Bayesian Methods for DSGE models Lecture 1 Macro models as data generating processes

Transcription

1 Bayesian Methods for DSGE models Lecture 1 Macro models as data generating processes Kristoffer Nimark CREI, Universitat Pompeu Fabra and Barcelona GSE June 30, 2014

2 Bayesian Methods for DSGE models Course admin Lectures: Mon-Fri, 9-11, Room Practicas: Mon-Thur, , Room To contact me: Office (4th floor, silver building between Econ Dept and the park)

3 Bayesian Methods for DSGE models Course overview 1. Macro models as data generating processes 2. State space models and likelihood based estimation 3. Bayesian estimation of DSGE models 4. Bayesian analysis of DSGE models 5. Structural empirical models of news, noise and imperfect information

4 Macro models as data generating processes Today: Introduction to Bayesian estimation Theoretical models as likelihood functions Micro founded macro models Solving linear rational expectations models

5 Bayesian statistics

6 Bayesian statistics Bayesians used to be fringe types, but Bayesian methods now dominate applied macro economics This is largely due to increased computing power Bayesian methods have several advantages: Facilitates incorporating information from outside of sample Easy to compute confidence/probabilty intervals of functions of parameters Good (i.e. known) small sample properties

7 Why probabilistic models? Is the world characterized by randomness? Is the weather random? Is a coin flip random? ECB interest rates? It is difficult to say with certainty whether something is truly random.

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9 Two schools of statistics

10 Probability and statistics: What s the difference? Probability is a branch of mathematics There is little disagreement about whether the theorems follow from the axioms Statistics is an inversion problem: What is a good probabilistic description of the world, given the observed outcomes? There is some disagreement about how to interpret probabilistic statements and about how we should interpret data when we make inference about unobservable parameters

11 Two schools of thought What is the meaning of probability, randomness and uncertainty? The classical (or frequentist) view is that probability corresponds to the frequency of occurrence in repeated experiments The Bayesian view is that probabilities are statements about our state of knowledge, i.e. a subjective view. The difference has implications for how we interpret estimated statistical models and there is no general agreement about which approach is better.

12 Frequentist vs Bayesian statistics Variance of estimator vs variance of parameter Bayesians think of parameters as having distributions while frequentist conduct inference by thinking about the variance of an estimator Frequentist confidence intervals: If point estimates are the truth and with repeated draws from the population of equal sample length, what is the interval that θ lies in 95% of the time? Bayesian probability intervals: Conditional on the observed data, a prior distribution of θ and a functional form (i.e. model), what is the shortest interval that with 95% contains θ?

13 Coin flip example After flipping a coin 10 times and counting the number of heads, what is the probability that the next flip will come up heads? A Bayesian with a uniform prior would say that the probability of the next flip coming up heads is x/10 and where x is the number of times the flip came up heads in the observed sample A frequentist cannot answer this question: A frequentist makes probabilistic statements about the likelihood of observing a particular realized sample given a null hypothesis about the parameter that governs the probability of heads vs tails

14 The subjective view of probability Important: Probabilities can be viewed as statements about our knowledge of a parameter, even if the parameter is a constant E.g. treating risk aversion as a random variable does not imply that we think of it as varying across time. Subjective does not mean that anything goes!

15 The main components in Bayesian inference

16 Bayes Rule Deriving Bayes Rule Symmetrically Implying or p (A, B) = p(a B)p(B) p (A, B) = p(b A)p(A) p(a B)p(B) = p(b A)p(A) p(a B) = which is known as Bayes Rule. p(b A)p(A) p(b)

17 Data and parameters The purpose of Bayesian analysis is to use the data y to learn about the parameters θ Parameters of a statistical model Or anything not directly observed Replace A and B in Bayes rule with θ and y to get p(θ y) = p(y θ)p(θ) p(y) The probability density p(θ y) then describes what we know about θ, given the data.

18 Parameters as random variables Since p(θ y) is a density function, we thus treat θ as a random variable with a distribution. This is the subjective view of probability Important: the randomness of a parameter is subjective in the sense of depending on what information is available to the person making the statement. The density p(θ y) is proportional to the prior times the likelihood function p(θ y) p(y θ) }{{}}{{} posterior p(θ) }{{} likelihood function prior But what exactly is a prior and a likelihood function?

