Introduction: structural econometrics. Jean-Marc Robin
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1 Introduction: structural econometrics Jean-Marc Robin
2 Abstract 1. Descriptive vs structural models 2. Correlation is not causality a. Simultaneity b. Heterogeneity c. Selectivity
3 Descriptive models Consider a sample of N observations of a couple of variables S = f(y i ; x i ); i = 1; :::; Ng. A descriptive model is a statistical restriction (a list of assumptions) on the distribution of S. It describes the statistical link between x i and y i, not the causal relationship between these variables. Example: the linear model. 0 Assume x i 2 R K and X = x T 1. 1 C A is full column rank (the columns of X are linearly indep.). x T N Then, the symmetric matrix X T X = P N i=1 x ix T i is invertible and the Ordinary Least Squares (OLS) estimator exists: 0 1 " N # 1 X X N b = x i x T i x i y i = X T X 1 X T y where y = i=1 i=1 y 1. y N C A : 1
4 Assuming moreover iid observations and E(x i x T i ) non singular, the Law of Large Numbers implies that plim b = E(xi x T i ) 1 E(xi y i ) b: N!1 Parameter b is the coe cient of the theoretical regression of y i on x i. It describes the correlations between y i and x i in the whole population whereas b describes the correlations between y i and x i in the sample. b b and b always exist under these assumptions......even if the true Data Generating Process is not a linear model. Examples: it may very well be that the true DGP is y i N cx 2 i ; 2, y i ; x i 2 R. y i 2 f0; 1g and Pr fy i = 1jx i g = x T i (Probit model). 2
5 Correlation is not causality In general, however, correlations can be pointing in the wrong direction of causality. This happens in particular when x i is endogenous: either because of simultaneity biases, or because of unobserved heterogeneity. when the sample is endogenous (selectivity biases). 3
6 Aggregate demand: D i = a bp i + u i. Aggregate supply: S i = + p i + v i. Example of simultaneity bias: the supply-demand model Assume demand shock u i and supply shock v i uncorrelated. At the equilibrium, supply equals demand equals exchanged quantity: 8 ( a + b y i = a bp i + u >< y i = i ) b + + u i + bv i b + y i = + p i + v i >: p i = a b + + u i v i b + : Regressing y i on p i yields = Cov (y i; p i ) Var (p i ) which is a weighted average of b and. = 2 u b 2 v 2 u + 2 v 4
7 Instrumental variables Demand curves and supply curves are identi able if there exist observed supply and demand shocks: ( u i = x T i c + " i v i = z T i + i Under the assumptions that Cov (x i ; i ) = 0 Cov (z i ; " i ) = 0 regressing y i on p i by Two Stage Least Squares (2SLS) using z i (supply shocks) to instrument p i yields consistent estimates of a and b (demand curve); regressing y i on p i by Two Stage Least Squares (2SLS) using x i (demand shocks) to instrument p i yields consistent estimates of and (supply curve). 5
8 Example of heterogeneity bias: convergence and growth Do LDCs grow faster than developed countries so that their wealths will converge? Idea: regress q it1 q it0 on q it0 q t0, for a sample f(q it0 ; q it1 ) ; i = 1; :::; Ng where q it is per capita GDP (in log) of country i measured at two di erent times t 0 and t 1 (for example t 0 = 1960 and t 1 = 1990) and q t = 1 N P N i=1 q it. The OLS estimate of b, b b, is found signi cantly negative, which seems to imply that the countries starting from a high GDP value relative to the mean (q it0 q t0 > 0) have a lower growth rate than the others. Structural model. Assume that the GDP of each country uctuates around the same international trend but with di erent levels: q it = q t + i + v it : where E i = 0 and Var i = 2, and v it is an iid shock, independent of i and q t, with mean 0 and Var v it = 2 v (white noise). 6
