2.
   2023
/  
 
(


- VI)
: 3     :100
:    -    ,        
             
 -
I.  -  (OLS)  '   X      
        :

50
45
70
75
90
55
100
120
135
130

60
80
100
130
140
160
180
200
220
240
II.(a)   ,      
()       
 - 
III.(a) -    
(b) R²           
IV.(a)     ?        120   
        
  100    ' 
2.
      15       116    

?   5%    
()  (     ) 
-
V.    ?    
  ,     ?
VL (a)        ?
()         
-
VII. (a)        
()    -       
 
VIII. (a)       
()    -        

2.
   2023
/  
 
(


- VI)
: 3     :100
:    -    ,        
             
 -
I.  -  (OLS)  '   X      
        :

50
45
70
75
90
55
100
120
135
130

60
80
100
130
140
160
180
200
220
240
: 1:  -  (OLS)  
             :
     
 

  
 , 
         ,  
   
    
 :
󷷑󷷒󷷓󷷔    ,          ?
        (OLS)   
2.
OLS   
  
           
 
  :

:
=  (Y = 0  ' X  )
=  (Y  1  ' X   )
       '   
        
 ( X   X  )
OLS  :
󷷑󷷒󷷓󷷔         - 
  
?

:
     

  '    
      
  OLS    :
"                
 "
 2:   
 X     X   Y '      

50
45
75
90
55
100
120

60
80
130
140
160
180
200
      :

2.
 3: Y  X    
OLS  :
󰇛
󰇜󰇛
󰇜
󰇛
󰇜

   :
X  
Y  

 1:    

X   = 50+45+70+75+90+55+100+120+135+130 = 870
 
 Y:
Y   = 60+80+100+130+140+160+180+200+220+240 = 1510
 
 2:     

  :

Y   
( )
2.
󰇛
󰇜󰇛
󰇜 
󰇛
󰇜

 3:    



 4:    

 󰇛 󰇜
 

  
 
 4:  (   )
         
󷄧󷄫 (0.501)
 Y 1   ,  X  0.5   
  X, Y       
󷄧󷄬 (11.34)
 Y = 0,  X ≈ 11.34    
( Y  
      
 ,    '  
  )
2.
 5: OLS  
 
OLS     :
   
 
 
 
:
  
  
  
 6:  
       '  :
  ' Y
  ' X
     OLS         
  

-    '   
 7: OLS   
       :
1. -   = 0
2.  X   =  X  
3.     
󰇛
󰇜
4.       
5.      
  ( )
 -  (OLS) :
OLS                
               
2.
     -         
      
 
    X     , Y  X    :

:
󰇛
󰇜󰇛
󰇜
󰇛
󰇜



 ,    :
 
II.(a)   ,      
()       
:(a)   ,   
1.   
       -       ,
          ,    
     
    ( , "       ")
    ( ,  )
            
 ,    
 -   ,     
  
2.   
    ,         :
2.
  : ,     
 ?
:     GDP , ,    

 :         
 :    ,   , 
          
 :  ,         
 
 ,    ""  -      ,  
  ,      
3.   
         :
1.    
o      :   '   
o    '  : . 󰇛󰇜
2.    
o       :

   
3.   
o -   (    )  
4.   
o      (  )    
5.  
o         
   ?  
 
 ?
6.    
o               
 
7. 
o            
,   
:        10%    8%   ,
            
()    
2.
       ,        
  
1. 
            :
   (  
   )
   (   
      )
  ':

=   ( , )
=   (  )
=  (  )
=  (1   ' )
=   (          )
2. 
-        (OLS)    
 
OLS               
 
  ,      
"    "  
3. 
  
 
 :
 
 ():    ,  50  (  ) 
 ():      ,  0.8    
          
4.   
      ,     :
   
2.
     
      
    ()
  
       ,        
5. 
 :    
: '   
:  '   
:     

             
              
  -     ,  ,    , 
  
      
      
,               , 
 ,       
 

III.(a) -    
(b) R²           
:III.(a) -  -  
   
          '   
       
         
               
    :
󷷑󷷒󷷓󷷔         
  ?
󷷑󷷒󷷓󷷔         ?
2.
     -  
 
󽆤  ?–    (  )
  :
         ,   
 (OLS)        
() 
  - 
    ()”  -- 
󷄧󷄫
      (- )       :

, 
       
󷄧󷄬
     , ,     
  :  
  -      '   
   -  
 ,        
  OLS        


󷄧󷄭  
 "  "    - 
2.
      -       
   -   ( ),   

-   :
󷷑󷷒󷷓󷷔    , OLS      
󷷑󷷒󷷓󷷔      
󽆤
  ()
-            :
1.    
2.    = 0 
3.       ()
4.  - 
5.   - 
     OLS  
󽆤   
        
:
 OLS     
 
     OLS   '  
     
 
   

 
󷷑󷷒󷷓󷷔-  ,   
   OLS    
        :
"     ,  OLS       
  "
III.(b) R²   
     
2.
 