19 The prior The prior summarizes the investigators knowledge about parameters θ before observing the data y. In practice, we need to choose functional forms and select hyper parameters that reflect this our prior knowledge. Example: Let s say that θ is 2 dimensional θ 1 U[ 1, 1] θ 2 N(µ θ2, σ 2 θ 2 ) -1,1,µ θ2 and σ 2 θ 2 would then be the hyper parameters

20 The likelihood function The likelihood function is the probability density of observing the data y conditional on a statistical model and the parameters θ. For instance if the data y conditional on θ is normally distributed we have that p(y θ) = (2π) T /2 Ω 1 [ 1/2 exp 1 ] 2 (y µ) Ω 1 (y µ) where Ω and µ are either elements or functions of θ.

21 The prior and the likelihood function Remember how we derived Bayes rule: p(θ, y) = p(y θ)p(θ) The model determines both the likelihood function and what the the prior is about.

22 The posterior density p(θ y) The posterior density is often the object of fundamental interest in Bayesian estimation. The posterior is the result. We are interested in the entire conditional distribution of the parameters θ

23 Model comparison Bayesian statistics generally do not reject models Instead: assign probabilities to different models We can index different models by i = 1, 2,...m p(θ y, M i ) = p(y θ, M i)p(θ M i ) p(y M i )

24 Model comparison The posterior model probability is given by p(m i y) = p(y M i)p(m i ) p(y) where p(y M i ) is called the marginal likelihood. It can be computed from p(y θ, Mi )p(θ, M i ) p(θ y, M i )dθ = dθ p(y M i ) by using that p(θ y, M i )dθ = 1 so that p(y M i ) = p(y θ, M i )p(θ, M i )dθ

25 The posterior odds ratio The posterior odds ratio is the relative probabilities of two models conditional on the data It is made up of The prior odds ratio p(m i ) p(m j ) The Bayes factor p(y M i ) p(y M j ) p(m i y) p(m j y) = p(y M i)p(m i ) p(y M j )p(m j )

26 Concepts of Bayesian analysis Prior densities, likelihood functions, posterior densities, posterior odds ratios, Bayes factors etc are part of almost all Bayesian analysis. Data, choice of prior densities and the likelihood function are the inputs into the analysis Posterior densities etc are the outputs The rest of the course is about how to choose these inputs and how to compute the outputs

27 Bayesian computation

28 Bayesian computation More specifics about the outputs Posterior mean E(θ y) = θp(θ y)dθ Posterior variance var(θ y) = E [[θ E(θ y)] y] 2 Posterior prob(θ i > 0) These objects can all be written in the form E(g (θ) y) = g (θ) p(θ y)dθ where g(y) is the function of interest.

29 Posterior simulation There are only a few cases when the expected value of functions of interest can be derived analytically. Instead, we rely on posterior simulation and Monte Carlo integration. Posterior simulation consists of constructing a sample from the posterior distribution p (θ y) Monte carlo integration then uses that ĝ S = 1 S S g s=1 (θ (s)) and that lim S ĝ S = E(g (θ) y) where θ (s) is a draw from the posterior distribution.

30 The end-product of Bayesian statistics Most of Bayesian econometrics consists of simulating distributions of parameters using numerical methods. A simulated posterior is a numerical approximation to p(θ y) We rely on ergodicity, i.e. that the moments of the constructed sample correspond to the moments of the distribution p(θ y) The most popular (and general) procedure to simulate the posterior is called the Random Walk Metropolis Algorithm

31 Bayesian DSGE models

32 Modern macro models (Kocherlakota) 1. Specifies constraints for households, technologies for firms and resource constraints for the whole economy 2. Specifies household preferences and firm objectives 3. Assumes forward looking behavior 4. Includes stochastic processes 5. Are models of the entire economy: No prices are exogenous

33 Bayesian DSGE models A strategy for estimating structural macro models: Specify problems of representative agent and firm and the technologies at their disposal Log-linearize model to find approximate first order conditions and budget constraints Estimate parameters of model using likelihood based methods Likelihood based means that we specify a complete statistical model that is assumed to be the data generating process Bayesian estimation Combine prior and sample information in a transparent manner

