How to get the coefficient and t-statistic of the first-order autoregressive coefficient produced by the Cochrane Orcutt procedure in Stata?

[Mathias Nielsen]

How to get the coefficient and t-statistic of the first-order autoregressive coefficient produced by the Cochrane Orcutt procedure in Stata?

Is rho equal to the first-order autoregressive coefficient produced by the Cochrane Orcutt procedure in Stata? If so, how do we get the t-statistics of that coefficient?
See our Stata output and an example of a similar study below? (Please note: The coefficient (ψ) is an estimate of the first-order autoregressive coefficient produced by the Cochrane Orcutt procedure.

[Nick Cox]

Cross-posted at http://stackoverflow.com/questions/3...cutt-statistic

Please note our cross-posting policy http://www.statalist.org/forums/help#crossposting which is that you tell us about it.

[Mathias Nielsen]

Sorry. I am new at using this forum. But really sorry.

[Emad Shehata]

Return to this post

http://www.stata.com/statalist/archi.../msg00647.html

[Emad Shehata]

clear all
 input t y x1 x2
 1  99.2  96.7   101
 2    99  98.1 100.1
 3   100   100   100
 4 111.6 104.9  90.6
 5 122.2 104.9  86.5
 6 117.6 109.5  89.7
 7 121.1 110.8  90.6
 8   136 112.3  82.8
 9 154.2 109.3  70.1
10 153.6 105.3  65.4
11 158.5 101.7  61.3
12 140.6  95.4  62.5
13 136.2  96.4  63.6
14   168  97.6  52.6
15 154.3 102.4  59.7
16   149 101.6  59.5
17 165.5 103.8  61.3
 end

 tsset t
 prais y x1 x2 , corc
 matrix b=e(b)'
 matrix v=e(V)
 matrix v = diag(vecdiag(v))
 matrix se = vecdiag(cholesky(v))'
 mata: b= st_matrix("b")
 mata: se= st_matrix("se")
 mata: t=  b:/se
 mata: t=st_matrix("t",t)
 matrix stat = b , t
 matlist stat

.  prais y x1 x2 , corc

Iteration 0:  rho = 0.0000
Iteration 1:  rho = -0.1824
Iteration 2:  rho = -0.1603
Iteration 3:  rho = -0.1564
Iteration 4:  rho = -0.1557
Iteration 5:  rho = -0.1556
Iteration 6:  rho = -0.1555
Iteration 7:  rho = -0.1555
Iteration 8:  rho = -0.1555

Cochrane-Orcutt AR(1) regression -- iterated estimates

      Source |       SS       df       MS              Number of obs =      16
-------------+------------------------------           F(  2,    13) =  160.30
       Model |   9404.0186     2   4702.0093           Prob > F      =  0.0000
    Residual |  381.325374    13  29.3327211           R-squared     =  0.9610
-------------+------------------------------           Adj R-squared =  0.9550
       Total |  9785.34397    15  652.356265           Root MSE      =   5.416

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
          x1 |   1.168129   .2487314     4.70   0.000     .6307779    1.705481
          x2 |  -1.413293   .0790578   -17.88   0.000    -1.584087   -1.242499
       _cons |   121.6113   24.50442     4.96   0.000     68.67271    174.5499
-------------+----------------------------------------------------------------
         rho |  -.1555319
------------------------------------------------------------------------------
Durbin-Watson statistic (original)    2.018549
Durbin-Watson statistic (transformed) 2.027891
.  matrix b=e(b)'
.  matrix v=e(V)
.  matrix v = diag(vecdiag(v))
.  matrix se = vecdiag(cholesky(v))'
.  mata: b= st_matrix("b")
.  mata: se= st_matrix("se")
.  mata: t=  b:/se
.  mata: t=st_matrix("t",t)
.  matrix stat = b , t
.  matlist stat

             |        y1         c1
-------------+----------------------
          x1 |  1.168129   4.696349
          x2 | -1.413293  -17.87671
       _cons |  121.6113    4.96283

[Mathias Nielsen]

Thanks for your answer. However, I do not really see, how to get the t-statistics on the first order autoregressive coefficient (rho?) from that output?

