Instrumental Variables Regression (2SLS)

2SLS IV regressions by group with weights, clustering, and HDFE

Important

Run gtools, upgrade to update gtools to the latest stable version.

Warning

givregress is in beta and meant for testing; use in production NOT recommended. (To enable beta features, define global GTOOLS_BETA = 1.)

givregress computes fast IV regression coefficients and standard errors by group. Its basic functionality is similar to that of the user-written rangestat (reg) or regressby, except that it computes 2SLS instead of OLS; in addition, givregress allows weights, clustering, and HDFE by group. This program is not intended as a substitute for ivregress, ivreghdfe, or similar commands. Support for some estimation operations are planned; however, givregress does not compute any significance tests and no post-estimation commands are available.

Syntax

givregress depvar (endogenous = instruments) [exogenous] ///
    [if] [in] [weight] [, by() absorb() options]

By default, results are saved into a mata class object named GtoolsIV. Run mata GtoolsIV.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).

Note that extended varlist syntax is not supported. Further, fweights behave differently than other weighting schemes; that is, this assumes that the weight referes to the number of available observations.

Options

Save Results

  • mata(name, [nob nose]) Specify name of output mata object and whether to save b and se

  • gen(...) Specify any of b(varlist), se(varlist), and hdfe(varlist). One per covariate is required (hdfe() also requires one for the dependent variable).

  • prefix(...) Specify any of b(str), se(str), and hdfe(str). A single prefix is allowed.

  • replace Allow replacing existing variables.

Options

  • by(varlist) Group statistics by variable.
  • robust Robust SE.
  • cluster(varlist) One-way or nested cluster SE.
  • absorb(varlist) Multi-way high-dimensional fixed effects.
  • hdfetol(real) Tolerance level for HDFE algoritm (default 1e-8).
  • noconstant Whether to add a constant (cannot be combined with absorb()).

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 read-only 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 or bench(level) prints how long in seconds various parts of the program take to execute. Level 1 is the same as benchmark. 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 does gtools will try to use native commands. The user can specify it throw an error instead by passing oncollision(error).

Results

givregress estimates a linear IV model via 2SLS, optionally weighted, by group, with cluster SE, and/or with multi-way high-dimensional fixed effects. The results are by default saved into a mata object (default GtoolsIV). Run mata GtoolsIV.desc() for details; the following data is stored:

regression info
---------------

    string scalar caller
        model used; should be "givregress"

    real scalar kx
        number of (non-absorbed) 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 GtoolsIV.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

IV is computed using via 2SLS in a two-step process. Let

  • $Y$ be the dependent variable.

  • $X$ be a matrix with endogenous covariates and exogenous covariates.

  • $Z$ be a matrix with exogenous covariates and instruments.

Note that the exogenous covariates that appear in $X$ and $Z$ are the same. Now we project $X$ onto $Z$ to obtain $\widehat{X}$ and then we project $Y$ onto $\widehat{X}$ to obtain the coefficients. In particular $$ \begin{align} \widehat{\Gamma} & = (Z^\prime Z)^{-1} Z^\prime X \\ \widehat{X} & = Z \widehat{\Gamma} \end{align} $$

We then essentially run OLS on $Y$ and $\widehat{X}$: $$ \widehat{\beta} = \left(\widehat{X}^\prime \widehat{X}\right)^{-1} \widehat{X}^\prime Y $$

A column of ones is automatically appended to the exogenous covariates (both in $X$ and $Z$) unless the option noconstant is passed or absorb(varlist) is requested.

Identification, Collinearity, and Inverses

givregress is not here to judge you. Whether your instrument is good, bad, or anything in between does not matter for the algorithm so long as the steps required for 2SLS can be run. As such, the only sanity check is a basic identification criterion: The number of instruments must be weakly greater than the number of endogenous (instrumented) covariates. (This is an important difference from, for example ivregress 2sls, which will also exit with error if the dependent variable is collinear with any other variable.)

Let $C$ be a matrix with endogenous covariates, exogenous covariates, and instruments. $C^\prime C$ is scaled by the inverse of $M = \max_{ij} C^\prime C$ and subsequently decomposed into $L D L^\prime$, with $L$ lower triangular and $D$ diagonal (note $C^\prime C$ is a symmetric positive semi-definite 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 $C$ and $2.22\mathrm{e}{-16}$ is the machine epsilon in 64-bit systems).

