Exact interpretation of dummy variables in semilogarithmic equations

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1 Econometrics Journal (2002), volume 5, pp Exact interpretation of dummy variables in semilogarithmic equations KEES JAN VAN GARDEREN AND CHANDRA SHAH Department of Quantitative Economics, Faculty of Economics and Econometrics, University of Amsterdam, 1018 WB, The Netherlands and Department of Economics, University of Bristol Monash University-ACER Centre for the Economics of Education and Training, Faculty of Education, Monash University, Victoria 3800, Australia Received: July 2001 Summary This paper considers the percentage impact of a dummy variable regressor on the level of the dependent variable in a semilogarithmic regression equation with normal disturbances. We derive an exact unbiased estimator, its variance, and an exact unbiased estimator of the variance. The main practical contribution lies in a convenient approximation for the unbiased estimator of the variance, which can be reported together with Kennedy s approximate unbiased estimator of the percentage change. The two approximations are very simple, yet highly reliable. The results are applied to teacher earnings and further illustrated by examples from the literature. Keywords: Loglinear model, Dummy variable, Percentage change, Exact distribution theory, Hypergeometric function, Teacher earnings. 1. INTRODUCTION Halvorsen and Palmquist (1980) pointed out a common error in the interpretation of the coefficients of dummy variables in semilogarithmic regression equations, and gave specific examples from the literature on discrimination, education, and income to illustrate the point. Unlike that of a continuous variable, the coefficient of a dummy variable, multiplied by 100, is not the usual percentage effect of that variable on the dependent variable. In a typical model of this kind with dummy variables D j and continuous variables X i : ln Y = a + i b i X i + j c j D j + ε, (1.1) the percentage effect on Y of a small change in X i is simply 100b i since p i = Y Y = 100 ln Y = 100b i. (1.2) X i X i. Published by Blackwell Publishers Ltd, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA, 02148, USA.

2 150 Kees Jan van Garderen and Chandra Shah In the case of a dummy variable, the percentage change in Y, from Y 0 to Y 1, say, for a discrete change in D j from 0 to 1, should be calculated instead as 100(Y 1 Y 0 )/Y 0, which, by straightforward use of (1.1), leads to: 1 p j = 100 (exp {c j } 1). (1.3) This is the case if the value of c j is known. In practice, however, c j is unknown and has to be estimated. Kennedy (1981) pointed out that the use of transformation (1.3) results in a biased estimator for p j. This is true whether the estimator of c (the subscript is dropped for clarity) is biased or not due to the randomness of the estimator and the nonlinearity of transformation (1.3). If the error term in (1.1) is assumed to be normally distributed then the ordinary least squares (OLS) estimator of, ĉ, is efficient and unbiased. Goldberger (1968) noted that the expected value of exp {ĉ} is exp {c V (ĉ)}, where V (ĉ) is the variance of ĉ. This led Kennedy to suggest ˆp = 100 (exp {ĉ 1 2 ˆV (ĉ)} 1), (1.4) for estimating p, where ˆV (ĉ) is the OLS estimate of the variance of ĉ. In essence, Kennedy s approach is similar to that of Halvorsen and Palmquist (HP), in that unknown quantities in the function of interest are replaced by their estimates and the estimator is still biased. This raises three important issues. First, although (1.4) was likely to be an improvement on simply substituting ĉ for c in (1.3), it was unclear at the time how much better the approximation was. This issue was addressed by Giles (1982), who gave a series expression for the exact unbiased estimator of p based on results from Goldberger (1968). He showed that in many practical situations the difference between Kennedy s and the unbiased estimator was negligible, and because of the computational advantage, concluded that Kennedy s estimator had the advantage in practical applications. Second, it is important in the context of estimation to indicate a measure of spread, such as the standard error (SE), of the estimator. This issue has not been addressed previously. Third, and related, is the issue of how good the estimators are in terms of being distributed closest to the true value. Although unbiasedness is a desirable property it is not generally true that unbiased estimators are more tightly distributed around the true parameter value, even if the minimum variance unbiased estimator is used. It is still possible that a biased estimator exists with a smaller expected loss; for example, with a smaller mean squared error (MSE). In general, there does not exist an estimator that minimizes the MSE for all parameter values. 2 Therefore, the class of estimators has to be restricted, and, like Kennedy, we have chosen unbiasedness as the criterion to restrict the class of estimators. It would nevertheless be interesting to know if other standard estimators, which are biased, have smaller MSE. In this paper we address these three issues, and also derive the exact minimum variance unbiased estimator for p that is an improvement on both (1.3) and (1.4). The estimator is numerically 1 In case of only one dummy variable we would simply have: { }/ { Y 1 /Y 0 1 = exp a + j b j X j + c 1 + ε exp a + j b j X j + c 0 + ε } 1 = exp {c} 1. 2 The nonsense estimator, θ = 0.1, has zero expected loss when θ = 0.1. Any other estimator would have expected (convex) loss greater than zero for θ = 0.1. Unbiasedness is one way of restricting the estimators to a class from which an optimal one can be selected. Alternative restrictions are discussed in, e.g. Lehmann (1986, p. 12).

