GMM estimation. Fabio Canova EUI and CEPR September 2014

Transcription

1 GMM estimation Fabio Canova EUI and CEPR September 2014

2 Outline GMM and GIV. Choice of weighting matrix. Constructing standard errors of the estimates. Testing restrictions. Examples.

3 References Hamilton, J., (1994), Applied Time Series Analysis, Princeton University Press. Hayashi, F. (2002), Econometrics, Princeton University Press. Gali, J and Gertler, M. (1999), "Ination Dynamics: A Structural Econometric Analysis, Journal of Monetary Economics, 44, Hansen, L.P., (1982), "Large Sample Properties of GMM Estimators", Econometrica, 50, Hansen, L.P. and Singleton, K., (1982), "Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models", Econometrica, 50, (corrigenda, 1984). Hansen, L. and Hodrick, R. (1980), "Forward Exchange Rates as Optimal Predictors of Future Spot Rates: An Econometric Analysis, Journal of Political Economy, 88,

4 Hall, A., (1992), "Some Aspects of Generalized Method of Moment Estimators", in G.S. Maddala, C.R. Rao and H.D. Vinod (eds.), Handbook of Statistics, vol. 11, Elsevier Science. Hall, A. and Sen, A. (1999) "Structural Stability testing in models estimated by GMM", Journal of Business and Economic Statistics, 17(3), Harding, D. and Pagan, A. (2006) "Synchronization of cycles", Journal of Econometrics, 132(1), Ogaki, M., (1993), "GMM: Econometric Applications", in G.S. Maddala, C.R. Rao, and H.D. Vinod, eds., Handbook of Statistics, vol. 11, Elsevier Science. Newey, W. (1990) "Ecient Instrumental variable estimation of nonlinear models", Econometrica, 58, Newey, W. and West, K., (1987), "A Simple, Positive Semi-Denite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix", Econometrica, 55,

5 Pagan, A. and Yoon, Y., (1993), "Understanding Some Failures of Instrumental Variable Estimators", University of Rochester, manuscript. Christiano, L. and den Haan, W. (1996) "Small Sample Properties of GMM for Business Cycle Analysis", Journal of Business and Economic Statistics, 14, Burnside, C. and Eichenbaum, M. (1996) "Small Sample properties of GMM-based Wald Tests", Journal of Business and Economic Statistics, 14, Furher, J. Moore, J. and Schuh, S. (1995) "Estimating the Linear Quadratic Inventory model, ML vs GMM", Journal of Monetary Economics, 35,

6 1 Introduction Dynamic rational expectation models are dicult to estimate: - There are expectations of future variables. - The decision rules are often non-linear in the parameters. - If they are of general equilibrium type, need to solve the model (to obtain a "nal form") and use (full information) maximum likelihood techniques.

7 Can we estimate the parameters without computing a nal form? - Yes, there is a class of distance estimators which can be used to estimate the parameters of a structural model using just the rst order conditions. - These estimators can be used also in many other situations. - Very general econometric framework

8 2 Denition of GMM estimator GMM minimizes distance between sample and population functions. If there are m conditions, we need weights to transform the vector of m conditions into an index. g 1 () vector of population (theoretical) functions. g T () vector of sample functions (sample size is T). GMM = argmin[g T () 0 W T g T () g 1 () 0 W g 1 ()] (1) where W T is a m m full rank matrix, converging in probability to some W.

9 In many economic models the g function are "orthogonality" conditions: g 1 () = E t [g(y t ; ) C] = 0 where is a set of parameters, y t are observable data, C a constant and E t (:) denotes conditional expectation operators. Hence GMM solves: GMM = argmin[g T () 0 W T g T ()] (2)

10 2.1 Examples of economic orthogonality conditions Example 2.1 Model of consumption and savings. max fc t ;sa t+1 g 1 t=0 E 0 X t t u(c t ) (3) c t + sa t+1 w t + (1 + r)sa t (4) w t = labor income, sa t+1 = savings, and r t = r 8t. Letting U c;t t )=@c t, and t the Lagrangian on the constraint at t, the FOC are U c;t = t (5) t = E t t+1 (1 + r) (6)

11 Combining the two equations leads to the Euler equation: E t [(1 + r) U c;t+1 U c;t 1] = 0 (7) (7) is an orthogonality condition for g(y t ; ) = [(1 + r)u c;t+1 =U c;t ] and C = 1. Suppose u(c t ) = ln c t. If we assume that r is given, (7) is one condition in one unknown parameter (just-identied case).

