Econometrics Master in Business and Quantitative Methods Helena Veiga Universidad Carlos III de Madrid
Models with discrete dependent variables and applications of panel data methods in all fields of economics have become increasingly important. This course starts with a brief review of concepts previously carried out in other courses of the program and focuses afterwards mainly on the methodological and empirical issues concerning the analysis of cross section and panel data in the specific context of economic models. Selected topics in time series analysis, especially topics of importance for the panel data analysis of dynamic models, will also be discussed. On satisfactory completion of this course, students will be provided with a number of sophisticated econometric tools which are of use in advanced empirical research or professional work.
Chapter1. Endogeneity of Regressors 1.1 Stochastic regressors and the properties of OLS estimators 1.2 Measurement errors in the variables 1.3 Simultaneous equation bias 1.4 Instrumental variables 1.5 Testing for endogeneity Chapter 2. Models with Discrete Dependent Variables 2.1 Models for binary choice 2.2 Estimation and inference in binary choice models 2.3 Multinomial models 2.4 A Poisson model for count data
Chapter 3. Limited Dependent Variable Models 3.1 The truncated regression model 3.2 The censored regression model 3.3 Sample selection Chapter 4. Panel Data 4.1 Basic panel data models 4.2 Estimation and testing methods for random and fixed effect models 4.3 Limited dependent variable panel models
The final mark of the course is based on: A final exam (60%) Four problem sets that will require theoretical and computational work (compulsory) A project (30%). The project consists of reproducing some empirical results of a published paper or a student idea structured in a research paper An exercise done in class (10%)
C. Cameron and Trivedi, Microeconometrics, Cambridge University Press, 2005. F. Hayashi, Econometrics, Princeton University Press, 2000. J. Wooldridge, Econometric Analysis of Cross Section and Panel Data, MIT Press, 2002. M. Creel: Econometrics. Available at http://pareto.uab.es/mcreel/econometrics/econometrics.pdf M. Arellano, Panel Data Econometrics, Oxford University Press, 2005. W.H. Greene, Econometric Analysis, Prentice-Hall, 2008.
1.1 Stochastic regressors and the properties of OLS estimators The stochastic setting is the appropriated one for the kinds of cross section and panel data sets collected for most econometric applications. For much of our classes we adopt a random sampling assumption. More precisely, we assume that: a population model has been specified and an independent, identically distributed (i.i.d.) sample can be drawn from the population. An important virtue of the random sampling assumption is that it allows us to separate the sampling assumption from the assumptions made on the population model.
Consider a linear model for cross section data, written for each observation i as: y i = X i β +u i i = 1,...,N where X i and β are 1 K and K 1 vectors, respectively. Some of the classical assumptions for this model are E(u i ) = 0 and Var(u i ) = σ 2 and often the u i are also assumed to be normally distributed. The problem with this statement is that it omits the most important consideration: What is assumed about the relationship between u i and X i? If the X i are taken as nonrandom then u i and X i are independent of one another.
In nonexperimental environments this assumption rules out too many situations of interest. Some important questions, such as efficiency comparisons across models with different explanatory variables, cannot even be asked in the context of fixed regressors. In a random sampling context, the u i are always independent and identically distributed, regardless of how they are related to the X i. Assuming that the population mean of the error is zero is without loss of generality when an intercept is included in the model. Thus, the statement The errors u i i = 1,...N are i.i.d. with E(u i ) = 0 and Var(u i ) = σ 2 is vacuous in a random sampling context. Viewing the X i as random draws along with y i forces us to think about the relationship between the error and the explanatory variables in the population.
For example, in the population model y = Xβ +u, is the expected value of u given X equal to zero? Is u correlated with one or more elements of X? Is the variance of u given X constant, or does it depend on X? These are the assumptions that are relevant for estimating β and for determining how to perform statistical inference. Therefore, consider formally that y = Xβ +u, where y is a n 1 vector, X is a n K matrix, β is a K 1 and u is n 1 vector.
The classical assumptions are: The relationship between the dependent variable y and the independent variable(s) X is linear E(u X) = 0 Var(u X) = σ 2 I The errors are uncorrelated such that Cov(u i,u j X) = 0 i j The regressors are stochastic and uncorrelated with the error term The rank of X is K < n
Figure :
Note that ˆβ = (X X) 1 X y. It is obtained from the minimization of the following problem min (y X β) (y X β) with β Θ and Θ is the space of the parameters. Are the OLS estimators BLUE (best linear unbiased estimators) under this new assumption that the regressors are stochastic? There are BLUE if they are unbiased, consistent and have the minimum variance in the class of linear and unbiased estimators.