[Emad Shehata]

Dear Mathias

However, I do not really see, how to get the t-statistics on the first order autoregressive coefficient (rho?) from that output?

Now I think you dont know the difference between Iterative Cochrane-Orcutt transformation and MLE of Cochrane-Orcutt
Rho value in the example above not parameter coefficient but a calculated scalar.

[Emad Shehata]

any how if you are interested in to know the t-test of Rho
follow the next steps:

prais y x1 x2 , corc
predict double e , res
reg  e L.e , noconst

 prais y x1 x2 , corc

Iteration 0:  rho = 0.0000
Iteration 1:  rho = -0.1824
Iteration 2:  rho = -0.1603
Iteration 3:  rho = -0.1564
Iteration 4:  rho = -0.1557
Iteration 5:  rho = -0.1556
Iteration 6:  rho = -0.1555
Iteration 7:  rho = -0.1555
Iteration 8:  rho = -0.1555

Cochrane-Orcutt AR(1) regression -- iterated estimates

      Source |       SS       df       MS              Number of obs =      16
-------------+------------------------------           F(  2,    13) =  160.30
       Model |   9404.0186     2   4702.0093           Prob > F      =  0.0000
    Residual |  381.325374    13  29.3327211           R-squared     =  0.9610
-------------+------------------------------           Adj R-squared =  0.9550
       Total |  9785.34397    15  652.356265           Root MSE      =   5.416

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
          x1 |   1.168129   .2487314     4.70   0.000     .6307779    1.705481
          x2 |  -1.413293   .0790578   -17.88   0.000    -1.584087   -1.242499
       _cons |   121.6113   24.50442     4.96   0.000     68.67271    174.5499
-------------+----------------------------------------------------------------
         rho |  -.1555319
------------------------------------------------------------------------------
Durbin-Watson statistic (original)    2.018549
Durbin-Watson statistic (transformed) 2.027891

. predict double e , res

. reg  e L.e , noconst

      Source |       SS       df       MS              Number of obs =      16
-------------+------------------------------           F(  1,    15) =    0.34
       Model |  8.66958701     1  8.66958701           Prob > F      =  0.5679
    Residual |  381.325374    15  25.4216916           R-squared     =  0.0222
-------------+------------------------------           Adj R-squared = -0.0430
       Total |  389.994961    16  24.3746851           Root MSE      =   5.042

------------------------------------------------------------------------------
           e |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           e |
         L1. |  -.1555317   .2663312    -0.58   0.568    -.7232032    .4121397
------------------------------------------------------------------------------

.

[Mathias Nielsen]

You are right, I am new at doing this - but I think the proposed method is exactly, what I needed. THANKS for helping!

[Emad Shehata]

You can use also NLS estimationas follows:

clear all
 input t y x1 x2
 1  99.2  96.7   101
 2    99  98.1 100.1
 3   100   100   100
 4 111.6 104.9  90.6
 5 122.2 104.9  86.5
 6 117.6 109.5  89.7
 7 121.1 110.8  90.6
 8   136 112.3  82.8
 9 154.2 109.3  70.1
10 153.6 105.3  65.4
11 158.5 101.7  61.3
12 140.6  95.4  62.5
13 136.2  96.4  63.6
14   168  97.6  52.6
15 154.3 102.4  59.7
16   149 101.6  59.5
17 165.5 103.8  61.3
 end

 tsset t
 gen y1=L.y
 gen x11=L.x1
 gen x21=L.x2
 prais y x1 x2 , corc
 nl (y=({B1}*x1+{B2}*x2+{B0}-{R}*({B1}*x11+{B2}*x21+{B0}))+{R}*y1) in 2/17

