Identification of Average Marginal Effects Under Misspecification when Covariates are Normal∗ Jos´e Ignacio Cuesta† Jonathan M.V. Davis‡ Andrew Gianou§ Alejandro Hoyos¶

Abstract A previously known result in the econometrics literature is that when covariates of an underlying data generating process are jointly normally distributed, estimates from a nonlinear model that is misspecified as linear can be interpreted as average marginal effects. This has been shown for models with exogenous covariates and separability between covariates and errors. In this paper, we extend this identification result to a variety of more general cases, in particular for combinations of separable and non-separable models under both exogeneity and endogeneity. So long as the underlying model belongs to one of these large classes of data generating processes, our results show that nothing else must be known about the true DGP –beyond normality of observable data, a testable assumption– in order for linear estimators to be interpretable as average marginal effects. We demonstrate via simulation the performance of these estimators from the misspecified linear model and indeed show that they perform well when the data is normal but can perform poorly when this is not the case. JEL codes: C14, C21, C26 Keywords: misspecification, identification, marginal effects

∗

This version: January, 2017. We would like to thank Dan Black, Stephane Bonhomme, Azeem Shaikh, and Daniel Wilhelm for useful comments and suggestions.. † Department of Economics, University of Chicago. Email: [email protected]. ‡ Department of Economics, University of Chicago. Email: [email protected]. § Department of Economics, University of Chicago. Email: [email protected]. ¶ Department of Economics and Booth School of Business, University of Chicago. Email: [email protected].

1

1

Introduction

The vast majority of empirical work in economics uses linear specifications in order to study causal relationships between variables. However, in many cases there is no a priori reason to impose such structure and theory might actually predict nonlinear relationships among variables of interest. Previous studies have shown that when a linear model is used but the true model is actually nonlinear, the estimate will yield a weighted average of the underlying marginal effects: Yitzhaki (1996) derived the weights for the OLS case, while Angrist and Imbens (1995) and then Heckman et al. (2006) derived the corresponding weights for the IV case. Importantly, the estimated effects under this form of misspecification cannot be interpreted as the average marginal effect in the population (Lochner and Moretti, 2014; Loken et al., 2012). Non-parametric methods can be used to estimate average marginal effects (Heckman et al., 1997; Newey et al., 1999; Imbens and Newey, 2009), but their use is not yet widespread and, in some cases, they require large support assumptions that are unlikely to be satisfied in practice. Non-parameteric instrumental variables can be particularly difficult because it is an ill-posed inverse problem (Newey and Powell, 2003). In this paper, we build upon a previously known result in the econometrics literature: that when covariates are normally distributed, ordinary least squares estimates can be interpreted as average marginal effects even when the true model is non-linear. Concretely, when the covariates included in a regression are jointly normally distributed, the coefficients from a linear regression are consistent estimates of the average marginal effect when: (1) the true model is separable in observables and unobservables, and the regressors are uncorrelated with the unobservables. The contribution of this paper is to show that this result also holds in cases in which: (2) the true model is non-separable in observables and unobservables, and the observables and unobservables are independent; (3) a set of valid instruments is available and they are jointly normally distributed with observables of a true model that is separable in observables and unobservables; and (4) a set of valid instruments is available and they are jointly normally distributed with observables and unobservables of a true model that is nonseparable in observables and unobservables. While the first of these results has already been established (Yitzhaki, 1996; Angrist and Krueger, 1999), we also provide a new proof. To the best of our knowledge, the second, third and fourth results are novel. As noted above, the main requirement for these results to hold is that covariates in a regression or covariates and instruments in a two-stage least squares regression must be normally distributed. Notably, this condition can be verified since covariates and instruments are observable. We do not take a stand on whether these results are an endorsement or an indictment of linear models. On the one hand, certain variables utilized in the fields of labor economics, health economics and finance are approximately normally or log-normally distributed. Moreover, some variables are constructed to be normally distributed, such as latent ability measures. In these cases, we show that a misspecified linear regression can nonetheless be a powerful tool 2

in identifying a particular quantity of interest, namely the average marginal effects. On the other hand, many variables of interest are not normally distributed. In a series of simulations, we show that the coefficients from a misspecified linear model do not correspond to average marginal effects in general.1 The remainder of the paper is organized as follows. Section 2 presents our results, section 3 illustrates the results using simple simulations, and section 4 concludes.

