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Alternative Errors-in-Variable Models and Their Applications in Finance Research*

Alternative Errors-in-Variable Models and Their Applications in Finance Research*. Cheng-Few Lee Rutgers University , USA Hong-Yi Chen National Central University , Taiwan. Outline. Introduction 2. Effects of Errors-in-Variables in Different Cases 2.1 Bivariate Normal Case

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Alternative Errors-in-Variable Models and Their Applications in Finance Research*

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  1. Alternative Errors-in-Variable Models and Their Applications in Finance Research* Cheng-Few Lee Rutgers University , USA Hong-Yi Chen National Central University , Taiwan

  2. Outline • Introduction 2. Effects of Errors-in-Variables in Different Cases 2.1 Bivariate Normal Case 2.2 Multivariate Case 2.2.1 The Classical Case 2.2.2 The Constrained Classical Case 3. Six Different Estimation Methods When Variables are Subject to Error 3.1 Classical Estimation Method 3.1.1 The Classical Method to a Simple Regression Analysis 3.1.2 The Classical Method to a Multiple Regression Analysis 3.1.3 The Constrained Classical Method 3.2 Grouping Method 3.3 Instrumental Variable Method

  3. Outline 3.4 Mathematical Method 3.4.1 Bivariate Case 3.4.2 Multivariate Case 3.5 Maximum Likelihood Method 3.6 LISREL and MIMIC Methods 3.6.1 Structural model (LISEREL model) 3.6.2 MIMIC Model 4. Applications of Errors-in-Variables Models in Finance Research 4.1 Cost of Capital 4.2 Capital Asset Pricing Model 4.3 Capital Structure 4.4 Investment and Capital Asset Allocation 5. Conclusion

  4. Abstract The main purposes of this paper are (i) to review and extend existing errors-in-variables (EIV) estimation methods; and (ii) to discuss how these estimation methods have been used in finance research. We first show how EIV problems affect estimators in the regression model. We further extend six alternative estimation methods dealing with EIV problem. These six methods are (i) classical method, (ii) grouping method, (iii) instrumental variable method, (iv) mathematical programming method, (v) maximum likelihood method, and (vi) LISREL method will be discussed and extended from the original simple regression case to multiple regression case. Finally, we investigate how EIV estimation methods have been used to determine cost of capital and capital structure, and test capital asset pricing model.

  5. 1. Introduction • This paper studies and extents exiting EIV estimation methods and to discuss how these estimation methods have been used in finance research. We first show how EIV problems affect estimators in the regression model. We further provide six alternative estimation methods dealing with EIV problem. Classical method, grouping method, instrumental variable method, mathematical programming method, maxima likelihood method, and LISREL method will be discussed and extended from the original simple regression case to multiple regression case. Finally, we investigate how EIV problem associate to estimated beta can affect the second-pass regression in testing capital asset pricing model.

  6. 2. Effects of Errors-in-Variables in Different Cases • Bivariate Normal Case • Multivariate Case

  7. Two variate structural relationship where Then 2.1 Bivariate Normal Case

  8. Trivariate structural relationship where Then 2.2 Multivariate Case

  9. 3. Six Different Estimation Methods When Variables are Subject to Error • Classical Method • Grouping Method • Instrumental Variable Method • Mathematical Programming Method • Maxima Likelihood Method • LISREL and MIMIC Methods

  10. 3.1 Classical Method

  11. 3.2 Grouping Method • Wald’s (1940) grouping method (2 groups) in deal with errors-in-variable problem • - using groups instead of individual securities can minimize measurement error. • CAPM test: (k-groups) • Black et al. (1972), Blume and Friend (1973), Fame and MacBath (1973), Lizenberger and Ramaswamy (1979), etc. • , where

