COMPUTING MARGINAL EFFECTS IN THE BOX COX MODEL

Transcription

1 ECONOMETRIC REVIEWS Vol. 21, No. 3, pp , 2002 COMPUTING MARGINAL EFFECTS IN THE BOX COX MODEL Jason Abrevaya Department of Economics, Purdue University, Indiana ABSTRACT This paper considers computation of fitted values and marginal effects in the Box Cox regression model. Two methods, (1) the smearing technique suggested by Duan (see Ref. 10) and (2) direct numerical integration, are examined and compared with the naive method often used in econometrics. Key Words: Marginal effects; Box Cox model JEL Classification: C13, C21 1 INTRODUCTION The Box Cox model [1] has been widely used in the statistics and econometrics literature to model a non-negative dependent variable. The model is given by y lþ i ¼ x 0 i b þ e i i ¼ 1;...; nþ 1Þ where 8 < y l 1 y lþ ¼ if l 6¼ 0 l 2Þ : ln y if l ¼ 0 If e is assumed to be normally distributed e N0; s 2 ÞÞ, maximum likelihood estimation can be used to estimate b; l; sþ. It is well known that the normality assumption is at best an approximation since it can not be strictly true DOI: =ETC Copyright # 2002 by Marcel Dekker, Inc (Print); (Online)

2 384 ABREVAYA for l 6¼ 0. Poirier [2] has suggested the use of a truncated normal distribution in the likelihood function to explicitly restrict e i to be large enough that y i must be positive. Others [3] have proposed alternative transformation models in which normality could theoretically occur. If e is non-normal, the maximum likelihood estimates of b; l; sþ are generally inconsistent. As long as e is independent of x, however, Amemiya and Powell [4] have shown that non-linear instrumental variables estimation consistently estimates b and l. This paper has nothing new to add with regard to estimation of b and l in the Box Cox model. Instead, the focus is on drawing useful inference from these parameters. As a result, we assume that the researcher has obtained consistent estimates ^b and ^l (and, under the normality assumption, ^s). In the discussion that follows, we assume that l > 0. 1 Poirier and Melino [5] derived expressions for the conditional expectation and marginal effects in the Box Cox model (under the normality assumption). This note focuses on obtaining valid predictions, marginal effects, and (perhaps most importantly) standard errors based on such expressions. Although it has long been known that the error disturbance does not disappear from such expressions in the Box Cox model (except when l ¼ 1), the standard practice in empirical work is to ignore the error disturbance. To make inference about y, first re-write (1) to isolate y on the left-hand-side: y ¼½lx 0 b þ eþþ1š 1=l 3Þ The conditional expectation of y given x follows immediately: E½ yjxš ¼Ef½lx 0 b þ eþþ1š 1=l j xg 4Þ where the expectation is taken over the distribution of e. The conditional expectation can also be written in terms of the c.d.f. F of the error disturbance, E½ y j xš ¼ ½lx 0 b þ eþþ1š 1=l dfe j xþ 5Þ Unlike standard linear regression problems, the error disturbance does not drop out of the expectation and must be accounted for in making conditional predictions. Several authors have acknowledged this problem. In fact, Wooldridge [6] has proposed an alternative model to the Box Cox model which still allows for flexible functional forms but specifies that the error disturbance does not appear in the conditional expectation. This modeling technique is interesting, and we do not aim to argue whether the Box Cox model or the model of Wooldridge [6] is more appropriate (see Ref. 7 for empirical comparisons of the two approaches). Instead, given the prevalence of Box Cox estimation, we focus on how inference should be done when the Box Cox model is chosen. Strangely, the standard practice is to ignore the error 1 If l < 0, the re-transformed variable may not have finite expectation in Eq. (4). See Poirier and Melino [5] for further discussion.

