Motivation Non-linear Rational Expectations The Permanent Income Hypothesis The Log of Gravity Non-linear IV Estimation Summary.

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1 Econometrics I Department of Economics Universidad Carlos III de Madrid Master in Industrial Economics and Markets

2 Outline Motivation 1 Motivation

3 Motivation

4 Hansen's contributions GMM was developed by Lars Peter Hansen in 1982 as a generalization of the method of moments one important advantage of GMM is that it requires less restrictive assumptions than those needed for maximum likelihood estimation this is specially interesting in the context of semiparametric models, where the parameter of interest is finite-dimensional, whereas the full shape of the distribution function of the data may not be known GMM is currently applied in numerous fields, including labor economics, international finance, finance and macroeconomics today we are going to review three applications in finance, macroeconomics, and trade

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6 Hansen-Singleton (1982) Hansen and Singleton (1982) explore the econometric implications of dynamic rational expectations in the context of multi-period asset pricing If economic agents solve quadratic optimization problems subject to linear constraints, it is possible to fully characterize the equilibrium dynamic paths in more general settings, closed-form solutions for the equilibrium time paths of the variables of interest cannot be obtained Hansen-Singleton (1982) avoid this problem by the use of GMM

7 The Representative Consumer Plan the objective of the consumer is to maximize ] E 0 [ t=0 β t U (C t ) where C t is consumption at t and β is the discount factor in every time period t, the consumer has the choice of investing in 2 assets: Q 1 t and Q 2 t a feasible consumer plan must satisfy C t + P 1 t Q1 t + P 2 t Q2 t R 1 t Q1 t 1 + R 2 t Q2 t 1 + W t note that future prices P j τ and payos R j τ are random variables at t = 0

8 First Order Conditions rst-order necessary conditions for the consumer's problem [ P 1 t U (C t ) = β E t R 1 t+1 U (C ] t+1) [ P 2 t U (C t ) = β E t R 2 t+1 U (C ] t+1) Hence [ E β r 1 U ] (C t+1) t U (C t ) Z t = 1 [ E β r 2 U ] (C t+1) t U (C t ) Z t = 1 where r 1 t = R1 t+1 P 1 t variables and r 2 t = R2 t+1 P 2 t and Z t is composed of lags of all

9 CRRA Utility if Motivation U (C t ) = (C t) 1+α 1 + α, γ < 1 (Constant Relative Risk Aversion) then U ( ) α (C t+1) Ct+1 U = (C t ) and E E [ β r 1 t [ β r 2 t ( Ct+1 C t ( Ct+1 C t C t ) α Z t ] = 1 ) α Z t ] = 1

10 GMM conditions Motivation ( m 1 (b,a) = E [Z t br 1 t ( [Z t m 2 (b,a) = E br 2 t ( Ct+1 C t ( Ct+1 C t ) a 1)] ) a 1)] GMM provides consistent estimates of the time discount β and the relative risk aversion coecient α Since we have over-identifying restrictions, GMM also provides a test for the adequacy of the rational expectations model

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12 Motivation The permanent income hypothesis predicts that there is a structural and positive relation between current consumption and permanent income we let y represent permanent income i we observe c i, current consumption, and y i, current income if we regress c i on y i the OLS estimate will not be consistent if (y y i i ) is correlated with y i

13 Structural Equations we are interested in the model c i = β y i + v i where v i captures variations in consumption unrelated to permanent income we know that y i = y i + u i where u i represents a transitory shock in income to simplify things, let us assume σ 2 Var ({u i,v i,y }) = u 0 0 i 0 σ 2 v σ 2

14 Moment conditions since y i = y + u i i and c i = β y + v i i then E [y i y i ] = σ 2 + σ 2 u E [y i c i ] = βσ 2 E [c i c i ] = βσ 2 + σ 2 u that is, we have four parameters ( β,σ 2 u,σ 2 v,σ 2 ) but only three moment conditions the system is not identied and GMM cannot be implemented

15 Additional information Suppose now that we have additional information on, say, housing expenditure h i = γy i + ε i where ε i is uncorrelated then E [h i y i ] = γσ 2 E [h i c i ] = γβσ 2 E [h i h i ] = γ 2 σ 2 + σ 2 ε we now have six parameters ( β,σ 2 u,σ 2 v,σ 2,γ,σ 2 ε ) and six moment conditions the system is just-identied and β = E[h i c i ] E[h i y i ]

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17 The Traditional Gravity Equation many trade theories predict a special pattern in bilateral trade ows between countries that some economists call gravity laws becauseso they saythese laws are analogous to Newton's law of universal gravitation any way, regardless of how they call them, the relations are of the form where T ij = α 0 Y α i Y β D γ j ij T ij : trade ow from country i to country j Y i : GDP of country i Y j : GDP of country j D ij : a broad dention of distance between the two countries

18 The econometric model the gravity laws in economics however do not hold exactly, so empirical economists include a random term then, assume that T ij = α 0 Y α 1 i Y α 2 D α 3 j ij η ij E [log (η ij ) log (Y i ),log (Y j ),log (D ij )] = 0 one can then estimate the equation in logs

19 The equation in logs the equation in logs takes the form log (T ij ) = β 0 + α 1 log (Y i ) + α 2 log (Y j ) + α 3 log (D ij ) + log (η ij ) there are, however, two problems rst, many countries do not trade with each other, so T ij = 0 and we cannot take the logarithm this can be overcome by nonlinear methods second, if the model in levels is heteroskedastic, the assumption is NOT true E [log (η ij ) log (Y i ),log (Y j ),log (D ij )] = 0

20 Moment Conditions Note that T ij = exp (β 0 + α 1 log (Y i ) + α 2 log (Y j ) + α 3 log (D ij ))η ij or, equivalently, such that T ij = exp (x i β)η ij E [T ij exp (x i β) x i ] = 0 if we impose these population conditions in the sample we obtain an MM estimator that is equivalent to assuming that the conditional variance is proportional to the conditional mean

21 Non-linear IV Estimator

22 Pseudo PML estimation the MM estimator is dened using the moment conditions m (b) = (T ij exp (x ij β))x ij ij this estimator is equivalent to assuming that the conditional variance is proportional to the conditional mean it is sometimes referred to as the Poisson Maximum Likelihood estimator, although strictly speaking, one does not need to assume the Poisson distribution to conduct inference, one has to take into account that the model is heteroskedastic

23 GMM Motivation one way to improve the estimation is by adding new exogenous variables so that m (b) = (T ij exp (x ij β))z ij ij GMM in this context provides consistent estimates for non-linear IV estimation

24 Motivation one important advantage of GMM is that it requires less restrictive assumptions than those needed for maximum likelihood estimation this is specially interesting in the context of semiparametric models, where the parameter of interest is finite-dimensional, whereas the full shape of the distribution function of the data may not be known we have reviewed several examples from the fields of finance, macroeconomics, and trade