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					Lecture :Apply Gauss Markov Modeling
Regression with One Explanator

    (Chapter 3.1–3.5, 3.7
    Chapter 4.1–4.4)




                                       5-1
Agenda

• Finding a good estimator for a
  straight line through the origin:
  Chapter 3.1–3.5, 3.7

• Finding a good estimator for a straight
  line with an intercept: Chapter 4.1–4.4




                                            5-2
Where Are We? (範例)

• We wish to uncover quantitative features
  of an underlying process, such as the
  relationship between family income and
  financial aid.
• 更精準些 How much less aid will I
  receive on average for each dollar of
  additional family income?
• DATA, a sample of the process, for example
  observations on 10,000 students’ aid awards
  and family incomes.

                                                5-3
隨機項

• Other factors (e), such as number of siblings,
  influence any individual student’s aid, so we
  cannot directly observe the relationship
  between income and aid.

• We need a rule for making a good guess
  about the relationship between income and
  financial aid, based on the data.



                                                   5-4
Guess

• A good guess is a guess which is right
  on average.

• We also desire a guess which will have
  a low variance around the true value.




                                           5-5
估計式

• Our rule is called an “estimator.”
• We started by brainstorming a number
  of estimators and then comparing their
  performances in a series of computer
  simulations.
• We found that the Ordinary Least Squares
  estimator dominated the other estimators.
• Why is Ordinary Least Squares so good?

                                              5-6
工具

• To make more general statements, we
  need to move beyond the computer and
  into the world of mathematics.

• Last time, we reviewed a number of
  mathematical tools: summations,
  descriptive statistics, expectations,
  variances, and covariances.

                                          5-7
DGP

• As a starting place, we need to write down all
  our assumptions about the way the
  underlying process works, and about how that
  process led to our data.
• These assumptions are called the “Data
  Generating Process.”
• Then we can derive estimators that have
  good properties for the Data Generating
  Process we have assumed.

                                                   5-8
Model

• The DGP is a model to approximate
  reality. We trade off realism to gain
  parsimony and tractability.

• Models are to be used, not believed.




                                          5-9
DGP assumptions

• Much of this course focuses on different
  types of DGP assumptions that you can
  make, giving you many options as you
  trade realism for tractability.




                                             5-10
Two Ways to Screw Up in Econometrics



  – Your Data Generating Process assumptions
    missed a fundamental aspect of reality (your DGP
    is not a useful approximation); or
  – Your estimator did a bad job for your DGP.

• Today we focus on picking a good estimator
  for your DGP.



                                                       5-11
GMT

• Today, we will focus on deriving the
  properties of an estimator for a simple
  DGP: the Gauss–Markov Assumptions.
• First we will find the expectations and
  variances of any linear estimator under
  the DGP.
• Then we will derive the Best Linear
  Unbiased Estimator (BLUE).

                                            5-12
Our Baseline DGP: Gauss–Markov
(Chapter 3)

• Y = bX +e
• E(ei ) = 0
• Var(ei ) = s 2
• Cov(ei ,ej ) = 0, for i ≠ j
• X ’s fixed across samples (so we can
  treat them like constants).
• We want to estimate b
                                         5-13
A Strategy for Inference

•   The DGP tells us the assumed relationships
    between the data we observe and the
    underlying process of interest.

•   Using the assumptions of the DGP and the
    algebra of expectations, variances, and
    covariances, we can derive key properties
    of our estimators, and search for estimators
    with desirable properties.


                                                   5-14
 An Example: bg1

Yi  b X i  e i
E(e i )  0
Var(e i )  s 2
Cov(e i ,e j )  0, for i  j
X's fixed across samples (so we can treat it as a constant).
      1 n Yi
b g1  
      n i1 X i
In our simulations, b g1 appeared to give estimates close to b.
Was this an accident, or does b g1 on average give us b ?