3 BOX COX MODEL 385 disturbance in (4) altogether and estimate E½y j xš by ½^lx 0 ^bþþ1š 1=^l (see, for example, Refs. 8 and 9). 2 For instance, the statistical package Stata ignores the error disturbance in computing fitted values from a Box Cox model. Instead, one can estimate the conditional expectation by plugging the estimates ^b and ^l into (4) and numerically integrating over the distribution of e. This approach, which can be applied to marginal-effect estimation as well, is the focus of this paper. Since the distribution of e is not known with certainty, an estimated error distribution must be used in the numerical integration. Under the normality assumption, integration using the normal p.d.f. can be used. More generally, the smearing technique of Duan, [10] which uses estimated residuals to approximate the error distribution, can be used. The smearing technique proves to be an extremely simple method to employ in practice, though it has not been previously considered in the economics literature. The marginal effects of the covariates in the Box Cox model are not readily discernible from the parameter estimates ^l and ^b (unlike the standard linear regression model). Consider the marginal effect of a covariate x j on the conditional expectation of y given x, qe½ y j xš qx j ¼ b j ½lx 0 b þ eþþ1š 1 lþ=l dfeþ which is obtained by differentation of (5). The parameter b j is the element of b corresponding to covariate x j. Again, numerical integration will be needed to estimate marginal effects based on (6). Ignoring e results in biased estimates of the marginal effects. 6Þ 2 FITTED VALUES AND MARGINAL EFFECTS Let y denote the full vector of parameters in the Box Cox model (which, depending on the estimation technique, may contain parameters describing the error distribution F). Let ^y denote the estimate of y. We first consider estimation of the conditional expectation of y given x. Based on (4) and (5), an estimator of the conditional expectation is given by ^yx; ^yþ ¼ ½^lx 0 ^b þ eþþ1š 1=^l d ^FeÞ 7Þ where ^F is an estimate of the true error distribution F. We consider two alternative assumptions on the error disturbance: 3 2 Showalter [9] accounts for the error disturbance in the true conditional mean function, but uses the fitted value formula without the error disturbance. As he notes, a forecast bias results from the omission of the disturbance from the fitted value formula. 3 Like the normality assumption, Assumption 2 can also not be literally true unless the support of x and e are restricted in some way.

4 386 ABREVAYA where Assumption 1 The distribution of e, conditional on x, isn0; s 2 Þ for all x. Assumption 2 e and x are independent. Under Assumption 1 (normality), an estimator (proposed by Berndt et al. [7] )is ^y 1 x; ^yþ ¼ ½^lx 0 ^b þ eþþ1š 1=^l f t;^seþ de 8Þ e^l;^b;xþ e^l; ^b; xþ fe: ^lx 0 ^b þ eþþ1 > 0g 9Þ where f t;^s is the p.d.f. of a normal distribution with standard deviation ^s truncated by the limits of integration. The limits of integration are required for (8) to be welldefined. Note that maximum likelihood estimation of the Box Cox model does not require the truncated normal restriction, but the restriction is implicit in the model from (1) since y i is a non-negative number. Under the weaker Assumption 2, an estimator suggested by Duan [10] is given by ^y 2 x; ^yþ ¼ 1 n X n i¼1 ½^lx 0 ^b þ ^e i Þþ1Š 1=^l where n is the number of observations for which ^e i 2 e^l; ^b; xþ and the summation is taken over these observations. If the Box Cox model is appropriate, there should be very few (if any) observations excluded from the summation in (10). In most empirical examples of interest for which the Box Cox model is used, the size of the index values x 0 b is much larger in magnitude than the size of the error disturbances. If many observations have ^e i 62 e^l; ^b; xþ, the researcher should determine whether or not the Box Cox model is appropriate. 4 Equation (10) can also be written in terms of the observed data: ^y 2 x; ^yþ ¼ 1 n ¼ 1 n X n i¼1 X n i¼1 f^y½x 0 ^b þy i ^lþ x 0 i ^bþš þ 1g 1=^l f^l½y i ^lþ þx x i Þ 0 bšþ1g 1=^l 10Þ 11Þ The estimators in (8) and (10) yield point estimates of the true conditional expectation E½y j xš. The asymptotic variance of the estimators can be derived by application of the delta method : 4 For ^l > 0, an alternative way to ensure that (10) is well-defined is to consider those observations for which x 0 ^b i > x 0 ^b.