  12. 3.2 Grouping Method - continued • Advantages • is intuitive and easy to implement with real data. • is convenient to incorporate other estimation methods or examine model misspecification (multi-factor models). • Limitations: • shrinks number or observations and the range of beta risk in the second-pass => the variance of the estimator becomes larger and reduce statistic power. • The formation of portfolios for the second-pass estimation might cause a loss of valuable information about cross-sectional behavior among individual securities, since the cross-sectional variations would be smoothed out. • may get different results by using different portfolio grouping methods. (e.g., Ahn et al., 2009)

  13. 3.3 Instrumental Variable Method Durbin (1953) proposes an instrumental variable method to deal with the errors-in-variables problem in a regression model. If , then instrumental variable method will reduce to Wald’s 2-group method. • Grouping method is a special case of instrumental method. • Instrumental variable method is more generalized.

  14. 3.4 Mathematical Programming Method

  15. 3.4 Mathematical Programming Method • Deming (1943), York (1966) and Clutton-Brock (1967)

  16. We can obtain a “least-square cubic” 3.4 Mathematical Programming Method

  17. If no error in ; subject to errors=> => weighted regression of Y on X. If no error in ; subject to errors=> => weighted regression of X on Y. 3.4 Mathematical Programming Method

  18. If both and subject to errors => numerical method to find . Two independent variable case: 3.4 Mathematical Programming Method

  19. Kim (1995, 1997, 2010) The second-pass regression and the measurement error can be jointly presented as is a linear function of past idiosyncratic error terms prior to the cross-sectional regression at time . 3.5 Maxima likelihood Method Additional information

  20. Conditional on the market return, the ratio of the idiosyncratic error variance matrix to the measurement error variance matrix is , and The closed form solution 3.5 Maxima likelihood Method - continued

  21. 3.6 LISREL and MIMIC Methods • The linear simultaneous equation system is widely used in finance and accounting related research. However, a serious limitation of the simultaneous equation approach is EIV problem. For example, the theoretical determinants of capital structure in corporate finance can be attributed to unobservable constructs that are usually measured in empirical studies by a variety of observable indicators or proxies. These observable indicators or proxies can then be viewed as measures of latent variables with measurement errors. Maddala and Nimalendran (1996) show that the use of of these indicators as theoretical explanatory variables may cause EIV problems. Bentler  (1983) also emphasizes the estimated results of traditional simultaneous equation model has no meaning when variables have measurement errors. Therefore, the latent variable covariance structure model is provided and applied in corporate finance. Titman and Wessels (1988), Chang et al. (2009), and Yang et al. (2009), to mitigates the measurement problems of proxy variables, apply structure equation models (e.g. LISREL model and MIMIC model) to determining capital structure decision. Maddala and Nimalendran (1996) use structure equation model to examine the effect of earnings surprises on stock prices, trading volumes, and bid-ask spreads.

  22. 3.6 LISREL and MIMIC Methods - continued • Jöreskog and Goldberger (1975) develop  a structure equation model with multiple indicators and multiple causes of a single latent variable, MIMIC mode, and obtain maximum likelihood estimates of parameters. Figure 1 shows the path diagram that depicts a simplified MIMIC model in which variables in a rectangular box denote observable variables, while variables in an oval box are latent constructs. In this diagram, observable variables X1, X2, and X3 are causes of the latent variable η, while Y1, Y2, and Y3 are indicators of η. In our study, X’s are determinants of capital structure (η), which are then measured by Y’s. • Jöreskog and Sörbom (1989) show that the full structural equation (LIEREL model) can be restricted to be a MIMIC model. We first discuss with specifying the structural model, then we show how structural model can be restricted to a MIMIC model.

  23. 3.6.1 Structural model (LISEREL model) • A structural equation model is composed of two sub-models--structural sub-model and measurement sub-model. The structural model can be defined as  =  X + , (93) Y = y, (94) where Y is a vector of indicators of the latent variable , and X is a vector of causes of . • The latent variable  is linearly determined by a set of observable exogenous causes, X = (x1, x2, …, xq)’, and a disturbance . The latent variable , in turn, linearly determines a set of observable endogenous indicators, Y = (y1, y2, …, yp)’ and a corresponding set of disturbance,  = (1, 2, …, p)’. Stapleton (1978) further develops MIMIC with more latent variables.