                                                             5-15
An Example: bg1 (OK on average)
               1 n Yi    1 n   Yi  1 n   b X i  ei
 E( b g1 )  E(  )   E( )   E(                 )
               n i1 X i n i1 X i n i1    Xi
           1 n        1 n 1
            E( b )   E(e i )
           n i1      n i1 X i
         1
        nb  0  b
         n
 On average, b g1  b.
 E( b g1 )  b
 Using the DGP and the algebra of expectations,
     we conclude that b g1 is unbiased.
                                                        5-16
Checking Understanding

              1 n Yi    1 n   Yi  1 n   b X i  ei
E( b g1 )  E(  )   E( )   E(                 )
              n i1 X i n i1 X i n i1    Xi
         1 n        1 n 1
          E( b )   E(e i )
         n i1      n i1 X i
            1
           nb  0  b
            n
E( b g1 )  b
Question: which DGP assumptions did we need to use?


                                                       5-17
Which assumption used?
               1 n Yi    1 n   Yi  1 n   b X i  ei
 E( b g1 )  E(  )   E( )   E(                 )
               n i1 X i n i1 X i n i1    Xi
   Here we used Yi  b X i  e i
         1 n        1 n 1
          E( b )   E(e i )
         n i1      n i1 X i
   Here we used the assumption that X's
      are fixed across samples.
        1
        nb  0  b
        n
  Here we used E(e i )  0
                                                        5-18
Checking Point 2:


We did NOT use the assumptions about
 the variance and covariances of e i .


We will use these assumptions when we
 calculate the variance of the estimator.


                                         5-19
Linear Estimators

•   bg1 is unbiased. Can we generalize?
• We will focus on linear estimators.
• Linear estimator: a weighted sum of the Y ’s.


            ˆ
            b         wY     i i


                                                  5-20
Linear Estimators (weighted sum)

• Linear estimator:
    b   wYi
    ˆ
           i


• Example: bg1 is a linear estimator.
         1  Yi
   b g1  
         n  Xi
         1
   wi 
        nX i
   b g1   wiYi
                                        5-21
A class of Linear Estimators
  1) Mean of Ratios:       3) Mean of Ratio of Changes:
               Yi                                  Yi  Yi1
     b g1                                    
           1                               1
                                 b g3 
           n Xi                         n 1 X i  X i1
             1                     1        1            1     
     wi                    wi                               
            nX i                 n 1  X i  X i1 X i1  X i 
   2) Ratio of Means:      4) Ordinary Least Squares:

     b g2   
              Y   i
                                   b g4   
                                            Y X  i       i

              X       i                    X        j
                                                          2


             1                      Xi
     wi                    wi 
            Xj                    X     j
                                              2




• All of our “best guesses” are linear estimators!
                                                                    5-22
Expectation of Linear Estimators

Yi  b X i  e i           E (e i )  0
Var (e i )  s 2           Cov(e i , e j )  0, for i  j
X 's fixed across samples (so we can treat it as a constant).
       n
b   wiYi
ˆ
      i 1
                     n            n              n
E ( b )  E ( wiYi )   wi E (Yi )  wi E ( b X i  e i )
    ˆ
                    i 1         i 1           i 1
              n                                   n
         wi [ E ( b X i )  E (e i )] b  wi X i
             i 1                                i 1




                                                               5-23
Condition for Unbias

      n
b   wiYi
ˆ
     i 1



             n
E ( b )  b  wi X i
    ˆ
            i 1



                                     n
A linear estimator is unbiased if   w X
                                    i 1
                                           i   i    1.

                                                          5-24
Check others

• A linear estimator is unbiased if SwiXi = 1
• Are bg2 and bg4 unbiased?
    2) Ratio of Means:           4) Ordinary Least Squares:

      b g2   
               Y  i
                                             b g4   
                                                      Y X  i       i

               X      i                              X        j
                                                                    2


            1                                 Xi
      wi                         wi 
           Xj                           X         j
                                                        2


                                                            Xi
      wi X i                   w X  
                   1
                      Xi                                                        Xi
                  Xj                    i     i
                                                            X          j
                                                                            2




                       Xi  1                              X i2  1
                   1                                 1
                                              
                  Xj                                X j2
                                                                                     5-25
Better unbiased estimator

• Similar calculations hold for bg3

• All 4 of our “best guesses” are unbiased.

