Chapter 6. Maximum Likelihood Analysis of Dynamic Stochastic General Equilibrium (DSGE) Models

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1 Chapter 6. Maximum Likelihood Analysis of Dynamic Stochastic General Equilibrium (DSGE) Models Fall 22 Contents Introduction 2. An illustrative example Discussion The main steps in analyzing DSGE models Solving the model 5 2. The example The general case Business cycle implications 9 3. Impulse response function Variance decomposition Estimating DSGE models 4. The log likelihood function Bayesian inference

2 2. Introduction DSGE models provide a uni ed framework for analyzing business cycles, evaluating monetary policies, and forecasting. The paper by Lucas and Sargent (978) is a prelude to the literature. An and Schorfheide (27) provide a comprehensive survey of the application of Bayesian methods to the estimation and comparison of DSGE models... An illustrative example We use the following example, taken from Lubik and Schorfheide (24), to illustrate the main steps in analyzing DSGE models: x t = E t [x t+ ] (R t E t [ t+ ]) + g t () t = E t [ t+ ] + (x t z t ) (2) R t = t + e t ; (3) where g t i:i:d:n(; 2 g); z t i:i:d:n(; 2 z); e t i:i:d:n(; 2 e): The equations: The rst equation is an intertemporal Euler equation obtained from the households optimal choice of consumption and bond holdings (i.e., the IS-equation). The second equation is an expectational Phillips curve. The third equation is a monetary policy rule. Endogenous variables: x t is real spending or output, t is in ation, R t is the nominal interest rate. Shocks: g t is an aggregate demand shock, z t is exogenous change in marginal costs of production, e t is an exogenous disturbance to the interest rate. In more calibrated

3 3 models, g t, z t and e t are usually serially correlated processes. Such models can only be solved numerically. Parameters: = (; ; ; ; 2 g; 2 z; 2 e) with > ; < < ; > :.2. Discussion The above three equations are present in almost all DSGE models. An insightful discussion of them can be found in King (2). The forward-looking IS equation makes current real spending x t depend on the expected future level of real spending E t y t+ and the real interest rate R t E t [ t+ ]. There is also an aggregate demand shock g t : a positive g t raises aggregate spending at given levels of the endogenous determinants E t [x t+ ] and (R t E t [ t+ ]). The parameters > determines the e ect of the real interest rate on aggregate demand: If is larger then a given rise in the real interest rate causes a larger decline in real demand. This IS equation is described as forward-looking because E t [x t+ ] enters on the right-hand side. The expectational Phillips curve relates the current in ation rate t to expected future in ation E t [ t+ ], the current output x t, and an in ation shock z t. The parameter > governs how in ation responds to deviations of output from the capacity level. If there is a larger value of then there is a greater e ect of output on in ation; in this sense, prices may be described as adjusting faster being more exible if is greater. The monetary policy rule contains a systematic component t and a shock component e t. In more calibrated models, it often also involves deviations of output from

4 4 the capacity level. Here, it provides one additional restriction on the comovement of the in ation and interest rates. Question: How are DSGE models di er from Keynesian models analyzed in 96s? Answer: In one sentence, DSGE models describe general equilibrium relationships derived from consistently posed dynamic optimization problems. More speci cally:. All variables are determined endogenously in DSGE models, while in Keynesian models they are classi ed somewhat arbitrarily into endogenous and exogenous variables; 2. The expectations are determined from optimizing behaviors and are rational while in Keynesian models they are handled informally; 3. The parameters are identi ed through cross equation restrictions, while in Keynesian models they are identi ed through ad hoc exclusion restrictions. In addition:. DSGE models are dynamic. Both the past and the expectation about the future a ect the state of the economy; 2. The equilibrium relationships in DSGE models are stochastic. The economy uctuates while remaining in the equilibrium..3. The main steps in analyzing DSGE models. Obtain the optimality conditions that characterize the equilibrium; 2. Approximate the dynamics of the model using log linearization or higher order approximations around the steady state; 3. Solve the model with forward/backward recursions; 4. Construct the likelihood function. Maximize it numerically (rarely practical) or apply a Bayesian procedure for estimation and inference.