  24. 3.6.2 MIMIC Model • Substituting Eq. (93) into Eq. (94), we obtain a reduced form: (95) • In structural equation modeling, the total effect of a cause variable on an indicator can be measured as the sum of the direct effect and the indirect effect. Since a MIMIC model is a reduced form of a structural equation model, the total effect of MIMIC model, denoted as ’ in Eq. (95), comes merely from the indirect effect. • Since the scale of the latent variable is unknown, the factor indeterminacy is a common problem in the MIMIC model, as in other structure equation models. We can obtain infinite parameter estimates from the reduced form by arbitrarily changing the scale of the latent variables. However, by fixing the scales of latent variables, one can solve the indeterminacy problem. Two methods are usually adopted to fix the scale of latent variables. One method is normalization in which a unit variance is assigned to each latent variable, while another method is to fix a nonzero coefficient at unity for each latent variable.

  25. 3.6.2 MIMIC Model - continued • In terms of estimation of the parameters, Jöreskog and Goldberger (1975) adopt the normalization method to deal with the factor indeterminacy problem and use maximum likelihood estimation method in structural equation modeling to estimate parameters. The maximum likelihood estimates for the parameters of the model are obtained at the minimization of the fit function as follows: F = log |||| + tr(S-1) – log||S|| - (p + q) (96) where  is the population covariance matrix; S is the model-implied covariance matrix; p is the number of exogenous observable variables; and q is the number of endogenous observable variables. Minimization of the fit function can be done by LISREL program provided by Jöreskog and Goldberger (1981).

  26. 4. Applications of Errors-in-Variables Models in Finance Research • Cost of Capital • Capital Asset Pricing Model • Capital Structure • Investment and Capital Asset Allocation

  27. 4.1 Cost of Capital The true relation between value and anticipated earnings, when replaced by the observable estimates, implies a simultaneous system of relationships: Where = (the true anticipated earnings); ui and wi are regression residuals; vi = Measurement errors associated with current earnings; Xi = Observable estimate of earnings derived from the accounting statements; Zij = Other relevant variables determining earnings.

  28. 4.2 Capital Asset Pricing Model • Theoretical Model - Sharpe (1964), Lintner (1965), and Mossian (1966) the expected returns on securities are a positive linear function of their market betas, and market betas have sufficient power to explain the cross-sectional expected returns. • Empirical Tests (Two-passes regression) fails to explain the cross-sectional average returns

  29. 4.2 Capital Asset Pricing Model - continued Why beta fails? Model misspecification Fama & French (1992), Carhart (1996), Chordia & Shivakumar (2006) suggest new risk factors can explain cross-sectional average returns. (size, B/M, price momentum factor, and earnings momentum factor.) Errors-in-variables problem Roll (1969 and 1977), Lee (1984) The errors-in-variables problem underestimates the market beta which is suffered measurement error and overestimates other risk factors with no measurement error. Correction: Lee (1973), Gibbons (1985), Shanken (1992), Kim (1995, 1997, 2011), etc.

  30. 4.2 Capital Asset Pricing Model - continued This study: • We investigate the effect of errors-in-variables problem on asset pricing test. - The errors-in-variables problem underestimates the market beta which is suffered measurement error and overestimates other risk factors with no measurement error. 2. We provide three alternative estimation methods for errors-in-variable problem. • (1) Grouping method; (2) Instrumental variable method; (3) Maxima likelihood method 3. We will reexamine asset pricing models after correcting errors-in-variable problem. • Market beta does not fail

