Simultaneous Equations

Transcription

1 Simultaneous Equations 1 Motivation and Examples Now we relax the assumption that EX u 0 his wil require new techniques: 1 Instrumental variables 2 2- and 3-stage least squares 3 Limited LIML and full FIML information maximum likelihood Also it is no longer clear if you can estimate parameters at all identification problem his is the 11 Example 1: Consumption c t α + βy t + u t ; y t c t + i t he second equation is called an identity because there is no error he two equations together are the structure of the model or the structural equations In this model, c t and y t are endogenous because they are determined inside the model, and i t and u t are exogenous because they are determined outside the model In the first equation, 1 and y t are explanatory variables If c t 1 were in the model, it would be predetermined and therefore maybe have many of the characteristics of exogenous variables, but it would not be exogenous In this model, there is correlation between y t one of the explanatory variables and u t the error Why? We can write our structure as c t α + βy t + u t 1 α + β c t + i t + u t α 1 β + β 1 β i t β u t; y t c t + i t 2 α 1 β + β 1 β i t + i t β u t α 1 β β i t β u t Equations 1 and 2 are called reduced form equations because they represent each endogenous variable as a function of only exogenous variables We can write the reduced form equations as c t π 01 + π 11 i t + v 1t y t π 02 + π 12 i t + v 2t 1

2 where π 01 π 11 π 02 π 12 α 1 β, β 1 β, α 1 β, 1 1 β are reduced form parameters and v 1t v 2t 1 1 β u t are reduced form errors Note the restrictions on the reduced form parameters and errors such as but not limited to π 01 π 02 ; v 1t v 2t Also note that we can solve for possibly nonunique values of the structural parameters in terms of the reduced form parameters For example, β π 11 ; 3 π 12 α π 01 1 π 11 π 12 So a feasible estimation approach is to estimate π using OLS OLS provides a consistent estimate of π Why? hen solve for α, β using equation 3 his is called the indirect least squares estimate of α, β, and it is consistent Why? However, α, β is biased why? and nonunique why? If, instead, we used OLS on the structural equation for c t, we would get β t ỹt c t t ỹt βỹ t + ũ t β + t ỹtũ t t ỹ2 t t ỹ2 t t ỹ2 t But ỹ t 1 1 β ĩt β ũt from the reduced form equation So 1 Eỹ t ũ t E 1 β ĩt β ũt ũ t 1 1 β Eũ2 t σ2 u 1 β 2

3 hus, plim β β + plim t ỹtũ t t ỹ2 t β + plim 1 t ỹtũ t plim 1 t ỹ2 t σ 2 u 1 β β β σ 2 i β 2 σ 2 u β + 1 β σ2 u σ 2 i + σ2 u > β 12 Example 2: Bananas Consider the structural model below describing the supply and demand for bananas: q d t α + βp t + e t qt s γ + δp t + u t qt d qt s We never observe supply or demand; we observe only the equilibrium price and quantity Consider the scatter plot of combinations of equilibrium prices and 3

4 quantities It is not at all obvious how to fit a supply and demand curve to them especially when both are moving around over time because of the errors e t and u t We try to analyze this problem more rigorously below so we don t slip on the banana problem First define q t qt d qt s, and rewrite the structural equations as q t α + βp t + e t q t γ + δp t + u t If we regress q t as a function of p t, what are we estimating? reduced form of the model: Consider the We can estimate and α + βp t + e t γ + δp t + u t p t γ α β δ + u t e t β δ q t α + β γ α β δ + e t + β u t e t β δ π p γ α β δ π q α + β γ α β δ 4