Are the OLS estimators unbiased? The estimator ˆβ is unbiased if E(ˆβ) = β. Proof. [unbiasness of the OLS estimator] E(ˆβ X) = E ( (X X) 1 X y X ) = E ( (X X) 1 X (Xβ +u) X ) = β +(X X) 1 E(X u X) = β and by the law of iterated expectations (LIE) E X (E(ˆβ X)) = E(ˆβ) = β where E X indicates the expectation over the values of X.
The conditional covariance matrix of ˆβ is equal to Var(ˆβ X) = σ 2 (X X) 1 Proof. [conditional covariance matrix of the OLS estimator] [ ] Var(ˆβ X) = E (ˆβ β)(ˆβ β) X = E [ (X X) 1 X uu X(X X) 1 X) ] = (X X) 1 X E(uu X)X(X X) 1 = σ 2 (X X) 1.
The Var(ˆβ) = E X (Var(ˆβ X))+Var X (E(ˆβ X)). The second term is zero since E(ˆβ X) = β for all X, therefore Var(ˆβ) = E X ( σ 2 (X X) 1) = σ 2 E(X X) 1. The estimator ˆβ is efficient if it is the minimum variance estimator in the class of linear and unbiased estimators.
Theorem [GAUSS-MARKOV] For any vector of constants, w, the minimum variance linear unbiased estimator of w β in the classical regression model is w ˆβ, where ˆβ is the least squares estimator. Proof. Let β = Cy be another linear unbiased estimator of β, where C is a K n matrix equal to C = D+(X X) 1 X. If β is unbiased, E(Cy X) = E(CXβ +Cu X) = β, which implies that CX = I and the covariance matrix of β is Var( β X) = σ 2 CC.
Proof. [Cont.] Replacing C, Var( β X) = σ 2 [(D+(X X) 1 X )(D+(X X) 1 X ) ]. Note that since CX = I = DX+(X X) 1 (X X) this implies that DX = 0. Finally, Var( β X) = σ 2 DD +σ 2 (X X) 1 = Var(ˆβ X)+σ 2 DD.
The estimator ˆβ is consistent if it converges in probability to β, that is plimˆβ = β. Let x n be a random variable indexed by the size of the sample. Definition [Convergence in probability] x n converges in probability to c if lim n P ( x n c > ε) = 0 for any positive ε.
Theorem [Convergence in mean square] If x n has mean µ n and variance σn 2 such that the ordinary limits of µ n and σn 2 are c and 0, respectively, then x n converges in mean square to c and plim x n = c. Convergence in mean square implies that the distribution of x n collapses to a spike at plim x n. Moreover, according to the Chebyshev inequality any random variable satisfies the following inequality: P ( x n µ n > ε) σ2 n ε 2.
Taking the limits of µ n and σ 2 n in the previous expression, we see that if then lim n E(x n ) = c and lim n Var(x n ) = 0, plimx n = c Therefore, we can prove that ˆβ is consistent by proving that lim n E(ˆβ) = β and lim n Var(ˆβ) = 0.
Proof. Let ˆβ be equal to ˆβ ) 1 = β +( X X X u n n then ( ) 1 ( ) lim n E(ˆβ) = β +lim n E X X n limn E X u n = β +Q 1 0 = β, ) where Q = plim( X X n is a positive definite matrix and ( lim n Var(ˆβ) = lim n {E ( = Q 1 1 lim n E n ) X 1 ( X n E X uu X n 2 ) ) X uu X Q 1 = 0 n ( E ) } X 1 X n
Figure : Consistency 30 Kernel Density (beta=0.8, n=100) 25 20 15 10 5 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 BETA_100
Figure : Consistency 30 Kernel Density (beta=0.8, n=1000) 25 20 15 10 5 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 BETA_1000
Figure : Consistency 30 Kernel Density (beta=0.8, n=5000) 25 20 15 10 5 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 BETA_5000
1.2 Measurement errors in the variables Measurement errors arise when we observe the variables of interest imperfectly that is: y = y +v v N(0,σv 2 I), (1) where y is the variable that we are interested in explaining and v is the measurement error or X = X +ε ε N(0,σ 2 εi), (2) where X is the independent variable matrix that we observe with error ε. We assume that the errors v and ε are independent of each other and independent of y and X.
