Computational Macroeconomics. Prof. Dr. Maik Wolters Friedrich Schiller University Jena

Transcription

1 Computational Macroeconomics Prof. Dr. Maik Wolters Friedrich Schiller University Jena

2 Overview Objective: Learn doing empirical and applied theoretical work in monetary macroeconomics Implementing macroeconomic models on the computer Doing policy simulations, estimating models, forecasting with models Model Class: Vector Autoregressions (VAR) Dynamic Stochastic General Equilibrium (DSGE) Models Methods and Tools: VARs: OLS, Bayesian estimation, structural identification DSGE: Derivation, stochastic simulation, deterministic simulation, Bayesian estimation 2

3 Lecturers Lectures: Prof. Dr. Maik Wolters, jena.de Exercise sessions / Research Project Lars Other, Office 4.162, lars.other@uni jena.de Josefine Quast, Office 4.148, josefine.quast@uni jena.de Office Hours By appointment (send an ) 3

4 Outline Part 1: Vector Autoregressions Part 2: The baseline New Keynesian model Derivation and stochastic simulations Part 3: Medium scale DSGE models Stochastic simulations Part 4: Deterministic simulations Fiscal policy applications Part 5: Estimation of DSGE models, Forecasting 4

5 Literature Macroeconomic Theory: Walsh, Carl (2010). Monetary Theory and Policy, The MIT Press, Third Edition. Galí, Jordi (2015). Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework and Its Applications, Princeton Press, Second Edition. Romer, David, (2012). Advanced Macroeconomics, McGraw Hill, Fourth Edition. Wickens, Michael (2011). Macroeconomic Theory. A Dynamic General Equilibrium Approach, Princeton University Press, Second Edition. Heijdra, Ben j. (2017). Foundations of Modern Macroeconomics, Oxford University Press, Third Edition. Methodology: Canova, F. (2007). Methods for Applied Macroeconomic Research, Princeton University Press. DeJong, David N., and Chentan Dave (2011). Structural Macroeconometrics, Princeton University Press, Second Edition. Lütkepohl, H. (2007). New Introduction to Multiple Time Series Analysis, Springer. Kilian, L. and H. Lütkepohl (2017). Structural Vector Autoregressive Analysis, Cambridge University Press, personal.umich.edu/~lkilian/book.html Dynare User Guide, Dynare Manual. Academic papers: Will be announced during the course. 5

6 Part 1 Vector Autoregressions 6

7 Data in monetary macroeconomics Time series of aggregate data Examples: GDP, CPI inflation, short and long term interest rates, exchange rates, Frequency: typically quarterly and sometimes monthly Data availability: only data for few recent decades available Many structural breaks (different policy regimes, structural change in the economy, etc.) Typical time series: observations in quarterly frequency 7

8 Core Macro Time Series (US Data) Real GDP growth (year on year) Unemployment Rate CPI Inflation (year on year) 18 Fed Funds Rate

9 Trends and persistence Some time series show trends and keep growing: GDP, CPI, Real GDP (level) CPI (level)

10 Trends and persistence Some time series do not show a trend, but are highly persistent: interest rates, unemployment rate, year treasury bond 11 Unemployment Rate

11 Trends and persistence Some time series hardly show any persistence: GDP growth, investment growth, consumption growth, industrial production growth, GDP growth (q on q) Investment growth (q on q) Consumption growth (q on q)

12 Collinearity and dynamic correlation Macroeconomic datasets are characterized by collinearity GDP, consumption, investment, industrial production, Inflation, interest rates, Dynamic interaction of macroeconomic variables Phillips curve: unemployment rate and inflation IS curve: real interest rate and output Taylor rule: output gap, inflation and interest rate We do not study univariate time series models (AR, ARMA, ARIMA, ARCH, GARCH, ), but focus on multivariate models (VAR) 12

13 Data transformation Seasonal adjustment Source: Leamer, 2009, Chapter 2 13

14 Data transformation Take logs of series not expressed in rates Real GDP Log real GDP 9,5 9 8,5 8 7,5 7 6,5 6 5, IfGDP grows at a constant rate g, then the log representation is a straight line With Vector Autoregressions we can include variables in log levels. If the number of lags is sufficiently large error term stationary Don t regress level variables in other models on each other. Spurious regression unless there is cointegration! Regressing variables in logs on each other: regression coefficients are interpreted as elasticities. By how much in percent does the dependent variable change, if we change a dependent variable by one percent? 14