5 BOX COX MODEL 387 Var ½ ^y x; ^yþš ¼ ½H y^y x; ^yþšvar^yþ½h y^y x; ^yþš 0 for ¼ 1; 2 12Þ where H y denotes the gradient with respect to the parameter vector. To consistently estimate the asymptotic variance, a consistent estimate of the asymptotic variance for ^y can be plugged in for Var^yÞ and the derivatives in (12) can be computed numerically. Estimating marginal effects is similar to estimating conditional expectations. Based on (6), an estimator of the marginal effect of covariate x j on the conditional expectation of y given x is ^m j x; ^yþ ¼^b j ½^lx 0 ^b þ eþþ1š 1 ^lþ=^l d ^FeÞ 13Þ where ^F is an estimate of the true error distribution F. For normal disturbances and general disturbances, the analogues to (8) and (11) for marginal effects are ^m j;1 x; ^yþ ¼^b j ½^lx 0 ^b þ eþþ1š 1 ^lþ=^l f t;^seþ de 14Þ and e^l;^b;xþ ^m j;2 x; ^yþ ^b j X n ¼ f^l½y ^lþ i þx x i Þ 0 ^bšþ1g 1 ^lþ=^l 15Þ n i¼1 The asymptotic variance of ^m j; x; ^yþ for ¼ 1; 2, given by the delta method, is Var ½ ^m j; x; ^yþš ¼ ½H y ^m j; x; ^yþš Var^yÞ½H y ^m j; x; ^yþš 0 ; which can be consistently estimated. 5 16Þ 3 EMPIRICAL EXAMPLES In this section, the techniques described in the previous sections are applied to two datasets for which the Box Cox model has been used. These datasets were analyzed by Berndt et al. [7] 6 The first dataset, called CPS78, is a random sample of 550 observations from the May 1978 Current Population Survey. The dependent variable is hourly wage (WAGE); the independent variables are years of education (EDUC), years of experience (EXP, defined as age minus EDUC minus 6), experience squared (EXP2), a race indicator variable (RACE, de.fined as one if the individual is non-white and non-hispanic), and a gender indicator variable (FEMALE). The second dataset, called COLE, consists of 91 observations pertaining to mainframe computer hard disk drives between 1972 and 1984 that 5 Alternatively, one could use the bootstrap in order to estimate standard errors for either the fitted values or the marginal effects. 6 The data are available on the companion diskette to Berndt. [12]