5 5 Applying Steps and 2 to a micro-founded dynamic general equilibrium model, say that of Woodford (23), leads to equations () to (3). We skip these steps. Our subsequent discussion focuses on Steps 3 and Solving the model We express the endogenous variables as a function of their lagged values and the exogenous shocks. In other words, the conditional expectations need to be solved out of the system. 2.. The example The system can be represented as x t = E t [x t+ ] ( t E t [ t+ ]) + g t e t (4) t = E t [ t+ ] + (x t z t ) It now involves only two endogenous variables. Equivalently {z } A E t [x t+ ] E t [ t+ ]! = {z } B x t t! + {z } C e t g t z t C A ; Or E t [x t+ ] E t [ t+ ]! = A 8 >< >: B x t t! + C e t g t z t 9 >= C A >; : Algebra shows that the eigenvalues of A B are greater than. Therefore, the only way to obtain a stable solution is to have B x t t! + C e t g t z t C A = :

6 6 This delivers Because R t x t t R t = = + + R t = t + e t ; h i e t In summary, the law of motion for output, in ation and interest rate is given by x t e t B C 4 t 5 = 4 g + t A : Question: Will an OLS regression of R t on t consistently estimate the policy reaction function? e t g t z t g t z t C A : C A : z t 2.2. The general case In general, particularly when the exogenous shocks and serially correlated, the models can only be solved numerically. A variety of algorithms are available. Here we brie y discuss the GENSYS algorithm of Sims (22). We only consider the case where the model has a unique stable solution. The starting point is the following canonical form S t = S t + t + t ; where S t is a vector of model variables that includes () the endogenous variables, (2) the conditional expectations, (3) variables from exogenous processes if they are serially correlated. t is a vector of exogenous disturbances.

7 7 t is a vector of expectation errors satisfying E t t = for all t. ; ; and are coe cient matrices. Note that in the working example, we have the following correspondence S t = ( t ; x t ; R t ; E t [ t+ ] ; E t [x t+ ]) ; t = (g t ; z t ; e t ) ; t = ( t E t [ t ] ; x t E t [x t ]) : The canonical system can be transformed using a generalized complex Schur decomposition of and. Speci cally, there exists Q, Z, and satisfying Q Z = ; Q Z = ; where stands for conjugate transpose, Q, Z are unitary matrices satisfying QQ = ZZ = I, and and are upper triangular matrices. Let w t = Z S t and premultiply the canonical form by Q, leading to 2 w ;t 2 = 22 w 2;t 22 w ;t w 2;t + Q : Q 2: ( t + t ) ; (5) where an ordering has been imposed such that the diagonal elements of ( 22 ) are greater (smaller) than those of ( 22 ) in absolute values. Assume the zero diagonal elements of lie on di erent rows from those of such that 22 is invertible. Because the generalized eigenvalues corresponding to 22 and 22 are unstable, the block of equations corresponding to w 2;t has a stable solution if and only if w 2; = and Q 2: t = Q 2: t for all t > : (6)

8 8 The rst condition, w 2; =, requires Z S = ; therefore determining the initial condition of the system. The second condition determines Q 2: t. Meanwhile, the system also involves a di erent linear transformation of t ; Q : t. If the solution exists and is unique, the row space of Q : must be contained in that of Q 2:. In this case, we can write Q : = Q 2: for some matrix. Premultiplying (5) by [I; ] gives us a new set of equations, free of references to t, that can be combined with the second block of (5) to give us 2 22 w ;t I w 2;t 2 22 w ;t Q : Q 2: = + t : w 2;t Note that t no longer appears in the system. By construction, the rst matrix on the left hand side is invertible. Thus, w ;t w 2;t = I 2 22 Q : Q 2: + t : I Finally, multiple both sides of the above equation by Z : w ;t w 2;t S t = Z Z S t I 2 22 Q : Q 2: +Z t : I More concisely, we can write S t = S t + t : It is important to recall from the de nition of S t that it includes terms such as the conditional expectations and variables from exogenous shock processes. Such variables

9 9 are not observable to the econometrician. To conduct estimation and inference, we can de ne a selection matrix C; which includes zeros and ones, to single out the observables. This delivers the following representation for dynamics of the observables Y t = CS t (7) S t = S t + t : The rst equation is a measurement equation. The second is a transition equation. 3. Business cycle implications The system (7) has a vector moving average representation: Y t = C ( L) t = C t + C t + C 2 t 2 + ::: 3.. Impulse response function = C ; = C ; = C 2 ; :::: t 2 Therefore, C is the impact e ect of t on the endogenous variable, C is the e ect of t one period later, C 2 is two periods later, and so on Variance decomposition As in Chapter 5, The h-step ahead forecast of Y t is Y t=t h = E Y t j f s g t h s= : The h-step ahead forecast error is Xh e t=t h = Y t Y t=t h = C k t For small values of h, e t=t h can be interpreted as errors in predicting short-run movements in Y t, while for large values of h, e t=t h can be interpreted as long-run k= k