12 Example 2.2 Lucas Asset Pricing Model (endogenous r t ). Agents choose fc t ; B t+1 ; S t+1 g 1 t=0 to maximize E 0 P t t u(c t ) subject to c t + B t+1 + p s ts t+1 w t + (1 + r t )B t + (p s t + sd t )S t (8) stock prices. Op- B t (S t ) are bond (stock) holdings ; sd t dividends and p s t timality implies: E t [ U c;t+1 U c;t (p s t+1 + sd t+1) p s t 1] = 0 E t [ U c;t+1 U c;t r t ] = 0 (9)

13 (9) are two orthogonality conditions for g 1 (y t ; ) = U c;t+1 p s t+1 +sd t+1 U c;t p s t g 2 (y t ; ) = (1 + r t ) U c;t+1 U and C = 1. c;t, and If u(c t ) = ln c t, (9) are two conditions in one unknown parameter. (overidentied case). How do we nd in this case? - Could use either one of the two conditions. - Could also combine assigning weights w 1 and w 2, i.e. min (w 1T g 2 1T ()+ w 2T g 2 2T ()). In this case estimation becomes more ecient (we use all available information).

14 Example 2.3 Real Business Cycle Model X max E 0 t u(c t ; N t ) (10) fc t ;N t ;K t+1 g 1 t=0 t c t + K t+1 f(k t ; N t ; t ) + (1 )K t (11) N t = hours, 0 N t 1; K t = capital and t = technology disturbance. The Euler equation for capital accumulation is E t ( U c;t+1 U c;t [f K;t+1 + (1 )] 1]) = 0 (12) where U c;t t;n t ; f t (12) is an orthogonality condition t for g(y t ; ) = U c;t+1 U (f c;t K + (1 )) and C = 1. If u(c t ) = ln c t + V (1 N t ), we have one condition, and at least two parameters (; ) to estimate (under-identied case). Unless we add other orthogonality conditions, we can not estimate both parameters.

15 Conclusion: Dynamic models where agents have rational expectations produce orthogonality conditions as the result of optimization. Parameters entering these conditions can be estimated with GMM. How do we add conditions if we are as in a situation like example 2.3? - Use the law of iterated expectations: E(E t g(y t ; )) = Eg(y t ; ) = 0. Hence, we can substitute conditional expectations E t with unconditional ones. This extends the set of conditions to which GMM can be applied. In fact: GMM be applied to static optimality conditions, e.g. in example 2.3 one extra condition is U N;t U = f c;t N. Since this holds for every t, it must be that it holds on average, i.e. E( U N;t U c;t f N ) = 0. GMM can be applied to identities: e.g. budget constraint c t + k t+1 (1 )k t = y t implies E(c t + k t+1 (1 )k t y t ) = 0.

16 2.2 Denition of GIV estimators In example 2.3 there are more parameters than orthogonality conditions. We could jointly estimate the Euler equation and the intratemporal conditions if ; entered there but they don't. How do we proceed in this case? Let z t be any variables which is observable at time t. If E t [g(y t ; ) C] = 0 then Ef[g(y t ; ) C]z t g = 0 and GIV = argmin[z 0 T g T () 0 W T g T ()z T ] (13)

17 In example 2.3, let u(c t ) = ln c t, then: E t ( c t c t+1 [f K;t+1 + (1 )] 1]) = 0 (14) If c t 1 ; f K;t belong to the information set of agents, to estimate ; we could use E( c t c t+1 [f K;t+1 + (1 )] 1]) c t 1 c t = 0 (15) E( c t c t+1 [f K;t+1 + (1 )] 1])f K;t = 0 (16) - If u(c t ) has additional parameters, we could use c t 2 c t 1 ; f k;t 1, as instruments to "increase" the number of orthogonality conditions.

18 - How do we choose z t? Dicult! - How many z t should I use? If, for example, I add to the above E( c t c t+1 [f K;t+1 + (1 )] 1]) c t 2 c t 1 = 0 We now have three equations in two unknowns. Need to use a weighting matrix whenever dim(z)> dim().