15 Reduced form VAR A VAR in reduced form describes the dynamics of a vector of variables System of equations in which each variable depends on its own past observations and on past observations of other variables in the system Example: 2 variable VAR with two lags (VAR(2)),,,,,,,,,, Σ,,,, Estimation: ordinary least squares (OLS), no endogeneity problem, seemingly unrelated regressions (SUR) 15

16 VAR notation VAR(2),,,,,,,,,, Σ,,,, Write the VAR(2) in Matrix notation,,,,,,,,,, Σ,,,, 16

17 VAR notation VAR(2) in Matrix notation,,,,,,,,,, Σ,,,, Define: Y VAR(p) in Matrix notation: Short notation with lag operator:, Σ can include of course more than 2 variables 17

18 Origin of VARs in Economics VARs were popularized by Sims (1980) in his classic paper, Macroeconomics and Reality, Econometrica, 48, He got the nobel price in 2011 for this work Response to the failure of structural assumptions in Keynesian large scale econometric models The connection between... models and reality the style in which identification is achieved for these models is inappropriate, to the pointat which claims for identification in these models cannot be taken seriously. (Sims, 1980) So far, we have a system of equations that requires just two assumptions 1. Which variables to include (usually based on economic theory) 2. How many lags to include (guided by practical considerations: uncorrelated residuals, cover dynamics of a certain time span, information criterion...) 18

19 Usage of VARs 1. Describe and summarize macroeconomic data; find stylized facts in the data that structural models should generate No further modifications of our VAR system required 2. Make macroeconomic forecasts Need Bayesian methods to avoid in sample over fit and poor out of sample forecasts 3. Structural analysis; example: effects of a monetary policy shock Need additional identifying assumptions 19

20 1. Desriptive analysis with VARs Granger causality Granger causality can be useful, but it is not strictly the same as economic causality Definition: If a variable canhelpforecast, then does Granger cause The MSE of the forecast,,0 is smaller than the MSE of the forecast, 0 Test with an F test (example):,,,,,,,,,, If, 0,, 0then fails to Granger cause General case:, 0, 1,, 20

21 Granger causality: example with a bivariate VAR RBC models with nominal neutrality imply that money has no effect on the real variables Classical dichotomy (real and nominal variables can be analyzed separately) We can use a VAR to check whether money in fact does not Grangercause real variables Sims (1972) finds that output does not Granger cause money, but that money Granger causes output His interpretation was that money supply is exogenous (set by the Fed) and thus is not influenced by output On the other hand money has real effects (rejection of the classical dichotomy) Here, a combination of two Granger causality tests has been used to make an economic interpretation 21

22 Granger causality: example with a 3 variable VAR Stock and Watson (2001) use a VAR with inflation, unemployment and the federal funds rate Table 1 shows p values of F tests VAR can be roughly be interpreted as Phillips Curve IS equation (however, inflation not significant) Taylor Rule 22

23 2. Forecasting with VARs Univariate case Autoregression: AR(p) Forecast is simply obtained by iterating forward One step forecast: Two step forecast: h step in general: Multivariate model Consider the following model: One step forecast: What to use here? Two step forecast: VAR(p) model: Joint forecasting model for all variables that can simply be iterated forward:,,,,,,,, 23

24 Forecasting with VARs: Example 1. Decide on variables to include in the VAR 2. Decide on number of lags 3. Estimate with OLS 4. Iterate forward to get forecasts Source: Stock and Watson (2001) 24

25 Problem when forecasting with VARs Many parameters need to be estimated Example: VAR with 3 variables and 4 lags 3 constants + 36 lag parameters + 9 variance covariance terms 48 parameters Typical sample of 20 years of data: 80 observations Typically extremely good in sample fit, but poor out of sample forecasting performance Sometimes estimation is infeasible: VAR(4) with 5 variables, but 80 obs. Solution: Use Bayesian methods to shrink the parameters towards zero need a prior (shrinkage = prior) 25

26 The Bayesian approach Frequentist approach: Inference by means of hypothesis testing, confidence intervals Probability viewed as long run frequency Unknown true parameters are constant Bayesian approach: Inference by means of posterior distributions Probability viewed as subjective belief Unknown parameters are treated as random variables with a probability distribution For forecasting the Bayesian approach can also simply be viewed as a pragmatic tool to increase forecasting accuracy 26

