Linear Regression (OLS)
OLS linear regressions by group with weights, clustering, and HDFE
Important
Run gtools, upgrade
to update gtools
to the latest stable version.
Warning
gregress
is in beta and meant for testing; use in production NOT recommended. (To enable beta features, define global GTOOLS_BETA = 1
.)
gregress
computes fast OLS regression coefficients and standard errors
by group. Its basic functionality is similar to that of the userwritten
rangestat (reg)
or regressby
; in addition, gregress
allows weights,
clustering, and HDFE by group.
This program is not intended as a substitute for regress
,
reghdfe
, or similar commands. Support for some estimation operations
are planned; however, gregress
does not compute any significance tests
and no postestimation commands are available. For nongrouped OLS, in
fact, Stata's regress
is faster (unless clustering). For nongrouped
OLS with HDFE, ftools
' reghdfe
is more stable and offers more
features.
Syntax
gregress depvar indepvars [if] [in] [weight] [, ///
by() absorb() options]
By default, results are saved into a mata class object named
GtoolsRegress
. Run mata GtoolsRegress.desc()
for details; the name
and contents can be modified via mata()
. The results can also be
saved into variables via gen()
or prefix()
(either can be combined
with mata()
, but not each other).
Extended varlist syntax is not supported. Further, fweights
behave differently than other weighting schemes; specifically,
this assumes that the weight refers to the number of available
observations. Other weights run WLS; default weights are aweights
.
Options
Save Results

mata(name, [nob nose])
Specify name of output mata object and whether to saveb
andse

gen(...)
Specify any ofb(varlist)
,se(varlist)
, andhdfe(varlist)
. One per covariate is required (hdfe()
also requires one for the dependent variable). 
prefix(...)
Specify any ofb(str)
,se(str)
, andhdfe(str)
. A single prefix is allowed. 
replace
Allow replacing existing variables.
Options
by(varlist)
Group statistics by variable.robust
Robust SE.cluster(varlist)
Oneway or nested cluster SE.absorb(varlist)
Multiway highdimensional fixed effects.hdfetol(real)
Tolerance level for HDFE algoritm (default 1e8).algorithm(str)
Algorithm used to absorb HDFE: CG (conjugate gradient; default) MAP (alternating projections), SQUAREM (squared extrapolation), IT (Irons and Tuck).maxiter(int)
Maximum number of algorithm iterations (default 100,000). Pass.
for unlimited iterations.tolerance(real)
Convergence tolerance (default 1e8). Note the convergence criterion isX(k + 1)  X(k) < tol
for thek
th iteration, with
the sup norm (i.e. largest element). This is a tighter criteria than the squared norm and setting the tolerance too low might negatively impact performance or with some algorithms run into numerical precision problems.traceiter
Trace algorithm iterations.standardize
Standardize variables before algorithm (may be faster but is slighty less precise).noconstant
Whether to add a constant (cannot be combined withabsorb()
).
Gtools options
(Note: These are common to every gtools command.)