19 2.3 Examples of econometric orthogonality conditions OLS: E(x 0 t e t) = 0. Then g(x t ; ) = (x 0 t y t x 0 t x0 t ) = x0 t e t and C = 0. IV: E(z 0 t e t) = 0. Then g(x t ; z t ; ) = (z 0 t y t z 0 t x0 t ) = z0 t e t and C = 0. ML: L = Q t f(y t ; ); C = ln L( ML = 0, g(x t ; ) ln f t(y t and Conclusion: linear models featuring (unconditional) orthogonality conditions among the identifying assumptions can be estimated with GMM.

20 3 Mechanics of GMM in linear models - Model: y t = x t + e t ; E(e t jx t ) 6= 0; E(e t e 0 t ) = 2 : - We have available z t such that E t (e t jz t ) = 0; E t (x t jz t ) 6= 0. Want: min Q T () = (e() 0 z)w T (e() 0 z) 0 where e() 0 z T 1 P t(e t () 0 z t ) and it is assumed to converge to E(e t z t ) = 0 as T becomes large, and W T is a weighting matrix. The F.O.C. are: x 0 T z T W T z 0 T y T = x 0 T z T W T z 0 T x T (17)

21 Two cases: a) If dim() = dim(z); z 0 T x T W T is a square matrix and GMM = (z 0 T x T ) 1 z 0 T y T. This is the standard IV estimator. b) If dim() < dim(z); GMM = (x 0 T z T W T z 0 T x T ) 1 (x 0 T z T W T z 0 T y T ). This is because z 0 T x T W T is rectangular and z 0 T x T W T Z T e T = 0 does not imply Z T e T = 0 (only dim() conditions are set to zero, dim(z) dim() are unrestricted).

22 - z must correlated with x otherwise (z 0 T x T ) 1 may not be computable and variance of estimator may be large. Two results: GMM estimators are consistent under general conditions, i.e. as T! 1; GMM! with probability one. GMM estimators are asymptotically normal i.e. T 0:5 ( GMM ) D! N(0; ) where = 1 z;x 2 z;z 1 x;z case a) (18) = ( x;z W z;x ) 1 ( x;z W 2 z;z W z;x )( x;z W z;x ) 10 case b) (19)

23 Covariance matrix in (19) depends on W. How to choose W? Choose W to minimize (asymptotic eciency). Solution: ^W = 2 1 z;z. Then ( ^W ) = 1 z;x 2 z;z 10 x;z (20) GMM ( ^W ) GMM = (x 0 T ^x T ) 1 (^x 0 T y T ) (21) where ^x T = z 0 T (z0 T z T ) 1 z 0 T x T.

24 Implications: i) Optimal W is proportional to the covariance matrix of z! use as weights the relative variability of instruments. ii) GMM with the optimal W is nothing else than 2SLS. If the model is nonlinear in variables and parameters what can we say about the properties of GMM estimators?

25 - In nonlinear model GMM is also consistent and asymptotically normal To obtain this result we need: i) y t must be covariance stationary and ergodic. ii) The function g must be martingale dierence, e.g. E t [g t jf t ] = 0. iii) E(g t (y t ; ) C) = 0 must have a unique solution. iv) A set of technical conditions (e.g. true parameter is not on the boundary of parameter space; space of is compact, etc.).

26 Summary of the results a) Just identied case (dim(g) = dim()): - GMM P! 0. - GMM D! N(0 ; T 1 B 1 AB 10 ) where B = E 0 0 ; A = E t (g t ( 0 )g t ( 0 ) 0 ). Since A,B are unknown, we can use consistent estimators in the formulas. Test of hypothesis are standard, i.e. for a t-test on 1;GMM use ( 1;GMM 1 )=(T 1 B 1 AB 10 ) 11 0:5 and compare it to a t-distribution table.

27 b) Overidentied case (dim(g) > dim()). Assume W T converges to some xed W a m m full rank matrix as T! 1. Then: - GMM P! 0 - GMM D! N(0 ; T 1 B 1 AB 10 ) if W T is any matrix converging to A 1 as T! 1 (this is the optimal weighting matrix in the non-linear case).

