Estimating the Expected Natural Interest Rate Using Affine Term









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

          


                
                 









           
             
          
 
       
              
           
         

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

       
      

     

           
              
   
            


               
            



            
         
        

             

               
             
  
              

                
             

                 


               
             



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



             
             
             

               
              
          



                
               








                
               


                   




 
             
        

             
          
        
              

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



              
                


                
          

                
   



               
 


                
            
 
         
            

              
          
              
          



 
             
           

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                 

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         
               
               
              
            

                
 

             

                
             



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
              
   
                    




              
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 

        
 

               
              
               
             
              



               
      
 
 






 






                 
 

                






  

    


 


 


  
  

    


  
 
  




 
  

   


                    

              


  


 

 

   



                   

 

    





 





                


 

   


                








 
  
    
   

 
   
  


  





   
   


  







    




 
 




   



  
 
 

         

    




 
 
  

                
  

  
                
                




               
                 




             


  
                  


 
               


 





  

  


   
    



 
  








  






  
   


 

   
   


 
   

              













 


  
  
  

  






 
  







  


  






 
 


  


    
 


    


   

 
   










 

 
   



               

                



              
 
 
               
              
 
              

             
              
          
        
            

                 



            


               
        

                 

 
 

  



     



             
              

             


 

                 

 


                  


               
             
               
         
     

                


            
 
                




           
                





 

 




 

 

 

               




               

              

 











 







          


      



             
































           


        
           
     
 
          
   




 
 







              

              
               
             


  
 
 
 





                   

 
  

  


  












              
               

 
 
 







 


  










  


  


               
            


         
              

 
              
                

               
                 
             
 
                


               

             

 


  





  
  
   



        

          

             













               

        


             

     
             

 
   

     


 

   




 


  
 
  

  

   




     
 
 

  

 





       

   

    
 








    
  




 
















 

  
               
 
 



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
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
             

               
 
              


 


                
           
           

              
             




 
 
             
               
  
              

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
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


              
                
             
           
                   
            

          



               
               

               





 



  




              


 
 
              
             
             
          

 
             










 

 





 
 
 





 







 
































 
 
 









      

   



 
 


  




                     






               
           
        

 
 
 



  

           

               



           

     
            

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
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
                
             
             


              
                
              
              

                
              
           
  



               
                 
                    

                  
   
                
                 
               
                
  
 
                 
 
                 

             
                
               
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  


                
                

               

 
              
             
  




               

                   
             
              

 


                

                


 
                 

                
              










             
             































  



 














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
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


                 
                
          

 

             

                   

               
                
         

 
              
             


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
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            



               

                
               
            
 
               
               

             

                


             

       





              

                
             
              
            
            
             





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            






 
                  









 
 






              



 

                
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
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  
 



              

 
               
             
               
     
                   
             


              
             
            


                

 
 


              



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

























               

        
 























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


















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







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
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


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

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
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
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
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
 


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
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









                
 
               
               




                
                
        

  
              

           

              
             

 


                 

 

              
                 
               

 
  





      



                   





                 
                


 
                 
                  

              

 



 

                  
                
                

 

                
                


                


               
             
                



































































































































































 
                





























 


 
 






            





        
              

               

             

            

         

           


              



      
            


            
   






















































             
              
             
             
             

                 













 
                 
        
               
                  






 



 
  


  


 



  


  


 

 


   






               
                
                  
                 







           
 

               

               

                
                
   
 
                 



                

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
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


                


               


                
                


 
                


 

              
              


                
                
             
             
             
          

    
             
          
      
           

                 
   
              







             
        
                 

















        
          
          

               










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
           
            


              
               

           
            
                 

                
                


 

              
             
           
         
               

              
               










                
               
            
                











 






 

 

 


 
 

 
 
             
              

 
              


              

               





          

              
                
       
                



                


































 

   

  

            




            

              


             

              





               
           

           



   

              

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
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            

            



 

              

             



             







            



 

            

             



 

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
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

             

            




              

             
          

          

          

              





               

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