All sorts of measurement problems can come into the data that must be used in our analyses. Aggregate statistics such as GDP are only estimates of their theoretical counterparts, Some variables, such as depreciation and the interest rate, do not even exist in an agreed-upon theory. There also may be no physical measure corresponding to the variable in our model, such as: intelligence and permanent income. Nonetheless, they all have appeared in regression models.
1.2.1 Measurement error in the dependent variable We begin with the case where the dependent variable is the only variable measured with error, that is, the dependent variable, y, is defined by equation 1 and the relation of interest is given by the following model: y = X β +u. However, since y is measured with error we can only estimate the following model: y v = X β +u y = X β +(u+v) = X β +w,
where w = u+v. Given the properties of the measurement errors we can show E(X w) = 0, which means that the regressors are not correlated with the error w and consequently, exits exogeneity. In this case, the OLS estimator is still BLUE.
1.2.2 Measurement error in the independent variables In this case the independent variables are measured with error, that is, the independent variables, X, are defined by equation 2 and the relation of interest is given by the following model: y = X β +u. Nevertheless, since X is measured with error we can only estimate the following model: y = (X ε)β +u y = Xβ +(u εβ) = = Xβ +r,
where r = u εβ. Note that ε is a matrix of dimension n K. For a single affected variable ε = 0 0 ε 1 0. 0 0 0 ε 2 0. 0...... 0 0 ε n 0. 0
In this case, the regressors are correlated with the new error r since E(X r) = E ( (X +ε) (u εβ) ) 0 which will affect the properties of the OLS estimators. In fact these estimators are going to be biased and inconsistent. Proof. [Inconsistency] ( ) 1 ( ) lim n E(ˆβ) = β +lim n E X X n limn E X r n = = β [Q +Σ εε ] 1 Σ εε β. where Q and Σ εε are positive definite matrices.
Proof. [Cont.] For the case that only a single variable is affected Σ εε = ( lim n Var(ˆβ) = lim n E = 0. 0 0 0 0. 0 0 0 0 0. 0 0 0 σ 2 ε 0. 0...... 0 0 0 0. 0 ) X 1 ( X 1 limn E n n ) ( X rr X E n ) X 1 X n
Proof. [Cont.] Therefore, plim ˆβ = β [Q +Σ εε ] 1 Σ εε β, which means that ˆβ is inconsistent.
1.3 Simultaneous equation bias Consider the following example of a system of simultaneous equations: Demand equation: q d,t = α 1 p t +α 2 X t +ε d,t, Supply equation: q s,t = β 1 p t +ε s,t, Equilibrium equation: q d,t = q s,t = q t. Since the model is one of the joint determination of price and quantity, they are labeled jointly dependent or endogenous variables. Income X is assumed to be determined outside of the model, which makes it exogenous.
All three equations are needed to determine the equilibrium price and quantity, so the system is interdependent. Moreover, these equations are structural equations since they are derived from theory and in each equation we have endogenous variables that are explained by endogenous variables. Finally, since an equilibrium solution for price and quantity in terms of income and the disturbances is, indeed, implied, the system is said to be a complete system of equations. The completeness of the system requires that the number of equations equal the number of endogenous variables. As a general rule, it is not possible to estimate all the parameters of incomplete systems (although it may be possible to estimate some of them).
For simplicity, assume that ε d and ε s are well behaved, classical disturbances with E(ε d,t X t ) = E(ε s,t X t ) = 0, E(ε 2 d,t X t) = σ 2 d, E(ε2 s,t X t ) = σ 2 s, E(ε d,t ε s,t X t ) = 0 and E(ε d,t X t ) = E(ε s,t X t ) = 0
All variables are mutually uncorrelated with observations at different time periods. Price, quantity, and income are measured in logarithms in deviations from their sample means. Solving the equations for p and q in terms of X, and ε d and ε s produces the reduced form of the model p t = α 2X t β 1 α 1 + ε d,t ε s,t β 1 α 1 = π 1 X t +v 1,t, q t = β 1α 2 X t β 1 α 1 + β 1ε d,t α 1 ε s,t β 1 α 1 = π 2 X t +v 2,t.
The Cov(p t,ε d,t )=Cov(π 1 X t +v 1,t,ε d,t ) 0. Therefore, neither the demand nor the supply equation satisfies the assumptions of the classical regression model and the OLS estimators of the parameters of equations with endogenous variables on the right-hand side are inconsistent.