• But bg4 did much better than bg3. Not all
  unbiased estimators are created equal.

• We want an unbiased estimator with a low
  mean squared error.



                                              5-26
First: A Puzzle…..

• Suppose n = 1
  – Would you like a big X or a small X for
    that observation?
  – Why?




                                              5-27
What Observations
Receive More Weight?
1) Mean of Ratios:      3) Mean of Ratio of Changes:
             Yi                                 Yi  Yi1
   b g1                                   
         1                              1
                              b g3 
         n Xi                        n 1 X i  X i1
          1                     1        1            1     
  wi                    wi                               
         nX i                 n 1  X i  X i1 X i1  X i 
2) Ratio of Means:      4) Ordinary Least Squares:

  b g2   
           Y   i
                                b g4   
                                         Y X
                                           i       i

           X       i                    X    j
                                                   2


          1                       Xi
  wi                    wi 
         Xj                     X j2

                                                                 5-28
(Stat. significant)?

•   bg1 puts more weight on observations with low
    values of X.
•   bg3 puts more weight on observations with low
    values of X, relative to neighboring observations.
•   These estimators did very poorly in the simulations.


                   Yi                              Yi  Yi1
                                              
               1                          1
      b g1                      b g3 
               n Xi                     n 1 X i  X i1
             1                       1        1             1    
     wi                      wi                               
            nX i                   n 1  X i  X i1 X i1  X i 

                                                                      5-29
What Observations
Receive More Weight? (cont.)

• bg2 weights all observations equally.
• bg4 puts more weight on observations with high
  values of X.
• These observations did very well in the simulations.


    b g2   
             Yi
                                          b g4   
                                                   Y X i        i

             X    i                               X       j
                                                                2


          1                                     Xi
    wi                                   wi 
         Xj                                    X j2

                                                                     5-30
Why Weight More Heavily Observations
With High X ’s?

• Under our Gauss–Markov DGP the
  disturbances are drawn the same for all
  values of X….
• To compare a high X choice and a low X
  choice, ask what effect a given disturbance
  will have for each.




                                                5-31
Figure 3.1 Effects of a Disturbance for
Small and Large X




                                          5-32
Linear Estimators and Efficiency

• For our DGP, good estimators will place more
  weight on observations with high values of X
• Inferences from these observations are less
  sensitive to the effects of the same e
• Only one of our “best guesses” had this
  property.
• bg4 (a.k.a OLS) dominated the other
  estimators.
• Can we do even better?
                                                5-33
 Min. MSE

• Mean Squared Error = Variance + Bias2

• To have a low Mean Squared Error,
  we want two things: a low bias and a
  low variance.




                                          5-34
Need Variance

• An unbiased estimator with a low variance
  will tend to give answers close to the true
  value of b

• Using the algebra of variances and our
  DGP, we can calculate the variance of
  our estimators.




                                                5-35
Algebra of Variances

  (1) Var (k )  0
  (2) Var (kY )  k 2 ·Var (Y )
  (3) Var (k  Y )  Var (Y )
  (4) Var ( X  Y )  Var ( X )  Var (Y )  2Cov( X , Y )
              n         n              n    n
  (5) Var ( Yi )   Var (Yi )   Cov(Yi , Y j )
             i 1      i 1           i 1 j 1
                                           j i

• One virtue of independent observations is that
  Cov( Yi ,Yj ) = 0, killing all the cross-terms in the
  variance of the sum.
                                                             5-36
Back again to Our Baseline DGP
: Gauss–Markov

• Our benchmark DGP: Gauss–Markov
• Y = bX +e
• E(ei ) = 0
• Var(ei ) = s 2
• Cov(ei ,ej ) = 0, for i ≠ j
• X ’s fixed across samples
  We will refer to this DGP (very) frequently.


