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2015 | OriginalPaper | Chapter

57. Assessing the Performance of Estimators Dealing with Measurement Errors

Authors : Heitor Almeida, Murillo Campello, Antonio F. Galvao

Published in: Handbook of Financial Econometrics and Statistics

Publisher: Springer New York

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Abstract

We describe different procedures to deal with measurement error in linear models and assess their performance in finite samples using Monte Carlo simulations and data on corporate investment. We consider the standard instrumental variable approach proposed by Griliches and Hausman (Journal of Econometrics 31:93–118, 1986) as extended by Biorn (Econometric Reviews 19:391–424, 2000) [OLS-IV], the Arellano and Bond (Review of Economic Studies 58:277–297, 1991) instrumental variable estimator, and the higher-order moment estimator proposed by Erickson and Whited (Journal of Political Economy 108:1027–1057, 2000, Econometric Theory 18:776–799, 2002). Our analysis focuses on characterizing the conditions under which each of these estimators produce unbiased and efficient estimates in a standard “errors-in-variables” setting. In the presence of fixed effects, under heteroscedasticity, or in the absence of a very high degree of skewness in the data, the EW estimator is inefficient and returns biased estimates for mismeasured and perfectly measured regressors. In contrast to the EW estimator, IV-type estimators (OLS-IV and AB-GMM) easily handle individual effects, heteroscedastic errors, and different degrees of data skewness. The IV approach, however, requires assumptions about the autocorrelation structure of the mismeasured regressor and the measurement error. We illustrate the application of the different estimators using empirical investment models. Our results show that the EW estimator produces inconsistent results when applied to real-world investment data, while the IV estimators tend to return results that are consistent with theoretical priors.

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previous chapter Density and Conditional Distribution-Based Specification Analysis

next chapter Realized Distributions of Dynamic Conditional Correlation and Volatility Thresholds in the Crude Oil, Gold, and Dollar/Pound Currency Markets

Naturally, if extraneous instruments are available, they can help solve the identification problem. See Rauh (2006) for the use of discontinuities in pension contributions as a source of variation in cash flows in an investment model. Bond and Cummins (2000) use information contained in financial analysts’ forecasts to instrument for investment demand.

Lags of the well-measured variable may also be included in the instrument set if they are believed to also contain information about the mismeasured one.

See, among others, Biorn (2000), Wansbeek (2001), and Xiao et al. (2008).

Examples are Whited (2001, 2006), Hennessy (2004), and Colak and Whited (2007).

The results for the Arellano–Bond GMM estimator are similar to those of the OLS-IV estimator. To save space and because the OLS-IV estimator is easier to implement, we focus on this estimator.

See Hubbard (1998) and Stein (2003) for comprehensive reviews. We note that the presence of financing frictions does not necessarily imply that the cash flow coefficient should be positive. See Chirinko (1993) and Gomes (2001) for arguments suggesting that financing frictions are not sufficient to generate positive cash flow coefficients.

First, the measurement errors, the equation error, and all regressors have finite moments of sufficiently high order. Second, the regression error and the measurement error must be independent of each other and of all regressors. Third, the residuals from the population regression of the unobservable regressors on the perfectly measured regressors must have a nonnormal distribution.

More specifically, these conditions are as follows: (z _i, χ _i, u _i, ε _i) is an independent and identically distributed sequence; u _i and the elements of z _i, χ _i, and ε _i, have finite moments of every order; (u _i, ε _i) is independent of (z _i, χ _i), and the individual elements in (u _i, ε _i) are independent of each other; E(u _i) = 0 and E(ε _i) = 0; E[(z _i, χ _i)′(z _i, χ _i)] is positive definite; every element of β is nonzero; and the distribution of η satisfies E[(η _i c)³] ≠ 0 for every vector of constants c = (c ₁,⋯,c _J) having at least one nonzero element.

See Erickson and Whited (2002) Lemma 1 for the definition of their proposed influence function.

A more recent paper by Xiao et al. (2008) also shows how to relax the classical Griliches–Hausman assumptions for measurement error models.

Formally, one can show that C(x _it, x _iθ) = ∑ _tθ ^χχ + ∑ _tθ ^εε , E(x _it, y _iθ) = ∑ _tθ ^χχ β + ∑ _t ^χη , and E(y _it, y _iθ) = β′ ∑ _tθ ^χχ β + ∑ _t ^xn β + β ^H′(∑ _θ ^xn )′ + σ _tθ ^uu + σ ^ηη.

In particular, if ∣t − p∣, ∣θ − p∣ > τ, then (B1) and rank (E[χ _ip ^′ (Δχ _itθ)]) = K for some p ≠ t ≠ 0 ensure consistency of OLS–IV B, \( {\widehat{\beta}}_{xp\left( t\theta \right)} \), and (B2) and the same rank condition ensure consistency of \( {\widehat{\beta}}_{xy\left( t\theta \right)} \). In the same way, if ∣p − t∣, q − t > τ, (B1), (D1), (D2), and rank (E[(Δχ _ipq)′χ _it)]) = K for some p ≠ q ≠ t ensure consistency of OLS-IV B, \( {\widehat{\beta}}_{x(pq) t}, \) and (B2), (D1), (D2), and the same rank condition ensure consistency of \( {\widehat{\beta}}_{y(pq) t} \).