 
 , 
  :
󷷑󷷒󷷓󷷔     
 ?
 , 
    
󽆤²(  )
  :
     ,       
:  R² = 0.80 →     80%      
     
󽆤  ()
Explained variation
Total variation
: 0 ≤ R² ≤ 1
0 →    


1 →  
󽆤  ²
  :
󷷑󷷒󷷓󷷔 
           
 
:  =    +   
    ,     
     
󽆤  ² -  
2.
     
     :
  
 
               
 
󽆤 
    
      
    
󽆤   ²  ²

²
 

  %  

  %  

   
  
    
  







  
  
󽆤   
    
 
   
  
 
   
  
2.
 
  
      
    
   
󽆤   
   
   '      :




 
 
     :
󷷑󷷒󷷓󷷔
󷷑󷷒󷷓󷷔  
    
󽆤 
- :
        , OLS     
  (BLUE) ,         - 
      OLS    
    
    
 
²:
            
     
           
  R²:
               
               
   
2.
:
     ,     ,
             
 ,          
IV.(a)     ?        120   
        
  100    ' 
      15       116    

?   5%    
()  (     ) 
:(a)   
    ?
                   
    ()           ,
              
     
    
   :
  (H₀):   
   
  (H₁):  ,  
     H₀
    
 
               (5%
     )       
 ,  
H₀
  
  
:      120         

100   :
  = 116   
  = 15 
2.
  (n) = 100
  = 5%
 1:  
H₀: μ = 120 (   120 / )
H₁: μ ≠ 120 (   120 / 
)
  -    
 
        (   

)
 2:     

z-     
      (n = 100)
:
:
( )
(   )
( )

 







 3: 5%  '  
5%  ' -   :
 z- = ±1.96
 4: 
   z = -2.67
-2.67 < -1.96 ,     
 
2.
:
       
      120
      () 
 
  
     ,        
  '      ,      '    
        
()  -      
     : 
  ?
 ,  
   
  (     
), 
      :
  :-  (  )
:         
:,       
:           
       (OLS)     
OLS  
 (- )
-       (  ,  - 
,
   ,     )  , OLS    
   
:OLS       
:        
  :    , OLS      , 
     
     
  
 (X)   (Y)     
 :

OLS      
   ,      ( )
2.
       
        -  
 , OLS  

  :            ' 
          ,  120
/        
    (116) 5%  '
  
:  OLS   
         
(- ), ,   ,     
  
 
 ,              ,  
   
            ,
          
-
V.    ?    
  ,     ?
:   ?
   
            
      
        
 :

󰇛󰇜
󰇛󰇜
     ,      ,     ' 
      ,      
 ,            '  
 
      
      
 
󷷑󷷒󷷓󷷔   -      
   
2.
 
-              
       
        ,       
-   
 (-)
          -    
      
 ,     
   
  
-     :
󷷑󷷒󷷓󷷔     '  
 
-   ()
  
     '      
   
 
1.     
  
         
 
:



   '         
2.    
  
   
 
:
2.
 
 
- 

   =  + -,    ' - 
3.     
    ,        
:
...



      - -
4.   
        
: :
 = 1  
 = 1  
    -     
:  +  = 1

5.     
      ,       -  
 
-   ()
2.
         

1.     
       
:  :   = 2000  :   = 500
  -   
2.     
         ,   

 
:       -   
 
3.      

     ,        

4.    -
    

  :
  (   '     )
    

? 
         
5.  
2.

  
          

-  
1.  -
  
: X₃ = X₁ + X₂
    
  
2.  -
   
  ,    

    
-  
  : 
-      ?
1. -  
      
 - > 0.8  0.9 → -   
:
Corr(, ) = 0.92 → 
:       
 ,    
2.    (VIF)
2.
     
 :     -        
  :
VIF = 1 →  - 
VIF > 5 → 
VIF > 10 →  -
3.  
 = 1 / VIF
:  < 0.1 → - 
4.   t-
        -   
        
5.  /   
      
:   > 30 →  -
     
     󺋿󺋼󺌀󺋽󺋾
1. -    
       ,     
2.
2.  
    
: -  
3.    
  -  
 
4.   ( )   
- - 
 
5.  
            