3 Exact interpretation of dummy variables in semilogarithmic equations 151 the same as that of Giles, but rather than put it in terms of an infinite series like he does, we express it in terms of a standard function. Next, we derive the exact SE for the estimator, something that has not been done previously. We express it in terms of the same standard function and derive the exact unbiased estimator for the variance. Finally, we give a simple and highly reliable approximation for this unbiased variance estimator, using only the exponential function. We apply these results to teacher earnings and estimate the percentage effects of subjects taught, sex, and special responsibilities, and provide SEs and unbiased estimates for the variance of these estimated percentage changes. The estimates are compared with those obtained using the Naive estimator, 100ĉ, based on (1.2) and the estimates based on (1.3) and (1.4). Since these three estimators are biased, we use the MSE, which takes into account both the bias and the variance, to make the comparison. The main practical contribution of the paper is that it provides a simple estimator for the variance that can be used in conjunction with Kennedy s estimator in (1.4). Both estimators are very simple to implement and are practically unbiased. 2. EXACT MINIMUM VARIANCE UNBIASED ESTIMATOR Theorem 1 below is an extension of Propositions 2 and 3 in van Garderen (2001). The proof is relegated to the Appendix. Theorem 1. Let ĉ be the OLS estimate of the coefficient of a dummy variable, D, in a semilogarithmic regression model such as (1.1), and ˆV (ĉ) the OLS estimate of its variance. The exact minimum variance unbiased estimator of the percentage change in Y due to D changing from 0 to 1 is given by ˆp = 100{exp (ĉ) 0 F 1 (m; 1 2 m ˆV (ĉ)) 1}, (2.1) where m = (n k)/2, n is the sample size, k is the number of regressors (including intercept, dummy-, and continuous variates), and 0 F 1 is the hypergeometric function explained in the Appendix. The theoretical variance of ˆp is given by V ( ˆp) = exp (2c){exp {V (c)} 0 F 1 (m; [V (c)] 2 ) 1}. (2.2) The exact minimum variance unbiased estimator of the variance is ˆV ( ˆp) = exp (2ĉ){[ 0 F 1 (m; 1 2 m ˆV (ĉ))] 2 0 F 1 (m; 2m ˆV (ĉ))}. (2.3) Note that ˆV (ĉ) is calculated simply as the square of the SE of ĉ as reported by regression packages. The 0 F 1 term in (2.1) is the bias correction for parameter uncertainty and tends to one as the sample size increases and the SE of the estimate goes to zero. In moderate samples, the term is typically close to one, but can make a significant difference in small samples as shown below. It is important to note that this can happen even for small values of ĉ. Furthermore, since m appears in both arguments of the hypergeometric function, see (A.1), the term 0 F 1 (m; 1 2 m ˆV (ĉ)) tends to be close to exp { 1 2 ˆV (ĉ)} when the sample size is not too small. In fact, as m goes to infinity the confluence relation lim a 0 F 1 (m; ma) = 0 F 0 (a) = exp {a}, (e.g. Abadir (1999, p. 312)),

