O in order to conceive the idea behind the assumption

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o In order to conceive the idea behind the assumption of T>k+1 let us follow an example. Suppose that we want to understand the systematic relationship between Y and X given follows: 0 1 t t Y X β β = + . If we have only one observation ( T =1) such as X 1 =4 and Y 1 =15 then we can write: 0 1 4 15 β β + = Can we solve the equation above and obtain values of 0 β and 1 β ? The answer is no , since there are two unknowns ( k +1=2) but we have only one equation (since we have one observation, T =1), hence this is a pedagogic example of the T < k+1 situation 11 . It is also said that 0 β and 1 β are not identified . Now suppose that we have two observations ( T =2) such as; X 1 =4, Y 1 =15 and X 2 =8, Y 2 =23. Then, we can write: 0 1 4 15 β β + = 0 1 8 23 β β + = What about now? Can we solve the two equations above and obtain values of 0 β and 1 β ? The answer is yes , since there are still two unknowns ( k +1=2) but we have now two equations (or two observations, T =2), hence this is a pedagogic example of the T = k+1 situation. Note that, in this case, there is a unique 11 Note that this type of problems is called ill-posed problems . Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 12
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