2.
   2022
/   
     - VI
 : 3     :100
:     -    ,          
                
  -
I.               
   ?
II.        ?     , OLS   -
      :

6
8
12
14
18
20
 
8
10
6
8
10
12
  - 
III.       ?          
IV. R²          :

65
54
 
26
−7
      :

2.
     ,      5%  
 '   β ≠ 0   β = 0       
  - 
V.            
               
VI.           ?
  - 
VII.      -        
         '  
VIII.    ?         
2.
   2022
/   
     - VI
 : 3     :100
:     -    ,          
                
  -
I.               
   ?
:   
              
 ,     
  ,       
             ,  

            ?      

 
            
  
     -          
            ,  
    
       ,        
    
  
      : “” () ” ()  ,
       
1.     
2.
      
; ,      :
  -     ( ,       )
-            
-                
 
 ,       "    ,    "
                
   
2.          
   
      
  - 
 '    :
      ?
      ?
 -      ?
    ,  ,     
   
3.   '   
   ,     '   ,   
'   
 ,       "  
 ",  
     :
󷷑󷷒󷷓󷷔   10%      ?
     
 
4.     
               
             
 
        100%    -     
   
5.   -
2.
       
 
        :
   
    
  
    
-  ,     '  
 
    
          , 
   
1.       
                

 :
         ?
      ?
   ,           
2.         
       
    :
 
 
   
  
         - '  
  
3.    
2.
              :
     ?
     ?
       ?
 ,             
       
4.       
         
   :
      
     
     
  
        
5.     
          :
   
    
     
     
 
6.   
  ,        
         
    ,          

,      
    
  
      
  ,      
2.
   
      ,        
    
 : 2 + 2 = 4 ( )
,  -           
 :
   '  ;     '  
    ;      
    ;    
󷷑󷷒󷷓󷷔   ,        
       
   
      ,  ,      
        , ,     
 
   
 :
   '     ;    '   
  
      ;   (  )  
 
        ,       
󷷑󷷒󷷓󷷔 ,             
-          '    

          
   
               ,
2.
  
              
 
  , -,  -    , ,  
    ,            

 
        ,    
   
  '   '        ,   
 ,       
II.        ?    , OLS   -
      :

6
8
12
14
18
20
 
8
10
6
8
10
12
:        ?
               
    :
  (X):     
  (Y):   
     X  Y            :

:
=  (Y    X = 0  )
=  (X  -   Y  )
=   (X        )
  :    
"    "     
  
X  Y    
    
2.
   :

6
8
12
14
18
20

8
10
6
8
10
12
, X   ,  Y   
OLS ( - )        
OLS            Y    Y  
        
 1: X  Y       





, X   = 13, Y   = 9
 2:      (b)
:
󰇛
󰇜󰇛
󰇜
󰇛
󰇜
     :

 
 
󰇛
󰇜󰇛
󰇜

󰇛

󰇜
6
8
-7
-1
7
49
8
10
-5
1
-5
25
12
6
-1
-3
3
1
14
8
1
-1
-1
1
18
10
5
1
5
25
20
12
7
3
21
49
,    :
󰇛
󰇜󰇛
󰇜
󰇛
󰇜

 ,
2.



 3:  (a)   
:
󰇛󰇜
 4:  
 ,     :

   
 (6.4):  X = 0  ,  Y    6.4  
  (0.2): X   1-  , Y  0.2   
    X  Y        - X   Y  
 
   
   X        Y     
   :
   
(X=0),     6.4  ()    
     ,   0.2 ()  
,   -    (   ),      
  
 
   
 :         ( ,  
 ' )
  : 
         
 : ,  ,       

2.
               
      OLS     , 
    
     :

  
   X   Y        
 -
          ,      
  
  - 
III.       ?          
:  
             

   
        ,   , 

,             
   

             ?     
    (GLRM)     
    ,            
     
󷄧󼿒      ?
            (  
   )         (   
)         
󷷑󷷒󷷓󷷔  ,           :
         ?
         '   ?
         ?
   ,         
  
󹵙󹵚󹵛󹵜  
2.
   '     :
Y = β₀ + β₁X₁ + β₂X₂ + … + βₙXₙ + ε
    -        
   (
    ) :  

X₁, X₂, X₃…  (    ) :  
, 
β₀ ( )           
β₁, β₂…       X    Y   
ε ()   , ,     

 
󷷑󷷒󷷓󷷔 ,        :
"   =     +    "
     
?
󷘹󷘴󷘵󷘶󷘷󷘸 "" 
   ?