4 152 Kees Jan van Garderen and Chandra Shah Variance Estimate: ^V( ^c ) T-k =80 % Error T-k = 40 T-k = 20 Figure 1. Percentage error in the approximate unbiased variance estimator (2.4) relative to the exact unbiased variance estimator (2.3). shows that the terms are asymptotically equal. This explains why Kennedy s estimates are often close to optimal, in particular when m is large. The same argument suggests an approximate estimator of the variance of ˆp, since m appears in all arguments of the hypergeometric function in (2.3), and we therefore propose the following: Definition 1. Approximate unbiased variance estimator: Ṽ ( ˆp) = exp {2ĉ}[exp { ˆV (ĉ)} exp { 2 ˆV (ĉ)}]. (2.4) The approximation is simple, yet works very well in practice, as we will illustrate with examples below. Figure 1 shows the quality of the approximation by graphing the relative error, i.e. the percentage difference between the exact unbiased estimator (2.3) and the approximation (2.4), as a function of the estimate ˆV (ĉ) for various degrees of freedom. The relative error is always negative, but for common values of ˆV (ĉ), usually in the order of 0.1 or less, the approximation is very good, even in small samples. The relative error in Figure 1 does not depend on ĉ since the term exp {ĉ} appears multiplicatively in both the exact and approximate variance estimates. Published studies using Kennedy s estimator tend not to report SEs as correct and simple expressions were not available until now. An obvious approach for obtaining the variance of a non-linear function of parameter estimates, and the one typically used in standard econometric packages, is based on the Delta method. This leads to over-estimation of the variance in the present context. Although it performs reasonably well for small values of ˆV (ĉ), the method can result in serious overestimation for larger values, by over 25% when ˆV (ĉ) = 0.5 and T k = 40, and getting worse as ˆV (ĉ) increases. The better accuracy and ease of computation make approximation (2.4) preferable to the Delta method. 3. APPLICATION TO TEACHER EARNINGS In this section, we show the practical implications of Theorem 1 when analyzing the determinants of secondary school teacher earnings in Australia. The dependent variable in this application is the log of earnings, and the discrete explanatory variables are subjects taught, gender,

5 Exact interpretation of dummy variables in semilogarithmic equations 153 and whether the teacher has additional responsibilities or not. 3 See Shah (1998) for details and analysis. We concentrate on two typical schools with sufficient numbers of teachers. Table 1 summarizes the results of using four different estimators the Naive (100 ĉ), HP, Kennedy and the exact unbiased. For each dummy explanatory variable the estimated percentage change in earnings (in bold), the estimated SE, and the unbiased estimates of the MSE are presented. The SE estimates are not exactly unbiased since they are square roots of the unbiased estimates of the variance. The SE and MSE in columns 4 and 8 are based on our approximation in (2.4). Table 1 also reports a number of standard specification tests. None of the tests reject the model at the usual significance levels, and, in particular, normality of the disturbances cannot be rejected. One obvious feature in Table 1 is that Kennedy s estimates are practically indistinguishable from the exact unbiased ones. Given the explanation in Section 2 and Giles results, they were expected to be close, but in this application the differences are less than 0.01 percentage points. Table 1 shows that the other two estimators can give substantially different estimates of the percentage effect. In school B, for instance, the estimated gender effect differs by 2.3 percentage points between the HP and Naive estimates, and in school A there is a 2.3 percentage point difference between the exact unbiased and HP estimates of the effect of teaching Voc Ed, even though the estimate itself is only 12.9 percent. The HP estimator can be thought of as correcting the Naive estimator for the right interpretation. The exact unbiased estimator further corrects for the bias of the HP estimator. The interpretation correction, i.e. the difference between the HP and the Naive estimator, is always positive (or 0 if ĉ = 0), and the bias correction, i.e. the difference between the exact unbiased and the HP estimator, is always negative by virtue of Jensen s inequality and convexity of the HP transform (1.3). The bias correction can be larger, in absolute terms, than the interpretation correction as illustrated by the estimates for Lote in school A. The interpretation correction is 2.4 percentage points and the bias correction is 2.6 percentage points. The exact unbiased estimate is therefore smaller than the Naive estimate and the HP estimator seems to pull the Naive estimate in the wrong direction. The HP and exact unbiased estimates can also be very close, e.g. for the gender effect in school B they differ by less than 0.2 percentage points, but are both more than 2.1 percentage points different from the Naive estimate. The explanation is that the HP correction depends only on the size of the coefficient ĉ, whereas the bias correction is driven by the variance. The HP correction has very little effect if ĉ is small and ĉ has to be larger than 0.15 to induce more than a 1%-point change. The bias correction on the other hand, can be large even for small values of ĉ. A SE of roughly 0.1 will induce a change of at least 1%-point in the predicted effect, even if ĉ 0. Figure 2 shows the percentage change as a function of the coefficient estimate for the Naive, HP and unbiased estimators. It also shows the difference between the Naive and unbiased, and HP and unbiased estimators. The differences depend on the estimated variance which has been set to 0.1 for this graph. The HP estimator always over-predicts whereas the Naive predictor over-predicts around zero and under-predicts for large absolute values of ĉ. As a consequence, there are two values of ĉ, one positive and one negative for which the Naive estimator is unbiased and these values are further apart when the variance is larger. 3 The subjects taught are Arts, Health, LOTE (language other than English), Maths, Science, SOSE (study of society and the environment), Technology (Tech) and Vocational Education (Voc Ed), and English (the base subject not included in the regressions). Holding of additional responsibilities is indicated by the coordinator (Coord) variable.