2

Main results

We consider the case in which a researcher would like to estimate the average marginal effect of some covariates, X ∈ Rk , on an outcome variable Y ∈ R. Specifically, we assume the researcher estimates: Y = α + X 0β + ν (1) where ν is a regression error term. We present the four results introduced above, which show four different settings for which the coefficient estimates, βˆOLS or βˆIV , can be interpreted as average marginal effects. Each result assumes the data was generated by a different true model, in terms of both the separability and statistical relationship between observables and unobservables. Our key insight is that when covariates are normally distributed, Stein’s lemma (Stein, 1981), which is frequently used in finance applications, can be applied to write covariances in terms of marginal effects.2 The lemma is stated below for reference. Stein’s Lemma. If X and Y are bivariate normal, g (X) is differentiable and E [|∇g (X)|] < ∞,3 then: cov [Y, g (X)] = cov [Y, X] E [∇g (X)] Although the results presented below are fundamentally about identification, we discuss them in an estimation framework. Therefore, it will be useful at this point to introduce a series of assumptions needed to guarantee the consistency of the OLS estimator of β in equation 1. Assumption I. Assume that (i) (Y, X, ν) is a random vector, such that Y, ν ∈ R and X ∈ Rk , (ii) E[XX 0 ] < ∞, E[XY ] < ∞, and E[Y ] < ∞, (iii) var[X] is invertible, and (iv) {(Yi , Xi ), i = 1, . . . , n} is an independent and identically distributed sequence of random vectors. 1

The requirement of covariates being jointly normally distributed is similar to the main condition in Ruud (1983), which considers misspecification of the unobservable distribution when using maximum likelihood to estimate multinomial discrete choice models. While our paper instead considers misspecification of the functional relationship between the outcome of interest and the observables, Ruud (1983) relatedly finds that a maximum likelihood model is consistent under this alternative form of misspecification when covariates are multivariate normal. 2

Actually, as noted by Landsman and Neˇslehov´ a (2008), Stein’s lemma can be generalized to the family of elliptical distributions and thus all of our results apply for such family of distribution. For ease of exposition however, we refer only to the normal distribution throughout the paper. 3

∂g Where ∇g (X) = ( ∂X 1

∂g ∂X2

...

∂g T ) ∂Xk

is the gradient of g(X).

3

The first result assumes the true model is additively separable in exogenous observables, X, and unobservables, but makes no restrictions on the functional relationship between Y and X. For additively separable models, X is exogenous so long as it is uncorrelated with unobservables; statistical independence is not required. Regardless of whether the regression is correctly specified and reflective of the true model, if a researcher uses a linear model to estimate a regression with normally distributed covariates and the assumptions above hold, the estimates from the linear model are consistent estimates of the average marginal effects. As mentioned, this case of separability with exogenous covariates can be found in the works cited above. Lemma 1. Consider the model Y = m (X) + ε. Assume (1.1) X is normally distributed, (1.2) m (X) is differentiable with E [|∇m (X)|] < ∞, and (1.3) ε and X are uncorrelated. Then the OLS estimator of β in equation 1 is a consistent estimator of E [∇m (X)], the average marginal effect of X on Y . Proof. Under assumption I, the probability limit of the OLS estimator of β is: plim βˆOLS = var [X]−1 cov [X, Y ] where: cov [X, Y ] = cov [X, m (X) + ε] = cov [X, m (X)] = var [X] E [∇m (X)] where the second equality uses assumption 1.3, and the last equality follows from Stein’s lemma and assumptions 1.1 and 1.2. Therefore: plim βˆOLS = E [∇m (X)] and βˆOLS is a consistent estimator of the average marginal effect. The second result addresses the more general case in which we allow the true model to be non-separable in observables and unobservables, and shows that the same result as above is obtained if observables and unobservables are assumed to be statistically independent rather than simply uncorrelated. Note that these conditions would for instance be satisfied by a linear random coefficients model where the vector of random coefficients is assumed to be statistically independent of the covariates. Lemma 2. Consider the model Y = m (X, ε). Assume (2.1) X is normally distributed, (2.2) m (X) is differentiable in X with E [|∇x m (X, ε)|] < ∞, and (2.3) ε and X are statistically inde4