6 388 ABREVAYA were compiled by Cole et al. [11] The dependent variable is list price (PRICE); the independent variables are drive speed (SPEED), 7 drive capacity in megabytes (CAP), and year indicator variables. These data have been chosen since the Box Cox model is often used in conjunction with wage regressions and hedonic pricing regressions. CPS78 Results: Table 1 reports the parameter estimates for the CPS78 wage regression using three different estimators. The first two columns, MLE and NLIV, estimate the wage regression under the Box Cox specification. The MLE estimates are the standard maximum likelihood estimates under the normality assumption. The NLIV column gives the nonlinear instrumental variables estimates, [4] using EDUC 2, EXP 4,EDUC* EXP, and EDUC * EXP 2 as instruments. 8 Both methods yield similar parameter estimates. The estimated ^l is not significantly different from zero (log-wage specification) in either case. As a comparison, then, OLS estimates from a regression of log(wage) on the independent variables are reported in the third column. The final rows of Table 1 give pseudo R-squared values for the different methods of computing predicted values of y ¼ WAGE. Following the recommendation by Berndt et al., [7] we define the pseudo R-squared value as Pi ~R 2 1 y i ^y i Þ 2 Pi y i yþ 2 : 17Þ Since the fitted values depend on the method used, multiple pseudo R-squared values are reported for each column in the table. This measure is a useful comparison between the methods since it gives no particular advantage to any of Table 1. MLE CPS78 Parameter Estimates NLIV OLS (Log-Linear) l (0.1034) (0.3318) Const (0.0973) (0.1037) (0.0973) EDUC (0.0065) (0.0079) (0.0066) EXP (0.0041) (0.0051) (0.0048) EXP (0.0001) (0.0001) (0.0001) RACE (0.0658) (0.0565) (0.0561) FEMALE (0.0360) (0.0355) (0.0352) s (0.0136) R 2 (naive) 28.8% 28.7% 28.8% R 2 (normal) 31.5% 31.4% R 2 (indep.) 30.7% 30.5% 7 SPEED is defined as ACT=[ACCT þ ROTD þ (2000=TR)], where ACT is the number of actuators (read=write heads), ACCT is the average access time (in milliseconds), ROTD is the average rotation delay (in milliseconds), and TR is the transfer rate (in kilobytes per second). See Cole et al. [11] for further details. 8 Amemiya and Powell [4] recommend using cross-products and squares of variables as instruments.

7 BOX COX MODEL 389 them. The three methods in the table are: naive (error disturbance ignored), normal (Assumption 1; integration using the normal distribution), and indep. (Assumption 2; smearing method). For the OLS regression, the pseudo R-squared value for the normal method is reported. For this method, fitted values were based on the conditional expectation, E½ y j xš ¼expx 0 b þ s 2 =2Þ. As expected, the naive method for computing fitted values is outperformed by the methods discussed in the previous section. In this empirical example, the normal method for the MLE estimates does a slightly better job in explaining the variation in wages than the smearing technique. Table 2 reports fitted-value and marginal-effect estimates based on the parameter estimates from Table 1. Since these estimates depend on the values of the independent variables, four different cases are considered: Case 1: EDUC ¼ 12, EXP ¼ 18, RACE ¼ 0, FEMALE ¼ 0 Case 2: EDUC ¼ 12, EXP ¼ 18, RACE ¼ 0, FEMALE ¼ 1 Case 3: EDUC ¼ 12, EXP ¼ 18, RACE ¼ 1, FEMALE ¼ 0 Case 4: EDUC ¼ 12, EXP ¼ 18, RACE ¼ 1, FEMALE ¼ 1 These cases correspond to 36-year-old high-school graduates with the four possible combinations of the RACE and FEMALE indicator variables. 9 The second column of the table describes the method used to calculate the estimates. MLE=NLIV=OLS indicates which of the parameter estimates from Table 1 is used. Normal=indep.=naive refers to the treatment of the error disturbance. For instance, the three MLE estimates of the fitted value are given by Eq. (8) (normal), Eq. (10) 0 1=^l (indep.), and ½^lx ^bþþ1š (naive). The OLS estimates are based on the conditional expectation from above. The naive estimates are, except for a few instances in Case 4, about 5 10% lower than the estimates that take into account the error disturbance. For instance, the Case 1 fitted values (for hourly wage) are $6:72 6:75 across the methods that account for the error disturbance and $6:24 6:40 across the naive methods. Interestingly, the indep. method yields quite similar results to the normal method for the MLE estimates; the only differences seem to be in Case 4, which is toward the extreme of the estimated index values. COLE Results: Table 3 reports the parameter estimates from MLE and NLIV estimation of the Box Cox model on the COLE data. The NLIV method used SPEED 2, CAP 2, and SPEED * CAP as instruments. Unlike the CPS78 results, the L estimates are somewhat different between the two methods. Since the MLE ^l is not significantly different from one (linear model), ignoring the error disturbance in calculating fitted values and marginal effects may not matter much. This intuition is supported by the results in Table 4, which reports fitted values and marginal effects at the means of SPEED and CAP (0.06 and 570, respectively) for 9 Fitted values and marginal effects were computed at all possible values of the independent variables. The qualitative results were the same as those reported here. Only four specific cases are reported in the interest of space.

