Stacked linear regression analysis to facilitate testing of hypotheses across OLS regressions

Abstract

In empirical work, researchers frequently test hypotheses of parallel form in several regressions, which raises concerns about multiple testing. One way to address the multiple-testing issue is to jointly test the hypotheses (for example, Pei, Pischke, and Schwandt [2019, Journal of Business & Economic Statistics 37: 205–216] and Lee and Lemieux [2010, Journal of Economic Literature 48: 281–355]). While the existing commands suest (Weesie, 1999, Stata Technical Bulletin Reprints 9: 231–248) and mvreg enable Stata users to follow this approach, both are limited in several dimensions. For instance, mvreg assumes homoskedasticity and uncorrelatedness across sampling units, and neither command is designed to be used with panel data. In this article, we introduce the new community-contributed command stackreg, which overcomes the aforementioned limitations and allows for some settings and features that go beyond the capabilities of the existing commands. To achieve this, stackreg runs an ordinary least-squares regression in which the regression equations are stacked as described, for instance, in Wooldridge (2010, Econometric Analysis of Cross Section and Panel Data, p. 166–173, MIT Press) and applies cluster–robust variance–covariance estimation.

1 Introduction

In empirical work, researchers often test hypotheses of parallel form in several regressions. Examples are regression-based balancing tests for multiple independent variables (for example, following Lee and Lemieux [2010] and Pei, Pischke, and Schwandt [2019]) as well as studies that examine the relationships between multiple dependent variables and the same set of independent variables. Numerous researchers, including Lee and Lemieux and Pei, Pischke, and Schwandt, point out the need for joint testing across regression equations in such settings; otherwise, statistical inference may be invalid due to the multiple-comparisons problem. However, independently testing numerous parallel hypotheses without taking multiple-testing issues into account seems to still be common in applied research; see, for instance, the discussions in Anderson (2008) and List, Shaikh, and Xu (2019).

The Stata command mvreg and the more general sureg address such situations by joint estimation; however, they are subject to several limitations. For instance, they are based on a generalized least-squares approach that implies strong assumptions regarding the form of the error-correlation structure, they do not handle panel data in a satisfying way, and they have to rely on the bootstrap to obtain (cluster) robust standard errors. The originally community-contributed command suest (introduced by Weesie [1999]) overcomes some limitations of mvreg. Specifically, suest allows for (cluster) robust inference and, in addition, is highly flexible because it is able to combine different model types—say, ordinary least squares (OLS) and probit—and test across these models. This flexibility, however, comes at the cost of suest not covering all aspects of OLS regressions, the workhorse of much empirical research.

In this article, we introduce stackreg, a community-contributed command for joint testing of hypotheses across OLS regressions. stackreg offers three advantages over suest: first, it implements fixed-effects estimations; second, it allows for multiway clustering building upon the community-contributed command cgmreg (Gelbach and Miller 2009); and third, it enables cross-equation constraints. Furthermore, for convenience, stackreg allows users to specify factor variables as dependent variables.

In essence, stackreg stacks the data used in multiple OLS regressions and runs one regression on these stacked data. Stacking regressions is a conceptually simple way to do joint estimations that is laid out in, for instance, Wooldridge (2010, 166–173) and proposed for regression-based balancing tests for multiple independent variables in Lee and Lemieux (2010) and Pei, Pischke, and Schwandt (2019). However, stacking regressions is computationally more demanding than the residual-based approach used by suest.

Section 2 sketches the underlying econometric idea. Section 3 explains stackreg‘s implementation in Stata and compares it with alternative Stata commands. Section 4 describes the syntax used for stackreg, and section 5 presents two applications of stackreg. Section 6 concludes.

2 Stacked regression analysis

Consider a set of G regressions, where the dependent variables y _{1 i},…, y_Gi are regressed on the same set of independent variables x _i for sampling units i = 1,…, N:

$\begin{array}{c}{y}_{1i}=x_i{\beta }_1+{\epsilon }_{1i}\\ ⋮\\ y_{Gi}=x_i{\beta }_G+{\epsilon }_{Gi}\end{array}$

1

This description accommodates a wide range of applications. Examples include balancing tests of covariates in experiments (with x _i comprising only a treatment indicator) and in regression discontinuity designs ( x _i now including a function of the forcing variable) as well as sets of regressions that examine the relationship between several dependent variables and a fixed set of independent variables. In all such applications, statistical inference about β ₁ ,…, β_G should ideally account for the multitude of statistical tests. The loss of statistical power associated with a Bonferroni correction is, however, problematic, and it is thus not particularly attractive in many applications.

