b.)

A  two  sample  t-­‐test  for  QM1  and  QM2  scores  

c.)

d.)  

talent  and  QM  scores  

A  paired  sample  t-­‐test  for  the  difference  in  QM1  and  QM2  grades  

ANOVA  with  the  students  as  the  levels  and  the  QM  scores  as  the  response  

Forget  about  the  claim  and  focus  on  something  different.  Some  people  claim  that  

male  and  female  students  are  not  equally  successful  in  QM1.  Unfortunately,  it  is   10  


Uniseminar  –  Quantitative  Methods  II    

 

Practice  Exam  1  

not  clear  yet  who  of  the  two  actually  performs  better.  Therefore  we  could  test  the  

following  set  of  hypothesis  about  the  mean  difference  in  the  QM1  bonus  scores  of   female  and  male  students.  

đ??ťđ??ť : đ?œ‡đ?œ‡ − đ?œ‡đ?œ‡ = 0                                                    đ??ťđ??ť : đ?œ‡đ?œ‡ − đ?œ‡đ?œ‡ ≠ 0  

2 2 . ) To   test   this   claim,   consider   the   descriptive   statistics   table   on   the   right.    

Against  the  state  alternative,  the  p-­�value  is...  

a.)

between  5%  and  10%.  

c.)

between  1%  and  2.5%.  

b.)

d.)  

between  2.5%  and  5%.  

smaller  than  1%.  

Obviously  gender  is  not  the  only  characteristic  that  impacts  the  grade.  Even  more  

academic   discussion   focuses   on   the   difference   in   the   performance   of   IB,  

Economics  and  Fiscal  Economics  students.  With  the  help  of  a  one  way  ANOVA  we   can  test  the  following  hypothesis:  

đ??ťđ??ť : đ?œ‡đ?œ‡ = đ?œ‡đ?œ‡ = đ?œ‡đ?œ‡  

The  results  of  this  ANOVA  test  are  given  in  the  Excel  report.  

2 3 . ) Which  of  the  following  is  n n o t  true  for  this  ANOVA  test?  

 

11  


Seminar

Extras

E


Extras  

    Quantitative  Methods  II  (EC)   Academic  Year  2012/2013,  Block  3  

               

   

 

 


Extras    

E x t r a s  

 

Uniseminar  –  Quantitative  Methods  II  

In   this   part   you   find   several   extras   that   will   be   very   helpful   for   your   exam  

preparation.  In  this  course  you  will  find  an  extra  explanation  of  how  to  read-­‐off   the  critical  values  of  the  statistics  tables  and  the  formula  sheets  for  math  as  well  

as   statistics.   We   decided   to   give   you   the   original   formula   sheets,   which   you   will   also   have   during   your   exam,   as   it   is   essential   to   get   used   to   the   exam’s  

circumstances.   However   during   the   seminar   the   tutor   will   discuss   the   formula   sheets  and  will  highlight  or  extend  several  sections.    

T a b l e    o o f    C C o n t e n t s    

F o r m u l a    SS h e e t s  

1  

H o w    tt o    rr e a d    ss t a t i s t i c s    tt a b l e s  

9  

     

 


Uniseminar  –  Quantitative  Methods  II  

 

 

H o w    tt o    rr e a d    tt h e    SS t a t i s t i c s    tt a b l e s  

 

 

 

                         Extras  

z -­‐-­‐tt a b l e  

The   𝑧𝑧-­‐scores   are   given   on   the   outside   of   the   table.   The   numbers   in   the   inside   correspond  to  the  area  under  the  curve,  which  represents  the  probability.          