  
󷷑󷷒󷷓󷷔-- =     -
󷷑󷷒󷷓󷷔         
󷷑󷷒󷷓󷷔,      
󷷑󷷒󷷓󷷔-, VIF, ,    
VL (a)        ?
()         
:(a)      
   ?
            
       

   ,   
2.
     
    ,    ,  
       

   
1.   
o           
o :      ,     '
   
o :            
    ,      
2.    
o        

o :         
o :          ( )
3.   
o      
o :           
o :     ,     

4.  
o       
o :           

o :        
5.     
o     
o :      ,      
 
o :      
 
   
 :   
 :  ,  
  :-  -     
 :    

  :         
 :    ,     


()      
2.
  ?
  ,          
  

:      ( )
:        
:     ,        
  
   
?
OLS    ,     

   
 ,       
     (t, F)    
  
1.  
o      
o         ,    

2. - 
o             
o     
 
3.   
o            

o    -    
 
4. - 
o             
o              
 
1.  
o          
o :       
2.       (WLS)
o         
o       
3.   
o      
2.
o    ,        
4.  
o              
o         

 :         
      ,      ,
     ,        : 
,  ,    
:          
  OLS
     

 -, ', 
-         ,   
  ,   - ,     

 ,               
            
  
     ,     
-
VII. (a)        
()    -       
 
:VII (a)   -   
   
            
  :         ?     
            ?
󷷑󷷒󷷓󷷔               
  
 
   :
  (DLM)
-  (AR)
2.
    
 
1.   (DLM)
              
       '   
  :
󷷑󷷒󷷓󷷔"       '  
   '    "

  
         
      
 
         
           
         '     -
    
  ' ( ):
ₜ = a + b₀Adₜ + b₁Adₜ₋₁ + b₂Adₜ₋₂ + 
:
ₜ =  
ₜ₋₁ =    
ₜ₋₂ =     
󷷑󷷒󷷓󷷔        
2. -  (AR)
      :
  
            
 '   ? 
-    '    
     ₹50,000  ,         
   
2.
 :
󷷑󷷒󷷓󷷔"       '   "
 -    
 :
ₜ = a + b₁ₜ₋₁ + b₂ₜ₋₂ + 
:
    '   
      
 
  ( )

 
- 

       
  '   
    
 '   

      
   

   '  

   '
  

 , ,  
   

   
 
󷷑󷷒󷷓󷷔   :
  = X   Y   
- = Y   Y   
VII ()   -     
      - -
-    :
  
        ,       

      
 -      
2.
  (  )   
󷷑󷷒󷷓󷷔 (): -            
  
-   ()
     
 
1.   
    Y         
:          '     
           ,   
   

        -  ,    - 
 
2.   
   -   
   ,    
 
:    ,  
      
 -  
3.    
    GDP, ,   -     
    '  
:         
4.   - 
2.
-     :
...





         
5.  
          ,   - ' 
 
:           
 
-   
?
 -  :
OLS     
      
   
 
     
       
-   
    /       
1.     
    -    ,   
:              

2.
   - 
 
2.     
     '   ,   
: ₜ = a + bₜ + cₜ₋₁
      
3.   ()
    
 : 
:
Δ = ₜ − ₜ₋₁
     
 
-    
4.    (GLS)
 -   ,  OLS     GLS   - 
  
 :
- 
- 
  -      
5.     
    
 
2.
:           
  
        :
        
-        
-       
  ,  , - 
  ,    ,  , GLS
   
  ₜ₋₁ 
-ₜ₋₁ 
-  ₜ₋₁
VIII. (a)       
()    -        

:(a)    
   ?
              
               0
 1    
: 
           , 

     :
 = 1
 = 0
 ,  (  )         
2.
   
1.     
o      , , ,     
      
o    ,     
2.   
o            
o :  (1)   (0)     
3.   
o   , ,     
o :           (1 =
  , 0 =   )
4.     
o        
o : Q1, Q2, Q3, Q4       
5.  
o           
 
o :    '         ;
     
 

    ,         
   
         
,        
()    -      
-  ?
-            ,   
  , -  
  ( - 
):  
   (-):       
   
:   ,          
  ,  -   
-   
?
2.
OLS    ,     

    ,      

   -    
-     
1.  
o     
o   (   )   ,  -    
2. - 
o -  -        
o   0  4   :
 2 →  - 
2    -
2    -
3. - 
o   , - -  
o   ' -     '
4.  
 
o -     
o       (/   ) -
    
-  
1.  
o         
2.    (GLS)
o   -       
3. - 
o -  -     
4. -  
o -           

 :        
     ,  ,    
         ,   -
   
-:              
   OLS      

 -
, -,            ,
    GLS, -,      
2.
"              
 
       ,       
 "