  31. The Original CAPM Sharpe (1964), Lintner (1965),and Mossin (1966) Behavioral Finance Kahneman and Tversky (1979, Econometrica) Tversky and Kahneman (1992, Journal of Risk and Uncertainty) Levy (2010, European Financial Management) The Static CAPM (single-period) Existence of Equilibrium Hart (1974, Journal of Economic Theory) Nielsen (1989, Review of Economic Studies) The Dynamic CAPM (multi-period) Dividend and Taxation Effect Models Miller and Modigliani(1961,Journal of Business) Brennan (1970, National Tax Journal) Black and Scholes (1974,Journal of Financial Economics) Sasson and Kolodny(1976, The Review of Economics and Statistics) Miller and Scholes (1978,Journal of Financial Economics) Litzenberger and Ramaswamy (1979, Journal of Financial Economics) Morgan (1982,The Journal of Finance) Litzenberger and Ramaswamy (1982,The Journal of Finance) Hagiwara and Herce (1997, The American Economic Review) Equilibrium Models with Heterogeneity Investment Horizon Lee (1976, The Review of Economics and Statistics) Levhari and Levy (1977, The Review of Economics and Statistics) Lee, Wu, and Wei (1990, Journal of Financial and Quantitative Analysis) Supply-Side Effect Models Black (1976, American Economic Review) Grinols (1984, Journal of Finance) Lee, Tsai, and Lee (2009, Quarterly Review of Economics and Finance) Intertemporal CAPM－Merton Model Merton (1973,Econometrica) International CAPM Stulz (1981a, Journal of Finance) Stulz (1981b, Journal of Financial Economics) Stulz (1982, Journal of International Economics) Stulz (1984, Journal of International Business Studies) Equilibrium Models with Heterogeneity Beliefs and Investors Constantinides (1982, Journal of Business) Constantinides and Duffie (1996, Journal of Political Economy) Brav, Constantinides, and Geczy (2002, Journal of Political Economy) Basak (2005, Journal of Banking and Finance) Levy, Levy, and Benita (2006, Journal of Business) Intertemporal CAPM－Consumption-based Models Breeden (1979,Journal of Financial Economics) Campbell (1993, American Economic Review) Campbell and Cochrane (1999, Journal of Political Economy) Jagannathan and Wang (1996, Journal of Finance) Lettau and Ludvigson (2001a, Journal of Finance) Lettau and Ludvigson (2001b, Journal of Political Economy) Lewellen and Nagel (2006, Journal of Financial Economics) Balvers and Huang (2009, Journal of Financial and Quantitative Analysis) Skewness Effect Models Borch (1969, Review of Economics Studies) Feldstein (1969, Review of Economics Studies) Jean (1971,Journal of Financial and Quantitative Analysis) Tsiang (1972, American Economic Review) Ingersoll (1975,Journal of Financial and Quantitative Analysis) Schweser (1978,Journal of Financial and Quantitative Analysis) Liquidity-based Models Pastor and Stambaugh (2003, Journal of Political Economy) Acharya and Pedersen (2005, Journal of Financial Economics) Yoel(2009,working paper) Intertemporal CAPM－Production-based Models Balvers, Cosimano, and McDonald (1990, Journal of Finance) Cochrane (1991, Journal of Finance) (1996, Journal of Political Economy) Balvers and Huang (2007, Journal of Financial Economics) Figure1. Flow Chart

  32. The Dynamic CAPM (multi-period) Supply-Side Effect Models Black (1976, American Economic Review) Grinols (1984, Journal of Finance) Lee, Tsai, and Lee (2009, Quarterly Review of Economics and Finance) Intertemporal CAPM－Merton Model Merton (1973,Econometrica) International CAPM Stulz (1981a,Journal of Finance) Stulz (1981b,Journal of Financial Economics) Stulz (1982,Journal of International Economics) Stulz (1984,Journal of International Business Studies) Chang and Hung (2000, Review of Quantitative Finance and Accounting) Intertemporal CAPM－Consumption-based Models Breeden (1979,Journal of Financial Economics) Campbell (1993, American Economic Review) Campbell and Cochrane (1999, Journal of Political Economy) Jagannathan and Wang (1996, Journal of Finance) Lettau and Ludvigson (2001a, Journal of Finance) Lettau and Ludvigson (2001b, Journal of Political Economy) Lewellen and Nagel (2006, Journal of Financial Economics) Balvers and Huang (2009, Journal of Financial and Quantitative Analysis) Intertemporal CAPM－Production-based Models Balvers, Cosimano, and McDonald (1990, Journal of Finance) Cochrane (1991, Journal of Finance) (1996, Journal of Political Economy) Balvers and Huang (2007, Journal of Financial Economics) Figure2. The Dynamic CAPM