One alternative way to adjust statistical inference is to jointly estimate the regression equations in (1). Stacking the G regressions is a conceptually simple approach to estimate the regressions jointly (described in, for example, Wooldridge [2010, 166–173]). Using this approach, the statistical inference can account for possible crossequation correlations of the errors ε_g without imposing additional structure. Defining y _i = (y _{1 i},…, y_Gi ) ^′ , ε_i = (ε _{1 i},…, ε_Gi ) ^′ , β = (β ₁ ,…, β_G ) ^′ , and

$X_i=\left[\begin{array}{cccc}{x}_i& 0& \dots & 0\\ 0& x_i& & 0\\ ⋮& & \ddots & ⋮\\ 0& 0& \dots & x_i\end{array}\right]$

the stacked regression reads

$y_i=X_i\beta +{\epsilon }_i$

2

The stacked regression (2) mechanically gives regression coefficients that are identical to those from the separate regressions in (1). By clustering standard errors and adjusting the degrees of freedom (explained in more detail in section 3.4), the stacked regression additionally yields identical standard errors for the regression coefficients and still allows for joint testing of hypotheses across equations. The next section describes how stackreg implements this stacking approach and adjusts the degrees of freedom when computing standard errors.

3 Implementation in Stata

The core of the stacked regression procedure is temporarily reshaping the estimation sample from wide format to long format using reshape long or, optionally, sreshape long (Simons 2016). ¹ This transforms the G original outcome variables y _{1 i},…, y_Gi into a single left-hand-side variable. At the same time, a subobservation identifier— denoted by j() in the syntax of reshape—is generated that serves as a key to the original outcome variables. Subsequently, stackreg calls regress to run a regression of the stacked left-hand-side variable on a saturated set of interactions of x _i with the dummy-expanded subobservation identifier. While this yields coefficient estimates that are identical to equation-by-equation estimation, it allows for estimating cross-equation coefficient covariances, which is required for testing hypotheses that involve coefficients from more than one equation. Analogously to an unbalanced panel setting, which is handled by reshape, this procedure accommodates settings in which the pattern of observed outcomes varies across the original sample units. This will be the case if, for instance, the different outcome variables exhibit different patterns of missing values. Even if clustering is not explicitly requested by the user, stackreg routinely applies cluster–robust variance–covariance estimation with clustering at the level of the original sampling units, because each contributes several (sub)observations to the stacked regression. The output from stackreg—displayed and stored in e()—is arranged to mimic the output from mvreg, which allows for inference after stackreg in the same way as after mvreg.

3.1 Panel data and fixed-effects estimation

stackreg is accompanied by the xtstackreg command, which implements fixed-effects panel estimation. ² That is, rather than using the original variables in the stacked regression, the data are first temporarily within-transformed using xtdata. If the estimating samples are heterogeneous across the different outcomes, the within-transformation is applied equation by equation. This guarantees that xtstackreg yields exactly the same coefficient estimates one gets from equationwise applying xtreg, fe or areg, absorb( panelvar ). Fully equivalent to stackreg, fe, the key objective of xtstackreg is, hence, to facilitate postestimation inference that involves coefficients from more than one regression equation. As any Stata xt command, xtstackreg requires the data to be declared as panel data using xtset.

3.2 Higher-level and multiway clustering

Cluster–robust variance–covariance matrix estimation is a key feature of stackreg. The procedure can be adapted to grouped data by considering a level of clustering higher than the original sample unit. Extending this approach to multiway clustering is straightforward. If multiway clustering is requested—via a varlist entered in the cluster() option as an argument—then instead of regress, stackreg calls the community-contributed command cgmreg(Gelbach and Miller 2009), which implements multiway clustering as suggested in Cameron, Gelbach, and Miller (2011).

3.3 Constrained estimation

stackreg allows for imposing linear constraints on the estimated coefficients, including cross-equation constraints. This, in the usual fashion, requires specifying the option constraints(). Constrained estimation can be used, for instance, to specify sets of explanatory variables that vary across the different equations. Technically, constrained estimation is implemented by calling cnsreg instead of regress.