We then run 2SLS as described in the previous section using only the variables that are determined to be linearly independent using this method. The identification check is also conducted after the collinearity check. If the model is not identified after collinear variables have been removed then both the coefficient and the standard errors are set to missing (.).

Note that variables columns that appear earlier in the matrix $C$ are favored to be kept in the collienarity check. For example, if an instrument is collinear with an exogenous variable, the exogenous variable will be kept and the instrument will be dropped. As noted, the order in which variables appear in $C$ is: Endogenous, exogenous, instruments. Hence instrumented variables are only dropped if they are collinear with other instrumented variables; instruments, however, are dropped if they are collinear with any other variable.

The coefficients for collinear columns are coded as $0$ and their standard errors are coded as missing (.). Finally, inverses in each step are computed directly, since the collinearity check is conducted at the start and the 2SLS computation is conducted only on linearly independent columns. (Though note the program will still print a warning if the determinant is numerically zero, that is, $< 2.22\mathrm{e}{-16}$.)

Standard Errors

The standard error of the $i$th coefficient is given by $$ SE_i = \sqrt{\frac{n}{n - k} \widehat{V}_{ii}} $$

where $n$ is the number of observations, $k$ is the number of covariates (both endogenous and exogenous), and $\frac{n}{n - k}$ is a small-sample adjustment and $n \widehat{V}$ is a consistent estimator of the asymptotic variance of $\widehat{\beta}$. (Note that givregress always computes this small-sample adjustment; Stata's ivregress 2sls, for example, only does so with the option small.) The standard error of collinear columns is coded as missing (.).

By default, homoskedasticity-consistent standard errors are computed: $$ \begin{align} \widehat{V} & = (\widehat{X}^\prime \widehat{X})^{-1} \widehat{\sigma} \\ \widehat{\sigma} & = \widehat{\varepsilon}^\prime \widehat{\varepsilon} / n \end{align} $$

where $$ \widehat{\varepsilon} = Y - X \widehat{\beta} $$

is the error of the 2SLS fit (note that $X$ is used here instead of $\widehat{X}$). If robust is passed then White heteroskedascitity-consistent standard errors are computed instead: $$ \begin{align} \widehat{\Sigma} & = \text{diag}\{\widehat{\varepsilon}_1^2, \ldots, \widehat{\varepsilon}_n^2\} \\ \widehat{V} & = (\widehat{X}^\prime \widehat{X})^{-1} \widehat{X}^\prime \widehat{\Sigma} \widehat{X} (\widehat{X}^\prime \widehat{X})^{-1} \end{align} $$

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 $$ \begin{align} \widehat{V} & = (\widehat{X}^\prime \widehat{X})^{-1} \left( \sum_{j = 1}^J \widehat{u}_j \widehat{u}_j^\prime \right) (\widehat{X}^\prime \widehat{X})^{-1} \\ \widehat{u}_j & = \widehat{X}_j^\prime \widehat{\varepsilon}_j \end{align} $$

with $\widehat{X}_j^\prime$ the matrix of projected 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. $$ \begin{align} \widehat{X} & = Z (Z^\prime W Z)^{-1} Z^\prime W X \\ \widehat{\beta} & = \left(\widehat{X}^\prime W \widehat{X}\right)^{-1} \widehat{X}^\prime W Y \end{align} $$

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 small-sample adjustment for the standard errors, and $$ \begin{align} W & = \text{diag}\{w_1, \ldots, w_n\} \\ \widehat{V} & = (\widehat{X}^\prime W \widehat{X})^{-1} \widehat{X}^\prime W \widehat{\Sigma} \widehat{X} (\widehat{X}^\prime W \widehat{X})^{-1} \end{align} $$

is used for robust standard errors. In contrast, for other weights (aweights being the default), $n$ is used to compute the small-sample adjustment, and $n \widehat{V}$ estimates the asymptotic variance of the WLS estimator. That is, $$ \begin{align} \widehat{V} & = (\widehat{X}^\prime W \widehat{X})^{-1} \widehat{X}^\prime W \widehat{\Sigma} W \widehat{X} (\widehat{X}^\prime W \widehat{X})^{-1} \end{align} $$

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 small-sample adjustment is computed.

Finally, with weights and HDFE, the iterative de-meaning (see below) uses the weighted mean.