6 154 Kees Jan van Garderen and Chandra Shah Table 1. Comparing alternative estimators of percentage impact of dummy variables on teacher earnings. School A (n = 27) School B (n = 34) Naive HP Kennedy/ Exact Naive HP Kennedy/ Exact Covariate (100 ĉ) Approx a unbiased (100 ĉ) Approx a unbiased Coord SE MSE Female SE MSE Arts na na na na SE MSE Health SE MSE Lote SE MSE Maths na na na na SE MSE Science SE MSE Sose SE MSE Tech SE MSE Voc Ed SE MSE Test: Value test Approximate Value test Approximate statistic p-value statistic p-value Breusch-Pagan b Ramsey RESET c Bera-Jarque Shapiro-Wilk Notes: Least squares estimates and associated SEs for the original log-linear regression are obtained by dividing the relevant entries in columns 2 and 6 by 100. a Bold entries are based on Kennedy s (1981) approximation (1.4). The estimates for the SE and MSE in these columns are based on our approximation (2.4). b Breusch Pagan test using the explanatory variables. c RESET test using powers 2, 3, and 4 of the fitted values.

7 Exact interpretation of dummy variables in semilogarithmic equations 155 Predicted Change 100 % 80% 60% 40% 20% % -40% -60% -80% -100% HP-prediction Coefficient Estimate c Exact unbiasedprediction Naïveprediction HP Unbiased (difference) Naive Unbiased (difference) Figure 2. Predicted percentage change (solid lines) as a function of the estimate ĉ for three different predictors Naive, HP, and exact unbiased. Dashed lines are the differences (HP unbiased) and (Naive unbiased). Differences depend on the estimated variance, set equal to ˆσ 2 = 0.1 for the purpose of this graph. Although the differences in the estimates for p obtained using different estimators can be large in absolute terms, the comparison should include a consideration of the uncertainty in the estimates, for example, in terms of their SEs. The SE s in Table 1 show that the uncertainty can be extremely large relative to the estimated effect. For example, the estimated percentage effect of additional responsibilities, Coord, is 20.5 per cent with estimated SE of 10.5 per cent (despite a t-statistic of 2.19 in the original regression). One should therefore be cautious in interpreting the percentage effect when the estimated variance ˆσ 2 is large, even if the coefficient ĉ is statistically significant. Table 1 also shows that the unbiased estimator has smaller SEs than the HP estimator in all cases, but sometimes it is the Naive estimator that has the smallest SE. The SE is not the proper way to compare alternative estimators because of the implicit trade-off between bias and variance. Since unbiased estimation is generally motivated by a squared error loss function, the

