`I am trying to replicate an example in Principles of Econometrics by
``Hill, Griffiths and Lim on estimating a model with AR(1) on p. 236 using
``non-linear least squares.
`

`There is actually an example of this problem in Using Stata for
``Principles of Econometrics on p. 218. I am able to replicate the example
``on p 218 exactly but these results are not in agreement with those
``presented in the text on p. 236.
`

`The example starts by estimating a simple model using OLS. They it uses
``newey-west standard errors. Here are those results.
`
. newey la lp, lag(3)

`Regression with Newey-West standard errors Number of obs
``= 34
``maximum lag: 3 F( 1, 32)
``= 4.21
`` Prob > F =
``0.0484
`
------------------------------------------------------------------------------
| Newey-West

` la | Coef. Std. Err. t P>|t| [95% Conf.
``Interval]
`-------------+----------------------------------------------------------------

` lp | .7761187 .3782067 2.05 0.048 .0057369
``1.546501
`` _cons | 3.893256 .0624443 62.35 0.000 3.766061
``4.020451
`------------------------------------------------------------------------------

`Next the text claims to use non-linear least squares estimates but the
``results presented in the text cannot be gotten using nlls.
`
Here are the results for non-linear least squares.

`. nl (la = {b1}*(1-{rho}) + {b2}*lp_1+{rho}*la_1-{rho}*{b2}*(lp_1)),
``variables(lp la la_1 lp_1)
`(obs = 33)
Iteration 0: residual SS = 29.15276
Iteration 1: residual SS = 2.979124
Iteration 2: residual SS = 2.979124
Source | SS df MS

`-------------+------------------------------ Number of obs
``= 33
`` Model | .404285358 2 .202142679 R-squared =
``0.1195
`` Residual | 2.97912389 30 .09930413 Adj R-squared =
``0.0608
``-------------+------------------------------ Root MSE =
``.3151256
`` Total | 3.38340925 32 .105731539 Res. dev. =
``14.28896
`
------------------------------------------------------------------------------

` la | Coef. Std. Err. t P>|t| [95% Conf.
``Interval]
`-------------+----------------------------------------------------------------

` /b1 | 4.101714 .1141125 35.94 0.000 3.868666
``4.334763
`` /rho | .3327122 .182082 1.83 0.078 -.0391488
``.7045733
`` /b2 | -.7779345 .614835 -1.27 0.216 -2.033595
``.4777261
`------------------------------------------------------------------------------
Parameter b1 taken as constant term in model & ANOVA table

`Not only do these results not agree with the text, but the sign on b2 is
``the wrong sign and it is not statistically significant. Also the
``estimate of rho does not agree with rho on p. 236.
`

`The closest I can get to the results in the textbook is using
``Cochrane-Orcutt minimizing the SSE.
`
Cochrane-Orcutt AR(1) regression -- SSE search estimates

` Source | SS df MS Number of obs
``= 33
``-------------+------------------------------ F( 1, 31) =
``12.33
`` Model | .971527978 1 .971527978 Prob > F =
``0.0014
`` Residual | 2.4435749 31 .078824997 R-squared =
``0.2845
``-------------+------------------------------ Adj R-squared =
``0.2614
`` Total | 3.41510288 32 .106721965 Root MSE =
``.28076
`
------------------------------------------------------------------------------

` la | Coef. Std. Err. t P>|t| [95% Conf.
``Interval]
`-------------+----------------------------------------------------------------

` lp | .8883711 .2530456 3.51 0.001 .3722812
``1.404461
`` _cons | 3.898771 .0906242 43.02 0.000 3.713942
``4.0836
`-------------+----------------------------------------------------------------
rho | .4221394
------------------------------------------------------------------------------
Durbin-Watson statistic (original) 1.168987
Durbin-Watson statistic (transformed) 1.820559

`The textbook actually says that non-linear least squares are equivalent
``to Cochrane-Orcutt so this result is not particularly surprising and it
``almost matches the example on p. 236.
`

`So the question is why are the non-linear least squares results so
``different?
`

`The text also suggests that you can run the non-linear least squares
``model using OLS and then test the restriction. When one does this the
``magnitude of the estimate of b2 is the same as nlls but the sign is the
``opposite. In fact, using the nlls model and estimating it with OLS and
``testing the restriction lead to approximately the same estimate as is
``obtained with Newey-West estimates.
`
Rudy
--
Rudy Fichtenbaum
Professor of Economics
Chief Negotiator AAUP-WSU
Wright State University
Dayton, OH 45435-0001
937-775-3085
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