34 A minimalistic DSGE model

35 The building blocks of a minimalistic DSGE model A representative agent that consumes and supply labor A representative firm that sets prices to maximize profit A central bank that sets nominal interest rates Exogenous shocks (or wedges )

36 Preferences and technology

37 The representative agent s objective In period t the representative agent maximizes expected future utility { } E t β s U (C t+s (i), N t+s (i)) s=0 where C t is consumption, N t is supplied hours of labor. Agents are assumed to be forward looking, but also to value the present more than the future (0 < β < 1).

38 The representative agent s objective Period t utility U (C t (i), N t (i)) = (C t(i)) 1 γ (1 γ) N t(i) 1+ϕ 1 + ϕ Main assumptions: Households like to consume but they do not like to work. Decreasing marginal utility of consumption Increasing marginal disutility of working But we also impose specific functional forms.

39 The decisions made by the representative agent How much to consume and how much to save (Euler-equation) How much to work: U c (C t ) = βe t R t P t U c (C t+1 ) P t+1 W t P t U c (C t ) = N ϕ t

40 Production of goods Output in firm j (0, 1) is produced using labor as the sole input Y t (j) = exp(a t )N t (j) where a t is an exogenous labor augmenting technology.

41 Price setting Prices are reset only infrequently (with prob = 1 θ). This results in a (linearized) Phillips curve where x is potential output. π t = βe t π t+1 + κ(y t x t ) Monopolistic competition: Each firm produces a unique good and has some market power.

42 Monetary policy Monetary policy is often represented by a simple Taylor-type rule r t = φ y y t + φ π π t + φ r r t 1 + u r t Can be viewed as a reduced form representation of an optimizing central bank and fits the data quite well. We will simplify the model further by assuming that φ y = φ r = 0.

43 The linearized structural system After linearizing, the main model equations are given by π t = βe t π t+1 + κ(y t y t ) y t = E t y t+1 σ (i t E t π t+1 ) i t = φπ t x t = ρx t 1 + ut x : ut x N(0, σu) 2 where π t, y t, y t, i t are inflation, output, potential output and nominal interest rate respectively. We want to estimate the parameters θ = {ρ, γ, κ, φ, σ x, σ y, σ π, }

44 Solving the model

45 3 ways to solve a linear model Solving a model using full information rational expectations as the equilibrium concept involves integrating out expectations terms from the structural equations of the model by replacing agents expectations with the mathematical expectation, conditional on the state of the model. Three different ways of doing this. 1. Method of undetermined coefficients, can be very quick when feasible and illustrates the fixed point nature of the rational expectations solution. 2. Replacing expectations with linear projections onto observable variables 3. Decouple the stable and unstable dynamics of the model and set the unstable part to zero.

46 The 3 equation NK model As a vehicle to demonstrate the different solution methods, we will use a simple New-Keynesian model π t = βe t π t+1 + κ(y t y t ) y t = E t y t+1 σ (i t E t π t+1 ) i t = φπ t x t = ρx t 1 + u t : u t N(0, σu) 2 where π t, y t, y t, i t are inflation, output, potential output and nominal interest rate respectively. Single variable, potential output x t, as the state.

47 Method I: Method of undetermined coefficients Pros Method is quick when feasible Illustrates well the fixed point nature of rational expectations equilibria. Cons Difficult to implement in larger models

48 Method of undetermined coefficients π t = βe t π t+1 + κ(y t x t ) y t = E t y t+1 σ (i t E t π t+1 ) i t = φπ t x t = ρx t 1 + u t : u t N(0, σu) 2 Start by substituting in the interest rate in the Euler equation x t = ρx t 1 + u x t y t = E t (y t+1 ) 1 γ [φ ππ t E t (π t+1 )] π t = E t (π t+1 ) + κ [y t x t ]

49 Solving model using method of undetermined coefficients Conjecture that model can be put in the form x t = ρx t 1 + ut x y t = ax t π t = bx t Why is this a good guess?