5 But knowledge of π p and π q will not provide us with any information about the structural parameters We say that all of the structural parameters are not identified Now consider changing the structural model to q d t α + βp t + θy t + e t qt s γ + δp t + u t qt d qt s In the new model, we know that the demand curve is shifting in a way predicted by the exogenous variable, y t his is going to allow us to plot a supply curve However we still won t be able to say much about the demand curve Consider the problem more rigorously First, as before, write q t α + βp t + θy t + e t q t γ + δp t + u t Now when we solve for the reduced form parameters, we get p t q t α γ δ β + θ δ β y t + et ut δ β δα γβ δ β π 0p + π 1p y t + v pt + δθ δ β y t + δet βut δ β π 0q + π 1q y t + v qt So, given consistent estimates of π 1p and π 1q eg, from OLS, we can estimate δ π1q / π 1p ; 5

6 thus δ is identified Similarly, we can set γ π 0q δ π 0p + π 1q he students should try at home the structural model q d t α + βp t + θy t + e t q s t γ + δp t + φz t + u t q d t q s t Note that, if the parameter is not identified, then no amount of data will allow us to estimate it hus, before we can even discuss estimation, we must first determine identification Over the rest of this topic, we will discuss: 1 Concepts and erminology 2 Identification 3 Estimation 4 Properties of Estimators 2 Concepts and erminology he general structural specification of our model is B y t C x t + u t n nn 1 n m m 1 n 1 By stacking equations over time next to each other, we get We assume that: 1 B is nonsingular; B Y C n nn n mm X + U n 2 All diagonal elements of B are 1 scaling convention; 3 u t iid 0, Ω with Ω positive definite; 4 x t are exogenous; 5 a priori from theory restrictions on B and C he reduced form of the model is Y B 1 CX + B 1 U AX + V 6

7 3 Identification We observe Â, but we want to observe B and Ĉ n2 n unknown parameters in B, nm unknown parameters in C, and nm known parameters in  We have nm equations in A B 1 C to explain n 2 n+nm parameters We can t do it; we need to place restrictions from theory on B and C to reduce the number of unknown parameters Possible restrictions: 1 B ij 0 or C ij 0 exclusion or zero restrictions; 2 γ ij B ij + i,j i,j δ ij C ij α; 3 Restrictions on Ω or nonlinear restrictions By far, the most popular restrictions are exclusion restrictions hings that can happen: 1 No structural parameters are identified; 2 Some structural parameters are identified; 3 Some structural parameters are identified, and there are restrictions on reduced form parameters overidentified; 4 All structural parameters are identified, and there are restrictions on reduced form parameters overidentified; 5 All structural parameters are identified just identified We also can discuss identifying an equation If all of the structural parameters in a structural equation are identified, then the equation is identified 31 Necessary and Suffi cient Conditions for Identification Let P [B C P 1 P 2 P n Write the jth restriction on the ith equation as P i φ ij 0 7

8 Assume R i restrictions on the ith equation, and let Φ i [ φ i1, φ i2,, φ iri hen we can write all of the restrictions on the ith equation as Example 1 Example: hen with restrictions P i Φ i 0 y 1t β 12 y 2t + u 1t y 2t γ 21 x 1t + γ 22 x 2t + u 2t P Φ 1 [ 1 β12 γ 11 γ 12 β 22 1 γ 21 γ 22 γ 11 γ 12 β ; Φ heorem 2 he ith equation is identified iff Rank P Φ i n 1 An implication of this theorem is that a necessary condition for the ith equation to be identified is R i n 1 his is called the order condition While it is not suffi cient for identification, it is necessary and very easy to verify Example 3 continued from Example 1 [ 1 β12 γ P Φ 1 11 γ 12 β 22 1 γ 21 γ 22 If the restrictions are true, then P Φ [ 0 0 γ 21 γ 22 Rank P Φ 1 1 [ γ11 γ 12 γ 21 γ 22 unless γ 21 γ 22 0 What would it mean for γ 21 γ 22 0? herefore the first equation is identified unless γ 21 γ 22 0 Also [ 1 [ 1 β12 γ P Φ 2 11 γ β 22 1 γ 21 γ 22 0 β 22 0 whose rank is 1 the second equation is identified 8