28 Implementation algorithm In the overidentied case, we need to know W to construct GMM and we need GMM to construct W. So we need an iterative approach. Assume g 1 = 0 1) Set W 0 T = I. 2) Solve 1 GMM = argmin[g T () 0 W 0 T g T ()] 3) Set W 1 T = T [P t g t ( 1 GMM )g t( 1 GMM )0 ] 1. 4) Repeat steps 2 and 3. until jjwt i WT i 1 jj < ; small,jj i i 1 jj <, or both.

29 When iterations are over, for each component i of the vector - i GMM N( 0; T 1 (( 1 T P i GMM 0 GMM 0 WT i 1 ( T 1 P i GMM 0 ) 1. GMM - The fully iterative algorithm may be costly to implement if is of large dimension. Alternative: two step estimator. Start from some 1 GMM, compute A; B, compute W 1, compute 2 GMM and stop. Properties of two step GMM are the same as those of a fully iterative GMM if initial 1 GMM is T 0:5 consistent.

30 3.1 Complications Covariance stationarity is necessary. Linear trends can be accommodated (see Ogaki (1993)) but not unit roots. In some economic model the g functions are not martingale dierence. Example 3.1 Consider a saving problem with two maturities: one and. The Euler equations are E t [ U c;t+1 U c;t r 1t ] = 0 (22) E t [ U c;t+ U c;t r t ] = 0 (23)

31 U c;t r t jt is the real interest rate quoted at t for bonds of maturity j. Then E t [ 1U c;t+ U c;t r 1t 1 + r t ] = 0 (24) i.e. expected (at t) forward rate for 1 periods must satisfy the no arbitrage condition (24). Log linearizing (24) around the steady state, y t+ ^c t+ + ^c t+1 ; x t r 1+r ^r t + r 1 1+r ^r 1 1t where ^: represents percentage deviations from the state, y t+ = x t + e t+ where e t+ satises E t [e t+ ] = 0 and 0 = 1. If sampling interval of the data dierent than, e t+ will be serially correlated, e.g. if data is monthly and is 12, e t+ will have MA(11) components. What happens if the (sample) orthogonality conditions are not martingale dierences?

32 A Few Results: - GMM estimators are consistent and normal in large samples even if g t is not a martingale dierence. However standard errors need to be corrected. In this case, in place of A, use S(! = 0) = P 1 = 1 1 T Pt g t g t = P 1= 1 ACF g (). - Newey-West (1987) Problem: S(! = 0) may not be positive denite. To make sure this is the case need to weight ACF g () appropriately i.e. S(! = 0) = P 1 = 1 1 T Pj K(J(T ); )ACF g (). J(T ) is truncation point and K(J(T ); ) is a kernel.

33 - Small sample properties of estimators depend on the kernel used. - All kernel estimates converge very slowly to their true values. Hence, asymptotic approximations are very poor in small samples. - Approximations may be so bad that it may be preferable to use incorrect standard errors than poorly estimated but correct ones. g may also be heteroschedastic. If the form of heteroschedasticity is known, get the right expression for A. Otherwise, use a HAC (heteroschedastic consistent covariance) matrix (Newey-West (1987)). Again, it may be better not to correct for heteroschedasticity if sample is small.

34 4 Testing orthogonality restrictions a) if dim() = dim(z) no testing is possible. b) if dim() = q 1 < dim(z) = q: To estimate need only q 1 conditions: q q 1 are left free. If model is correct, also these q q 1 conditions must be close to zero once GMM is plugged in. Hence, under H 0. J T = T [g T ( GMM ) 0 ]W T g T ( GMM )] D 2 (q q 1 ) where W T P! A 1 = E(g t ()g t () 0 ) 1.

35 Intuition: z 1 ; z 2 instruments, is a scalar. y = X + e Approximately: i) get ^ using z1 ; ii) compute e(^); iii) calculate J T using (z 2 e(^)) 2. If T is large J T is 2 (1).

