**[TS] prais** -- Prais-Winsten and Cochrane-Orcutt regression

__Syntax__

**prais** *depvar* [*indepvars*] [*if*] [*in*] [**,** *options*]

*options* Description
-------------------------------------------------------------------------
Model
__rho__**type(**__reg__**ress)** base rho on single-lag OLS of residuals; the
default
__rho__**type(freg)** base rho on single-lead OLS of residuals
__rho__**type(**__tsc__**orr)** base rho on autocorrelation of residuals
__rho__**type(dw)** base rho on autocorrelation based on Durbin-Watson
__rho__**type(**__th__**eil)** base rho on adjusted autocorrelation
__rho__**type(**__nag__**ar)** base rho on adjusted Durbin-Watson
**corc** use Cochrane-Orcutt transformation
__sse__**search** search for rho that minimizes SSE
__two__**step** stop after the first iteration
__nocons__**tant** suppress constant term
__h__**ascons** has user-defined constant
__save__**space** conserve memory during estimation

SE/Robust
**vce(***vcetype***)** *vcetype* may be **ols**, __r__**obust**, __cl__**uster** *clustvar*, **hc2**,
or **hc3**

Reporting
__l__**evel(***#***)** set confidence level; default is **level(95)**
**nodw** do not report the Durbin-Watson statistic
*display_options* control columns and column formats, row spacing,
line width, display of omitted variables and base
and empty cells, and factor-variable labeling

Optimization
*optimize_options* control the optimization process; seldom used

__coefl__**egend** display legend instead of statistics
-------------------------------------------------------------------------
You must **tsset** your data before using **prais**; see **[TS] tsset**.
*indepvars* may contain factor variables; see fvvarlist.
*depvar* and *indepvars* may contain time-series operators; see tsvarlist.
**by**, **fp**, **rolling**, and **statsby** are allowed; see prefix.
**coeflegend** does not appear in the dialog box.
See **[TS] prais postestimation** for features available after estimation.

__Menu__

**Statistics > Time series > Prais-Winsten regression**

__Description__

**prais** uses the generalized least-squares method to estimate the
parameters in a linear regression model in which the errors are serially
correlated. Specifically, the errors are assumed to follow a first-order
autoregressive process.

__Options__

+-------+
----+ Model +------------------------------------------------------------

**rhotype(***rhomethod***)** selects a specific computation for the autocorrelation
parameter rho, where *rhomethod* can be

__reg__**ress** rho_reg = B from the residual regression e_t = B *
e_(t-1)
**freg** rho_freg = B from the residual regression e_t = B *
e_(t+1)
__tsc__**orr** rho_tscorr = e'e_(t-1)/e'e, where e is the vector of
residuals
**dw** rho_dw = 1 - dw/2, where dw is the Durbin-Watson d
statistic
__th__**eil** rho_theil = rho_tscorr * (N - k)/N
__nag__**ar** rho_nagar = (rho_dw * N^2 + k^2)/(N^2 - k^2)

The **prais** estimator can use any consistent estimate of rho to
transform the equation, and each of these estimates meets that
requirement. The default is **regress**, which produces the minimum
sum-of-squares solution (**ssesearch** option) for the Cochrane-Orcutt
transformation -- none of these computations will produce the minimum
sum-of-square solution for the full Prais-Winsten transformation.
See Judge et al. (1985) for a discussion of each estimate of rho.

**corc** specifies that the Cochrane-Orcutt transformation be used to
estimate the equation. With this option, the Prais-Winsten
transformation of the first observation is not performed, and the
first observation is dropped when estimating the transformed
equation; see *Methods and formulas* in **[TS] prais**.

**ssesearch** specifies that a search be performed for the value of rho that
minimizes the sum-of-squared errors of the transformed equation
(Cochrane-Orcutt or Prais-Winsten transformation). The search method
is a combination of quadratic and modified bisection searches using
golden sections.

**twostep** specifies that **prais** stop on the first iteration after the
equation is transformed by rho -- the two-step efficient estimator.
Although iterating these estimators to convergence is customary, they
are efficient at each step.

**noconstant**; see **[R] estimation options**.

**hascons** indicates that a user-defined constant, or a set of variables
that in linear combination forms a constant, has been included in the
regression. For some computational concerns, see the discussion in
**[R] regress**.

**savespace** specifies that **prais** attempt to save as much space as possible
by retaining only those variables required for estimation. The
original data are restored after estimation. This option is rarely
used and should be used only if there is insufficient space to fit a
model without the option.

+-----------+
----+ SE/Robust +--------------------------------------------------------

**vce(***vcetype***)** specifies the estimator for the variance-covariance matrix
of the estimator; see **[R] ***vce_options*.