     ,         
:  =    + 
           '   
              
     ""    -      '   
     :
  (  )
 (  )
 (     )
 (  )
 (  )
 ,          
2.
󷄧󼿒         
    ,          

        -       ,   
 
1. 
         
󷷑󷷒󷷓󷷔   
         ,   
       
:      ,    '   
    (    ),       


2.    
 (ε)    
 
  :
󷷑󷷒󷷓󷷔        
 
    ' -      ,    
    '  
 
3.  (  )
   -    
       X          
   
   :
         
2.
          '    
    
  
4.     
     (    )     
  
?

          '   
 ,   ,      '   
5.   -  
      !
            
 
 :
        
                 
      ,           
    
󷄧󼿒     
        
 
1.        ( )
 -   ,    :
2.
󷄧󼿒   -  
󷄧󼿒     '
󷄧󼿒 ,      
               
2.   
              
 β₁ = 5,    :
󷷑󷷒󷷓󷷔X₁     Y 5    
-      
3.   
         
:
     
    
    
     '     
4. 
     :
󷄧󼿒 
󷄧󼿒  (  : /)
󷄧󼿒 
󷄧󼿒 
   -     
2.
5.      
     -      '  

    
GLRM   ,  
      
   
󷄧󼿒
          -      
       
  
         ;    
      ,    , 
,         
   
    ,          -
, ,    
,             
    
IV.        :

65
54
 
26
−7
      :

     ,      5%  
 '   β ≠ 0   β = 0       
: 1: R²   
R² (   ):       (Y)    
 (X)    
2.
o R² 0  1    
o             

o : R² = 0.80     Y  80%  X  
 
  :         
              
   
o :
Adjusted
󰇧
󰇛
󰇜󰇛
󰇜
󰇨
 =   , =     
  ( ) ,   '   , 
   
  " "  ,    R² " "     
 
 2:    

65
57
57
54
66

26
13
16
-7
27

      :

 3: X  Y       



󰇛󰇜


 4:   (β)    
:
󰇛
󰇜󰇛
󰇜
󰇛
󰇜
     :
2.

 
 

65
26
5.2
11
57.2
27.04
57
13
-2.8
-2
5.6
.
57
16
-2.8
1
-2.8
.
54
-7
-5.8
-22
127.6
33.64
66
27
6.2
12
74.4
38.44
   = 57.2 + 5.6 - 2.8 + 127.6 + 74.4 = 262
   = 27.04 + 7.84 + 7.84 + 33.64 + 38.44 = 114.8
 ,



 5:     (α)
󰇛󰇜
 6:  

 7: R²   
:


:
SST =   =󰇛
󰇜
SSR =   =󰇛
󰇜
  :
SST = 5, 5, 1, 484, 144 →  = 639
... ≈ 615.7
 ,



2.
    Y   96.4%  X    
 8:  
, , .
Adjusted
󰇛
󰇜󰇛
󰇜


 ,  R² = 95.2%
 9: β    

  :
-  :
󰆹
󰇛
󰆹
󰇜
     ,     (0.964),     5%
 ', df = n-2 = 3  ,  t ≈ 3.182    t (≈ 9.1)   
 , 
H₀       '  

   :

R² = 0.964 →   Y  96.4%     
  R² = 0.952 →    ,    
  
   β ≠ 0 → X  Y     

  :
²            
          
2.
  ,     ,    
 (β)   '  ,      X, Y  
 
  
       
 -    
      
  - 
V.            
               
:1.   ?
  ,    (OLS)      
           
     ,  
  ( = )  
         ,  

 ( = )  
   :
              -
()    
 
2.     

  
        
      '         
 
-          ,  
     
 , -   ,            
  :
X →      '
2.
X →     '
    OLS     
3.    
?
        :
 OLS      
 
,      
 

  
(a)     
  (    )     ,    
, :
  -  

OLS     
()   
     

    :
-  
 
-    
          
 ,
   :
        

           
 ,      
4. 
      ?
2.
            

A.   
(i)     
Y- ' -  X- '    
 -    
    
 -    ,  ,   
  
 
         
B.  
       '      '  
1. - 
                 ' 
 
    :
       ,  
 
 ():
1.   
2.     
3.   '  -  
4. -    
 :
       
2.
2.  
      