8 156 Kees Jan van Garderen and Chandra Shah third entry for each coefficient reports the MSE for each estimate, which only equals the variance if the estimator is unbiased. The MSE estimates in Table 1 indicate that even after taking into account the bias and the variance, the (minimum variance) unbiased estimator can have a larger MSE than the Naive estimator. Furthermore, the MSE for the exact unbiased estimator is always less than that for the HP estimator. Finally, just like Kennedy s estimator for the percentage impact, our simple approximation (2.4) for the unbiased variance estimator is very close to the exact results in all cases considered. 4. FURTHER EXAMPLES In this section we present further examples from the literature showing that the results are not specific to the above application. There are other well-known studies with more extreme parameter values that lead to even larger differences between the estimates. Despite the more extreme parameter values, Kennedy s estimator and our estimator for the variance are still very reliable. Examples used by Halvorsen and Palmquist (1980) and Kennedy (1981) are complemented by providing the SEs, and we have added two recent applications. In the first example, Hanushek and Quigley (1978) reported that a postgraduate degree increases the wages of a black worker by 64 per cent. Using the HP correction leads to a drastically different estimate of 91 per cent. The exact unbiased estimate is 89 per cent, the same as Kennedy s. The SEs (root mean squared errors, RMSE), of the HP estimator and the unbiased estimator are 27.0(27.1) and 26.7(26.7) percent, respectively. The approximation in (2.4) results in SEs that are smaller than their exact counterparts by only 0.04 percentage points. In the second example Solnick and Hemenway (1996) use a semilogarithmic equation to explain the equivalent value of a gift received by a person as a function of the cost of the gift and who the giver of the gift was. The estimated percentage effects of the dummy variables using the different estimators are given in Table 2. The estimates obtained using Kennedy s estimator are very close to the exact unbiased ones and the approximate SEs based on (2.4) differ from their exact counterparts by less than The HP estimates have substantial upward bias. The differences in SEs and in RMSE between the HP and the exact unbiased estimator are relatively small for all variables. As in the teacher earnings application, the SEs are smallest for the Naive estimator and even after taking into account the bias, the RMSE is still smallest for the Naive estimator in most cases. The reason for this is that in small samples the variance term dominates the bias term. In large samples, however, the RMSE will tend to zero for all but the inconsistent Naive estimator. In the final example, Johnson (1996) estimated the effect on pension income for workers who participated in company training programs using the HP estimator to be 52 per cent higher than for similar workers who did not participate. The (unreported) SE is estimated as 14.1 per cent. Both the exact unbiased and Kennedy s estimator, estimate the effect on pension income to be 51 per cent with SEs of 14.0 percent. The Naive estimate is substantially different at 42 per cent. 5. CONCLUSION In this paper we have derived the exact minimum variance unbiased estimator for the percentage impact of a dummy variable in a semilogarithmic equation. We have also derived the exact minc Royal Economic Society 2002

9 Exact interpretation of dummy variables in semilogarithmic equations 157 Table 2. Comparing alternative estimators of percentage effects of dummy variables on gift value (cf. Solnick and Hemenway (1996)). Estimator Presenter of gift Naive HP Kennedy/Approx a Exact unbiased Parent Estimate SE RMSE Sibling Estimate SE RMSE Spouse Estimate SE RMSE Friend Estimate SE RMSE a Note: The SE and RMSE are based on approximation (2.4). imum variance unbiased estimator of its variance, and more importantly, a simple and practical approximation to this variance estimator. The estimator for the percentage impact is compared with three alternatives that are commonly used but which are biased using an application on teacher earnings and three examples from the literature. The HP estimator provides the correct interpretation when parameter values are known, but gives biased results in practice when parameters have to be estimated. Due to their random nature and the non-linear convex transformation, the estimates are always upwardly biased. Kennedy (1981) suggested an alternative estimator to correct for the bias. Giles (1982) pointed out that it gives estimates that are very close to those given by the exact unbiased estimator for reasonable sample sizes. Our examples confirm this and we have provided a formal explanation of why this occurs in terms of the hypergeometric function used to calculate the exact predictor. The most important practical contribution of this paper is the approximate unbiased variance estimator (2.4) which can be used in conjunction with Kennedy s estimator (1.4). Both give results that are very close to the exact unbiased estimates for all the examples considered and are very simple to calculate. The exact formulas are straightforward to use, but in most applications Kennedy s estimator together with our estimator for the variance are more than adequate. ACKNOWLEDGEMENTS We would like to thank Karim Abadir, two referees, Simon Burgess, Richard Smith, and seminar participants in Bristol for their comments. The research of Kees Jan van Garderen has been supported by a fellowship of the Royal Netherlands Academy of Arts and Sciences.