50 Solving model using method of undetermined coefficients Substitute in conjectured form of solution into structural equation ax t = aρx t 1 γ [φ πbx t bρx t ] bx t = bρx t + κ [ax t x t ] where we used that x t = ρx t 1 + u x t implies that E[x t+1 x t ] = ρx t

51 Solving model using method of undetermined coefficients Equate coefficients on right and left hand side a = aρ 1 γ φ πb + 1 γ bρ b = bρ + κ [a 1] or [ (1 ρ) 1 γ (φ π ρ) κ (1 ρ) ] [ a b ] = [ 0 κ ]

52 Solving model using method of undetermined coefficients Solve for a and b [ ] a = b [ (1 ρ) 1 γ (φ π ρ) κ (1 ρ) ] 1 [ 0 κ ] or [ a b ] [ = κ φ ρ c κγ 1 ρ c ] where c = γ κρ 2γρ + κφ + γρ 2 < 0

53 The solved model The solved model is of the form x t = ρx t 1 + u x t y t = κ ρ φ π x t c π t = κγ ρ 1 x t c where c = γ κρ 2γρ + κφ + γρ 2 < 0

54 Method II: Replacing expectations with linear projections The second method uses that projections of the future values of variables on observables gives optimal expectations (in the sense of minimum error variance) if the observables span the space of the state. How does it work? Replace E t π t+1 and E t y t+1 with linear projections of these variables on current inflation. There is nothing special about inflation. Projecting onto current output would also work. But first we need to know more about projections.

55 Inner product spaces An real vector space H is said to be an inner product space if for each pair of elements x and y in H there is a number x, y called the inner product of x and y such that x, y = y, x x + y, z = x, z + y, z for all x, y, z H αx, y = α x, y for all x, y H and α R x, x 0 for all x H x, x = 0 if and only if x = 0

56 The projection theorem If M is a closed subspace of the Hilbert Space H and x H, then (i) there is a unique element x M such that x x = inf x y y M and (ii) x M and x x = inf y M x y if and only if x M and (x x) M where M is the orthogonal complement to M in H. is defined as, The element x is called the orthogonal projection of x onto M.

57 The space L 2 (Ω, F, P) The space L 2 (Ω, F, P) consists of all collections C of random variables X defined on the probability space (Ω, F, P) satisfying the condition EX 2 = X 2 (ω)p (dω) < and define the inner product of this space as X, Y = E (XY ) for any X, Y C Ω

58 Least squares estimation via the projection theorem The inner product of inner product space L 2 corresponds to a covariance orthogonal projection will be the minimum variance estimate since x x = E (x x) (x x) Both the information set and the variables we are trying to predict must be elements of the relevant space

59 Least squares estimation via the projection theorem To find the estimate x as a linear function of y simply use that x βy, y = E [ (x βy) y ] = 0 and solve for β β = E ( xy ) [ E ( yy )] 1 The advantage of this approach is that once you have made sure that the variables y and x are in a well defined inner product space, there is no need to minimize the variance directly. The projection theorem ensures that an estimate with orthogonal errors is the (linear) minimum variance estimate.

60 Two useful properties of linear projections 1. If two random variables X and Y are Gaussian, then the projection of Y onto X coincides withe the conditional expectation E(Y X ). 2. If X and Y are not Gaussian, the linear projection of Y onto X is the minimum variance linear prediction of Y given X.

61 Replacing expectations with linear projections We will use that in equilibrium E (π t+1 π t ) = cov(π t, π t+1 ) π t var (π t ) E (y t+1 π t ) = cov(π t, y t+1 ) π t var (π t ) if the innovations u t to x t are Gaussian.