.  tsset t
.  gen y1=L.y
(1 missing value generated)

.  gen x11=L.x1
(1 missing value generated)

.  gen x21=L.x2
(1 missing value generated)

.  prais y x1 x2 , corc

Iteration 0:  rho = 0.0000
Iteration 1:  rho = -0.1824
Iteration 2:  rho = -0.1603
Iteration 3:  rho = -0.1564
Iteration 4:  rho = -0.1557
Iteration 5:  rho = -0.1556
Iteration 6:  rho = -0.1555
Iteration 7:  rho = -0.1555
Iteration 8:  rho = -0.1555

Cochrane-Orcutt AR(1) regression -- iterated estimates

      Source |       SS       df       MS              Number of obs =      16
-------------+------------------------------           F(  2,    13) =  160.30
       Model |   9404.0186     2   4702.0093           Prob > F      =  0.0000
    Residual |  381.325374    13  29.3327211           R-squared     =  0.9610
-------------+------------------------------           Adj R-squared =  0.9550
       Total |  9785.34397    15  652.356265           Root MSE      =   5.416

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
          x1 |   1.168129   .2487314     4.70   0.000     .6307779    1.705481
          x2 |  -1.413293   .0790578   -17.88   0.000    -1.584087   -1.242499
       _cons |   121.6113   24.50442     4.96   0.000     68.67271    174.5499
-------------+----------------------------------------------------------------
         rho |  -.1555319
------------------------------------------------------------------------------
Durbin-Watson statistic (original)    2.018549
Durbin-Watson statistic (transformed) 2.027891

.  nl (y=({B1}*x1+{B2}*x2+{B0}-{R}*({B1}*x11+{B2}*x21+{B0}))+{R}*y1) in 2/17
(obs = 16)

Iteration 0:  residual SS =  9926.351
Iteration 1:  residual SS =  424.7784
Iteration 2:  residual SS =  381.5585
Iteration 3:  residual SS =  381.3257
Iteration 4:  residual SS =  381.3254
Iteration 5:  residual SS =  381.3254
Iteration 6:  residual SS =  381.3254
Iteration 7:  residual SS =  381.3254

      Source |       SS       df       MS
-------------+------------------------------         Number of obs =        16
       Model |  7188.51265     3  2396.17088         R-squared     =    0.9496
    Residual |  381.325374    12  31.7771145         Adj R-squared =    0.9370
-------------+------------------------------         Root MSE      =  5.637119
       Total |  7569.83803    15  504.655869         Res. dev.     =  96.14306

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         /B1 |   1.168129   .2705726     4.32   0.001     .5786022    1.757657
         /B2 |  -1.413293   .0881198   -16.04   0.000    -1.605289   -1.221296
         /B0 |   121.6113   26.14065     4.65   0.001     64.65571    178.5669
          /R |  -.1555315   .3240643    -0.48   0.640     -.861607    .5505439
------------------------------------------------------------------------------
  Parameter B0 taken as constant term in model & ANOVA table

[Eric de Souza]

Re your reply at #9: this is not what you wanted because it does take into account that the residuals are estimated and not observed.
The NLS method in #10 is preferable because it jointly estimates the beta coefficients and rho.

[Emad Shehata]

Dear Eric
You are right

Either NLS or MLE methods are preferable for joint estimation of beta coefficients and rho.
In addition MLE takes into account estimation of Sigma

[Gosia Aslan]

What is the difference between Prais-Winsten and Cochrane-Orcutt?

[Emad Shehata]

The only difference between Prais-Winsten and Cochrane-Orcutt according to autoregressive least squares method is the last drops the first observation and the estimation starts from the second observation.

[Gosia Aslan]

Thank you dear Emad Shehata

Either NLS or MLE methods are preferable for joint estimation of beta coefficients and rho.
In addition MLE takes into account estimation of Sigma

where I can find MLE method in Stata for Cochrane-Orcutt ?
thanks