9 Regression to the mean 1 OLS estimator of the regression of q it1 q t1 on q it0 q t0 : P Pi q it1 q t1 qit0 q t0 i b 10 = 2 = ( i + v it1 ) ( i + v it0 ) Pi q it0 q t0 Pi ( i + v it0 ) 2 P! Cov ( i + v it1 ; i + v it0 ) Var ( i + v it0 ) = v = < 1: OLS estimator of the regression of q it0 q t0 on q it1 q t1 : P Pi q it0 q t0 qit1 q t1 i b 01 = 2 = ( i + v it1 ) ( i + v it0 ) Pi q it1 q t1 Pi ( i + v it1 ) 2 P! Cov ( i + v it1 ; i + v it0 ) Var ( i + v it1 ) = v = < 1: Hence the regressing q it1 q t1 on q it0 q t0 or q it0 q t0 on q it1 q t1 yields two estimators b 10 and b 01 which converge to the same value < 1!!!! 7
10 Regression to the mean 2 Lastly, q it1 q t1 = b 10 q it0 q t0 + bui, q it1 q it0 = q t1 q t0 + b10 1 {z } =b<0 q it0 q t0 + bui : So, one obtains b b < 0 although all countries follow parallel GDP trajectories, which therefore cannot converge! This result is known since Galton ( ) who observed that regressing the sizes of sons on the sizes of fathers or the sizes of fathers on the sizes of sons produced a coe cient less than one. This is the regression to the mean phenomenon. 8
11 Alternative model Construct a dynamic model of each country s GDP with heterogeneous levels: q it = i + q i;t 1 + v it ; i = 1; :::; N; t = 1; :::; T: If < 1 and (v it ) is stationary, each country tends to uctuate around a speci c target i q it q i;t 1 = (1 ) i q i;t 1 + v it : 1 1 as Clubs of convergence: countries with similar values of i. 9
12 Example of selectivity bias: productivity and wages Based on Heckman, Ecta, I will develop the di erent steps from the construction of an economic model, to the construction of the econometric model. Individuals determine labour supply by trading o consumption for leisure: ( c = h + max U(c; `) s.t. c;` 0 < c; 0 < ` T where utility function U(c; `) is increasing in consumption c and leisure `, is the wage (equal to labour productivity), is nonlabour income. Let `( ; + T ) = arg max fu(c; `)jc + ` = T + ; 0 < c; 0 < `g += c;` be the Marshallian demand for leisure (interior solution to optimization programme, i.e. without ` T ). The solution to (1) is ` (; + T ) = 10 ` if ` < T T if ` T : (1)
13 Proof The Lagrangian is L(c; `; 1 ; 2 ) = U(c; `) + 1 (T + c `) + 2 (T `); where 1 and 2 are the Lagrange multipliers. A solution (c; `) is such that `; 1 ; `; 1 ; 2 = `) 2 0; 2(T `) = 0; c + ` = T + ; 1 = 0 c > 0; T ` > 0: 1 2 = 0 11
14 Interior solution. An interior solution, such that c > 0; T > ` > 0, has 2 = 0, `) / c + ` = T + : Note that duality theory implies that the solution to this system is where ` = ` (; T + ) (; T + ) (; T + ) =@y V (; y) = max U(c; `) s.t. c + ` = y; c>0;`>0 is the indirect utility function associated to U. 12
15 Corner solution. A corner solution has ` = T; c =, T ) = = : as 1 > 0 as U(c; `) is strictly increasing wrt c. However, 2 can be equal to 0. Lastly, by de nition, ` (; T + ), satis ; `) = ; c + ` = T + : The being a decreasing function (of `) when U is concave, the T ) holds i ` (; T + ) T. (Draw a picture that shows that you want to be at the corner when ` (; T + ) T.) 13
16 Speci cation of labour supply function The next step is to specify the labour supply functions. First way. Choose a speci cation for U or V and determine `. Example: where y = T +. By Roy s identity, ` (; y) = V (; y) = y (; (; 1 hy ; 2 i h i 2 2 = = y : (LS1) 14