36 5 Examples 5.1 Estimating linear monetary and scal rules i t = b 1 t + b 2 gap t + b 3 i t 1 + u t (25) d t = a 0 + a 1 d t 1 + a 2 (DD t gap t ) + + a 3 ((1 DD t ) gap t ) + a 4 debt t + e t (26) DD t = dummy variable equal to one if the gap is positive at t (i.e. we are in an expansion) and zero if the gap at t is negative (i.e. we are in a contraction), d t = decit, i t = nominal interest rate. Treat u t ; e t as expectational errors, i.e. E(u t jf t ) = E(e t jf t ) = 0.

37 Orthogonality conditions (using unconditional expectations): E(i t b 1 t b 2 gap t b 3 i t 1 ) = 0 (27) E(d t a 0 a 1 d t 1 a 2 (DD t gap t ) a 3 ((1 DD t )gap t ) a 4 debt t ) = 0 (28) Sample counterparts: 1 T 1 T X (i t b 1 t b 2 gap t b 3 i t 1 ) = 0 (29) t X (d t a 0 a 1 d t 1 a 2 (DD t gap t ) a 3 ((1 DD t )gap t ) a 4 debt t ) = 0(30) t Parameters to be estimated (a 0 ; a 1 ; a 2 ; a 3 ; a 4 ; b 1 ; b 2 ; b 3 ). 2 equations, 8 unknowns. No more equations in theory. Need to use GIV.

38 Issues of interest: What are the estimates of b 1 ; b 2 and a 4? Is monetary policy active (i.e. jb 1 j > 1) and scal policy passive a 4 < 0 or viceversa? (Leeper (1991)) Is scal policy reacting to output symmetrically over the cycle? Do results change with the sample? Use US data, 1967:1-2004:2. Output gap constructed as actual minus potential output (as reported by the FREDII dataset).

39 1) Instruments (dierent for dierent equations): one lag of output gap, ination and the nominal interest rate for equation 1; a constant, two lags of decit, of debt and of ination and one lag of the interacted variable DD t gap t for equation 2 (total of 11 instruments) b' se Ttest Jtest a a p value a a a equality test b E 06 b p value b Equality test is test of symmetric reaction of decit to cycle.

40 2) Use two lags of each instrument in the monetary policy equation. No change in the scal equation. b' se Ttest Jtest a a p value a a a equality test b E 05 b p value b

41 3) Omit the sample. b' se Ttest Jtest a a p value a a a equality test b E 10 b p value b Here both scal and monetary policy are passive jb 1 j < 1 and a 4 < 0 (insignicant). They act as if they have to balance the budget.

42 4) Eliminate but use two lags of ination. b' se Ttest Jtest a a p value a a a equality test b E 06 b p value b Estimates unstable. Dicult to draw useful conclusions. Rejection of model may come from rst equation (compare 1 and 2).

43 How do you check instrument relevance? i.e. that the instruments are correlated with the regressors? Use concentration (F)-statistics. y t = x t + e t x t = z t + v t Construct an F statistics for the hypothesis that = 0. If can't reject, instruments are irrelevant to explain regressors. Can't use this statistics if the model is nonlinear in the parameters (the distribution of this statistics is unknown in this case).

44 5.2 Estimating a New Keynesian Phillips curve t = E t t+1 + (1 & p)(1 & p ) & p mc t (31) where mc t = N tw t GDP are real marginal costs, & t p is the probability of not changing the prices, the discount factor and t is the ination rate. Transform this equation in an orthogonality condition: Sample counterpart: 1 T E( t t+1 (1 & p )(1 & p ) & p mc t ) = 0 (32) X t ( t t+1 (1 & p )(1 & p ) & p mc t ) = 0 (33)

45 Model is a single equation, nonlinear; parameters of interest (; & p ). Reduced form orthogonality condition: E t ( t a t+1 bmc t ) = 0 (34) Sample counterpart in reduced form linear estimation: 1 T X t ( t a t+1 bmc t ) = 0 (35) Real marginal costs not observable, need a proxy. Could use labor share (LS), or the fact, that in the model, GDP gap (Gap) is proportional to real marginal costs. Use a constant and up to 5 lags of ination and marginal costs as instruments. Report the result most favorable to the theory