1.4 Instrumental variables Consider the model y i = βx i +u i, where X i is observed with error such that X i = X i +ε i ε i N(0,σ 2 ε ). Then, the model that we can estimate is y i = β(x i ε i )+u i = βx i +(u i βε i ) = βx i +r i
We know from before that in this situation the OLS estimator of β is inconsistent since E(X i r i ) 0 i = 1,...,n. Suppose, however, that there exists a variable Z such that Z is correlated with X, but not with r. For example, in surveys of families, income is notoriously badly reported, partly deliberately and partly because respondents often neglect some minor sources. Suppose, however, that one could determine the total amount of checks written by the head(s) of the household. It is quite likely that this Z would be highly correlated with income, but perhaps not significantly correlated with the error.
In a multiple regression framework, if only a single variable is measured with error, then the preceding can be applied to that variable and the remaining variables can serve as their own instruments. In this case Z will be a n K matrix Z = Z 1 X 21. X K1 Z 2 X 22. X K2 Z 3 X 23. X K3.... Z n X 2n. X Kn
satisfying the following assumptions: and plim Z X n 0 plim Z r n = 0. Then, the IV estimator of β, ˆβ IV = (Z X) 1 Z y is consistent.
Proof. plimˆβ IV = β +plim( Z X n = β +Q 1 ZX 0 = β, ) 1plim ( ) Z r n where Q ZX is a finite matrix of rank K.
Theorem (Central limit theorem) If x 1,...,x n are a random sample from any probability distribution with finite mean µ and finite variance σ 2 and x n = 1/n i x i, n( xn µ) d N(0,σ 2 ) Theorem (Lindberg-Levy central limit theorem) If x 1,...,x n are a random sample from a multivariate distribution with finite mean vector µ and finite positive definite covariance matrix, Q, then n( xn µ) d N(0,Q), where x n = 1 n i x i
Theorem (Lindberg-Feller central limit theorem) Suppose that x 1,...,x n are a sample of random vectors such that E(x i ) = µ Var(x i ) = Q i, and all third moments of the multivariate distribution are finite. let We assume that Q n = 1 Q i. n lim Q n = Q, n where Q is a positive definite matrix, and for every i, i
Theorem (cont.) ( ) 1Q ( ) 1Qi lim n Qn = lim Q i i = 0. n n The second assumption states that individual components of the sum must diminish in significance and the sum of matrices is nonsingular. Therefore, n( xn µ) d N(0,Q). i
The distribution of OLS Estimator Given the previous results we want to know the distribution of ( X ) X 1 ( 1 n(ˆβ β) = n )X u, (3) n ) 1 and since the plim( X X n = Q 1, the distribution of 3 is the same as that of ( 1 Q 1 n )X u. Thus, we must establish the distribution of ( 1 n )X u = n( w E( w)), where, in this case, E( w) = 0. For obtaining the distribution of n w, we use the Lindberg-Feller version of the limit theorem and
define w = 1 x i u i n as the average of n independent random vectors, with means zero and variances Var(x i u i ) = σ 2 E(x i x i ) = σ2 Q i. Therefore, the variance of n w is σ 2 Q n = σ 21 n (Q 1 +Q 2 +...+Q n ) = σ 21 n ( i E(x ) ix i ) = σ 2 E X X n i
As long as the sum is not dominated by any particular ) term, the regressors are well behaved and the plim( X X n = Q lim n σ2 E ( X ) X = σ 2 Q. n We may apply the Lindberg-Feller central limit theorem that says if The errors all have the same distribution; ) The elements of X are bounded and the plim( X X n = Q, then ( 1 n )X u d N(0,σ 2 Q). It follows that n(ˆβ β) = Q 1( 1 n )X u d N(0,σ 2 Q 1 ).
Definition (Asymptotic distribution) An asymptotic distribution is a distribution that is used to approximate the true finite sample distribution of a random variable. Extending this definition, if ˆθ is a estimator of the parameter vector θ, the asymptotic distribution is obtained from the limiting distribution: n(ˆθ θ) d N(0,V) and it is ˆθ a N (θ, 1n V ), where the asymptotic covariance matrix is denoted Asy.Var(ˆθ) = 1 n V.