  33. The Static CAPM (single-period) Dividend and Taxation Effect Models Miller and Modigliani(1961,Journal of Business) Brennan (1970, National Tax Journal) Black and Scholes (1974,Journal of Financial Economics) Sasson and Kolodny(1976, The Review of Economics and Statistics) Miller and Scholes (1978,Journal of Financial Economics) Litzenberger and Ramaswamy (1979, Journal of Financial Economics) Morgan (1982,The Journal of Finance) Litzenberger and Ramaswamy (1982,The Journal of Finance) Hagiwara and Herce (1997, The American Economic Review) Equilibrium Models with Heterogeneity Investment Horizon Lee (1976, The Review of Economics and Statistics) Levhari and Levy (1977, The Review of Economics and Statistics) Lee, Wu, and Wei (1990, Journal of Financial and Quantitative Analysis) Equilibrium Models with Heterogeneity Beliefs and Investors Constantinides (1982, Journal of Business) Constantinides and Duffie (1996, Journal of Political Economy) Brav, Constantinides, and Geczy (2002, Journal of Political Economy) Basak (2005, Journal of Banking and Finance) Levy, Levy, and Benita (2006, Journal of Business) Skewness Effect Models Borch (1969, Review of Economics Studies) Feldstein (1969, Review of Economics Studies) Jean (1971,Journal of Financial and Quantitative Analysis) Tsiang (1972, American Economic Review) Ingersoll (1975,Journal of Financial and Quantitative Analysis) Schweser (1978,Journal of Financial and Quantitative Analysis) Liquidity-based Models Pastor and Stambaugh (2003, Journal of Political Economy) Acharya and Pedersen (2005, Journal of Financial Economics) Yoel(2009,working paper) Figure 3. The Static CAPM

  34. In testing CAPM, However, beta is unknown. Two passes regression method to test CAPM First-pass: estimate beta coefficient (market model). Second-pass: using estimated beta coefficient to test CAPM. If independent variable is measured with error, the estimated coefficient (market premium) by OLS is underestimated. => Empirical CAPM test reject the linear relationship between expected return and beta risk might be due to errors-in-variable problem. Errors-in-Variables Problem in testing CAPM

  35. 4.3Capital Structure • Titman and Wessels (1988), Chang et al. (2009), and Yang et al. (2009) use structure equation models (e.g. LISREL model and MIMIC model) to mitigates the measurement problems of proxy variables when working on capital structure theory. Titman and Wessel (1998) use LISREL method to investigate determinates of capital structure. In the structure equation model, they use 15 indicators associated with eight latent variables and set 105 restrictions on the coefficient matrix. Empirical results, however, do not support for four of eight propositions on the determinants of capital structure.Specifically, their results show that a firm’s capital structure is not significantly related to its non-debt tax shields, volatility of earnings, collateral value of assets, and future growth. One possible reason for the poor results is that the indicators used in the empirical study do not adequately reflect the nature of the attributes suggested by financial theory.

