William Lott

							Economics 310

         Lecture 15
       Autocorrelation
Autocorrelation
   Correlation between members of series
    of observations order in time or space.
   For our classic model, we have
    E(ij)0 for i j.
   Some times use the term serial
    correlation for autocorrelation.
Autocorrelation Model


  t  t 1  t   1    1
  t ~ I .I .D.
Rho = 0.9
   2

  1.5

   1

  0.5
                                mu
   0
        0   10   20   30   40
 -0.5

   -1

 -1.5
Plot error and error lagged
         2

        1.5

         1

        0.5
                      Mu lagged
         0
 -2           0   2
       -0.5

         -1

       -1.5
Rho = -0.9
  3

  2

  1

  0                           mu
      0   10   20   30   40
 -1

 -2

 -3
Plot of mu and mu lagged
           3

           2

           1

           0                Mu lagged
 -4   -2        0   2   4
           -1

           -2

           -3
Rho = 0.0 (Pure Random
Error)
 1.5

   1

 0.5

   0                            mu
        0   10   20   30   40
 -0.5

  -1

 -1.5
Plot of mu and mu lagged
           1.5

             1

           0.5

             0                Mu lagged
 -2   -1          0   1   2
           -0.5

            -1

           -1.5
        Causes of Autocorrelation
   Inertia
   Specification bias: excluded variables case
   Specification bias: incorrect functional form
   Cobweb Phenomenon
   Lags
   Manipulation of data
       interpolation
       extrapolation
Models of Autocorrelation
 first - order autoregressive scheme(AR(1))
  t   t 1   t
 E ( t )  0
 Var( t )   2
 Cov( t  t  s )  0 s  0
 First - order moving Average (MA(1))
  t   t  t 1  1    1
 Autoregressive moving average (ARMI(1,1))
  t   t 1   t  t 1
OLS Estimation with AR(1)
Error
Model : Yt  1   2 X t   t
            t   t 1   t


                                 n 1             n2
                                                                                      
                           2 2   t t 1          xt xt  2
                                        xx                                            
                    2                                                    n 1 x1 xn
Var(b2 ) AR (1)                 t 1 2   2   t 1
                                                                  ...              
                     xt2  xt2   xt               x   2
                                                          t                    xt 2

                                
                                                                                     
                                                                                      
Normally
                  2
Var(b2 ) 
             x        2
                       t
OLS Estimation Disregarding
Autocorrelation
   The residual variance is likely to
    underestimate the true variance.
   R2 is likely to be overestimated.
   Estimate of variance of b2 is likely to
    underestimate the true variance of b2.
   t and F tests are no longer valid.
Variance estimate is biased.
   ˆ   2
            
              
               ˆ          t
                           2


        n2
   When the error is AR(1)
                        2 n  2 /(1     2 r 
   E( ) 
      ˆ     2
                                                        2
                                     n2
   Where
            n 1

            x x       t t 1
   r       t 1
                n
                                the sample correlation coefficient
                x
                t 1
                         2
                         t


   between successivevalues of the X' s.
Methods of Detecting
Autocorrelation
   Graphic Method
   Runs Test
   Durbin-Watson d test
   Breusch-Godfrey test of higher-order
    autocorrelation
Durbin-Watson d Test

           n

           
             ˆ    t      t 1 
                          ˆ     2


     d   t 2
                  n

                 
                  ˆ
                 t 1
                          t
                           2
Durbin-Watson d Test
Assumptions
   Regression model includes an intercept
   The explanatory variables, the X’s are
    nonstochastic.
   The disturbances are generated by a
    AR(1) process.
   Model includes no lagged values of
    dependent variable.
   There are no missing observations.
Durbin-Watson d statistic
       n

       
         ˆ       t    t 1 
                       ˆ       2

                                           2  
                                       ˆ     ˆ
                                             2
                                                    ˆ ˆ  2
                                                        t 1            t 1
 d   t 2
              n
                                           t
                                                    n
                                                                    t


             
              ˆ
             t 1
                          t
                           2
                                             ˆ
                                                   t 1
                                                               t
                                                                2



 Since  t2 and
         ˆ                     
                                ˆ    2
                                    t 1   differ by only one observation, they
 are approximately equal and hence we get
        t t 1 
              ˆ ˆ
 d  2 1 -         
      
            t 
                ˆ 2 


 If we define

 
 ˆ     
        ˆ ˆ  t         t 1

        ˆ          t
                      2


 then d becomes
 d  21 -  
           ˆ
 sin ce  1    1, then 0  d  4
               ˆ
    Distribution Durbin-Watson
    Statistic
                   d


     dL                      dU




0              2                 4
           Decision Regions
           Durbin-Watson d
       H0: no positive                                                                  H0*: no negative
       autocorrelation                                                                  autocorrelaton


    Reject H0               Zone of                                                Zone of       Reject H0*
    Evidence of             indecision                                             indecision    Evidence of
    positive                                                                                     negative
    auto-correlation                          Do not reject H0 or H0* or                         auto-
                                              both.                                              correlation




                                                                                                                   d
0                      dL                dU              2                 4- du             4- dL             4
Durbin-Watson Decision Rules
 Null Hypothesis                 Decision        If

 No positive autocorrelation     Reject          0<d<dL


 No positve autocorrelation      No decision     dL<d<dU


 No negative autocorrelation     Reject          4-dL<d<4


 No negative autocorrelation     No decision     4-dU<d<4-dL


 No autocorrelation, positive or Do not reject   dU<d<4-dU
 negative
Bid-Ask Spread an Example
   The spread between the bid price for US
    currency and the ask price for US currency in
    the Brazilian blackmarket is function of
    opportunity cost of holding currency and the
    risk of holding currency.
   Opportunity cost is interest rate
   risk is the rate of variability in exchange rate
           Example Durbin-Watson
|_Ols spread interest sigma / dw resid=e

DURBIN-WATSON STATISTIC    =     1.51549
DURBIN-WATSON P-VALUE =        0.019933

 R-SQUARE =   0.6124     R-SQUARE ADJUSTED =  0.5983
VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.57694
STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.75956
SUM OF SQUARED ERRORS-SSE=   31.732
MEAN OF DEPENDENT VARIABLE =   3.4959
LOG OF THE LIKELIHOOD FUNCTION = -64.8077

VARIABLE    ESTIMATED    STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
  NAME     COEFFICIENT     ERROR      55 DF   P-VALUE CORR. COEFFICIENT AT MEANS
INTEREST   0.19908       0.4677E-01   4.256     0.000 0.498     0.4563     0.3327
SIGMA      0.39287       0.1021       3.847     0.000 0.460     0.4124     0.2798
CONSTANT    1.3547       0.2507       5.404     0.000 0.589     0.0000     0.3875