Then the asymptotic distribution of ˆβ is ) ˆβ a N (β, σ2 n Q 1 Regarding the ˆβ IV ( Z ) X 1 1 n(ˆβiv β) = n Z r, n which converges in distribution to ) n(ˆβiv β) d N(0,σr 2 Q 1 ZX Q ZZQ 1 XZ.
In the case of simultaneous equation bias we use the most efficient IV estimator, the two stage least squares (2SLS). The system of simultaneous equations is Demand equation: q d,t = α 1 p t +α 2 X t +ε d,t, Supply equation: q s,t = β 1 p t +ε s,t, Equilibrium equation: q d,t = q s,t = q t. The first equation of the system can be written as a function of the equilibrium quantity as q t = α 1 p t +α 2 X t +ε d,t and can be estimated consistently by 2SLS replacing the variable that is causing endogeneity (p t ) by an instrument variable that is correlated with it but not with ε d,t.
The instrument is ˆp t obtained by estimating the reduced form of the system by OLS: q t = α 1ˆp t +α 2 X t +ε d,t, then we estimate the previous equation by OLS. The two steps consists of the 2SLS estimation.
1.5 Testing for endogeneity It might not be obvious that the regressors in the model are correlated with the disturbances or that the regressors are measured with error. If not, there would be some benefit from using the least squares estimator rather than the IV estimator. Consider a comparison of the two covariance matrices under the hypothesis that both are consistent, that is, assuming plim X r n = 0.
The difference between the asymptotic covariance matrices of the two estimators is ( ) AsyVar(ˆβ IV ) AsyVar(ˆβ) = σ2 plim (X Z)(Z Z) 1 (Z 1 ( X) n n σ 2 plim n [ ((X = σ2 plim n Z)(Z Z) 1 (Z X) ) ] 1 n (X X) 1 ) X 1 X n.
To compare the two matrices in the brackets, we can compare their inverses. The inverse of the first is X Z(Z Z) 1 Z X that is equal to X (I M z )X = X X X M z X. M z is equal to M z = I Z(Z Z) 1 Z and it is therefore a nonnegative definite matrix, which implies that X M z X is also. So, X Z(Z Z) 1 Z X is smaller than X X and consequently its inverse is larger. Under the null hypothesis, the asymptotic covariance matrix of the OLS estimator is never larger than that of the IV estimator. Thus, we have established that if OLS is consistent it is a preferred estimator.
The logic of Hausman s approach is as follows: Under the null hypothesis, we have two consistent estimators of β, ˆβ and ˆβ IV. Under the alternative hypothesis, only one of these, ˆβ IV, is consistent. The idea of the test is to examine d = ˆβ IV ˆβ. Note that under the null hypothesis, plim d = 0, whereas under the alternative, plim d 0. Therefore, we might test the hypothesis of endogeneity with a Wald statistic,
H = d (Est.Asy.Var(d)) 1 d and Asy Var(d) = Asy Var(ˆβ IV )+AsyVar(ˆβ) 2Asy Cov(ˆβ IV, ˆβ). According to Hausman: the covariance between an efficient estimator of a parameter vector, β, and its difference from an inefficient estimator of the same parameter vector is zero.
Therefore, Cov(ˆβ, ˆβ ˆβ IV ) = 0, which implies that Cov(ˆβ IV, ˆβ) = Var(ˆβ). So, Asy Var(d) = Asy Var(ˆβ IV ) Asy Var(ˆβ). Finally, the Hausman statistic becomes H = (ˆβ IV ˆβ) ( Est.Asy Var(ˆβ IV ) Est.Asy Var(ˆβ)) 1(ˆβIV ˆβ) d χ 2 (J).
An easy alternative to this statistic is the one proposed by Wu (1973). It consists in estimating the model y = Xβ + ˆXγ +u, and testing the significance of ˆX in the previous regression. Note that ˆX are the fitted values in regressions of the variables X on Z.
APPLICATION (GREEN): Consider a consumption function of the form C t = α+βy t +u t. It is estimated using 204 observations on aggregate U.S. consumption and disposable personal income. Suppose that there is a possibility of bias due to correlation between Y t and u t. Consider instrumental variables estimation using Y t 1 as the instrument for Y t, and, of course, the constant term is its own instrument. One observation is lost because of the lagged values, so the results are based on 203 quarterly observations. Estimate the model using the instrumental variables method and compare it with the OLS estimation Test for the possible endogeneity using the Hausman test (version Wu)