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The Nature of Econometrics and Economic Data - ISET

VIEWS: 10 PAGES: 11

									Statistical Properties of OLS
Estimators
                                                              
To derive statistical properties of OLS estimators 0 , 1
  let’s first summarize all the assumptions that
  we have made so far
Assumption SLR1: Our empirical model is linear
  in parameters yi   0  1 xi  ui
Assumption SLR2: We have a random sample
  of n observations on y and x: {( xi , yi ) : i  1,..., n}
Assumption SLR3: Values of x are not all the
  same,  xi and x j such that xi  x j
Assumption SLR4: The expected value of the
  error term given x is zero, E(ui|x)=0
b1 is unbiased
We can write OLS slope estimator as
             n                             n

             ( x  x ) y  ( x  x )(
                           i   i                        i      0    1 xi  ui )
     b1    i 1
               n
                                         i 1
                                                                                    
            
                                                             SSTx
              ( xi  x ) 2
             i 1
                                                 SST
                    0 
                                            x
                     n                            n                    n
             0  ( xi  x )  1  ( xi  x ) xi   ( xi  x )ui
                   i 1                         i 1                 i 1
                                                                                        
                                                      SSTx
                                    n
        1  (1 / SSTx ) ( xi  x )ui  E (b1 | x)  1  0  1
                                   i 1
Which means that b1 is unbiased, conditional on x.
Note that E (b1 )  E x E (b1 | x)   E x 1   1
b0 is unbiased
Since
b0  y  b1 x  ( 0  1 x  u )  b1 x   0  ( 1  b1 ) x  u
 E (b0 | x)  E ( 0  x ( 1  b1 )  u | x)   0
Finally, as above:
          E (b0 )  E x E (b0 | x)   E x  0    0

And it follows that OLS intercept estimator is an
  unbiased estimator of population intercept.
Variance of b0 and b1
To derive the variance of b0 and b1 we need to
  make an additional assumption:
Assumption SLR5: The error u has the same
  variance given any value of x:  V u | x    2

This is known as homoskedasticity or “constant
  variance” assumption.
Note: This assumption is NOT needed to show
  unbiasedness of OLS estimators b0 and b1!
Variance of b0 and b1                               
                                                       2

Because  2  V u | x   E (u 2 | x)   E (u | x) 
                                           
                                                   0     
             E (u 2 | x)   2
             E (u 2 )  E x E (u 2 | x)    2
             V (u )  E (u   2
                                  ) E (u ) 2  E (u 2 )   2
Hence, σ2 is often called the error variance or the
  disturbance variance.
Also, σ is the standard deviation of the error.
 Variance of b0 and b1
Assumptions 4 and 5 are also written in the following way:
          E (u | x)  0    E ( y | x)   0  1 x
                       2
                           
         V (u | x)          V ( y | x)   2
The conditional expectation of y given x is linear
The conditional variance of y given x is constant
When V (u | x)   2 and it depends on x, the error term is
 said to be heteroskedastic (HS) or non-constant variance.
Because V (u | x)  V ( y | x) HS is present whenever
                             ,
  V(u|x) and hence V(y|x) are a function of x.
Variance of b0 and b1
Under assumptions SLR.1-5
                                                  2
                                                                      2
             V (b1 | x)         n
                                                               
                                (x  x)                   2       SSTx
                                               i
                                i 1
and                              n

                                x
                                               2
                                           i                           n

                                                                    x
                                                                                   2
                             2 i 1
                                                              2               i
        V (b0 | x)                    n                           i 1
                        n

                       
                                                           SSTx            n
                         ( xi  x ) 2
                       i 1

Proof: HW and/or quiz 
Analyzing the formulas
Let’s look at the formula for V(b1):
                                   2           2
             V (b1 | x)     n
                                            
                             (x  x)   2       SSTx
                                   i
                            i 1
It depends on:
σ2 – the larger is the variation in the unobserved factors,
    the harder it is to pinpoint the line
SSTx – the larger variation in the regressor, the easier it is
    to fit the line. As the sample size increases, so will SSTx=
     n
 =i 1
        ( xi  x ) 2 unless the new values of x are equal to x
Problem: σ is unknown…
Since σ2 is an unknown parameter, we can’t calculate
  V(b0) and V(b1).
We need to estimate σ2 and then use that estimate in the
  variance formulas.
The formula used to estimate σ2 is:
                          n
                   1              1
            s           ui  (n  2) SSR
                             ˆ
             2                 2

                (n  2) i 1

We divide by (n-2) to make sure the estimator is unbiased.
 Reason: there are (n-2) degrees of freedom in OLS
                        n           n
 residuals, since
                         u  0,
                          ˆ
                       i 1
                              i      u x 0
                                   i 1
                                       ˆ  i i
Using s2
It can be shown that s2 is an unbiased estimator of σ2
    (HW …).
In the literature s  s is called the standard error of
                         2

    the regression (model standard error - MSE).
Now, we have an estimate of σ2, which we can use to
    estimate the variance and the standard errors of OLS
    estimators:                   2
                                s
                  EstV (b1 ) 
                               SSTx
                               s2
                  se(b1 )          s / SSTx
                              SSTx
Regression through the Origin
Sometimes we need to impose  0  0 : if x=0 => y=0
In this case the econometric model is simply:
                          yi  1 xi  ui
We can use the OLS approach to find b1 from:
                              n

                  min 
                     b1
                        ( yi  b1 xi ) 2
                             i 1
                     n

                    
The F.O.C. gives  2 ( y  b x )x  0
  solution is:      i 1
                         i
                           n
                             1 i i
                                                   and the

                               x y        i   i
                      b1         i 1
                                     n

                                   xi2
                                    i 1

								
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