pendent. Then the OLS estimator of β in equation 1 is a consistent estimator of E [∇x m (X, ε)], the average marginal effect of X on Y . Proof. As before, under assumption I, the probability limit of the OLS estimator of β is: plim βˆOLS = var [X]−1 cov [X, Y ] where: cov [X, Y ] = cov [X, m (X, ε)] = E [cov [X, m [X, ε] |ε]] + cov [E [X|ε] , E [m (X, ε) |ε]] = E [var [X|ε] E [∇x m (X, ε) |ε]] + cov [E [X|ε] , E [m (X, ε) |ε]] = var [X] E [∇x m (X, ε)] where the second equality uses the law of total covariance, the third equality follows from Stein’s Lemma applied conditional on ε and assumptions 2.1, 2.2 and 2.3, and the last equality follows from assumption 2.3. Therefore: plim βˆOLS = E [∇x m (X, ε)] and βˆOLS is a consistent estimator of the average marginal effect. We move now to the cases of models in which X is endogenous and the researcher utilizes a linear instrumental variables approach for estimation. It will therefore be useful at this point to introduce a series of assumptions needed to guarantee the consistency of the IV estimator of β in equation 1. Assumption II. Assume that (i) (Y, X, Z, ν) is a random vector, such that Y, ν ∈ R, and X, Z ∈ Rk , (ii) E[ZX 0 ] < ∞, E[ZY ] < ∞, E[Y ] < ∞, E[X] < ∞, and E[Z] < ∞, (iii) cov[Z, X] is invertible, and (iv) {(Yi , Xi , Zi ), i = 1, . . . , n} is an independent and identically distributed sequence of random vectors. The third result considers the case in which the true model is additively separable in observables and unobservables, but X is endogenous, so that observables and unobservables are allowed to be arbitrarily correlated. This result shows that as long as valid instrumental variables Z are available, and the covariates and instrumental variables are jointly normally distributed, then the linear IV estimator of β can be interpreted as the average marginal effect. Lemma 3. Consider the model Y = m (X) + ε with cov [X, ε] 6= 0. Assume (3.1) X and Z are jointly normally distributed, (3.2) m (X) is differentiable with E [|∇m (X)|] < ∞, and (3.3) cov [Z, ε] = 0. Then the IV estimator of β in equation 1, where Z is used as an instrument for X, is a consistent estimator of E [∇m (X)], the average marginal effect of X on Y . 5

Proof. Under assumption II, the probability limit of βˆIV is: plim βˆIV = cov [Z, X]−1 cov [Z, Y ] where: cov [Z, Y ] = cov [Z, m (X) + ε] = cov [Z, m (X)] = cov [Z, X] E [∇m (X)] where the second equality uses assumption 3.3, and the last equality comes from applying Stein’s lemma and assumptions 3.1 and 3.2. Therefore: plim βˆIV = E [∇m (X)] and βˆIV is a consistent estimator of the average marginal effect. For completeness of the exposition, we include an additional result for the class of nonseparable models under endogeneity which requires unobservables to be jointly normally distributed with X and Z. While normality of covariates is a substantive assumption, it can be tested in the data. The same is not true for unobservables, which limits the empirical appeal of this result. Lemma 4. Consider the model Y = m (X, ε) with cov [X, ε] 6= 0. Assume (4.1) X, Z, and ε are jointly normally distributed, (4.2) m (X, ε) is differentiable in X and ε with E [|∇x m (X, ε)|] < ∞ and E [|∇ε m (X, ε)|] < ∞, and (4.3) cov [Z, ε] = 0. Then the IV estimator of β in equation 1, where Z is used as an instrument for X, is a consistent estimator of E [∇m (X)], the average marginal effect of X on Y . Proof. Under assumption II, the probability limit of βˆIV is: plim βˆIV = cov [Z, X]−1 cov [Z, Y ] where: cov [Z, Y ] = E [cov [Z, m [X, ε] |ε]] + cov [E [Z|ε] , E [m (X, ε) |ε]] = E [cov [Z, m [X, ε] |ε]] = cov [Z, X] E [∇x m (X, ε)] + cov [Z, ε] E [∇ε m (X, ε)] = cov [Z, X] E [∇x m (X, ε)] where the first equality uses the law of total covariance, the second equality follows because Z 6

and ε are mean independent, the third equality comes from applying Stein’s lemma conditionally on ε and assumptions 4.1 and 4.2, and the last equality follows from assumptions 4.2 and 4.3. Therefore: plim βˆIV = E [∇x m (X, ε)] and βˆIV is a consistent estimator of the average marginal effect.