z  

0 . 0 0  

0 . 0 1  

0 . 0 2  

0 . 0 3  

0 . 0 4  

0 . 0 5  

0 . 0 6  

0 . 0 7  

0 . 0 8  

0 . 0 9  

0 . 0   0.5000   0.5040   0.5080   0.5120   0.5160   0.5199   0.5239   0.5279   0.5319   0.5359   0 . 1   0.5398   0.5438   0.5478   0.5517   0.5557   0.5596   0.5636   0.5675   0.5714   0.5753   0 . 2   0.5793   0.5832   0.5871   0.5910   0.5948   0.5987   0.6026   0.6064   0.6103   0.6141   0 . 3   0.6179   0.6217   0.6255   0.6293   0.6331   0.6368   0.6406   0.6443   0.6480   0.6517   0 . 4   0.6554   0.6591   0.6628   0.6664   0.6700   0.6736   0.6772   0.6808   0.6844   0.6879   0 . 5   0.6915   0.6950   0.6985   0.7019   0.7054   0.7088   0.7123   0.7157   0.7190   0.7224   0 . 6   0.7257   0.7291   0.7324   0.7357   0.7389   0.7422   0.7454   0.7486   0.7517   0.7549   0 . 7   0.7580   0.7611   0.7642   0.7673   0.7704   0.7734   0.7764   0.7794   0.7823   0.7852   0 . 8   0.7881   0.7910   0.7939   0.7967   0.7995   0.8023   0.8051   0.8078   0.8106   0.8133   0 . 9   0.8159   0.8186   0.8212   0.8238   0.8264   0.8289   0.8315   0.8340   0.8365   0.8389   1 . 0   0.8413   0.8438   0.8461   0.8485   0.8508   0.8531   0.8554   0.8577   0.8599   0.8621   1 . 1   0.8643   0.8665   0.8686   0.8708   0.8729   0.8749   0.8770   0.8790   0.8810   0.8830   1 . 2   0.8849   0.8869   0.8888   0.8907   0.8925   0.8944   0.8962   0.8980   0.8997   0.9015   1 . 3   0.9032   0.9049   0.9066   0.9082   0.9099   0.9115   0.9131   0.9147   0.9162   0.9177   1 . 4   0.9192   0.9207   0.9222   0.9236   0.9251   0.9265   0.9279   0.9292   0.9306   0.9319   1 . 5   0.9332   0.9345   0.9357   0.9370   0.9382   0.9394   0.9406   0.9418   0.9429   0.9441   1 . 6   0.9452   0.9463   0.9474   0.9484   0.9495   0.9505   0.9515   0.9525   0.9535   0.9545   1 . 7   0.9554   0.9564   0.9573   0.9582   0.9591   0.9599   0.9608   0.9616   0.9625   0.9633   1 . 8   0.9641   0.9649   0.9656   0.9664   0.9671   0.9678   0.9686   0.9693   0.9699   0.9706   1 . 9   0.9713   0.9719   0.9726   0.9732   0.9738   0.9744   0.9750   0.9756   0.9761   0.9767   2 . 0   0.9772   0.9778   0.9783   0.9788   0.9793   0.9798   0.9803   0.9808   0.9812   0.9817   2 . 1   0.9821   0.9826   0.9830   0.9834   0.9838   0.9842   0.9846   0.9850   0.9854   0.9857   2 . 2   0.9861   0.9864   0.9868   0.9871   0.9875   0.9878   0.9881   0.9884   0.9887   0.9890   2 . 3   0.9893   0.9896   0.9898   0.9901   0.9904   0.9906   0.9909   0.9911   0.9913   0.9916   2 . 4   0.9918   0.9920   0.9922   0.9925   0.9927   0.9929   0.9931   0.9932   0.9934   0.9936   2 . 5   0.9938   0.9940   0.9941   0.9943   0.9945   0.9946   0.9948   0.9949   0.9951   0.9952   2 . 6   0.9953   0.9955   0.9956   0.9957   0.9959   0.9960   0.9961   0.9962   0.9963   0.9964   2 . 7   0.9965   0.9966   0.9967   0.9968   0.9969   0.9970   0.9971   0.9972   0.9973   0.9974   2 . 8   0.9974   0.9975   0.9976   0.9977   0.9977   0.9978   0.9979   0.9979   0.9980   0.9981   2 . 9   0.9981   0.9982   0.9982   0.9983   0.9984   0.9984   0.9985   0.9985   0.9986   0.9986   3 . 0   0.9987   0.9987   0.9987   0.9988   0.9988   0.9989   0.9989   0.9989   0.9990   0.9990  

9  



UM_QM2-EC_Ordner