                                                 5-37
Variance of OLS
                     X iYi 
      OLS )  Var  
      ˆ
Var ( b                   2 
                   X i 
                   X iYi        n      n
                                               X iYi X jY j
            Var      2 
                            2          Cov( X 2 , X 2
                   X i        i 1   j 1,
                                        j i
                                                  k     k

                           2
              Xi 
               2 
                      Var Yi   0
             X 
                k 
                           2
              Xi 
                  Var  b X i  e i 
             X 2 
                   
                k 

                                                              5-38
 Variance of OLS (cont.)
                          2
                 Xi 
      OLS )          Var  b X i  e i 
      ˆ
Var ( b
                X 2 
                      
                   k 
                          2
              Xi 
                  (0  Var  e i   0)
             X 2 
                   
                k 
                          2                    2
              Xi  2                 1                      s2
                 s s                          X i2 
                                     X 2 
                          2
             X 2                                           Xk2
                k                    k 




 • Note: the higher the Xk2 , the lower
   the variance.
                                                                    5-39
Variance of a Linear Estimator

• More generally:

Var ( wiYi )   Var ( wiYi )  2 Covariance Terms
    Var ( wiYi )  0   wi Var (Yi )
                                      2


    wi Var ( b X i  e i )
           2


    wi   2
               0  Var (e i )  0
  s   2
           w  i
                   2



                                                      5-40
Variance of a Linear Estimator (cont.)

• The algebras of expectations
  and variances allow us to get exact
  results where the Monte Carlos gave
  only approximations.

• The exact results apply to ANY
  model meeting our Gauss–Markov
  assumptions.

                                        5-41
Variance of a Linear Estimator (cont.)

• We now know mathematically that bg1–bg4
  are all unbiased estimators of b under our
  Gauss–Markov assumptions.
• We also think from our Monte Carlo models
  that bg4 is the best of these four estimators,
  in that it is more efficient than the others.
• They are all unbiased (we know from the
  algebra), but bg4 appears to have a smaller
  variance than the other 3.
                                                   5-42
Variance of a Linear Estimator (cont.)

• Is there an unbiased linear estimator
  better (i.e., more efficient) than bg4?
  – What is the Best, Linear, Unbiased
    Estimator?
  – How do we find the BLUE estimator?




                                            5-43
BLUE Estimators

• Mean Squared Error = Variance + Bias2

• An unbiased estimator is right
  “on average”

• In practice, we don’t get to average. We
  see only one draw from the DGP.



                                          5-44
BLUE Estimators (Trade-off ??)

• Some analysts would prefer an
  estimator with a small bias, if it gave
  them a large reduction in variance

• What good is being right on average if
  you’re likely to be very wrong in your
  one draw?



                                            5-45
BLUE Estimators (cont.)

• Mean Squared Error = Variance + Bias2
• In a particular application, there may be
  a favorable trade-off between accepting a
  little bias in return for a lot less variance.
• We will NOT look for these trade-offs.
• Only after we have made sure our
  estimator is unbiased will we try to make
  the variance small.

                                                   5-46
BLUE Estimators (cont.)

A Strategy for Finding the Best Linear
Unbiased Estimator:
   1. Start with linear estimators: wiYi
   2. Impose the unbiasedness condition wiXi=1
   3. Calculate the variance of a linear estimator:
      Var(wiYi) =s2wi2
       – Use calculus to find the wi that give the smallest
         variance subject to the unbiasedness condition

Result: the BLUE Estimator for Our DGP

                                                              5-47
BLUE Estimators (cont.)

                                    Xi
Using calculus, we would find wi 
                                    X j2
This formula is OLS!
OLS is the Best Linear Unbiased Estimator for
   the Gauss–Markov DGP.
This result is called the Gauss–Markov Theorem.



