Simple Formulas for Standard Errors that Cluster by Both Firm and Time
TL;DR: In this article, it is shown that it is easy to calculate standard errors that are robust to simultaneous correlation across both firms and time, and that any statistical package with a clustering command can be used to easily calculate these standard errors.
Abstract: When estimating finance panel regressions, it is common practice to adjust standard errors for correlation either across firms or across time. These procedures are valid only if the residuals are correlated either across time or across firms, but not across both. This note shows that it is very easy to calculate standard errors that are robust to simultaneous correlation across both firms and time. The covariance estimator is equal to the estimator that clusters by firm, plus the the estimator that clusters by time, minus the usual heteroskedasticity-robust OLS covariance matrix. Any statistical package with a clustering command can be used to easily calculate these standard errors.
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TL;DR: In this article, the authors examine the different methods used in the literature and explain when the different approaches yield the same (and correct) standard errors and when they diverge, and give researchers guidance for their use.
Abstract: In both corporate finance and asset pricing empirical work, researchers are often confronted with panel data. In these data sets, the residuals may be correlated across firms and across time, and OLS standard errors can be biased. Historically, the two literatures have used different solutions to this problem. Corporate finance has relied on clustered standard errors, while asset pricing has used the Fama-MacBeth procedure to estimate standard errors. This paper examines the different methods used in the literature and explains when the different methods yield the same (and correct) standard errors and when they diverge. The intent is to provide intuition as to why the different approaches sometimes give different answers and give researchers guidance for their use.
7,647 citations
TL;DR: This work considers statistical inference for regression when data are grouped into clusters, with regression model errors independent across clusters but correlated within clusters, when the number of clusters is large and default standard errors can greatly overstate estimator precision.
Abstract: We consider statistical inference for regression when data are grouped into clus- ters, with regression model errors independent across clusters but correlated within clusters. Examples include data on individuals with clustering on village or region or other category such as industry, and state-year dierences-in-dierences studies with clustering on state. In such settings default standard errors can greatly overstate es- timator precision. Instead, if the number of clusters is large, statistical inference after OLS should be based on cluster-robust standard errors. We outline the basic method as well as many complications that can arise in practice. These include cluster-specic �xed eects, few clusters, multi-way clustering, and estimators other than OLS.
3,236 citations
TL;DR: The authors proposed a variance estimator for the OLS estimator as well as for nonlinear estimators such as logit, probit, and GMM that enables cluster-robust inference when there is two-way or multiway clustering that is nonnested.
Abstract: In this article we propose a variance estimator for the OLS estimator as well as for nonlinear estimators such as logit, probit, and GMM. This variance estimator enables cluster-robust inference when there is two-way or multiway clustering that is nonnested. The variance estimator extends the standard cluster-robust variance estimator or sandwich estimator for one-way clustering (e.g., Liang and Zeger 1986; Arellano 1987) and relies on similar relatively weak distributional assumptions. Our method is easily implemented in statistical packages, such as Stata and SAS, that already offer cluster-robust standard errors when there is one-way clustering. The method is demonstrated by a Monte Carlo analysis for a two-way random effects model; a Monte Carlo analysis of a placebo law that extends the state–year effects example of Bertrand, Duflo, and Mullainathan (2004) to two dimensions; and by application to studies in the empirical literature where two-way clustering is present.
2,542 citations
Cites methods from "Simple Formulas for Standard Errors..."
...We thank Mitchell Petersen for sending us a copy of his paper and informing us of the econometrics paper by Thompson (2005, 2006) that provides some theory and Monte Carlo evidence for the two-way OLS case with panel data on firms....
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...Petersen (2009) compares a number of approaches for OLS estimation in a finance panel setting, using results by Thompson (2006) that provides some theory and Monte Carlo evidence for the two-way OLS case with panel data on firms....
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Posted Content•
TL;DR: The authors review and evaluate the methods commonly used in the accounting literature to correct for cross-sectional and time-series dependence and find that the extant methods are not robust to both forms of dependence.
Abstract: We review and evaluate the methods commonly used in the accounting literature to correct for cross-sectional and time-series dependence. While much of the accounting literature studies settings where variables are cross-sectionally and serially correlated, we find that the extant methods are not robust to both forms of dependence. Contrary to claims in the literature, we find that the Z2-statistic and Newey-West corrected Fama-MacBeth do not correct for both cross-sectional and time-series dependence. We show that extant methods produce misspecified test statistics in common accounting research settings, and that correcting for both forms of dependence substantially alters inferences reported in the literature. Specifically, several findings in the cost of equity capital literature, the cost of debt literature, and the conservatism literature appear not to be robust to the use of well-specified test statistics.
1,099 citations
Cites background or result from "Simple Formulas for Standard Errors..."
...18 This high correlation mirrors that observed in studies reporting both FM-t and Z2-t statistics (0.96 for Barth et al. 2001b and 0.99 for Wang 2006)....
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...Some studies claim that Z2 adjusts for cross-sectional and serial correlation (e.g., Aboody and Lev 1998; Barth et al. 1998, 2001b; Davis 2002; Wang 2006)....
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TL;DR: This article study the effect of the recent financial crisis on corporate investment and find that firms that have low cash reserves or high net short-term debt, are financially constrained, or operate in industries dependent on external finance are less likely to invest.