HDFE

Multi-way high-dimensional 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 well-known that with one fixed effect $\widehat{\beta}$ can be estimated via the within transformation (i.e. de-meaning the dependent variable and each covariate by the levels of the fixed effect; this can also be motivated via the Frisch-Waugh-Lovell theorem). That is, with one fixed effect we have the following algorithm:

  1. Compute $\overline{Y}$ and $\overline{C}$, the mean of $Y$ and $C$ by the levels of the fixed effect (recall $C$ is a matrix with all the variables: endogenous, exogenous, and instruments).

  2. Replace $Y$ and $C$ with $Y - \overline{Y}$ and $C - \overline{C}$, respectively.

  3. Compute 2SLS normally with $Y$ and $C$ de-meaned, making sure to include the number of fixed effects in the small-sample adjustment of the standard errors.

With multiple fixed effects, the same can be achieved by continuously de-meaning by the levels of each of the fixed effects. Following Correia (2017, p. 12), we have instead:

  1. 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 $m = 1$.

  2. Compute $\overline{Y}$ and $\overline{C}$ with the mean of $Y$ and $C$ by the levels of $\alpha_m$ (again, $C$ is a matrix with all the variables: endogenous, exogenous, and instruments).

  3. Replace $Y$ and $C$ with $Y - \overline{Y}$ and $C - \overline{C}$, respectively.

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

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

  6. Compute 2SLS normally with the iteratively de-meaned $Y$ and $X$, making sure to include the number of fixed effects across all fixed effect variables in the small-sample 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 $C$ 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 $C$ with $Q_m Y$ and $Q_m C$, respectively.)

Correia (2017) actually proposes several ways of accelerating the above algorithm; we have yet to explore any of his proposed modifications (see Correia's own reghdfe and ivreghdfe packages for an implementation of the methods discussed in his paper).

Finally, we note that in step 5 we detect "convergence" as the maximum element-wise absolute difference between $Y, C$ and $Q_m Y, Q_m C$, respectively (i.e. the $l_{\infty}$ norm). This is a tighter tolerance criterion than the one in Correia (2017, p. 12), which uses the $l_2$ norm, but by default we also use a tolerance of $1\mathrm{e}{-8}$. The trade-off is precision vs speed. The tolerance criterion is hard-coded 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 cross-platform 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. 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 multi-way HDFE. (This is specially important for small-sample 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 multi-way clustering.

  • Faster HDFE algorithm. At the moment the method of alternating projections (MAP) is used, which has very poor worst-case performance. While givregress is fast in our benchmarks, it does not have any safeguards against potential corner cases. (See Correia (2017) 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)

givregress price (mpg = gear_ratio) weight turn
givregress price (mpg = gear_ratio) _mpg, cluster(headroom)
mata GtoolsIV.print()

givregress price (mpg weight = gear_ratio turn displacement) _h*, absorb(rep78 headroom)
mata GtoolsIV.print()

givregress price (mpg = gear_ratio) weight [fw = rep78], absorb(headroom)
mata GtoolsIV.print()

givregress price (mpg = gear_ratio turn displacement) weight [aw = rep78], by(foreign)
mata GtoolsIV.print()

givregress price (mpg = gear_ratio turn) weight, by(foreign) mata(coefsOnly, nose) prefix(b(_b_) se(_se_))
givregress price (mpg weight = gear_ratio turn), mata(seOnly, nob) prefix(hdfe(_hdfe_))
givregress price (mpg weight = gear_ratio turn) displacement, mata(nothing, nob nose)

mata coefsOnly.print()
mata seOnly.print()
mata nothing.print()
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 on 1
givregress y (x1 x2 = x3 x4), absorb(g1 g2 g3) mata(greg)
timer off 1
mata greg.b', greg.se'
timer on 2
ivreghdfe y (x1 x2 = x3 x4), absorb(g1 g2 g3)
timer off 2

timer on 3
givregress y (x1 x2 = x3 x4), absorb(g1 g2 g3) cluster(g4) mata(greg)
timer off 3
mata greg.b', greg.se'
timer on 4
ivreghdfe y (x1 x2 = x3 x4), absorb(g1 g2 g3) cluster(g4)
timer off 4

timer list

   1:      2.44 /        1 =       2.4430
   2:     18.39 /        1 =      18.3870
   3:      2.44 /        1 =       2.4370
   4:     25.51 /        1 =      25.5070

References

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

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