                                              5-48
BLUE Estimators (cont.)

• OLS is a very good strategy for the
  Gauss–Markov DGP.
• OLS is unbiased: our guesses are right
  on average.
• OLS is efficient: it has a small variance
  (or at least the smallest possible variance
  for unbiased linear estimators).
• Our guesses will tend to be close to right (or
  at least as close to right as we can get; the
  minimum variance could still be pretty large!)
                                                   5-49
BLUE Estimator (cont.)

• According to the Gauss–Markov Theorem,
  OLS is the BLUE Estimator for the
  Gauss–Markov DGP.

• We will study other DGP’s. For any DGP,
  we can follow this same procedure:
  – Look at Linear Estimators
  – Impose the unbiasedness conditions
  – Minimize the variance of the estimator


                                             5-50
Example: Cobb–Douglas Production
Functions (Chapter 3.7)

• A classic production function in economics is
  the Cobb–Douglas function.

• Y = aLbK1-b

• If firms pay workers and capital their marginal
  product, then worker compensation equals a
  fraction b of total output (or national income).



                                                     5-51
Example: Cobb–Douglas

• To illustrate, we randomly pick 8 years
  between 1900 and 1995. For each year,
  we observe total worker compensation
  and national income.
• We use bg1, bg2, bg3, and bg4 to
  estimate
     Compensation = b·National Income +e


                                            5-52
TABLE 3.6 Estimates of the Cobb–Douglas
Parameter b, with Standard Errors




                                          5-53
TABLE 3.7
Outputs from
a Regression* of
Compensation on
National Income




                   5-54
Example: Cobb–Douglas

• All 4 of our estimators give very
  similar estimates.
• However, bg2 and bg4 have much smaller
  standard errors. (We will see the value of
  small standard errors when we cover
  hypothesis tests.)
• Using our estimate from bg4, 0.738, a
  1 billion dollar increase in National Income
  is predicted to increase total worker
  compensation by 0.738 billion dollars.
                                                 5-55
A New DGP

• Most lines do not go through the origin.
• Let’s add an intercept term and find the
  BLUE Estimator (from Chapter 4).




                                             5-56
Gauss–Markov with an Intercept

   Yi  b0  b1 X i  e i (i  1...n)
    E(e i )  0
   Var(e i )  s   2


   Cov(e i ,e j )  0, i  j
    X's fixed across samples.
    All we have done is add a b0 .
                                        5-57
Gauss–Markov with an Intercept (cont.)


• Example: let’s estimate the effect of income
  on college financial aid.

• Students whose families have 0 income do
  not receive 0 aid. They receive a lot of aid.

• E[financial aid | family income]
      = b0 + b1(family income)



                                                  5-58
Gauss–Markov with an Intercept (cont.)




                                         5-59
Gauss–Markov with an Intercept (cont.)

• How do we construct a BLUE Estimator?
• Step 1: focus on linear estimators.
• Step 2: calculate the expectation of a linear
  estimator for this DGP, and find the condition
  for the estimator to be unbiased.
• Step 3: calculate the variance of a linear
  estimator. Find the weights that minimize this
  variance subject to the unbiasedness
  constraint.

                                                   5-60
Expectation of a Linear Estimator


E ( b )  E   wYi    E ( wYi )
    ˆ
                 i             i

        wi E (Yi )   wi E ( b0  b1 X i  e i )
         wi E ( b 0 )  wi E ( b1 X i )  wi E (e i ) 
         b 0 wi  b1wi X i  0
       b0  wi  b1  wi X i

                                                        5-61
Checking Understanding


      E(b )  b0  wi  b1  wi X i
        ˆ


• Question: What are the conditions for an
  estimator of b1 to be unbiased? What
  are the conditions for an estimator of b0
  to be unbiased?


                                          5-62
Checking Understanding (cont.)