3

Simulation

In this section, we illustrate the results from Section 2 using a series of simulations. We demonstrate that, as expected, the coefficients from linear OLS and IV regressions yield average marginal effects when covariates are normally distributed but the true relationship is non-linear. While our theoretical results are limited to the case where covariates are normal, we also explore how similar OLS and IV estimates are to average marginal effects when covariates are not normally distributed. We use a simulation design that is nearly identical to the one used by Newey and Powell (2003) to show the performance of their non-parametric IV estimator. Specifically, we assume the following functional forms: Y

= log(|X − 1| + 1)sgn(X − 1) + U

X = Z +V Importantly, the relationship between Y and X is non-linear. Note that m(X) = log(|X − 1| + 1)sgn(X − 1), which we plot for reference in Figure 1. Hence, the marginal effect at a given value of X is given by:   1 if x < 1 ∇m(X) = 2−x 1 if x ≥ 1 x The average marginal effect in the population is: ˆ

1

E [∇m(x)] = −∞

1 fX (x)dx + 2−x

ˆ 1

∞

1 fX (x)dx x

where fX (x) is the marginal distribution of X. Our results imply that OLS and IV estimates from a regression of Y on X will be consistent estimators of the average marginal effect when fX (x) is a normal density. We simulate this data generating process by independently sampling (Zi , Vi , Ui ) from the relevant distributions. The simulation results are shown in Table 1. The first panel considers cases where X is exogenous, while the second panel considers cases in which X is endogenous. In each case, we simulate data generating processes using the Normal, Student’s t, Cauchy and

7

Gamma distributions. For reference, Figure 1 shows the shape of each density used in these simulations. For each data generating process, we compute the average marginal effect and the corresponding linear estimator, either OLS or IV, for samples of 100, 1, 000 and 10, 000 observations. Results reported in Table 1 correspond to average coefficients from linear regressions across 1, 000 replications of the simulation.

.4

4

.3

2

.2

0

.1

-2

0

m(x)

Density

Figure 1: Simulation densities and model

-4 -15

-10

-5

0 x

N(0,1) Cauchy(0,1) m(X)

5

10

15

T(5) Gamma(5,1)

Notes: This figure plots the four distributions used for the simulations displayed in Table 1. These are the distributions used for X in the case of exogeneity and for Z in the case of exogeneity. Additionally, this figure displays the non-linear model used for the simulations, m(X) = log(|X − 1| + 1)sgn(X − 1).

The first panel in Table 1 reports results for the case in which X is exogenous. In each of these exogenous cases, U is drawn from a standard normal distribution. The first row begins by considering the case where X is standard normal. Lemma 1 implies that the coefficients should equal the average marginal effect in this case. The first column shows that the average marginal effect is 0.528. The OLS coefficient estimates are remarkably close to the average marginal effect, regardless of the sample size. In the second row of Table 1 we consider the case of X being t-distributed. The t-distribution can reasonably be considered a small deviation from the normal distribution: it is symmetric but has thicker tails, and it actually converges to the normal distribution as the degrees of freedom increase. The average marginal effect when X is t-distributed with 5 degrees of freedom is 0.511. However, the OLS estimates are systematically smaller than the average marginal effect, at between 0.483 and 0.474 depending on the number of observations. In order to study how this result changes as the t-distribution converges to the normal, Figure 2 displays the evolution of

8

Table 1: Simulation Results

Desired Parameter

ˆ Estimated Parameter (β)

E[m0 (x)]

N=100

N=1,000

N=10,000

X ∼ N (0, 1)

0.528

0.529

0.526

0.528

X ∼ T (5)

0.511

0.483

0.475

0.474

X ∼ Cauchy(0, 1)

0.432

0.093

0.022

0.004

X ∼ Gamma(5, 1)

0.248

0.201

0.200

0.200

Z ∼ N (0, 0.5)

0.528

0.516

0.526

0.527

Z ∼ T (5)

0.497

0.465

0.464

0.462

Z ∼ Cauchy(0, 1)

0.425

0.094

0.022

0.004

Z ∼ Gamma(5, 1)

0.256

0.206

0.205

0.205

Data Generating Process X Exogenous

X Endogenous

ˆ are averaged over 1,000 replications of the simulation with the Notes: The estimated parameters, β, specified number of observations. Lemmas 1 and 3 show that βˆ should equal the average marginal effect, E[m0 (x)], in the first row of each panel of the  table, respectively.  In each of the exogenous cases, 0 1 0.5 U ∼ N (0, 1). In the endogenous cases, (U, V ) ∼ N , . 0 0.5 0.5

the ratio of the OLS coefficient to the average marginal effect as the degrees of freedom increase. As before, the OLS coefficient is smaller than the average marginal effect when there are few degrees of freedom, but the ratio quickly converges to 1 as the degrees of freedom increase. The ratio is 0.01, 0.98, and 0.99 with 1, 10, and 25 degrees of freedom respectively. The third and fourth rows of Table 1 display the same results for the cases when X follows a standard Cauchy distribution or a Gamma distribution with a shape parameter of 5 and a scale parameter of 1. The former has the characteristics of being symmetric but having thick tails, while the latter illustrates the effect of skewness in the distribution of X, since it is nonnegative and right skewed. In both cases, the estimated OLS coefficient is lower than the average marginal effect, with the difference being remarkably large for the Cauchy distribution. These results suggest interpreting OLS estimates as average marginal effects may be inappropriate when covariates are not normally distributed. The second panel in Table 1 shows results when X is endogenous. Rather than specifying the distributions of X and Z we consider several distributions for Z and assume (U, V ) ∼