8 390 ABREVAYA Table 2. CPS78 Fitted Values and Marginal Effects Case 1 Case 2 Case 3 Case 4 Fitted value MLE (normal) (0.1776) (0.1774) (0.3853) (0.2534) MLE (indep.) (0.1443) (0.1525) (0.3698) (0.3559) MLE (naive) (0.1726) (0.1689) (0.3559) (0.2605) NLIV (indep.) (0.1870) (0.1757) (0.3303) (0.2528) NLIV (naive) (0.2821) (0.2379) (0.3925) (0.2713) OLS (normal) (0.1992) (0.1707) (0.3376) (0.2430) OLS (naive) (0.1818) (0.1564) (0.3112) (0.2240) EDUC MLE (normal) (0.0440) (0.0330) (0.0467) (0.0315) MLE (indep.) (0.0420) (0.0318) (0.0453) (0.0325) MLE (naive) (0.0395) (0.0323) (0.0424) (0.0320) NLIV (indep.) (0.0421) (0.0440) (0.0422) (0.0523) NLIV (naive) (0.0409) (0.0570) (0.0503) (0.0646) OLS (normal) (0.0434) (0.0316) (0.0451) (0.0322) OLS (naive) (0.0401) (0.0292) (0.0416) (0.0297) EXP MLE (normal) (0.0117) (0.0087) (0.0112) (0.0075) MLE (indep.) (0.0111) (0.0083) (0.0107) (0.0077) MLE (naive) (0.0103) (0.0082) (0.0100) (0.0075) NLIV (indep.) (0.0111) (0.0118) (0.0106) (0.0130) NLIV (naive) (0.0112) (0.0146) (0.0126) (0.0158) OLS (normal) (0.0121) (0.0091) (0.0112) (0.0082) OLS (naive) (0.0112) (0.0084) (0.0103) (0.0076)

9 BOX COX MODEL 391 Table 3. COLE Parameter Estimates MLE NLIV l (0.1815) (0.4812) Const (0.1341) (0.1612) SPEED (3.8043) (4.3484) CAP (0.0002) (0.0005) s (0.0106) R 2 (naive) 76.7% 76.3% R 2 (normal) 77.1% R 2 (indep.) 77.1% 76.9% three different years (1972, 1976, 1980). The naive estimates are quite similar to the other estimates. The major differences in Table 4 are between the MLE and NLIV methods, rather than whether or not the naive method is used. The pseudo R-squared values are again lower for the naive method, as expected. The improvement in pseudo R-squared is not as large as in the CPS example since the model is (based on the MLE estimates) closer to linear for the COLE data. 4 CONCLUDING REMARKS This paper has considered estimation of fitted values and marginal effects in the Box Cox model, where the error disturbance does not disappear when expectations are taken. Since the methods proposed are very straightforward to Table 4. COLE Fitted Values and Marginal Effects Fitted value MLE (normal) 38,260 (4,757) 36,246 (3,705) 30,330 (1,908) MLE (indep.) 38,260 (4,262) 36,246 (3,348) 30,330 (2,007) MLE (naive) 38,138 (4,886) 36,120 (3,821) 30,184 (1,887) NLIV (indep.) 48,514 (3,577) 43,780 (3,498) 28,734 (6,047) NLIV (naive) 48,528 (3,661) 43,890 (3,606) 28,808 (5,285) SPEED (10 6 ) MLE (normal) 4.16 (1.24) 4.13 (1.20) 4.03 (1.16) MLE (indep.) 4.16 (1.24) 4.13 (1.20) 4.03 (1.16) MLE (naive) 4.17 (1.24) 4.14 (1.21) 4.05 (1.17) NLIV (indep.) 2.14 (1.01) 2.41 (1.11) 4.29 (3.27) NLIV (naive) 2.10 (1.00) 2.35 (1.09) 3.70 (1.53) CAP MLE (normal) 4.73 (7.36) 4.70 (7.35) 4.58 (7.22) MLE (indep.) 4.73 (7.36) 4.70 (7.35) 4.58 (7.22) MLE (naive) 4.74 (7.37) 4.71 (7.36) 4.60 (7.24) NLIV (indep.) (5.61) (6.76) (33.33) NLIV (naive) (5.47) (6.42) (17.89)