27 Priors andposteriors Standard VAR uses few a priori information: choice of variables and lag length The Bayesian approach combines information from estimation based on data with a prior belief on the parameters Example:, ~ 0, Frequentist approach maximizes likelihood to get estimates,, 2 / exp Yields the OLS estimate, and the ML estimate Bayesian approach: Combine prior belief with ML estimate Prior Distribution: description of uncertainty about model parameters before data is observed., ~,Σ Posterior: weighted average between prior belief about parameters before data is observed and information about parameters contained in observed data (likelihood) Posterior is obtained via Bayes law by combining prior and likelihood:,,, 27

28 Bayesian VARs Typical Bayesian analysis involves: 1. Formulation of probability model for the data (here: VAR) 2. Specification of prior distribution for unknown model parameters 3. Construction of likelihood from observed data 4. Combination of prior distribution and likelihood to obtain posterior distribution of model parameters 5. Bayesian Inference 28

29 Minnesota prior Prior developed to achieve a good forecasting model Achieve a parsimonous model by shrinking parameters towards zero Developed by Litterman and Sims at the University of Minnesota and the Federal Reserve Bank of Minneapolis VAR:,,,,,,,,,, Σ Need a prior for,,,, and Σ Minnesota prior simplifies by replacing Σ with an estimate Σ need only priors for,,,, Minnesota prior assumes that most time series can be described as a random walk (with drift) process: Coefficient of 1 on first lag of own variable Coefficient of 0 on all other lags of own variable Coefficient of 0 on other variables 29

30 Minnesota prior Prior: ~, Prior belief: randomwalk Coefficient of 1 on first lag of own variable ( 1,) Coefficient of 0 on all other lags of own variable ( 1,) Coefficient of 0 on other variables ( 1,,,), 1 1, 0 Prior is imposed less tight on more recent lags than on lags further in the past Higher probability that more recent observations provide more valuable information than observations further in the past ( in the denominator) Higher probability that lags of own variables provide more information than lags of other variables ( ) Variance around prior parameter:, 30

31 Illustration: Minnesota prior Source: Canova (2007) 31

32 Posterior with Minnesota prior Analytical solution is available: Posterior Variance: Σ Posterior Mean: Σ 32

33 Example: BVAR with Minnesota prior Forecasting German macroeconomic variables (Pirschel and Wolters, 2017) Variables: GDP, CPI, short term interest rate, unemployment rate Bayesian VAR: 4 lags, Normal Wishart Minnesota prior Unrestricted VAR: number of lags chosen with Bayesian Information Criterion Evaluation sample: 1994Q4 2013Q4 horizon BVAR: RMSFE VAR: RMSFE 1 (GDP) (GDP) (GDP) (CPI) (CPI) (CPI)

34 3. Structural VARs So far the reduced form VAR was restricted to dynamic interactions between variables This assumption is implausible for structural analysis. Variables depend on each other also in period and not only in 1,2, Structural form 2 variable VAR with 2 lags: SVAR(2),,,,,,,,,, D,,,, This could be a structural model derived from theory. Traditional Cowles commission models have this structure Structural parameters,, are different from the reduced form parameters, Σ. In particular the structural shocks, are different from the reduced form shocks,. 34

35 Structural VARs vs. reduced form VARs Reduced form VAR: So far, no identification necessary, just the choice of variables and the laglength. This increases the credibility of communicating the statistical results Structural VAR: F To interpret the VAR in an economically meaningful way, one needs to disentagle the reduced form shocks into structural shocks that have clearer interpretations Technology shocks, Monetary policy shocks, Fiscal policy shocks,... Once, we have the structural shocks we can compute impulse response functions to a monetary policy shock etc. 35

36 Need for identification Reduced form VAR allows to differenciate between shocks andsystematic (endogenous dynamics) movements The reduced form VAR can be viewed as a forecasting model, where the reduced form shocks represent unexpected movements, i.e. forecast errors Forecast errors of different variables are correlated with each other For example a forecasterroroftheinterestrate might be due to unexpected changes in the interest rate, i.e. a monetary policy shock, or due to other unexpected shocks (demand shocks, inflationary shocks etc.) if the interest rate responds to other variables within a given quarter Forecast errors or reduced form shocks cannot be regarded as fundamental, or structural, shocks to the economy. Instead, they should be viewed as a linear combination of these fundamental shocks 36

37 Mapping the structural form into the reduced form Reduced form VAR: SVAR: F Multiply both sides with where,,, Σ Key to understand the relation between the structural VAR and the reduced form VAR is the matrix controls how the endogenous variables are linked to each other contemporaneously Identification amounts to choosing an matrix 37