3.4 Degrees-of-freedom adjustment

stackreg is designed to exactly reproduce the robust standard errors one gets from separately regressing y _{1 i},…, y_Gi on x _i . ³ Because stackreg necessarily reproduces the coefficient estimates from separate regressions, the standard errors should also not deviate from what separate regressions yield. While this coincidence of the estimated standard errors is asymptotically guaranteed, it becomes an issue in finite samples, small ones in particular. In other words, the degrees-of-freedom correction regress initially applies when called by stackreg needs to be adjusted. In the most simple case of a cross-sectional estimating sample that is homogeneous across all regression equations, the correction factor that has to be applied to the initially estimated variance–covariance matrix is (N −1)/{N −(1/G)}. This adjustment also applies if panel data are used and one wants to reproduce the standard errors of xtreg, fe robust. More precisely, in this case the factor is (TN − 1)/{TN − (1/G)}, with T denoting the number of panel waves. ⁴ If the standard errors of areg, absorb( panelvar ) robust are to be reproduced, the adjustment factor is {(TN −K)/(TN −K−N +1)}[(TN −1)/{TN −(1/G)}].

Things get more involved if the estimation samples are heterogeneous across the dependent variables ⁵ or if restrictions are imposed on the coefficients; see section 3.3. In these cases, different adjustment factors must be applied to the different equations. For this reason, the initially estimated variance–covariance matrix is not adjusted by a single scalar factor but is adjusted element by element. The element-specific adjustment factors are $\sqrt{c_g\times c_h}$ , with c_g and c_h denoting the equation-specific adjustment factors and g and h indexing the equations 1 ,…, G. This approach to adjusting for degrees of freedom that are heterogeneous across equations parallels what sureg with option dfk does. ⁶

3.5 Comparison to existing Stata commands

stackreg is related to several existing Stata commands that also implement methods for statistical testing in a multiple-equation setting. Among these routines, the presumably most flexible is suest (Weesie 1999; StataCorp 2019a, 2578–2596). stackreg and suest share the idea of using cluster–robust variance–covariance estimation for valid inference in a multiple-equation setting. Unlike stackreg, however, suest is not confined to linear models and allows for testing joint hypotheses that involve various different linear and nonlinear models. Despite its confinement to the linear model, stackreg still accommodates features that go beyond suest. In detail, these are i) fixed-effects estimation (see section 3.1), ii) multiway clustering (see section 3.2), ⁷ iii) exact replication of the standard errors one gets from equation-by-equation estimation (see section 3.4), ⁸ and iv) imposing cross-equation restrictions (see section 3.3). ⁹ Finally, stackreg is more conveniently used, because it requires executing just one command and allows for factor variables in the list of dependent variables.

mvreg is another Stata routine to which stackreg is closely related. ¹⁰ Actually, in terms of the syntax and the output that appears on the screen, stackreg closely follows mvreg. From the perspective of econometric theory, the key difference between mvreg and stackreg is that mvreg implements a feasible generalized least-squares (FGLS) ¹¹ procedure to estimate cross-equation coefficient covariances, while stackreg uses cluster–robust variance–covariance estimation for this purpose. FGLS hinges on correctly specifying the structure of the error variance–covariance matrix, while cluster– robust standard error estimation does not require this. For this reason, the approach stackreg takes is more robust and more flexible than FGLS implemented by mvreg. Because of its greater flexibility—unlike mvreg—stackreg can deal with grouped data (that is, clustered standard error estimation), ¹² panel fixed-effects estimation, estimation samples that are not identical across outcomes, and constraint estimation. ¹³

Finally, combining Stata’s data management tool stack with regress can also be regarded as an alternative to stackreg. Weesie (1999), for instance, proposes implementing the stacked regression approach using stack. While this approach is relatively straightforward in basic applications, it becomes very cumbersome if the estimation procedure involves equation-specific data transformations, for example, the within-transformation to eliminate individual fixed effects. Moreover, using stack and afterwards regress, cluster() does not apply a degrees-of-freedom adjustment that takes into account the stacking of regressions.

4 The stackreg and xtstackreg commands

stackreg requires Stata 12 or higher. If multiway clustering is requested, stackreg requires the community-contributed command cgmreg. ¹⁴ The prefix commands by and svy are not allowed. The prefix commands bootstrap and jackknife are allowed. However, one may question whether bootstrapping stackreg makes much sense, because the prime benefit that the command provides is allowing for conventional—that is, no resampling or simulation based—inference in a multiple-testing framework. All weight types (aweight, fweight, iweight, and pweight) are allowed, with pweights being the default weight type.