      E(b )  b0  wi  b1  wi X i
        ˆ

• When is the expectation equal to b1?
   – When wi = 0 and wiXi = 1

• What if we were estimating b0? When is the
  expectation equal to b0?
   – When wi = 1 and wiXi = 0

• To estimate 1 parameter, we needed 1 unbiasedness
  condition. To estimate 2 parameters, we need 2
  unbiasedness conditions.

                                                  5-63
Variance of a Linear Estimator

Var ( b )  Var   wiYi    Var  wiYi   0
      ˆ

           wi Var  b 0  b1 X i  e i 
                  2


           wi   2
                      0  0  Var (e i )  0
           wi s 2     2


• Adding a constant to the DGP does NOT
  change the variance of the estimator.

                                                 5-64
BLUE Estimator

                                   ˆ
 To compute the BLUE estimator for b1, we want to
    minimize s 2  wi 2
   subject to the constraints
    w  0
        i

    w X 1
        i    i

 Solution:
    ˆ  ( X i  X )(Yi  Y )
    b1  n
             ( X j  X )2
                 j 1


                                                    5-65
BLUE Estimator of b1


            ˆ  ( X i  X )(Yi  Y )
            b1  n
                  (X j  X )
                      j 1
                                 2




• This estimator is OLS for the DGP with
  an intercept.
• It is the Best (minimum variance) Linear
  Unbiased Estimator for the Gauss–Markov
  DGP with an intercept.

                                             5-66
BLUE Estimator of b1 (cont.)

           ˆ  ( X i  X )(Yi  Y )
           b1  n
                 (X j  X )
                    j 1
                                2




• This formula is very similar to the
  formula for OLS without an intercept.
• However, now we subtract the mean
  values from both X and Y.

                                          5-67
BLUE Estimator of b1 (cont.)


              ˆ  ( X i  X )(Yi  Y )
              b1  n
                    (X j  X )
                       j 1
                                   2




• OLS places more weight on high values of:
          Xi  X
• Observations are more valuable if X is far
  away from its mean.

                                               5-68
BLUE Estimator of b1 (cont.)

                     Xi  X
            wi 
                    X       X
                                    2
                          j
                                                           2
                                                      
                                          Xi  X
      Var ( b1 )  s 2  wi 2  s 2  
            ˆ                                          
                                       X  X  
                                       j
                                                     2

                                                       
                               1
                 s 2                    ( X i  X )2
                       X j  X 
                                    2 2
                                        
                         s2
               
                    X       X
                                    2
                          j




                                                               5-69
BLUE Estimator of b0

• The easiest way to estimate the intercept:
                ˆ         ˆ
                b0  Y  b1 X
• Notice that the fitted regression line always
  goes through the point

                    ( X ,Y )
• Our fitted regression line passes through “the
  middle of the data.”

                                                   5-70
Example: The Phillips Curve

• Phillips argued that nations
  face a trade-off between inflation
  and unemployment.

• He used annual British data on wage
  inflation and unemployment from
  1861–1913 and 1914–1957 to regress
  inflation on unemployment.

                                        5-71
Example: The Phillips Curve (cont.)

• The fitted regression line for 1861–1913
  did a good job predicting the data from
  1914 to 1957.

• “Out of sample predictions” are a strong
  test of an econometric model.



                                             5-72
Example: The Phillips Curve (cont.)

• The US data from 1958–1969 also
  suggest a trade-off between inflation
  and unemployment.

   Unemploymentt  0.06 - 0.55·Inflationt

                ˆ
                b 0  0.06
                 ˆ
                b1  0.55
                                            5-73
Example: The Phillips Curve (cont.)

      Unemploymentt  0.06 - 0.55·Inflationt

• How do we interpret these numbers?
• If Inflation were 0, our best guess of
  Unemployment would be 0.06
  percentage points.
• A one percentage point increase of Inflation
  decreases our predicted Unemployment level
  by 0.55 percentage points.