9 Example 4 Consider a crime rate example developed by students in class 311 Some Formalism and Proof of the Identification heorem Definition 5 A structure of a model is a set of parameters and functions which completely define the stochastic relationships between the endogenous variables and the exogenous variables We can denote the true structure as B 0, C 0, f 0 where f is the joint density of the errors Definition 6 wo structures, B 0, C 0, f 0 and B 1, C 1, f 1 are equivalent if they imply the same joint distribution for the endogenous variables iff they have the same reduced form Definition 7 Suppose B 0, C 0, f 0 is the true structure hen if α is a parameter of the model and α α 0 in all structures equivalent to B 0, C 0, f 0, then α is identified heorem 8 he structure B 1, C 1, f 1 is equivalent to B 0, C 0, f 0 iff a nonsingular matrix D : B 1 DB 0, C 1 DC 0, and u 1 t has the same distribution as Du 0 t Proof Assume B 1 DB 0, C 1 DC 0, and u 1 t has the same distribution as Du 0 t hen B 1 C 1 DB 0 DC 0 B 0 C 0, and B 1 u 1 t B 0 u 0 t Assume B 0, C 0, f 0 and B 1, C 1, f 1 are equivalent structures hen they have the same reduced form B 1 C 1 B 0 C 0 C 1 DC 0 where D B 1 B 0 B 1 DB 0, u 1 t Du 0 t 9

10 heorem 9 Suppose the true structure is P 0 [ B 0 C 0 and the restriction matrix in the ith equation is Φ i identified iff Rank P 0 Φ i n 1 he the ith equation is Proof Suppose P 1 DP 0 is an equivalent structure satisfying Pi 1 Φ i 0 hen D ip 0 Φ i P 1 Φ i 0 D ip 1 Φ i 0 where D i is the ith row of D and e ip 0 Φ i e ip 1 Φ i 0 where e i is a vector with 1 in the ith element and 0 everywhere else e i and D i are both elements of Null [ [ P 0 Φ i But Null P 0 Φ i has dimension 1 because Rank P 0 Φ i n 1 D i λe i for some scalar constant λ Normalization of B ensures that λ 1 P 1 i P 0 i Let P 0 have the ith row identified, and let Rank P 0 Φ i < n 1 hen D i λe i : D i Null [ P 0 Φ i Define and note that P 1 P 0 Note that P 1 i D i P 0, D i e i P 0 0 P 0 has less than full rank his is not possible because B has full rank P has full rank 4 Instrumental Variables Consider the structural equation y X k β + Qγ + u where EX u 0 ie, X are endogenous explanatory variables find k regressors, Z, with the properties: We want to 10

11 Z 1 plim X is invertible; Z 2 plim u 0 Z Note that an implication of plim X invertible is that plim invertible Z satisfying these two conditions are called instruments Let be the set of exogenous variables Z Z Q Let X X Q be the set of explanatory variables Let β β γ Z Z is be the set of structural parameters ignoring covariance parameters hen our model can be written as y X β + u If we premultiply our equation by Z, we get Z y Z X β + Z u Consider the estimator that solves the orthogonality condition E [Z y Z X β 0 : β Z X Z y his is the instrumental variables estimator of β It is too diffi cult to derive the moments of β, and in many cases they do not even exist he trouble occurs because X in the denominator is endogenous and, therefore, random But we can derive the asymptotic properties of β β Z X Z y Z X Z X β + u Z X Z X β + Z X Z u β + Z X Z u plim β β + plim Z X Z u Z β X Z u + plim β + plim Z X plim Z u β + plim Z X 0 β 11