compress
Try to compress strL to str#. The Stata Plugin Interface has only limited support for strL variables. In Stata 13 and earlier (version 2.0) there is no support, and in Stata 14 and later (version 3.0) there is readonly support. The user can try to compress strL variables using this option. 
forcestrl
Skip binary variable check and force gtools to read strL variables (14 and above only). Gtools gives incorrect results when there is binary data in strL variables. This option was included because on some windows systems Stata detects binary data even when there is none. Only use this option if you are sure you do not have binary data in your strL variables. 
verbose
prints some useful debugging info to the console. 
benchmark
orbench(level)
prints how long in seconds various parts of the program take to execute. Level 1 is the same asbenchmark
. Levels 2 and 3 additionally prints benchmarks for internal plugin steps. 
hashmethod(str)
Hash method to use.default
automagically chooses the algorithm.biject
tries to biject the inputs into the natural numbers.spooky
hashes the data and then uses the hash. 
oncollision(str)
How to handle collisions. A collision should never happen but just in case it doesgtools
will try to use native commands. The user can specify it throw an error instead by passingoncollision(error)
.
Results
gregress
estimates a linear regression model via OLS, optionally
weighted, by group, with cluster SE, and/or with multiway
highdimensional fixed effects. The results are by default saved into a
mata object (default GtoolsRegress
). Run mata GtoolsRegress.desc()
for details; the following data is stored:
regression info  string scalar caller model used; should be "gregress" real scalar kx number of (nonabsorbed) covariates real scalar cons whether a constant was added automagically real scalar saveb whether b was stored real matrix b J by kx matrix with regression coefficients real scalar savese whether se was stored real matrix se J by kx matrix with corresponding standard errors string scalar setype type of SE computed (homoskedastic, robust, or cluster) real scalar absorb whether any FE were absorbed string colvector absorbvars variables absorbed as fixed effects string colvector clustervars cluster variables real scalar by whether there were any grouping variables string rowvector byvars grouping variable names real scalar J number of levels defined by grouping variables class GtoolsByLevels ByLevels grouping variable levels; see GtoolsRegress.ByLevels.desc() for details variable levels (empty if without by())  real scalar ByLevels.anyvars 1: any by variables; 0: no by variables real scalar ByLevels.anychar 1: any string by variables; 0: all numeric by variables string rowvector ByLevels.byvars by variable names real scalar ByLevels.kby number of by variables real scalar ByLevels.rowbytes number of bytes in one row of the internal by variable matrix real scalar ByLevels.J number of levels real matrix ByLevels.numx numeric by variables string matrix ByLevels.charx string by variables real scalar ByLevels.knum number of numeric by variables real scalar ByLevels.kchar number of string by variables real rowvector ByLevels.lens > 0: length of string by variables; <= 0: internal code for numeric variables real rowvector ByLevels.map map from index to numx and charx
Methods and Formulas
OLS is computed using the standard formla $$ \widehat{\beta} = (X^\prime X)^{1} X^\prime Y $$
where $Y$ is the dependent variable and $X$ is a matrix with $n$
rows, one for each set of observations, and $k$ columns, one for each
covariate. A column of ones is automatically appended to $X$ unless the
option noconstant
is passed or absorb(varlist)
is requested.
Collinearity and Inverse
$X^\prime X$ is scaled by the inverse of $M = \max_{ij} X^\prime X$ and subsequently decomposed into $L D L^\prime$, with $L$ lower triangular and $D$ diagonal (note $X^\prime X$ is a symmetric positive semidefinite matrix). If $D_{ii}$ is numerically zero then the $i$th column is flagged as collinear and subsequently excluded from all computations (specifically if $D_{ii} < k \cdot 2.22\mathrm{e}{16}$, where $k$ is the number of columns in $X$ and $2.22\mathrm{e}{16}$ is the machine epsilon in 64bit systems).