17 Second way. Determine U or V such as ` has the desired form. A good parametric speci cation is linear: ` (; y) = ln + y; (LS2) which is parametrically simpler as (LS1) (less parameters, e ects of and y additively separable). Is is possible to nd a utility function which rationalizes this function? Yes, one can show that where E 1 (t) = R 1 t V (; y) = e x x dx = Ei ( t), works. ln + y e E 1 () 15
18 Proof We search for a cost function y (; v) such that V (; y) = v (; (; Fix v and di erentiate V (; y) = v wrt (; + = ln + @ = (;y) or, assuming 6= (; v) = ln + This is a linear ODE, the solution of which is easily found to be where E 1 (t) = R 1 t y (; v) = ln + E 1 () + c (v) e x x dx = Ei ( t) is the E1-exponential-integral function and c (v) is an arbitrary function of v, that has to be increasing for y (; v) to be a cost function. Function c (v) is arbitrary, so one can take c (v) = v. By inverting y (; v) = y wrt v, one proves the announced result. 16 e
19 Theoretical predictions A model is useful if/as it allows to make predictions. Consumer theory tells us that the cost function y (; v) = ln + v + E 1 () e has to be increasing in v (true) and increasing and concave in (; Duality theory = ln + y (; v) > 0; = ` (; y (; v)). Hence ` (; y) > 0 2 y (; 2 = + ` (; y) < 0: (; These conditions cannot be imposed ex ante but should be veri ed over the whole support of the distribution of (; y) after estimation. Lastly, if leisure is a normal (; y) = > Moreover, we see that + ` < 0 and ` > 0 imply that > 0. Parameter can be positive or negative. 17 < 0:
20 Econometric model Now, we want to use the previous model to analyze a sample of individual data on female participation. There is an iid sample of N observations of weekly hours worked (h i = T `i), individual wage (w i ) if i does not work, then w i =, the number of years of education (Ed i ), the individual s age (Age i ) and the husband s wage ( i ). The theory yields the following model for (h i ; w i ): 8 h i w i! >< = >: h i i!! 0 if h i > 0; if h i 0; (selection rule) where h (; y) = T ` (; y) = T + ln y: An econometric model requires to consider the modelling of observed and unobserved heterogeneity. 18
21 Modelling heterogeneity We could assume that ( where x i = h i = at x i + ln i + u i ; ln i = b T z i + v i ; 1; Ed i ; Exp i ; (Exp i ) 2 ; i T ; a = (a1 ; a 2 ; a 3 ; a 4 ; a 5 ); (latent) z i = 1; Ed i ; Exp i ; (Exp i ) 2 ; Ed i Exp i T ; b = (b1 ; b 2 ; b 3 ; b 4 ; b 5 ); with Exp i = Age i Ed i 6, and! u i v i N 0 0! ; 2 u u v u v 2 v!! : The correlation between u i and v i re ects omitted variables determining both individual preferences and productivity. Note that education and experience/age are likely to determine both preferences and productivity. Note also that we have put some restrictions: i only conditions preferences and the interaction Ed i Exp i only conditions productivity. Always useful to have these sorts of restrictions as they provide instruments for potential endogeneity or selectivity problem. 19
22 The next step is to discuss identi cation. Identi cation Given that one has speci ed a fully parametric model, parametric identi cation is to show that two sets of parameters yielding the same likelihood value are necessarily equal. Here, one can show that the selection model makes no di erence. The model is identi ed if the latent model (latent) is identi ed. This requires the existence of variables determining ln i but not h i. Nonparametric identi cation holds when the distribution of observables picks only one set of parameters irrespective of the stochastic assumption on the distribution of u i and v i. Much more di cult to prove. We shall come back to this point later. 20