  36. 4.3 Capital Structure - continued • Chang et al. (2009), therefore, apply a MIMIC model with refined indicators to reexamine Titman and Wessel’s (1998) work on determinants of capital structure. Chang et al. (2009) examine seven indicator factors as follows: growth, profitability, collateral value, volatility, non-debt tax shields, uniqueness, and industry. Their empirical results show that the growth is the most influential determinant on capital structure, followed by profitability, and then collateral value. Under a simultaneous cause-effect framework, their seven constructs as determinants of capital structure have significant effects on capital structure decision. • Yang et al. (2009) apply a LISREL model to find determinants of capital structure and stock returns, and estimate the impact of unobservable attributes on capital structure decision and stock returns. Using leverage ratio and stock returns as two endogenous variables and 11 latent factors as exogenous variables, Yang et al. (2009) find that stock returns, expected growth, uniqueness, asset structure, profitability, and industry classification are main determinants of capital structure, while leverage, expected growth, profitability, firm value, and liquidity can explain stock returns. The capital structure and stock return, in addition, are mutually determined by each other.

  37. 4.4 Investment and Capital Asset Allocation • Modern q theory developed by Lucas and Prescott (1971) and Mussa (1977) shows that the shadow value of capital, marginal q, is the firm manager’s expectation of the marginal contribution of new capital goods to future profits. Marginal q, therefore, should summarize the effects of all factors relevant to the investment decision. However, empirical work in testing association between the investment decision and cash flow is inconsistent to the q theory (e.g. Fazzari et al., 1988). Lucas and Prescott (1971) and Hayashi (1982) show the equality of marginal q with average q under the assumptions of constant returns to scale and perfect competition. In addition, output, sales, and, measures of internal funds have statistically significant explanation in determining investment decision (e.g. Fazzari et al., 1988; Schaller (1990); Blundell et al., 1992; and Gilchrist and Himmelberg, 1995).

  38. 4.4 Investment and Capital Asset Allocation - continued • If financial markets valuation of the capital will be equal to the manager’s valuation, average q should equal an observable value, Tobin’s q, defined as the ratio of the market value to the replacement value. Most empirical studies therefore use Tobin’s q as a proxy for marginal q to test the q theory of investment. Erickson and Whited (2000) argue that the measurement error of marginal q can result different implications in empirical q models. They therefore incorporate an EIV model to reexamine the empirical work done by Fazzari et al. (1988). By using generalized method of moments (GMM), Erickson and Whited (2000) obtain consistent estimators that the information contained in the third- and higher-order moments of the joint distribution of the observed regression variables. The estimator precision and consistency can be increased by exploit the information afforded by an excess of moment equations over parameters. Results show that cash flow does not affect firms’ financial decision, even for financially constrained firms, and the q theory is held if measurement error is taken into account. • Almeida et al. (2010) use Monte Carlo simulations and real data to compare the performance of alternative approaches dealing with measurement error problem. In Monte Carlo simulations, they find estimators of GMM proposed by Erickson and Whited (2000) are biased for both mismeasured and well-measured regressors when the data have individual-fixed effects, heteroscedasticity, or no high degree of skewness.Erickson and Whited (2000). In contrast, the instrumental variable method results fairly unbiased estimators under those same conditions. Almeida et al. (2010) further examine empirical investment equation introduced by Fazzari et al. (1988) by using GMM and instrumental variable method. They conclude that estimators from GMM are unstable across different specifications and not economically meaningful, while estimators from a simple instrumental method are robust and conform to q theory.

  39. Conclusion • In this paper, we study how EIV problem affect the estimation of regression models, extent exiting EIV estimation methods, and discuss the empirical research applying these estimation methods in deal with measurement error problem. We first show how EIV problems affect estimators in the regression model. More specifically, in a multivariate regression, we show that the EIV leads an underestimation of the independent variable with measurement error and an overestimation of the independent variable without measurement error. We further provide six alternative estimation methods dealing with EIV problem. Classical method, grouping method, instrumental variable method, mathematical programming method, maxima likelihood method, and LISREL method are discussed in detailed. Finally, we investigate how EIV problem can affect empirical results in issues of cost of capital, asset pricing models, capital structure, and investment decision and how alternative EIV methods have been used to correct estimation bias in those issues. We suggest future studies should pay more efforts on dealing with EIV and obtain robust empirical results from EIV models.

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