4.1 Syntax

stackreg depvars = indepvars [ if ] [ in ] [ weight ] [ , fe noconstant constraints( numlist ) nocommon cluster( clustvarlist ) df(adjust | raw | areg) wald sreshape level( # ) edittozero( # ) omitted emptycells display_options ]

xtstackreg depvars = indepvars [ if ] [ in ] [ weight ] [ , noconstant

constraints( numlist ) nocommon cluster( clustvarlist ) df(adjust | raw | areg) wald sreshape level( # ) edittozero( # ) omitted emptycells display_options ]

The syntax for xtstackreg is exactly the same as the syntax for stackreg, except the fe option, which is automatically specified with xtstackreg. In other words, specifying xtstackreg is fully equivalent to specifying stackreg with option fe. We provide the separate xt command to make more salient that stackreg is designed to take the (possible) panel nature of the data into account.

depvars specifies the list of outcome variables, and indepvars specifies the list of explanatory variables. Factor variables are allowed in both indepvars and depvars; see [U] 1.4.3 Factor variables. Time-series operators such as L. and F. are also allowed.

4.2 Options

fe makes stackreg use within-transformed values of indepvars and depvars rather than their levels when estimating the stacked regression. That is, with the fe option (fixed effects), stackreg eliminates unobserved individual heterogeneity. fe requires that the data are declared as panel data by using xtset. stackreg with option fe is fully equivalent to xtstackreg (with and without option fe, that is, fe has no effect with xtstackreg). We provide the separate xt command to make more salient that stackreg can be used with panel data.

noconstant suppresses the constant terms in the stacked regression. noconstant drops the constant terms from all regression equations because stackreg considers the same set of explanatory variables for all equations.

constraints( numlist ) requests that stackreg apply the linear constraints specified by numlist, which must comply with Stata’s numlist syntax; see [U] 11.1.8 numlist. The specified constraints must be defined in advance by using constraint; see [R] constraint. The syntax for referring to a coefficient when defining constraints is [ depvar ] indepvar. To identify coefficients, both the equation and the explanatory variable are thereby specified. Factor-variables syntax is allowed for specifying constraints, for example, [health]1998.year = 0. Cross-equation constraints can be defined as usual, for example, [health]income = [happiness]income. The option constraints() cannot be combined with multiway clustering. If constraints() is specified and noconstant is not specified, then stackreg estimates an overall constant and drops the equation-specific constant from the final equation.

nocommon makes stackreg select the estimation sample on an equation-by-equation basis. That is, observations for which information on some variables in depvars is missing are used, and the number of observations thus may vary across the different equations. The default (common) is to only consider observations for which information is available for all variables in depvars. Whether or not the estimation sample is heterogeneous across equations is stored in e(common).

cluster( clustvarlist ) specifies how stackreg clusters the standard errors (and covariances) at a higher level than the original unit of observation. By default, an identifier of the original observations serves as clustvar, because stacking the regression makes each original sampling unit contribute several observations to the stacked regression analysis. stackreg accommodates multiway higher-level clustering; that is, clustvarlist may consist of more than one variable. Multiway clustering requires the community-contributed command cgmreg (by Gelbach and Miller [2009]) to be installed. stackreg has been tested with cgmreg version 3.0.0. Other versions of cgmreg may behave differently and might make stackreg fail or produce incorrect results.

df(adjust | raw | areg) specifies the type of degrees-of-freedom adjustment stackreg applies. The default is df(adjust). With df(adjust), stackreg adjusts the degrees-of-freedom correction such that the reported standard errors coincide with those one gets from separately regressing the elements of depvars on indepvars, using regress with option robust. This, depending on how the cluster() option is specified, likewise applies to the standard errors one gets from regress, cluster() and cgmreg, cluster(), respectively. In the most simple case (no higher-level clustering, no panel data, homogeneous number of observations across depvars), the initially estimated variance–covariance matrix is adjusted by the factor (N − 1)/(N − 1/G), with N denoting the genuine number of observations and G denoting the number of variables in depvars. ¹⁵

For xtstackreg, the default—that is, df(adjust)—is to adjust the degrees-offreedom correction such that the standard errors coincide with those from xtreg, fe robust and xtreg, fe cluster(), respectively. This implies that xtstackreg, by default, clusters the standard errors at the level of panelvar, which is the default with xtreg, fe robust. If df(areg) is specified, xtstackreg adjusts the degrees of freedom such that the standard errors match those from areg, absorb( panelvar ) robust and areg, absorb( panelvar ) cluster(), respectively. That is, with df(areg), stackreg does not cluster the standard errors at the level of panelvar unless this is explicitly requested with cluster( panelvar ). df(areg) is ignored by stackreg if the fe option is not specified. df(raw) prevents stackreg from adjusting the degrees-of-freedom correction to the stacked regression setting. See section 3.4 for further details of the degrees-of-freedom adjustment that stackreg applies.