The inverse is then computed as $(L^{1})^\prime D^{1} L^{1} M^{1}$,
excluding the columns flagged as collinear. If the determinant of
$X^\prime X$ is numerically zero ($< 2.22\mathrm{e}{16}$) despite
excluding collinear columns, a singularity warning is printed.
The coefficients for collinear columns are coded as $0$ and their
standard errors are coded as missing (.
).
Standard Errors
The standard error of the $i$th coefficient is given by $$ SE_i = \sqrt{\frac{n}{n  k} \widehat{V}_{ii}} $$
where $\frac{n}{n  k}$ is a smallsample adjustment and $n \widehat{V}$
is a consistent estimator of the asymptotic variance of $\widehat{\beta}$.
The standard error of collinear columns is coded as missing (.
).
By default, homoskedasticityconsistent standard errors are computed: $$ \begin{align} \widehat{V} & = (X^\prime X)^{1} \widehat{\sigma} \\ \widehat{\sigma} & = \widehat{\varepsilon}^\prime \widehat{\varepsilon} / n \end{align} $$
where $$ \widehat{\varepsilon} = Y  X \widehat{\beta} $$
is the error of the OLS fit. If robust
is passed then White
heteroskedascitityconsistent standard errors are computed instead:
$$
\begin{align}
\widehat{\Sigma} & = \text{diag}\{\widehat{\varepsilon}_1^2, \ldots, \widehat{\varepsilon}_n^2\} \\
\widehat{V} & = (X^\prime X)^{1} X^\prime \widehat{\Sigma} X (X^\prime X)^{1}
\end{align}
$$
Clustering
If cluster(varlist)
is passed then nested cluster standard errors are
computed (i.e. the rows of varlist
define the groups). Let $j$ denote
the $j$th group defined by varlist
and $J$ the number of groups. Then
$$
\begin{align}
\widehat{V} & =
(X^\prime X)^{1}
\left(
\sum_{j = 1}^J \widehat{u}_j \widehat{u}_j^\prime
\right)
(X^\prime X)^{1}
\\
\widehat{u}_j & = X_j^\prime \widehat{\varepsilon}_j
\end{align}
$$
with $X_j^\prime$ the matrix of covariates with observations from the $j$th group and $\widehat{\varepsilon}_j$ the vector with errors from the $j$th group. (Note another way to write the sum in $\widehat{V}$ is as $U^\prime U$, with $U^\prime = [u_1 ~~ \cdots ~~ u_J]$.) Finally, the standard error is given by
$$ SE_i = \sqrt{\frac{n  1}{n  k} \frac{J}{J  1} \widehat{V}_{ii}} $$
Weights
Let $w$ denote the weighting variable and $w_i$ the weight assigned to the $i$th observation. The weighted OLS estimator is $$ \widehat{\beta} = (X^\prime W X)^{1} X^\prime W Y $$
fweights
runs the regression as if there had been $w_i$ copies of the
$i$th observation. As such, $n_w = \sum_{i = 1}^n w_i$ is used instead
of $n$ to compute the smallsample adjustment, for the standard errors,
and
$$
\begin{align}
W & = \text{diag}\{w_1, \ldots, w_n\} \\
\widehat{V} & =
(X^\prime W X)^{1}
X^\prime W \widehat{\Sigma} X
(X^\prime W X)^{1}
\end{align}
$$
is used for robust standard errors. In contrast, for other weights
(aweights
being the default), $n$ is used to compute the smallsample
adjustment, and $n \widehat{V}$ estimates the asymptotic variance of the
WLS estimator. That is,
$$
\begin{align}
\widehat{V} & =
(X^\prime W X)^{1}
X^\prime W \widehat{\Sigma} W X
(X^\prime W X)^{1}
\end{align}
$$
With clustering, these two methods of computing $\widehat{V}$ will
actually coincide, and the only difference between fweights
and other
weights will be the way the smallsample adjustment is computed.
Finally, with weights and HDFE, the iterative demeaning (see below) uses the weighted mean.
HDFE
Multiway highdimensional fixed effects can be added to any regression
via absorb(varlist)
. That is, coefficients are computed as if the
levels of each variable in varlist
had been added to the regression
as fixed effects. It is wellknown that with one fixed effect
$\widehat{\beta}$ can be estimated via the within transformation (i.e.
demeaning the dependent variable and each covariate by the levels of
the fixed effect; this can also be motivated via the FrischWaughLovell
theorem). That is, with one fixed effect we have the following algorithm:

Compute $\overline{Y}$ and $\overline{X}$, the mean of $Y$ and $X$ by the levels of the fixed effect.

Replace $Y$ and $X$ with $Y  \overline{Y}$ and $X  \overline{X}$, respectively.

Compute OLS normally with $Y$ and $X$ demeaned, making sure to include the number of fixed effects in the smallsample adjustment of the standard errors.
With multiple fixed effects, the same can be achieved by continuously demeaning by the levels of each of the fixed effects. Following Correia (2017a, p. 12), we have instead:

Let $\alpha_m$ denote the $m$th fixed effect, $M$ the number of fixed effects (i.e. the number of variables to include as fixed effects), and $m = 1$.

Compute $\overline{Y}$ and $\overline{X}$ with the mean of $Y$ and $X$ by the levels of $\alpha_m$.

Replace $Y$ and $X$ with $Y  \overline{Y}$ and $X  \overline{X}$, respectively.

Repeat steps 2 and 3 for $m = 1$ through $M$.

Repeat steps 1 through 4 until convergence, that is, until neither $Y$ nor $X$ change across iterations.

Compute OLS normally with the iteratively demeaned $Y$ and $X$, making sure to include the number of fixed effects across all fixed effect variables in the smallsample adjustment of the standard errors.
This is known as the Method of Alternating Projections (MAP). Let $A_m$ be a matrix with dummy variables corresponding to each of the levels of $\alpha_m$, the $m$th fixed effect. MAP is so named because at each step, $Y$ and $X$ are projected into the null space of $A_m$ for $m = 1$ through $M$. (In particular, with $Q_m = I  A_m (A_m^\prime A_m)^{1} A_m^\prime$ the orthogonal projection matrix, steps 2 and 3 replace $Y$ and $X$ with $Q_m Y$ and $Q_m X$, respectively.)
Correia (2017a) actually
proposes several ways of accelerating the above algorithm; we have
yet to explore any of his proposed modifications (see Correia's own
reghdfe
package for an implementation of the methods discussed in his
paper).
Finally, we note that in step 5 we detect "convergence" as the
maximum elementwise absolute difference between $Y, X$ and $Q_m
Y, Q_m X$, respectively (i.e. the $l_{\infty}$ norm). This is
a tighter tolerance criterion than the one in
Correia (2017a, p. 12), which uses the $l_2$
norm, but by default we also use a tolerance of $1\mathrm{e}{8}$. The
tradeoff is precision vs speed. The tolerance criterion is hardcoded
but the level can be modified via hdfetol()
. A smaller tolerance will
converge faster but the point estimates will be less precise (and the
collinearity detection algorithm will be more susceptible to failure).
Technical Notes
Ideally I would have been keen to use a standard linear algebra library available for C. However, I was unable to find one that I could include as part of the plugin without running into crossplatform compatibility or installation issues (specifically I was unable to compile them on Windows or OSX; I do not have access to physical hardware running either OS, so adding external libraries is challenging). Hence I had to code all the linear algebra commands that I wished to use.
As far as I can tell, this is only noticeable when it comes to matrix
multiplication. I use a naive algorithm
with no optimizations. This is the main bottleneck in regression models
with multiple covariates (and the main reason regress
is faster without
groups or clustering). Suggestions on how to improve this algorithm are welcome.
Missing Features
This software will remain in beta at least until the following are added:

Option to iteratively remove singleton groups with HDFE (see Correia (2015) for notes on this issue)

Automatically detect and remove collinear groups with multiway HDFE. (This is specially important for smallsample standard error adjustment.)
In addition, some important features are missing:

Option to estimate the fixed effects (i.e. the coefficients of each HDFE group) included in the regression.

Option to estimate standard errors under multiway clustering.