12 β β Z X Z u Z X Z u [ D β β Z X Z u u Z X Z plim Z X Z uu Z X Z plim plim Z X plim Z uu Z plim X Z Note that EZ uu Z Z σ 2 IZ plim Z uu Z plim Z σ 2 IZ [ D β β σ plim 2 Z X plim Z Z plim X Z β β N [0, V with V σ 2 plim Z X plim Z Z Choice of instruments: 1 Excluded exogenous variables; 2 Lagged or leading regressors maybe; plim X Z 3 Excluded exogenous variables from implicit equations; 4 Out of thin air regressors; 5 2SLS 1st stage predictors 5 wo Stage Least Squares wo Stage Least Squares 2SLS is a special case of instrumental variables with some nice optimality properties Consider the model y X k β + Qγ + u 12

13 where X are endogenous explanatory variables with Rank Z k Let Z Z Q, X X Q, β β, γ Let Z be a set of instruments and define X i Z Z Z Z X i X Z Z Z Z X X Z Z Z Z X X satisfies the conditions for instruments for X and X for X as long as Z satisfies them Now consider the 2SLS estimator β X X X y First, this is equivalent to running a regression of o see this, write equation 4 as and y Xβ + Qγ + u 4 y X β + u β X X X y [ X Z Z Z Z Z Z Z Z X 1 X Z Z Z Z Z Z Z Z y [ X Z Z Z Z X 1 X Z Z Z Z y X X X y which is the instrumental variables estimator We can write β X X X y X X X X β + u β + X X X u 13

14 plim β β + plim X X X u X X X u β + plim β + plim X X plim X u Note that plim X X plim plim X X Q X X Q Q Q X Z Z Z Z X Q X plim X Z Z Z Z X plim Q X plim X Z plim Q X plim X Z plim Q X Z Z Z X t t plim Z Z t X Z Z Z Z Q Q Q plim X Z Z Z Z Q plim Q Q plim X Z plim Q Q plim Z X t Z Z Z Q plim X Z plim Q Q plim Z Z plim Z Q where all plim terms exist Also plim X u plim X u plim Q u plim X Z Z Z Z u plim Q u plim X Z plim Q u plim X Z plim Q u 0 0 Z Z Z u plim Z Z plim Z u 14

15 because Next, plim Z u plim Q u 0, 0 β β X X X u plim β β β β X X X u u X X X plim X X X uu X X X plim plim X X plim X uu X plim X X σ 2 plim X X plim X X plim X X β β N 0, V with V σ 2 plim X X plim X X plim X X Discuss testing Relationship Between 2SLS and IV 1 2SLS is a way to use k > k instruments optimally 2 If k k, then 2SLSIVILS 6 3SLS Basic idea: Do 2SLS and use residuals to estimate Ω consistently in GLS generalization of 2SLS Details: Consider hen use Ω y i X i β i + u i, i 1, 2,, n 15

16 with common instruments Z for each equation β i and residuals are [ Xi Z Z Z 1 [ Z Xi Xi Z Z Z Z y i X i P Z X i X i P Z y i, û i y i Xi β i [ I Xi P Z Xi Xi P Z y i Now stack equations: y 1 X1 0 0 y 2 0 X2 0 y n 0 0 Xn or Let be the instruments for X hen y X β + u I n Z β 1 β 2 β n + u 1 u 2 u n I n Z y I n Z X β + I n Z u 5 Note that D [ I n Z u I n Z D u I n Z I n Z Ω I I n Z Ω Z Z herefore, if we apply GLS to equation 5, we get β [ X I n Z Ω Z Z I n Z X 1 [ X I n Z Ω Z Z I n Z y [ X Ω 1 Z Z Z Z X 1 [ X Ω 1 Z Z Z Z y [ X Ω 1 P Z X 1 [ X Ω 1 P Z y Students should derive the asymptotic properties of 3SLS 16

17 7 FIML he basic idea in FIML is to write down the log likelihood function and maximize it over the parameters of interest We describe the details later Advantages and Disadvantages of FIML: 1 FIML allows for cross equation restrictions 2 FIML allows for nonlinearity more naturally 3 FIML is more sensitive to misspecification of the model 4 Small sample bias is larger for FIML than for IV than for OLS 5 All methods trade off effi ciency against robustness 17