46 Table: Estimates of NK Philips curve Linear estimates Nonlinear estimates Country/Proxy a b & p J-Test p-value US-Gap 0.993(16.83)-0.04 (-0.31)0.907 (10.35) (5.08) 2 (5)=0.15 US-LS 0.867(10.85) 0.001( 1.75) ( 7.74) (150.2) 2 (9)=0.54 UK-Gap 0.667(7.18) 0.528( 1.30) ( 4.96) (1.37) NA UK-LS 0.412(3.81) 0.004( 4.05) ( 4.07) (166.1) 2 (1)=0.25 GE-Gap 0.765(10.02)-0.01 (-0.22) (7.48) (0.03) NA GE-LS 0.491(3.34) 0.03 ( 1.85) ( 4.45) (7.18) 2 (1)=0.83

47 5.3 Estimating a RBC model Model has multiple equations and is nonlinear. max (ct ;K t+1 ;N t ) E 0 Pt tc1 ' t 1 ' + # N(1 N t ) subject to g t + c t + K t+1 = t Kt 1 Nt + (1 )k t = GDP t + (1 )K t (36) ln t = + z ln t 1 + 1t 1t (0; 2 z); ln g t = g + g ln g t 1 + 2t 2t (0; 2 g), k 0 given. Let g t be nanced with lump sum taxes. Optimality conditions: # N c ' t = t Kt 1 Nt 1 (37) 1 = E t ( c t+1 ) ' [(1 ) t+1 K c t+1 N t+1 + (1 )] (38) t

48 and w t = GDP t N t, r t = (1 ) GDP t K t + (1 ). 11 parameters: ve structural 1 = (; # N ; '; ; ) and 6 auxiliary ones 2 = ( ; g; z ; g ; g ; z ). Need at least 11 orthogonality conditions. E( 1 + k t+1 inv t ) = 0 (39) k t k t which determines if capital and investment data are available. E t ( c t+1 c t ) ' )[(1 ) t+1 K t+1 N t+1 + (1 )] = 0 (40) contains four parameters (; '; ; ). Since is "identied" from (39) transform (40) to produce at least three orthogonality conditions. Given

49 , the three parameters could be estimated using E(( c t+1 c t ) ' )[(1 ) t+1 K t+1 N t+1 + (1 )]) = 0 (41) E(( c t+1 c t ) ' )[(1 ) t+1 K t+1 N t+1 + (1 )] c t c t 1 ) = 0 (42) E(( c t+1 c t ) ' )[(1 ) t+1 K t+1 N t+1 + (1 )]r t) = 0 (43) The intratemporal condition E[c ' t t K 1 t N 1 t # N ] = 0 (44) involves ('; ; # N ). Given ('; ), it determines # N.

50 The auxiliary parameters can be estimated using E(ln t + z ln t 1 ) = 0 (45) E(ln t + z ln t 1 ) ln t 1 = 0 (46) E(ln t + z ln t 1 )) 2 2 z = 0 (47) E(ln g t g + g ln g t 1 ) = 0 (48) E(ln g t g + g ln g t 1 ) ln g t 1 = 0 (49) E(ln g t g + g ln g t 1 )) 2 2 g = 0 (50)

51 Government expenditure is observable, technological disturbances are not. One additional auxiliary condition needed. From production function, given estimates of ; ^z t = ln GDP t (1 ) ln K t ln N t. Sequential estimation: the last three conditions estimable separately from rst eight. (Need to correct for s.e. if you do this). Otherwise go joint. - Here use just identied system or a weakly overidentied one, without optimal weighting matrix. Overidentied estimates obtained adding lags of GDP t c K ; t t c, of investment or of the output labor ratio. t 1 Linearly detrended US quarterly data; 1956:1-1984:1.

52 Table: Estimates of a RBC model Parameter just identied over identied xing ' = (0.0002) 0.18 (0.0002) (0.0002) ' (0.043) (0.042) (0.038) 0.04 (0.039) (0.018) (0.033) (0.021) (0.068) # N (65.43) (72.32) (148.21) 4.42 (1.987) (0.001) (0.001) 0.001(0.001) (0.001) g 0.075(0.001) 0.076(0.001) (0.001) (0.001) z 1.038(0.054) (0.050) (0.050) (0.049) g (0.0005) (0.0006) (0.0006) (0.0006) 2 z ( ) ( ) ( ) ( ) 2 z ( ) ( ) ( ) ( ) 2 (6)

53 - Conditional on ', very low estimate of is obtained. - Structural parameters imprecisely estimated (except for ). - AR parameters in the near non-stationary region (both with just-identied or overidentied systems). - Estimates of are economically unreasonable except in the just-identied system. - Model strongly rejected in all cases ( 2 statistic smaller in last two cases). - Results are broadly independent of the value of ' chosen.