                                                 5-74
Figure 4.2 U.S. Unemployment and
Inflation, 1958–1969




                                   5-75
TABLE 4.1 The Phillips Curve




                               5-76
Example: The Phillips Curve

• We no longer need to assume our
  regression line goes through the origin.

• We have learned how to estimate
  an intercept.

• A straight line doesn’t seem to do a
  great job here. Can we do better?

                                             5-77
Review

• As a starting place, we need to write down all
  our assumptions about the way the
  underlying process works, and about how that
  process led to our data.
• These assumptions are called the “Data
  Generating Process.”
• Then we can derive estimators that have
  good properties for the Data Generating
  Process we have assumed.

                                               5-78
Review: The Gauss–Markov DGP

• Y = bX +e
• E(ei ) = 0
• Var(ei ) = s 2
• Cov(ei ,ej ) = 0, for i ≠ j
• X ’s fixed across samples (so we can
  treat them like constants).
• We want to estimate b
                                         5-79
Review

• We will focus on linear estimators.
• Linear estimator: a weighted sum of the Y ’s.



          ˆ
          b           i
                      wYi
                                                  5-80
Review (cont.)

 Yi  b X i  e i
 E (e i )  0
 Var (e i )  s 2
 Cov(e i , e j )  0, for i  j
 X 's fixed across samples (so we can treat it as a constant).
        n
 b   wiYi
 ˆ
       i 1
                 n
   ˆ
 E(b )  b      w X
                i 1
                       i   i

                                      n
 A linear estimator is unbiased if   w X
                                     i 1
                                            i   i    1.


                                                                 5-81
 Review (cont.)

Yi  b X i  e i
E(e i )  0
Var(e i )  s 2
Cov(e i ,e j )  0, for i  j
X's fixed across samples (so we can treat it as a constant).
                                    n
A linear estimator is unbiased if  wi X i  1.
                                   i1

Many linear estimators will be unbiased. How do I pick the "best"
linear unbiased estimator (BLUE)?


                                                               5-82
Review: BLUE Estimators

A Strategy for Finding the Best Linear
Unbiased Estimator:
   1. Start with linear estimators: wiYi
   2. Impose the unbiasedness condition wiXi = 1
   3. Use calculus to find the wi that give the smallest
      variance subject to the unbiasedness condition.

Result: The BLUE Estimator for our DGP



                                                           5-83
Review: BLUE Estimators (cont.)

• Ordinary Least Squares (OLS) is BLUE
  for our Gauss–Markov DGP.
• This result is called the “Gauss–Markov
  Theorem.”




                                            5-84
Review: BLUE Estimators (cont.)

• OLS is a very good strategy for the Gauss–
  Markov DGP.
• OLS is unbiased: our guesses are right
  on average.
• OLS is efficient: the smallest possible
  variance for unbiased linear estimators.
• Our guesses will tend to be close to right
  (or at least as close to right as we can get).
• Warning: the minimum variance could still be
  pretty large!
                                                   5-85
Gauss–Markov with an Intercept

   Yi  b0  b1 X i  e i (i  1...n)
   E(e i )  0
   Var(e i )  s   2


   Cov(e i ,e j )  0, i  j
   X's fixed across samples.
   All we have done is add a b0 .
                                        5-86
Review: BLUE Estimator of b1

             ˆ  ( X i  X )(Yi  Y )
             b1  n
                      ( X j  X )2
                      j 1



• This estimator is OLS for the DGP with
  an intercept.
• It is the Best (minimum variance) Linear
  Unbiased Estimator for the Gauss–Markov
  DGP with an intercept.

                                             5-87
BLUE Estimator of b0

• The easiest way to estimate the intercept:
               ˆ         ˆ
               b0  Y  b1 X
• Notice that the fitted regression line always
  goes through the point

                    ( X ,Y )
• Our fitted regression line passes through “the
  middle of the data.”

                                                   5-88

				
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