In models with exogenous explanatory variables, Z _i may consist of sub-matrices with the block diagonal (exploiting all or part of the moment restrictions), concatenated to straightforward one-column instruments.

See Propositions 1* and 2* in Biorn (2000) for a formal treatment of the conditions.

The mean squared error (MSE) of an estimator \( \widehat{\theta} \) incorporates a component measuring the variability of the estimator (precision) and another measuring its bias (accuracy). An estimator with good MSE properties has small combined variance and bias. The MSE of \( \widehat{\theta} \) can be defined as \( \mathrm{Var}\left(\widehat{\theta}\right)+{\left[\mathrm{Bias}\left(\widehat{\theta}\right)\right]}^2 \). The root mean squared error (RMSE) is simply the square root of the MSE. This is an easily interpretable statistic, since it has the same unit as the estimator \( \widehat{\theta} \). For an approximately unbiased estimator, the RMSE is just the square root of the variance, that is, the standard error.

Robustness checks show that the choice of a standard normal does not influence our results.

To our knowledge, all but one of the empirical applications of the EW model use the data in level form. In other words, firm-fixed effects are ignored outright in panel setting estimations of parameters influencing firm behavior.

The results using the median are similar.

We focus on the OLS-IV estimator hereinafter for the purpose of comparison with the EW estimator.

Since estimation biases have the same features across all well-measured regressors of a model, we restrict attention to the first well-measured regressor of each of the estimated models.

Our simulation results (available upon request) suggest that introducing heteroscedasticity makes the performance of the EW estimator even worse in these cases.

The results for w _it = z _it are quite similar to those we get from setting w _it = γ _i. We report only one set of graphs to save space.

We note that if the instrument set uses suitably long lags, then the OLS-IV results are robust to variations in the degree of correlation in the MA process. In unreported simulations under MA(1), we show that the OLS bias is nearly invariant to the parameter θ.

The empirical cumulative distribution function F _n is a step function with jumps i/n at observation values, where i is the number of tied observations at that value.

However, financial constraints are not sufficient to generate a strictly positive cash flow coefficient because the effect of financial constraints is capitalized in stock prices and may thus be captured by variations in q (Chirinko 1993; Gomes 2001).

Fama–MacBeth estimates are computed as a simple standard errors for yearly estimates. An alternative approach could use the Hall–Horowitz bootstrap. For completeness, we present in the appendix the actual yearly EW estimates.

In the next section, we examine the robustness of the results with respect to variation in the instrument set.

All of the F-statistics associated with the first-stage regressions have p-values that are close to zero. These statistics (reported in Table 57.10) suggest that we do not incur a weak instrument problem when we use longer lags in our instrumental set.

Agca, S., & Mozumdar, A. (2007). Investment-cash flow sensitivity: Myth or reality? Working paper, George Washington University.

Almeida, H., & Campello, M. (2007). Financial constraints, asset tangibility and corporate investment. Review of Financial Studies, 20, 1429–1460.CrossRef

Altonji, J., & Segal, L. (1996). Small-sample bias in GMM estimation of covariance structures. Journal of Business & Economic Statistics, 14, 353–366.

Arellano, M., & Bond, S. (1991). Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. Review of Economic Studies, 58, 277–297.CrossRef

Bakke, T., & Whited, T. (2009). What gives? A study of firms’ reactions to cash shortfalls. Working paper, University of Rochester.

Baltagi, B. (2005). Econometric analysis of panel data (3rd ed.). Chichester: Wiley.

Baum, C., Schaffer, M., & Stillman, S. (2003). Instrumental variables and GMM: Estimation and testing. Stata Journal, 3, 1–31.

Bertrand, M., & Mullainathan, S. (2005). Bidding for oil and gas leases in the Gulf of Mexico: A test of the free cash flow model?. mimeo, University of Chicago and MIT.

Bertrand, M., & Schoar, A. (2003). Managing with style: The effect of managers on firm policies. Quarterly Journal of Economics, 118, 1169–1208.CrossRef

Bertrand, M., Duflo, E., & Mullainathan, S. (2004). How much should we trust differences-in- differences estimates? Quarterly Journal of Economics, 119, 249–275.CrossRef

Biorn, E. (2000). Panel data with measurement errors: Instrumental variables and GMM procedures combining levels and differences. Econometric Reviews, 19, 391–424.CrossRef

Blanchard, O., Lopez-de-Silanes, F., & Shleifer, A. (1994). What do firms do with cash windfalls? Journal of Financial Economics, 36, 337–360.CrossRef

Blundell, R., Bond, S., Devereux, M., & Schiantarelli, F. (1992). Investment and Tobin’s Q: Evidence from company panel data. Journal of Econometrics, 51, 233–257.CrossRef

Bond, S., & Cummins, J. (2000). The stock market and investment in the new economy. Brookings Papers on Economic Activity, 13, 61–124.CrossRef

Chirinko, R. (1993). Business fixed investment spending: Modeling strategies, empirical results, and policy implications. Journal of Economic Literature, 31, 1875–1911.