**vce(ols)**, the default, uses the standard variance estimator for
ordinary least-squares regression.

**vce(robust)** specifies to use the Huber/White/sandwich estimator.

**vce(cluster** *clustvar***)** specifies to use the intragroup correlation
estimator.

**vce(hc2)** and **vce(hc3)** specify an alternative bias correction for the
**vce(robust)** variance calculation; for more information, see **[R]**
**regress**. You may specify only one of **vce(hc2)**, **vce(hc3)**, or
**vce(robust)**.

All estimates from **prais** are conditional on the estimated value of
rho. Robust variance estimates here are robust only to
heteroskedasticity and are not generally robust to misspecification
of the functional form or omitted variables. The estimation of the
functional form is intertwined with the estimation of rho, and all
estimates are conditional on rho. Thus estimates cannot be robust to
misspecification of functional form. For these reasons, it is
probably best to interpret **vce(robust)** in the spirit of White's
(1980) original paper on estimation of heteroskedastic-consistent
covariance matrices.

+-----------+
----+ Reporting +--------------------------------------------------------

**level(***#***)**; see **[R] estimation options**.

**nodw** suppresses reporting of the Durbin-Watson statistic.

*display_options*: **noci**, __nopv__**alues**, __noomit__**ted**, **vsquish**, __noempty__**cells**,
__base__**levels**, __allbase__**levels**, __nofvlab__**el**, **fvwrap(***#***)**, **fvwrapon(***style***)**,
**cformat(***%fmt***)**, **pformat(%***fmt***)**, **sformat(%***fmt***)**, and **nolstretch**; see **[R]**
**estimation options**.

+--------------+
----+ Optimization +-----------------------------------------------------

*optimize_options*: __iter__**ate(***#***)**, [__no__]__lo__**g**, __tol__**erance(***#***)**. **iterate()**
specifies the maximum number of iterations. **log**/**nolog** specifies
whether to show the iteration log. **tolerance()** specifies the
tolerance for the coefficient vector; **tolerance(1e-6)** is the default.
These options are seldom used.

The following option is available with **prais** but is not shown in the
dialog box:

**coeflegend**; see **[R] estimation options**.

__Examples__

---------------------------------------------------------------------------
Setup
**. webuse idle**
**. tsset t**

Perform Prais-Winsten AR(1) regression
**. prais usr idle**

Perform Cochrane-Orcutt AR(1) regression
**. prais usr idle, corc**

Same as above, but request robust standard errors
**. prais usr idle, corc vce(robust)**

---------------------------------------------------------------------------
Setup
**. webuse qsales**

Perform Cochrane-Orcutt AR(1) regression and search for rho that
minimizes SSE
**. prais csales isales, corc ssesearch**

Replay result with 99% confidence interval
**. prais, level(99)**
---------------------------------------------------------------------------

__Stored results__

**prais** stores the following in **e()**:

Scalars
**e(N)** number of observations
**e(N_gaps)** number of gaps
**e(mss)** model sum of squares
**e(df_m)** model degrees of freedom
**e(rss)** residual sum of squares
**e(df_r)** residual degrees of freedom
**e(r2)** R-squared
**e(r2_a)** adjusted R-squared
**e(F)** F statistic
**e(rmse)** root mean squared error
**e(ll)** log likelihood
**e(N_clust)** number of clusters
**e(rho)** autocorrelation parameter rho
**e(dw)** Durbin-Watson d statistic for transformed
regression
**e(dw_0)** Durbin-Watson d statistic for untransformed
regression
**e(rank)** rank of **e(V)**
**e(tol)** target tolerance
**e(max_ic)** maximum number of iterations
**e(ic)** number of iterations

Macros
**e(cmd)** **prais**
**e(cmdline)** command as typed
**e(depvar)** name of dependent variable
**e(title)** title in estimation output
**e(clustvar)** name of cluster variable
**e(cons)** **noconstant** or not reported
**e(method)** **twostep**, **iterated**, or **SSE search**
**e(tranmeth)** **corc** or **prais**
**e(rhotype)** method specified in **rhotype()** option
**e(vce)** *vcetype* specified in **vce()**
**e(vcetype)** title used to label Std. Err.
**e(properties)** **b V**
**e(predict)** program used to implement **predict**
**e(marginsok)** predictions allowed by **margins**
**e(asbalanced)** factor variables **fvset** as **asbalanced**
**e(asobserved)** factor variables **fvset** as **asobserved**

Matrices
**e(b)** coefficient vector
**e(V)** variance-covariance matrix of the estimators
**e(V_modelbased)** model-based variance

Functions
**e(sample)** estimation sample

__References__

Judge, G. G., W. E. Griffiths, R. C. Hill, H. Lütkepohl, and T.-C. Lee.
1985. *The Theory and Practice of Econometrics*. 2nd ed. New York:
Wiley.

White, H. 1980. A heteroskedasticity-consistent covariance matrix
estimator and a direct test for heteroskedasticity. *Econometrica* 48:
817-838.