Abstract: We study the effect of the recent financial crisis on corporate investment. The crisis represents an unexplored negative shock to the supply of external finance for non-financial firms. Corporate investment declines significantly following the onset of the crisis, controlling for firm fixed effects and time-varying measures of investment opportunities. Consistent with a causal effect of a supply shock, the decline is greatest for firms that have low cash reserves or high net short-term debt, are financially constrained, or operate in industries dependent on external finance. To address endogeneity concerns, we measure firms’ financial positions as much as four years prior to the crisis, and confirm that similar results do not follow placebo crises in the summers of 2003–2006. Nor do similar results follow the negative demand shock caused by September 11, 2001. The effects weaken considerably beginning in the third quarter of 2008, when the demand-side effects of the crisis became apparent. Additional analysis suggests an important precautionary savings motive for seemingly excess cash that is generally overlooked in the literature.
1,028 citations
Cites methods from "Simple Formulas for Standard Errors..."
...Column 6 of Table 3 shows that our main results in column 4 are robust to clustering standard errors by both firm and time (calendar quarter) using the method described in Thompson (2009) and Petersen (2009)....
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References
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TL;DR: In this article, a parameter covariance matrix estimator which is consistent even when the disturbances of a linear regression model are heteroskedastic is presented, which does not depend on a formal model of the structure of the heteroSkewedness.
Abstract: This paper presents a parameter covariance matrix estimator which is consistent even when the disturbances of a linear regression model are heteroskedastic. This estimator does not depend on a formal model of the structure of the heteroskedasticity. By comparing the elements of the new estimator to those of the usual covariance estimator, one obtains a direct test for heteroskedasticity, since in the absence of heteroskedasticity, the two estimators will be approximately equal, but will generally diverge otherwise. The test has an appealing least squares interpretation.
25,689 citations
TL;DR: In this article, a simple method of calculating a heteroskedasticity and autocorrelation consistent covariance matrix that is positive semi-definite by construction is described.
Abstract: This paper describes a simple method of calculating a heteroskedasticity and autocorrelation consistent covariance matrix that is positive semi-definite by construction. It also establishes consistency of the estimated covariance matrix under fairly general conditions.
18,117 citations
"Simple Formulas for Standard Errors..." refers background in this paper
...Autoregressive processes could be handled by allowing the lag length L to grow with the sample size (see for example Newey and West 1987)....
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TL;DR: In this article, the relationship between average return and risk for New York Stock Exchange common stocks was tested using a two-parameter portfolio model and models of market equilibrium derived from the two parameter portfolio model.
Abstract: This paper tests the relationship between average return and risk for New York Stock Exchange common stocks. The theoretical basis of the tests is the "two-parameter" portfolio model and models of market equilibrium derived from the two-parameter portfolio model. We cannot reject the hypothesis of these models that the pricing of common stocks reflects the attempts of risk-averse investors to hold portfolios that are "efficient" in terms of expected value and dispersion of return. Moreover, the observed "fair game" properties of the coefficients and residuals of the risk-return regressions are consistent with an "efficient capital market"--that is, a market where prices of securities
14,171 citations
"Simple Formulas for Standard Errors..." refers background or methods in this paper
...Fama and MacBeth (1973) proposed a sequential timeseries of cross-sections procedure that produces standard errors robust to correlation between firms at a moment in time....
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...Similarly, we could use the standard errors of Fama and MacBeth (1973), since they also solve the singleclustering problem (see Petersen (2009) for further explanation)....
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TL;DR: In this article, the authors examine the different methods used in the literature and explain when the different approaches yield the same (and correct) standard errors and when they diverge, and give researchers guidance for their use.
Abstract: In both corporate finance and asset pricing empirical work, researchers are often confronted with panel data. In these data sets, the residuals may be correlated across firms and across time, and OLS standard errors can be biased. Historically, the two literatures have used different solutions to this problem. Corporate finance has relied on clustered standard errors, while asset pricing has used the Fama-MacBeth procedure to estimate standard errors. This paper examines the different methods used in the literature and explains when the different methods yield the same (and correct) standard errors and when they diverge. The intent is to provide intuition as to why the different approaches sometimes give different answers and give researchers guidance for their use.
7,647 citations
TL;DR: This paper examined whether financial development facilitates economic growth by scrutinizing one rationale for such a relationship; that financial development reduces the costs of external finance to firms, and found that industrial sectors that are relatively more in need of foreign finance develop disproportionately faster in countries with more developed financial markets.
Abstract: Does finance affect economic growth? A number of studies have identified a positive correlation between the level of development of a country's financial sector and the rate of growth of its per capita income. As has been noted elsewhere, the observed correlation does not necessarily imply a causal relationship. This paper examines whether financial development facilitates economic growth by scrutinizing one rationale for such a relationship; that financial development reduces the costs of external finance to firms. Specifically, we ask whether industrial sectors that are relatively more in need of external finance develop disproportionately faster in countries with more developed financial markets. We find this to be true in a large sample of countries over the 1980s. We show this result is unlikely to be driven by omitted variables, outliers, or reverse causality.
6,815 citations
"Simple Formulas for Standard Errors..." refers methods in this paper
...Rajan and Zingales (1998) run regressions at the country and industry level, and use country and industry dummies to control for common effects....
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