10 158 Kees Jan van Garderen and Chandra Shah REFERENCES Abadir, K. M. (1999). An introduction to hypergeometric functions for economists. Econometric Reviews 18, Giles, D. E. A. (1982). The interpretation of dummy variables in semilogarithmic equations. Economics Letters 10, Goldberger, A. S. (1968). The interpretation and estimation of Cobb-Douglas functions. Econometrica 36, Halvorsen, R. and R. Palmquist (1980). The interpretation of dummy variables in semilogarithmic equations. American Economic Review 70, Hanushek, E. A. and J. M. Quigley (1978). Implicit investment profiles and intertemporal adjustments of relative wages. American Economic Review 68, Johnson, R. W. (1996). The impact of human capital investment on pension benefits. Journal of Labor Economics 14, Kennedy, P. E. (1981). Estimation with correctly interpreted dummy variables in semilogarithmic equations. American Economic Review 71, 801. Lehmann, E. L. (1986). Testing Statistical Hypothesis, 2nd edn. New York: Wiley. Shah, C. (1998). Recurrent teacher cost per student by key learning area: upper secondary schools, Victoria, Australia. Education Economics 6, Solnick and Hemenway (1996). The deadweight loss of Christmas: comment. American Economic Review 86, van Garderen, K. J. (2001). Optimal prediction in loglinear models. Journal of Econometrics 104, A. APPENDIX: MATHEMATICAL DETAILS The hypergeometric function 0 F 1 (m; x) is closely related to e x and can likewise be defined as an infinite sum: x i 0F 1 (m; x) =, (A.1) i!(m) i=0 i where (m) 0 = 1 and (m) i = m(m + 1)... (m + i 1). It can be found in many standard books on special functions. See also Abadir (1999) which includes applications in econom(etr)ics and the relation between 0 F 1 and 1 F 1 which has a number of useful asymptotic expansions. A number of software packages, e.g. Mathematica, include hypergeometric functions so that 0 F 1 [m, x] can be calculated directly. Simple GAUSS, TSP and Pascal routines are available from the authors. Proof of Theorem 1. We will use the following lemma, which is proved in van Garderen (2001). Lemma 1. Let (T k) ˆσ 2 /σ 2 χt 2 k, then for any real constant z we have E[ 0 F 1 (m; mz ˆσ 2 )] = exp {zσ 2 }, E[ 0 F 1 (m; mz ˆσ 2 ) 2 ] = exp {2zσ 2 } 0 F 1 (m; z 2 σ 4 ). Since ĉ N(c, V (ĉ)) and independent of ˆσ 2, we have for the expectation of ˆp, with V (ĉ) = σ 2 z: E[ ˆp] = 100E[exp {ĉ} 0 F 1 (m; 1 2 mz ˆσ 2 ) 1], = 100(exp {c V (ĉ)} exp { 1 2 σ 2 z} 1), = 100(exp {c} 1).

11 Exact interpretation of dummy variables in semilogarithmic equations 159 where we have used the moment generating function of the normal distribution: } MG F(r) = E[exp {ĉr}] = exp {rc + r22 V (ĉ), evaluated at r = 1. The predictor ˆp is minimum variance because ĉ and ˆσ 2 are minimal sufficient and complete statistics (jointly with the other OLS estimators for equation (1.1)). The formula for the variance of the predictor follows similarly from Lemma 1. To show that the variance estimator is unbiased write ˆV (ĉ) = z ˆσ 2 and note that Hence exp {2ĉ} 0 F 1 (m; mz ˆσ 2 ) 2 = ˆp 2, E[100 2 exp {2ĉ} 0 F 1 (m; 2mz ˆσ 2 )] = exp {2c + 2σ 2 } exp { 2σ 2 }, = (E[ ˆp]) 2. E[ ˆV (ĉ)] = E[ ˆp 2 ] (E[ ˆp]) 2, = var( ˆp).