  :
      

    :
    ,   ,  - '  

  :
           
 
 :
    ,         
  
3. -  
               
   
:
1.  X      
2.    
3.       
4.   
5.  F-       
 :
    '     
5.      ?
2.
       ,        
  
1. 
     
(a)   
       
:
Y     , log(Y)   
X  , log(X)   
 ,          
2.         (WLS)
      ,  WLS    
 :
      
      
          
3.   
        '     
   :
OLS     
      
  :
2.
      
    
 :
  
 ,  
4.       
      
:
     
   
           '   
6.   
    :
        
 OLS        
    
,      
         :
o   
o - 
o  
o - 
         :
o  
o      
o   
o   
2.
VI.           ?
:  ?
  ,    (OLS)      
     ()    ()       
   
       ( )
    X → Heteroscedasticity (  )   
(   )
   
   
 (X)   (Y)        -
  ,       ,        -
  ,    '   -     ,   
               
  ',  
X   -   :
  ""         
  ""  ""    ,  X   -
  
   
?
1.   :   OLS    ,     

(  -  
)
2.   :     ,  -  -
   
         
   
3.    :      
 
 ,       
,    
     
     
                :
1.   
          
2.
 -         ,  
  
2. - 
     -   '    
    ,    
3.   
           

       -       
 
4. -  
               
      ,       
5.    /  
-   (  , )      
 
   
 
   
      ',         
 :
1.  
  '      
: X  Y    , X  log(Y)     
  
2.         (WLS)
        
            

3.   
-      (     SE)
2.
  
      ,   
    
4.    
        
     
 (  ,   )
            
   
               
        ,      
 ()       '         ,
         -   
  
           - 
  ,
""         ,     '    



      -   
    
     


    , -,   , -,  
 
   ,      ,   ,
     
  - 
VII.      -        
         '  
:    :         
   
       
   ?  
     -     ,  ,  
  '   
   ,      
  
 :
2.
    
 ,        
    
 ,         

       ,        
 - - 
              
       - 
    ?
   (DLM)            
    
     
     '          , 
,               
     
   :
   =     
     :
   =      +       +    
   
          
        
1.     :   

2.    :          

3.     :         
    
              
  
 -       
2.
-    ?
     
 -  ( )   '        
     
,    '     , 
   
 '    
 :
      '   

? 
           ₹40,000 
 ,     ₹10,000           
  
-        
1.   '  :        

 
2.    :    
 
3.      :, ,  ,    
   '   
-  
       
  6   ,     

           
     -     
  
   ,     '  
1.   
  :          '
  
-  :      '   
2.
2. 
:        '   
  :     '   
3.  
:            '  

  :       '   
4. 
:             
   
  :            
 
5. 
                
-           
 , DLM         ,   AR  
  
   :     
              :
  
    
 
   ,         
      ,          
 -    -     
 :
          
         
2.
      
    ,           
       
       -   
           
     ,       
1. - 
       ,         
   
 
 ?

            
           -    
  
 :      
2.     
  ,      
             
 ,              
 -   
 
3.      
       '  
    

     :
2.
 ,  
    
 
   ,       ,      
4.   

    ,       

               
5.   
    ,           , 
      
   
'    

 ,    -    
             -   
-    
            
-       
          
        ,  , , 
,      
 
   :     
       -    
,             ,  
,    ,        
VIII.    ?         
2.
:   ?
             
          '   :
1     (  ),
0    (  )
  ,   ""          
     
 
  
   '          
 - 
   (/)      ,
   

:
 = 1,  = 0
              
     
?
         -    
, , ,              
  :
   
    
     
   ,      
    
1.     
      
 
:    ( = 1,  = 0)
               
   
2.     
2.
        
:   ,         
   (1)       (0)
       
 
3.       
   ,        
:    , 
Q1, Q2, Q3, Q4     
          
4.    
             
:    = 0,    = 1
  '    
 
5.   
  -   -       
   
:    '         
       , 
    
6.     
  ,     (   )
  
:  =     1,  0
 
           
     
    k  ,   "  " (
)    k-1    
       
         
 
   
2.
               
    ( )   (1)   (0)      
  -    '        

      
    , , ,       
 
     ,    ,   
,     ,    ,  
     
     '        
   
 
"                 
 
          ,         
 "