62 Replacing expectations with linear projections Let c 0 π t = E (π t+1 π t ) d 0 π t = E (y t+1 π t ) denote initial candidate projections of expected inflation and output on current inflation. We can then write the structural equations as π t = βc 0 π t + κ(y t x t ) y t = d 0 π t σ (φπ t c 0 π t )

63 Replacing expectations with linear projections Put the whole system in matrix form or x t π t y t = κ 1 βc 0 κ 0 d 0 + σφ σc κ 1 βc o κ 0 d 0 + σφ σc 0 1 X t = AX t 1 + Cu t 1 1 ρ u t x t 1 π t 1 y t 1

64 Solution algorithm 1. Make an initial guess of c 0 and d 0 in (1) 2. Compute the implied covariances of current inflation and future inflation and output using and E [X t X t ] = Σ XX Σ XX = AΣ XX A + CC E [X t+1 X t ] = AΣ XX 3. Replace the c s and d s with the c s+1 and d s+1 in ((1)) c s+1 = cov(π t, π t+1 ) var (π t ) d s+1 = cov(π t, y t+1 ) var (π t ) using the covariances from Step Repeat Step 2-3 until c s and d s converges.

65 Method III: Stable/unstable decoupling Originally due to Blanchard and Kahn (1980) Computational aspects of the method has been further developed by others, for instance Klein (2000). The most accessible reference is probably Soderlind (1999), who also has code posted on his web site. The method has several advantages: Fast Provides conditions for when a solution exists Provides conditions for when the solution is unique.

66 Method III: Stable/unstable decoupling Start by putting the model into matrix form or x 1 t = β 0 0 σ 1 ρ 0 0 κ 1 κ 0 σφ 1 A 0 [ x 1 t+1 E t x 2 t+1 ] x t+1 E t π t+1 E t y t+1 x t 1 π t y t = A 1 [ x 1 t x 2 t ] + C 1 u t+1 u t+1 is vector containing the pre-determined and/or exogenous variables (i.e. x t ) a vector containing the forward looking ( jump ) variables (i.e. E t y t+1 and E t π t+1 ). x 2 t

67 Method III: Stable/unstable decoupling Pre-multiply both sides of A 0 [ x 1 t+1 E t x 2 t+1 ] = A 1 [ x 1 t x 2 t ] + C 1 u t+1 by A 1 0 to get [ x 1 t+1 E t x 2 t+1 ] = A [ x 1 t x 2 t ] + Cu t+1 where A = A 1 0 A 1 and C = A 1 0 C 1. For the model to have unique stable solution the number of stable eigenvalues of A must be equal to the number of exogenous/pre-determined variables.

68 Method III: Stable/unstable decoupling Use a Schur decomposition to get A = ZTZ H where T is upper block triangular [ ] T11 T T = 12 0 T 22 and Z is a unitary matrix so that Z H Z = ZZ H = I ( = Z H = Z 1 ). For any square matrix W, W 1 AW is a so called similarity transformation of A. Similarity transformations do not change the eigenvalues of a matrix We can always choose Z and T so that the unstable eigenvalues of A are shared with T 22

69 Method III: Stable/unstable decoupling Define the auxiliary variables ] [ θt δ t = Z H [ x 1 t x 2 t ] We can then rewrite the system (67) as [ ] [ Z H x 1 t+1 E t xt+1 2 = Z H ZTZ H x 1 t xt 2 ] or equivalently E [ θt+1 δ t+1 ] [ T11 T = 12 0 T 22 ] [ θt δ t ] since Z H Z = I.

70 Method III: Stable/unstable decoupling For this system to be stable, the auxiliary variables associated with the unstable roots in T 22 must be zero for all t. (WHY?) Imposing δ t = 0 t reduces the relevant state dynamics to θ t = T 11 θ t 1 To get back the original variables we simply use that [ ] [ ] x 1 t Z11 = θ Z t 21 or [ x 1 t x 2 t x 2 t ] = [ Z11 Z 21 ] Z 1 11 x 1 t which is the solution to the model. It is in the form x 1 t = Mx 1 t 1 + ε t x 2 t = Gx 1 t where M = Z 11 T 11 Z 1 11 (= ρ in our example) and G = Z 21 Z 1 11.

71 The model as a state space system X t = AX t 1 + Cu t Z t = DX t where X t = x t, A = ρ, Cu t = ut x r t φ π Z t = π t, D = κγ 1 ρ c y t κ φπ ρ c

72 Additional references Bayesian econometrics: Bayesian Econometrics, Gary Koop, Wiley Contemporary Bayesian Econometrics and Statistics, John Geweke, Wiley New Keynesian macro models: Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework, Jordi Gali, Princeton 2008.