wald makes test and testparm apply a Wald rather than an F test after stackreg. This is achieved through preventing stackreg from saving the residual degrees of freedom in e(df_r). With multiway clustering, as with heterogeneous estimation samples across the different regression equations, e(df_r) is never stored, because there is no (universal) answer to the question of what the number of clusters is. Thus, test and testparm apply a Wald test in these cases, even if the wald option is not specified.

sreshape requests that stackreg call the community-contributed command sreshape (Simons 2016) instead of reshape. Because sreshape is much faster than reshape (Simons 2016) in many settings, specifying sreshape may speed up stackreg.

level( # ); see [R] Estimation options. The reported confidence level can be changed by retyping stackreg without arguments and only specifying the level( # ) option.

edittozero( # ) specifies how close to 0 an element of the estimated variance–covariance needs to be to set its value to 0. The specified value is passed through to the Mata function edittozero(); see [M-5] edittozero( ). The default is edittozero(1).

The different estimation commands that are alternatively called by stackreg may differ with respect to how estimated coefficient variances that are close to 0 are dealt with. Specifying edittozero() aligns their behaviors.

omitted specifies that variables that were omitted because of collinearity be displayed and labeled as (omitted). Unlike many Stata commands, the default is not to include in the results table any variables omitted because of collinearity. This is the default because stackreg regularly generates rather larger results tables due to depvars consisting of numerous variables. This applies in particular if factor variables are used. Hence, listing omitted variables may render the output hard to read.

emptycells specifies that empty cells for interactions of factor variables be displayed and labeled as (empty). The default is not to include them in the results table, for the same reason as the default for the omitted option.

display_options: noci, nopvalues, noomitted, vsquish, noemptycells, baselevels, allbaselevels, nofvlabel, fvwrap( # ), fvwrapon( style ), cformat(% fmt ), pformat(% fmt ), sformat(% fmt ), and nolstretch; see [R] Estimation options.

4.3 Stored results

stackreg and xtstackreg store the following results in e():

4.4 stackreg postestimation

Using postestimation commands after stackreg is essential; stackreg is hardly an estimation command in its own right but is meant to facilitate postestimation inference. The postestimation commands available after stackreg are those available after mvreg ¹⁶ (see [MV] mvreg postestimation). After stackreg, these commands behave in the same way as they behave after mvreg. The most important postestimation command is test (testparm, respectively). For instance, testparm * can be used for testing the joint null hypothesis that no variable in indepvars has explanatory power for any of the variables in depvars.

5 Applications

In this section, we present two applications of stackreg. The first application illustrates the syntax of stackreg in a cross-sectional setting that includes weighting and demonstrates that stackreg reproduces the original estimation results for each equation. The second application shows how stackreg handles panel data and multiway clustering.

5.1 The persistent effects of Peru’s mining Mita

To illustrate the application of stackreg, we in parts replicate and use data from Dell (2010). The data are available from the website of the Econometric Society and can be directly loaded—together with comprehensive documentation of Dell’s empirical work—into a new subfolder (name final; size 1.13 GB) of Stata’s current working directory. ¹⁷

Using a spatial regression discontinuity approach, Dell (2010) examines the long-run effects of a forced mining system in Peru and Bolivia, called Mita, that was in place in a clearly defined geographical area between 1573 and 1812. We focus on the regressions for which results are reported in table V, panel A, columns 6–8 (Dell 2010, 1886). Before carrying out the empirical analysis, some data preparation steps are necessary irrespective of stackreg. These steps and the replication of the original regressions are executed by quietly running dell_mita_prep.do. ¹⁸ The equation-by-equation replication results are stored as orig_Dell_sh_trib, orig_Dell_sh_boys, and orig_Dell_sh_women by using estimates store.

As a regression-based balancing check, Dell (2010) examines whether the population composition differed between Mita regions and control regions before Mita came into force. Specifically, Dell regresses the population shares of men, boys, and females (sh_trib, sh_boys, sh_women) separately on numerous control variables and a Mita indicator labeled pothuan_mita, where observations are weighted by the square root of the district’s total population. The estimation sample is restricted to districts that in terms of their capital are located no more than 50 kilometers distant from the boundary of the Mita area, with metropolitan Cusco being excluded from the sample. The output below displays the key results from these regressions, each equation being labeled with the name of its dependent variable. The coefficients of pothuan_mita are all close to 0 and statistically insignificant. However, separately testing the individual significance of these coefficients may not be an appropriate strategy for testing that Mita and non-Mita regions—conditional on controls—did not systematically differ prior to the implementation of Mita.

