Typically a linear relationship such as

  
    
      
        
          y
          
            i
          
        
        =
        
          a
          
            0
          
        
        +
        
          a
          
            1
          
        
        
          x
          
            1
            ,
            i
          
        
        +
        
          a
          
            2
          
        
        
          x
          
            2
            ,
            i
          
        
        +
        ⋯
        +
        
          a
          
            k
          
        
        
          x
          
            k
            ,
            i
          
        
        +
        
          e
          
            i
          
        
      
    
    {\displaystyle y_{i}=a_{0}+a_{1}x_{1,i}+a_{2}x_{2,i}+\dots +a_{k}x_{k,i}+e_{i}}
  is postulated, in which 
  
    
      
        
          y
          
            i
          
        
      
    
    {\displaystyle y_{i}}
   is the ith observation of the dependent variable (hypothesized to be the caused variable), 
  
    
      
        
          x
          
            j
            ,
            i
          
        
      
    
    {\displaystyle x_{j,i}}
   for j=1,...,k is the ith observation on the jth independent variable (hypothesized to be a causative variable), and 
  
    
      
        
          e
          
            i
          
        
      
    
    {\displaystyle e_{i}}
   is the error term for the ith observation (containing the combined effects of all other causative variables, which must be uncorrelated with the included independent variables).
Here the notion of causality is one of contributory causality as discussed above: If the true value 
  
    
      
        
          a
          
            j
          
        
        ≠
        0
      
    
    {\displaystyle a_{j}\neq 0}
  , then a change in 
  
    
      
        
          x
          
            j
          
        
      
    
    {\displaystyle x_{j}}
   will result in a change in y unless some other causative variable(s), either included in the regression or implicit in the error term, change in such a way as to exactly offset its effect; thus a change in 
  
    
      
        
          x
          
            j
          
        
      
    
    {\displaystyle x_{j}}
   is not sufficient to change y.