54 Economic analyses with GMM estimates - Tests of orthogonality conditions often not very informative about why the model is rejected. - What does rejection of orthogonality conditions imply in economic terms? Let h() be some statistics you can construct from the model (plug in estimate and solve it) and h T be the statistics you can construct in the data (with sample size T ). h=mean, variances, autocorrelations, etc.

55 Let m( T ) = h( T ) h T. Then T = 0 )( 0 ) + h. Under H 0 : T m( T ) 0 T 1 m( T )! D 2 (dim(h)). This test can be done for any subsets of statistics of interest. To construct h() need to solve and simulate model (GMM estimation does not require solving the model, just uses the FOCs).

56 Using just-identied estimates (after log-linearization). Economic Tests of the RBC model Moment P-value Moment P-value Moment P-value var(c)/var(gdp) 0.02 c-ar(1) 0.03 corr(c,gdp) 0.09 var(inv)/var(gdp) 0.00 inv-ar(1) 0.00 corr(inv,gdp) 0.01 var(n)/var(gdp) 0.00 N-AR(1) 0.00 corr(n,gdp) 0.00 gdp-ar(1) 0.00

57 6 Relationship GMM/Calibration - In GMM (just identied case) nd so that 1 T - In calibration nd so that 1 T P t g t () = 0. P t y t () = y (y are the steady states). If we set g t = y t y Calibration is equivalent to a just-identied GMM when the orthogonality condition is the deviation of the variable from the steady state. If we add conditions like var(c t ()) = var(c t ), then g t = [g 1t ; g 2t ]; g 1t = y t y and g 2t = var(c t ()) var(c t ). Calibration to the steady state with additional conditions is equivalent to just-identied GMM on [g 1t ; g 2t ].

58 7 General problems with GMM: - Choice of instruments: which one? How many? - Choice of orthogonality conditions? Which one do we select? - Small sample approximation? Generally bad. - What should we do to correct for deviation of data from ideal conditions? - Identication of parameters? B may be close to or exactly singular. - What if there are unobservable variables in orthogonality conditions?

59 8 Exercises 1) Consider estimating a log-linearized Euler equation with GIV when consumer's utility displays external habit persistence in consumption. In this case the Euler equation can be written as: 1 + c t = E tc t (1 + ) (i t E t t+1 ) (51) where c t is consumption growth, i t the nominal interest rate and t ination. There are two parameters: the risk aversion coecient and the habit parameter. Using both a just-identied and an overidentied system, test the relevance of consumption habit in matching the data of the country you are considering. (Hint: you will have to make a lot choices here: which instruments to use, which consumption series to use, how to deal with potential non-stationarities of the ination rate, etc. Make sure you are very transparent in specifying what you do. Most of the grade in this question depends on how well you explain what you are doing).

60 2) Consider three monetary policy rules of the form i t = i t 1 + (1 ) x x t + (1 ) p t + e t (52) i t = i t 1 + (1 ) x x t 1 + (1 ) p t 1 + e t (53) i t = i t 1 + (1 ) x E t x t+1 + (1 ) p E t t+1 + e t (54) The dierence between various specications is that the central bank looks at the past, the present or the future output gap (x t ) and ination rates ( t ) when setting interest rates. Estimate the parameters of these three rules by GIV using data for your favorite country. Which specication ts the data better? Why? Test the rst specication against the third (Hint: estimate a general model and test the three models as restricted specications). 3) Consider an extended Phillips curve equation of the form: t = 1 + p E t t+1 + p 1 + p t 1 + (1 & p)(1 & p ) & p mc t + e t (55) where mc t = N tw t GDP t are real marginal costs, & p is the probability of not changing the prices, t is the ination rate and p is an indexation parameter.

61 Estimate the parameters of this model by GIV. Test whether indexation is important. Repeat estimation xing = 0:99 (assuming you have quarterly data).