9

Figure 2: Convergence path when X is T-distributed 1

β OLS ⁄ E[ ∇ m(X)]

.8

.6

.4

.2

0 0

5

10 15 Degrees of freedom

20

25

Notes: This figure plots the ratio between the coefficient obtained from an OLS regression of Y on X, βˆOLS , and the parameter of interest, E[m0 (x)]. X is t-distributed with varying degrees of freedom. Each point in uses the the average βˆOLS over 1, 000 replications using 10, 000 observations.

N

0 1 0.5 , 0 0.5 0.5

! . Here, X is endogenous because V and U are correlated and Z is a valid

instrument because it is correlated with X but not U . Lemma 3 states that βˆIV should converge to the average marginal effect when X and Z are normally distributed. The first row in the second panel row of Table 1 confirms this result: the estimated IV coefficients converge to the average marginal effect as the number of observations increases. However, that is not the case when we simulate the results for other data generating processes. The second, third and fourth rows display results for the case in which U and V are jointly normally distributed but Z is distributed following a Student’s t, a Cauchy or a Gamma distribution respectively. The results in these cases show that when X and Z are not normally distributed, the IV coefficient may be far from the average marginal effect.

4

Conclusion

This paper shows that when the covariates in a model are normally distributed, the coefficients from a linear regression can be interpreted as average marginal effects even when the true model is non-linear. While average marginal effects may fail to uncover underlying heterogeneity in marginal effects, these results provide an interpretation of linear estimators under misspecification that connects them to the true model in a way that was only partially available in the literature and that we complement.

10

While other authors have demonstrated this result in the case of separable models under exogeneity, we show that it holds under varying sets of assumptions for a wide range of models including separable and non-separable models under exogeneity and endogeneity. Simulations provide evidence that, in general, this is not the case when covariates are not normally distributed. These results provide additional insight in to the applicability and interpretation of linear models in empirical research. In particular, the results follow from a limited set of assumptions, perhaps the strongest of which is normality, a feature of the observed covariates that can hence be tested for in the data.

References Angrist, J. D. and Imbens, G. W. (1995). Two-stage least squares estimation of average causal effects in models with variable treatment intensity. Journal of the American Statistical Association, 90(430):pp. 431–442. Angrist, J. D. and Krueger, A. B. (1999). Chapter 23 - empirical strategies in labor economics. volume 3, Part A of Handbook of Labor Economics, pages 1277 – 1366. Elsevier. Heckman, J., Urzua, S., and Vytlacil, E. (2006). Understanding instrumental variables models with essential heterogeneity. Review of Economic and Statistics, 88(3):389–432. Heckman, J. J., Ichimura, H., and Todd, P. E. (1997). Matching as an econometric evaluation estimator: Evidence from evaluating a job training programme. The Review of Economic Studies, 64(4):pp. 605–654. Imbens, G. W. and Newey, W. K. (2009). Identification and estimation of triangular simultaneous equations models without additivity. Econometrica, 77(5):pp. 1481–1512. Landsman, Z. and Neˇslehov´ a, J. (2008). Stein’s lemma for elliptical random vectors. Journal of Multivariate Analysis, 99:912–927. Lochner, L. and Moretti, E. (2014). Estimating and testing models with many treatment levels and limited instruments. Review of Economic and Statistics, 97(2):387–397. Loken, K. V., Mogstad, M., and Wiswall, M. (2012). What linear estimators miss: The effects of family income on child outcomes. American Economic Journal: Applied Economics, 4(2):1– 35. Newey, W. K. and Powell, J. L. (2003). Instrumental variable estimation of nonparametric models. Econometrica, 71(5):1565–1578. Newey, W. K., Powell, J. L., and Vella, F. (1999). Nonparametric estimation of triangular simultaneous equations models. Econometrica, 67(3):pp. 565–603. 11

Ruud, P. A. (1983). Sufficient conditions for the consistency of maximum likelihood estimation despite misspecificaton of distribution in multinomial discrete choice models. Econometrica, 51(1):225–228. Stein, C. M. (1981). Estimation of the mean of a multivariate normal distribution. The Annals of Statistics, 9(6):1565–1578. Yitzhaki, S. (1996). On using linear regressions in welfare economics. Journal of Business and Economic Statistics, 14(4):478–86.

12