23 Limited information models In general it is useful to search for model parts which are simpler to estimate. a. Participation model. Eliminate ln i out of h i : h i = a T x i + ln i + u i = a T x i + b T z i + u i + v i = c T q i + r i ; where q i = 1; Ed i ; Exp i ; (Exp i ) 2 ; Ed i Exp i ; i T ; c = (a 1 + b 1 ; a 2 + b 2 ; a 3 + b 3 ; a 4 + b 4 ; b 5 ; a 5 ) T ; r i = u i + v i ; and v i r i! N! 0 ; 0 2 v rv where 2 r = 2 u v + 2 u v and rv = 2 v + u v. 21 rv 2 r!! ;
24 Let i = ( 1 if c T q i + r i > 0 0 otherwise (Probit) Then Pr( i = 1jx i ; z i ) = Pr(h i > 0jx i ; z i ) = Pr(r i > c T q i jq i ) ri = Pr > ct q i jq i r r c T q i = 1 r c T q i = ; where () is a standard normal distribution function. The Probit model of participation identi es c= r. r 22
25 b. Reduced form labour supply model. Consider regressing h i on x i and z i using a TOBIT model: ( c T q i + r i if c T q i + r i > 0 h i = (Tobit) 0 otherwise This will separately identify c and r. c. Productivity equation. One has the following selection model for productivity: ( b T z i + v i if c T q i + r i > 0 ln w i = (Selection) otherwise One can (should) use ML to estimate this selection model and consistently estimate b; c= r ; v and. Alternatively, one can use Heckman s two-step estimator: E(ln i jr i ; x i ; z i ) = b T z i + E (v i jr i ; x i ; z i ) = b T z i + E (v i jr i ) = b T z i + rv r 2 i : r 23
26 Hence E(ln w i jh i > 0; x i ; z i ) = b T z i + rv 2 r E r i jr i > ri j r i r = b T z i + rv E > r r = b T z i + rv ' c T q i = r r (c T q i = r ) = b T z i + rv c T q i : r r c T q i ct q i r Note that c T q i e ectively corrects for selection only if there does not already exist some function of regressors in z i which looks like it. This is why, and also not to rely too much on the stochastic assumptions, in order to strengthen identi cation, a good practice is to require the existence of variables in q i (that is x i ) which are not in z i (here i ). (If (t) = E ri r j r i r > t is unknown, the regression is called a semiparametric regression. See Robinson (ECMA, 1988).) r 24
27 Semiparametric regression Consider the regression: y i = x T i + (z i ) + u i Then y i E (y i jx i ) = (z i ) E ( (z i ) jx i ) + u i m (x i ; z i ) + u i Function E (y i jx i ) is identi ed, and so is m (x i ; z i ) = E [y (x; = d (z) dz is identi ed. So (z) is identi ed up to an additive constant. E (y i jx i ) jx i ; z i ]. Hence: For estimation, replace conditional expectations by conditional mean estimators, like the Nadaraya- Watson kernel estimator: x 7! E b NX K x i x h (y i jx) = y i P N i=1 i=1 K x i x h where K h t is a function that puts a lot of weight at t = 0 et less and less when jtj gets away from 0 (like the density of a continuous distribution centered at 0). 25
28 Identi cation from reduced forms Reduced forms a, b and c su ce to identify all parameters. In particular, is identi ed from c 5 = b 5 ; and r ; v and rv r identify u and as ( 2 r = 2 u v + 2 u v rv = 2 v + u v Minimum distance can be used to recover structural parameters from the reduced form parameters. In general, suppose that a vector m can be estimated by a root-n consistent estimator bm. Suppose that 9 2 such that m = em (), where em() is injective. Parameter can be estimated by minimum distance (GMM): min [ em() 2 bm]t W 1 [ em() bm] ; where W is the positive semi-de nite matrix. The Minimum Distance estimator is consistent b P! 0 along the lines of consistency of M-estimators. The optimal choice of the weighting matrix is W 0 = AVar p N ( bm m). 26
29 The asymptotic variance of the GMM estimator with the optimal weighting matrix is AVar( b M0 W0 1 M T 1 0 ) = ; N where M 0 em( T. In this example, is related to m = b T ; c T ; T rv ; v ; r r = a T ; ; b T ; u ; v ; T by the set of nonlinear equations sketched above. 27