Colak, G., & Whited, T. (2007). Spin-offs, divestitures, and conglomerate investment. Review of Financial Studies, 20, 557–595.CrossRef

Cragg, J. (1997). Using higher moments to estimate the simple errors-in-variables model. RAND Journal of Economics, 28, 71–91.CrossRef

Doran, H., & Schmidt, P. (2006). GMM estimators with improved finite sample properties using principal components of the weighting matrix, with an application to the dynamic panel data model. Journal of Econometrics, 133, 387–409.CrossRef

Erickson, T., & Whited, T. (2000). Measurement error and the relationship between investment and Q. Journal of Political Economy, 108, 1027–1057.CrossRef

Erickson, T., & Whited, T. (2002). Two-step GMM estimation of the errors-in-variables model using high-order moments. Econometric Theory, 18, 776–799.CrossRef

Fazzari, S., Hubbard, R. G., & Petersen, B. (1988). Financing constraints and corporate investment. Brooking Papers on Economic Activity, 1, 141–195.CrossRef

Frees, E. (2004). Longitudinal and panel data analysis and applications in the social sciences. New York: Cambridge University Press.CrossRef

Gan, J. (2007). Financial constraints and corporate investment: Evidence from an exogenous shock to collateral. Journal of Financial Economics, 85, 709–734.CrossRef

Gomes, J. (2001). Financing investment. American Economic Review, 91, 1263–1285.CrossRef

Griliches, Z., & Hausman, J. A. (1986). Errors in variables in panel data. Journal of Econometrics, 31, 93–118.CrossRef

Hadlock, C. (1998). Ownership, liquidity, and investment. RAND Journal of Economics, 29, 487–508.CrossRef

Hayashi, F. (1982). Tobin’s marginal q and average q: A neoclassical interpretation. Econometrica, 50, 213–224.CrossRef

Hayashi, F. (2000). Econometrics. Princeton: Princeton University Press.

Hennessy, C. (2004). Tobin’s Q, debt overhang, and investment. Journal of Finance, 59, 1717–1742.CrossRef

Holtz-Eakin, D., Newey, W., & Rosen, H. S. (1988). Estimating vector autoregressions with panel data. Econometrica, 56, 1371–1395.CrossRef

Hoshi, T., Kashyap, A., & Scharfstein, D. (1991). Corporate structure, liquidity, and investment: Evidence from Japanese industrial groups. Quarterly Journal of Economics, 106, 33–60.CrossRef

Hubbard, R. G. (1998). Capital market imperfections and investment. Journal of Economic Literature, 36, 193–227.

Kaplan, S., & Zingales, L. (1997). Do financing constraints explain why investment is correlated with cash flow? Quarterly Journal of Economics, 112, 169–215.CrossRef

Lamont, O. (1997). Cash flow and investment: Evidence from internal capital markets. Journal of Finance, 52, 83–110.CrossRef

Lyandres, E. (2007). External financing costs, investment timing, and investment-cash flow sensitivity. Journal of Corporate Finance, 13, 959–980.CrossRef

Malmendier, U., & Tate, G. (2005). CEO overconfidence and corporate investment. Journal of Finance, 60, 2661–2700.CrossRef

Petersen, M. (2009). Estimating standard errors in finance panel data sets: Comparing approaches. Review of Financial Studies, 22, 435–480.CrossRef

Poterba, J. (1988). Financing constraints and corporate investment: Comment. Brookings Papers on Economic Activity, 1, 200–204.

Rauh, J. (2006). Investment and financing constraints: Evidence from the funding of corporate pension plans. Journal of Finance, 61, 33–71.CrossRef

Riddick, L., & Whited, T. (2009). The corporate propensity to save. Journal of Finance, 64, 1729–1766.CrossRef

Shin, H., & Stulz, R. (1998). Are internal capital markets efficient? Quarterly Journal of Economics, 113, 531–552.CrossRef

Stein, J. (2003). Agency information and corporate investment. In G. Constantinides, M. Harris, & R. Stulz (Eds.), Handbook of the economics of finance. Amsterdam: Elsevier/North-Holland.

Wansbeek, T. J. (2001). GMM estimation in panel data models with measurement error. Journal of Econometrics, 104, 259–268.CrossRef

Whited, T. (2001). Is it inefficient investment that causes the diversification discount? Journal of Finance, 56, 1667–1691.CrossRef

Whited, T. (2006). External finance constraints and the intertemporal pattern of intermittent investment. Journal of Financial Economics, 81, 467–502.CrossRef

Xiao, Z., Shao, J., & Palta, M. (2008). A unified theory for GMM estimation in panel data models with measurement error. Working paper, University of Wisconsin Madison.

Title: Assessing the Performance of Estimators Dealing with Measurement Errors
Authors: Heitor Almeida
Murillo Campello
Antonio F. Galvao
Publisher: Springer New York
Book: Handbook of Financial Econometrics and Statistics
Print ISBN: 978-1-4614-7749-5

Electronic ISBN: 978-1-4614-7750-1

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-1-4614-7750-1_57