We next use stackreg to set the foundation for carrying out a presumably more appropriate regression-based joint balancing test. The output below illustrates that in terms of the coefficients and standard errors, stackreg yields exactly the same results as equation-by-equation estimation. The header of the output shows that 65 original observations contribute into the stacked regression and that they stem from 65 clusters; that is, the cluster variable ubigeo uniquely identifies the observations.

Finally, after running stackreg, we test the joint hypothesis that none of the population shares differs between the two groups. This joint test is easily done using Stata’s test command.

The output from test does not provide any evidence for systematic pretreatment disparities between Mita and non-Mita regions.

5.2 One Mandarin benefits the whole clan

We next illustrate additional features of stackreg—in particular, xtstackreg—by replicating in parts and using data from Do, Nguyen, and Tran (2017b). These data are published as Do, Nguyen, and Tran (2017a). Downloading the data requires an account at the Inter-university Consortium for Political and Social Research (ICPSR).

Do, Nguyen, and Tran (2017b) examine how the promotion of Vietnamese officials affects their hometowns. In their table 3 (Do, Nguyen, and Tran 2017b, 18), which we will focus on, the authors present results for six different dependent variables, three of which measure infrastructure investments (Infras3yr_Productive, Infras3yr_Information, and Infras3yr_EduHealth) and the other three of which measure regional outcomes (F2logComAvgInc, F2logComAvgExp, and F2logComPop). They obtain these results from six separate regressions that include the same set of independent variables. Unlike the first example application, for which a multiple-testing issue arises in connection with balancing checks, Do, Nguyen, and Tran (2017b) resembles a classical multiple-testing problem, where a bunch of outcome variables are considered to test the comprehensive hypothesis that state and party officials favor their hometowns.

Together with data-preparation steps, we run the original regressions quietly in do_mandarin_prep.do ¹⁹ and store the results as orig_Do_*. ²⁰ To make the code easier to read, we place all control variables in the global macro controls. These regressions allow for commune fixed effects by using areg, absorb(ComID). Standard errors are clustered at the commune level, and the number of observations differs between equations. To accommodate including commune fixed effects, we declare the data to be panel data. Finally, we display the results of the one-to-one replication of Do, Nguyen, and Tran (2017b) as reference, using estimates table and focusing on the coefficients of key explanatory variable PowerCapital.

Then we use xtstackreg ²¹ for the replication. Because the original set of regressions used different observations, we now specify the nocommon option. The output’s header informs us that the stacked regression uses information from a total of 1,239 original observations. We furthermore specify the cluster(ComID) and df(areg) options because Do, Nguyen, and Tran (2017b) combine areg with standard errors clustered at the commune level. Using these options, xtstackreg yields the same point estimates and standard errors as the original code.

We can now perform a joint test of whether power capital is related to any of the outcome variables by using test.

Though tests on individual significance suggest that officials’ hometowns benefited in terms of higher investment in productive (Infras3yr_Productive) and information (Infras3yr_Information) infrastructure, on the basis of the joint test at the 10% level, we marginally cannot reject the null that it is immaterial for town-level outcomes, if people from that town reach high positions.

In this setting, multiway clustering may be an attractive alternative, say, because error terms are not only related within communes across years but also in each year across communes. ²² To use multiway clustering, we list the varlist of clustering variables as arguments in cluster().

After rerunning xtstackreg with multiway clustering, we carry out another joint test regarding the effects of PowerCapital on the six considered outcomes. The new test result is more informative: the null of no effects is clearly rejected.

6 Conclusions

In this article, we introduced the stackreg command, which offers a convenient way to test hypotheses across multiple OLS regressions. stackreg goes beyond similar Stata commands in three aspects: first, it implements fixed-effects estimations; second, it allows for multiway clustering; and third, it enables cross-equation constraints.

8 Programs and supplemental materials

Supplemental Material, sj-zip-1-stj-10.1177_1536867X211025801 - Stacked linear regression analysis to facilitate testing of hypotheses across OLS regressions

Supplemental Material, sj-zip-1-stj-10.1177_1536867X211025801 for Stacked linear regression analysis to facilitate testing of hypotheses across OLS regressions by Michael Oberfichtner and Harald Tauchmann in The Stata Journal