30 Minimum distance The MD estimator solves: H bm; em(b ) T W 1 em( b ) Taylor expansion in the neighboorhood of (m 0 em( 0 ); 0 ): H bm; b {z } =0 where i bm = 0: ' H (m 0 ; 0 ) (m 0; 0 ) ( bm m {z T 0 ) (m 0; 0 ) b T (m 0; 0 T b 0 (m 0; 0 T ( bm m 0 (m 0 ; 0 T em( T W 1 em(0 T M 0 W 1 M0 (m 0 ; 0 ) em( T W 1 = M 0 W 1 If matrix M 0 W 1 M0 T (i.e. em() injective) is invertible, the consistency and asymptotic normality of bm implies that of b : p N b L! 0 N (0; 0 (W )) 28
31 with 0 (W ) = M 0 W 1 M T 0 1 M0 W 1 W 0 W 1 M T 0 M 0 W 1 M T 0 1 which is minimal when W = W 0 : 0 (W ) 0 (W 0 ), 0 (W 0 ) 1 0 (W ) 1, M 0 W0 1 M0 T M 0 W 1 M0 T M 0 W 1 W 0 W 1 M0 T 1 M0 W 1 M0 T, W0 1 W 1 M0 T M 0 W 1 W 0 W 1 M0 T 1 M0 W 1, I W 1=2 0 W 1 M {z 0 T M } 0 W 1 W 0 W 1 M0 T 1 M {z } 0 W 1 W T=2 {z 0 } =A =(A T A) 1 =A T which is true as matrix A A T A 1 A T, for A = W 1=2 0 W 1 M T 0, is an orthogonal projector whose eigenvalues are zeroes and ones. 29
32 Full maximum likelihood yields e ciency. Full Maximum Likelihood Usually, involves numerical optimization techniques. Reduced forms are useful to provide starting values for the algorithms and control the results (easy to make programming mistakes; Monte Carlo simulations recommended). The likelihood is given by L() = Y h i >0 pdf(ln i = ln w i ; h i = h i ) Y h i =0 Pr(h i 0) = Y Pr(h i = h i j ln w i ) pdf(ln i = ln w i ) Y Pr(h i 0) h i >0 h i =0 = Y 1 p h i ' at x i + ln i + u v! ln w 2 i b T z i v p >0 u 1 2 u 1 2 Y c T q i 1 ; h i =0 r 1 ln i v v b T z i Notice that h i = h i when h i > 0 implies that h i > 0. So Pr (h i > 0) is embedded in pdf(ln i = ln w i ; h i = h i). 30
33 Do not maximize L() directly. Operate rst the change in variables: a! a u ;! u ; b! b v ; u! 1 u ; v! 1 v : The likelihood becomes more linear in the parameters, which makes numerical convergence easier to achieve. Moreover, parameter is usually di cult to recover. It is advisable to use a grid search for, i.e. maximize the likelihood wrt to a u ; u ; b v ; 1 u and 1 v for di erent values of in [ 1; 1]. This provides a rst approximation of b and the other estimates. Then maximize the likelihood wrt a u ; u ; b v ; 1 u ; 1 v and starting from this point. This allows to nd the right local maximum. The MLE provides an e cient estimator under the joint normality assumption. However, the joint normality may be a strong assumption. Multi-step methods are often more robust as they are consistent under less restrictive conditions. Moreover, the MLE is sometimes compositionally cumbersome. 31
34 Conclusion It is very di cult to properly analyze these issues of unobserved heterogeneity, simultaneity and selectivity without a proper economic model formatting the econometric equations. Ideally, the econometric model should be the economic model. That is, heterogeneity and shocks should be proper variables of the economic model, and the stochastic assumptions about their distributions should be embedded in the economic model itself. I will call this empirical approach, structural econometrics. Econometrics would then be only about the statistical techniques of inference on the model parameters, not about modelling. Unfortunately, this is not always the case. Papers start with a question. Then there is an economic interpretation. Sometimes, a formal economic model is developed to analyze this questions, but not always. Lastly, very often, a reduced form econometric study is provided which establishes a few correlations which are supposed to support the economic theory. The aim of the remaining lectures is to study exemplary structural econometric papers. 32
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