10 movements. In the limit as h!, e t=t h = Y t. The importance of a speci c shock can then be represented as the fraction of the variance in e t=t h that is explained by that shock; it can be calculated for short-run and long-run movements in Y t by varying h. In DSGE models, the elements of t are typically mutually independent. Then, the variance of the i-th element of e t=t h is Xn Xh 2 j ~c 2 ij;k j= k= where ~c ij;k is the (i; j)-th element of C k. The faction of the h-step-ahead forecast error variance in Y i;t attributed to j;t is given by ; R 2 ij;h = P n j= 2 P h j k= ~c2 ij;k h 2 j P h k= ~c2 ij;k i: It is called the variance decomposition of Y i;t at horizon h. 4. Estimating DSGE models The starting point is the DSGE solution: where Y t = CS t S t = S t + t ; t i:i:d:n(; Q): We continue to use to denote the structural parameters of the model. 4.. The log likelihood function The likelihood function is f (Y T ; :::; Y ; ) :

11 Recall: f (Y T ; :::; Y ; ) = f (Y T jy T ; :::; Y ; ) f (Y T ; :::; Y ; ) : Iterating this formula, we nd Under normality, where f (Y T ; :::; Y ; ) = TY f (Y t jy t ; :::; Y ; ) : t= L = log f (Y T ; :::; Y ; ) = constant 2 TX ln jf t j t= v t = Y t E(Y t jy t ; :::; Y ) F t = E(v t v tjy t ; :::; Y ): 2 TX t= v tf t v t ; Kalman lter provides an e cient way for computing v t and F t recursively. Initialization. Assume the initial condition satis es where S j and P j are some parameters. S N(S j ; P j ); Prediction. are Starting with S, the optimal prediction of S and the associated MSE S j = E [S ] = S j P j = E S S j )(S S j = P j + Q : The optimal prediction of Y is then Y j = CS j : This implies v = Y Y j = C S S j ; F = CP j C :

12 2 Updating. Upon observing Y, S can be updated using (This is proved in the appendix.) S j = E[S jy ] = S j + P j C F v (8) P j = E[(S S j )(S S j ) jy ] = P j P j C F CP j : Recursion. We continue the predicting and updating steps for t = 2; :::; T. As a matter of notation, let and S tjt = E [S t jy t ; :::; Y ] P tjt = E S t S tjt )(S t S tjt )jy t ; :::; Y S tjt = E[S t jy t ; :::; Y ] P tjt = E[(S t S tjt )(S t S tjt ) jy t ; :::; Y ]: Then, in each recursion, the optimal prediction given (Y t ; :::; Y ) satis es S tjt = S t jt P tjt = P t jt + Q : The forecast error given (Y t ; :::; Y ) satis es v t = Y t Y tjt = C S t S tjt ; F t = CP tjt C : And the updating formula given (Y t ; :::; Y ) satis es S tjt = S tjt + P tjt C F t v t (9) P tjt = P tjt P tjt C F t CP tjt :

13 Bayesian inference For very simple models, the likelihood function can be maximized numerically to deliver informative results. For most models considered in practice, however, such a procedure tends to be either unstable or uninformative. This is because the likelihood surface tends to display ridges or plateaus and can have multiple local maxima. Bayesian procedures are widely applied. The idea is to use information from economic theory or empirical evidence obtained elsewhere to narrow down the parameter values. Then, more informative results can be obtained (or we hope so). Here an excellent reference is An and Schorfheide (27).

14 A- Appendix Proof of (8) and (9). We have the following general result for multivariate Normal distributions: if!!! z m 2 N ; 2 22 then z 2 m 2 (z 2 jz ) N m (z m ) ; : From the de nitions, we have! v t S t Yt ;:::;Y N S tjt! ; F t CP tjt P tjt C P tjt Apply the general result to the above distribution. We obtain (S t jv t ; Y t ; :::; Y ) N S tjt + P tjt C F t v t ; P tjt P tjt C F t CP tjt : This in particular implies References S tjt = S tjt + P tjt C Ft v t P tjt = P tjt P tjt C Ft CP tjt : [] An, S., and F. Schorfheide (27): Bayesian Analysis of DSGE Models, Econometric Reviews, 26, [2] King, R.G. (2): The New IS-LM Model: Language, Logic, and Limits, Federal Reserve Bank of Richmond Economic Quarterly, 86, [3] Lubik, T.A. and F. Schorfheide (24): Testing for Indeterminacy: An Application to U.S. Monetary Policy, American Economic Review, 94, [4] Lucas, R.E., JR, and T.J. Sargent (978): After Keynesian Macroeconomics, in After the Phillips Curve: Persistence of High In ation and High Unemployment. Boston: Federal Reserve Bank of Boston, 978. [5] Sims, C. A. (22): Solving Linear Rational Expectations Models, Computational Economics, 2, -2. [6] Woodford, M. (23): Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press, Princeton.! :