10 392 ABREVAYA implement, these methods should be used by researchers in order to report empirical results from a Box Cox model (in particular, marginal effects along with their standard errors). The examples considered in the previous section demonstrate the improvement in fitted values (higher pseudo R-squared values) with the proposed methods. For the wage-regression example, there are substantial differences between the naive approach and the proposed methods in terms of the estimated fitted values and marginal effects. This finding was expected since the transformation of wages for which the regression model holds is quite non-linear (close to logarithmic based on the MLE results), which is when ignoring the error disturbance will result in a larger bias. Future work might extend the methods of this paper to models having heteroskedastic errors. With a parametrized model of the heteroskedasticity [e.g., e N0; z 0 gþš, the same approach described for normally distributed errors can be extended rather easily. With general heteroskedasticity, [13] a more complicated approach would be required. ACKNOWLEDGMENTS Financial support for this project was provided by the University of Chicago GSB faculty research fund. John Whitley provided outstanding research assistance. This paper has been greatly improved by the comments of the Editor (Esfandiar Maasoumi) and two referees. REFERENCES 1. Box, G.E.P.; Cox, D.R. An Analysis of Transformations (with Discussion). J. Roy. Statistical Society (Series B) 1964, 26, Poirier, D.J. The Use of the Box-Cox Transformation in Limited Dependent Variable Models. J. Amer. Statistical Assoc. 1978, 73, Bickel, P.J.; Doksum, K.A. An Analysis of Transformations Revisited. J. Amer. Statistical Assoc. 1981, 76, Amemiya, T.; Powell, J.L. A Comparison of the Box-Cox Maximum Likelihood Estimatior and the Non-linear Two Stage Least Squares Estimator. J. Econometrics 1981, 17, Poirier, D.J.; Melino, A. A Note on the Interpretation of Regression Coefficients within a Class of Truncated Distributions. Econometrica 1978, 46, Wooldridge, J.M. Some Alternatives to the Box-Cox Regression Model. Int. Econ. Rev. 1992, 33, Berndt, E.R.; Showalter, M.H.; Wooldridge, J.M. An Empirical Investigation of the Box-Cox Model and a Nonlinear Least Squares Alternative. Econometric Rev. 1993, 12, Greene, W.H. Econometric Analysis; Prentice-Hall: New Jersey, Showalter, M.H. A Monte Carlo Investigation of the Box-Cox Model and a Nonlinear Least Squares Alternative. Rev. Econ. Statist. 1994, 76,

11 BOX COX MODEL Duan, N. Smearing Estimate: A Nonparametric Retransformation Method. J. Amer. Statistical Assoc. 1983, 78, Cole, R.; Chen, Y.C.; Barquin-Stolleman, J.A.; Dulberger, E.; Helvacian, N.; Hodge, J.H. Quality-Adjusted Price Indexes from Computer Processors and Selected Peripheral Equipment. Sur. Curr. Bus. 1986, 66, Berndt, E.R. The Practice of Econometrics: Classic and Contemporary; Addison- Wesley: Massachusetts, Manning, W.G. The Logged Dependent Variable, Heteroskedasticity, and the Retransformation Problem. J. Health Econ. 1998, 17,