Faster HDFE algorithm. At the moment the method of alternating projections (MAP) is used, which has very poor worstcase performance. While
gregress
is fast in our benchmarks, it does not have any safeguards against potential corner cases. (See Correia (2017a) for notes on this issue.) 
Support for Stata's extended
varlist
syntax.
Examples
Note gregress
is in beta. To enable enable beta features, define global GTOOLS_BETA = 1
.
You can download the raw code for the examples below here
Showcase
sysuse auto, clear gen _mpg = mpg qui tab headroom, gen(_h) greg price mpg greg price mpg, by(foreign) robust greg price mpg _h* [fw = rep78] mata GtoolsRegress.print() greg price mpg _h* [fw = rep78], absorb(headroom) mata GtoolsRegress.print() greg price mpg _mpg, cluster(headroom) greg price mpg _mpg [aw = rep78], by(foreign) absorb(rep78 headroom) cluster(headroom) mata GtoolsRegress.print() greg price mpg, mata(coefsOnly, nose) greg price mpg, mata(seOnly, nob) greg price mpg, mata(nothing, nob nose) mata coefsOnly.print() mata seOnly.print() mata nothing.print() greg price mpg, prefix(b(_b_)) replace greg price mpg, prefix(se(_se_)) replace greg price mpg _mpg, absorb(rep78 headroom) prefix(b(_b_) se(_se_) hdfe(_hdfe_)) replace drop _* greg price mpg, gen(b(_b_mpg _b_cons)) greg price mpg, gen(se(_se_mpg _se_cons)) greg price mpg, absorb(rep78 headroom) gen(hdfe(_hdfe_price _hdfe_mpg))
Basic Benchmark
clear local N 1000000 local G 10000 set obs `N' gen g1 = int(runiform() * `G') gen g2 = int(runiform() * `G') gen g3 = int(runiform() * `G') gen g4 = int(runiform() * `G') gen x3 = runiform() gen x4 = runiform() gen x1 = x3 + runiform() gen x2 = x4 + runiform() gen y = 0.25 * x1  0.75 * x2 + g1 + g2 + g3 + g4 + 20 * rnormal() timer clear timer on 1 greg y x1 x2, absorb(g1 g2 g3) mata(greg) timer off 1 mata greg.print() timer on 2 reghdfe y x1 x2, absorb(g1 g2 g3) timer off 2 timer on 3 greg y x1 x2, absorb(g1 g2 g3) cluster(g4) mata(greg) timer off 3 mata greg.print() timer on 4 reghdfe y x1 x2, absorb(g1 g2 g3) vce(cluster g4) timer off 4 timer on 5 greg y x1 x2, by(g4) prefix(b(_b_)) timer off 5 drop _* timer on 6 asreg y x1 x2, by(g4) timer off 6 drop _* timer list 1: 0.64 / 1 = 0.6380 2: 11.77 / 1 = 11.7730 3: 0.91 / 1 = 0.9140 4: 15.74 / 1 = 15.7370 5: 0.46 / 1 = 0.4570 6: 2.09 / 1 = 2.0890
References
The idea for this function is from Correia (2017a). The conjugate gradient algorithm is from HernándezRamos, Escalante, and Raydan (2011) and implemented following Correia (2017b). The SQUAREM algorithm is from Varadhan and Roland (2008) and Varadhan (2016). Irons and Tuck (1969) method implemented following Ramière and Helfer (2015).

Correia, Sergio (2015). "Singletons, ClusterRobust Standard Errors and Fixed Effects: A Bad Mix" Working Paper. Accessed January 16th, 2020. Available at http://scorreia.com/research/singletons.pdf

Correia, Sergio (2017a). "Linear Models with HighDimensional Fixed Effects: An Efficient and Feasible Estimator" Working Paper. Accessed January 16th, 2020. Available at http://scorreia.com/research/hdfe.pdf

Correia Sergio (2017b). "reghdfe: Stata module for linear and instrumentalvariable/GMM regression absorbing multiple levels of fixed effects." Statistical Software Components S457874, Boston College Department of Economics. Accessed March 6th, 2022. Available at https://ideas.repec.org/c/boc/bocode/s457874.html

HernándezRamos, Luis M., René Escalante, and Marcos Raydan. 2011. "Unconstrained Optimization Techniques for the Acceleration of Alternating Projection Methods." Numerical Functional Analysis and Optimization, 32(10): 1041–66.

Varadhan, Ravi and Roland, Christophe. 2008. "Simple and Globally Convergent Methods for Accelerating the Convergence of Any EM Algorithm."" Scandinavian Journal of Statistics, 35(2): 335–353.

Varadhan, Ravi (2016). "SQUAREM: Squared Extrapolation Methods for Accelerating EMLike Monotone Algorithms." R package version 2016.82. https://CRAN.Rproject.org/package=SQUAREM

Bergé, Laurent (2016). "Efficient estimation of maximum likelihood models with multiple fixedeffects: the R package FENmlm." CREA Discussion Paper 201813. https://wwwen.uni.lu/content/download/110162/1299525/file/2018_13.

Irons, B. M., Tuck, R. C. (1969). "A version of the Aitken accelerator for computer iteration." International Journal for Numerical Methods in Engineering 1(3): 275–277.

Ramière, I., Helfer, T. (2015). "Iterative residualbased vector methods to accelerate fixed point iterations." Computers & Mathematics with Applications 70(9): 2210–2226