38 Structural impulse responses Impulse response: Reaction of the dynamic system of equations to an isolated one off shock. Example: Transmission of a monetary policy shock (while shutting off all other disturbances to the economy) Compute long run average or steady state by dropping time subscripts: Set,,, to steady state values, set all elements of to zero, set one element of to unity. Iterate the following equation forward, but putting all values,, to zero: where b, A F B Once we have defined a matrix impulse reponses to structural shocks can be easily computed (recall: ) 38

39 Structural identification Structural VAR F, Includes 1 parameters in,,,, parameters in and 1 /2in (it is symmetric) Reduced form VAR, Σ Includes parameters in,,, parameters in and 1 /2in Σ The SVAR includes more parameters than estimated in the reduced form VAR. We have to impose restrictions on the SVAR to have a clear one to one mapping between the parameters of the two VARs We have to impose restrictions on the structural parameters,,,,, to identifiy all of them 39

40 Recursive identification (Sims, 1980) Impose restrictions on the contemporaneous response of the different endogenous variables to the different structural shocks Example: interest rate reacts contemporaneously to monetary policy shocks and to inflationary shocks. Inflation reacts contemporaneously to inflationary shocks, but only with a lag to monetary policy shocks Implement this identification scheme via assuming that is lower triangular and being diagonal (diagonal elements are the variances of shocks): First variable canreacttolagsandthefirstshock Second variable canreacttolagsandthefirsttwoshocks Third variable can reacttolagsand thefirst threeshocks... Need to be careful how to order the variables (guided by economic theory) Number of restrictions on : 1/2 Number of restrictions on : 1/2 Overall number of restrictions: ² 40

41 Example of recursive identification Suppose the structural form is: =,, does not depent contemporaneous on and therefore not on the contemporaenous, does depend on contemporaneous The reduced Form VAR is is obtained by premultiplying = ,,,, 41

42 Example of recursive identification The reduced Form VAR is is obtained by premultiplying = ,,,,,,,,,,so the second reduced form shock is a linear combination of the first two structural shocks The covariance matrix is:,,,,,,,,,

43 Bring SVAR into a form with identity covariance matrix To recover the SVAR from the estimated reduced form VAR the Cholesky decomposition is used Rewrite the SVAR: instead of lower triangular with unity elements on the diagonal of and the diagonal of representing the variances of the structural shocks, we can write in general lower triangular form Example: premultiply the structural form by 1/, 0 0 1/, 1/, 0 /, 1/, = /, /, /, /,,/,, /, This structual form has a triangular matrix and a covariance matrix equal to an identity matrix 43

44 Cholesky decomposition We estimate Σ and want to find We know that is lower triangular Let Σ be a symmetric positive definite matrix. The Cholesky decomposition gives the unique lower triangular matrix such that Σ Step 1: From Σ (recall is assumed) the Cholesky decomposition recovers. Invert to get, which is triangular Step 2: Compute the structural parameters: a, 44

45 Application: Monetary policy transmission Study the transmission of a monetary policy shock on key macroeconomic aggregates Main reference: Christiano, L. J., M. Eichenbaum and C. L. Evans (1999). Monetary policy shocks: What have we learned and to what end? Handbook of Macroeconomics. Stylized facts Interest rates initially rise Aggregate price level initially responds very little. Price level decreases after 1 2 years Aggregate output initially falls with a zero long run effect 45

46 Set up a monetary SVAR Structural VAR with Cholesky identification to study the transmission of a structural monetary policy shock to the economy GDP, consumption, investment GDP deflator, commodity price index Fed Funds Rate, 10 year Rate, non borrowed reserves, total reserves 46

47 Identification: Ordering Assume that Federal Funds Rate responds contemporaneously to Output, consumption, investment, prices, commodity prices Assume that the federal funds rate responds with a lag of one quarter to 10 year rate, nonborrowed reserves, total reserves Specific assumptions are highly questionable check robustness or use other identification schemes 47

48 48

49 Monetary policy transmission 49

50 Summary VARs are one of the most important analysis tools in monetary macroeconomics Account for collinearity and dynamic correlation between variables Need a minimum number of assumptions let the data speak Three application areas 1. Desriptive analysis: need to choose number of variables and lags 2. Forecasting: need to shrink parameters towards zero BVAR 3. Structural analysis: need additional identification assumptions. VAR methodology is restricted to study the transmission of one time (temporary) suprise changes in policy. To study permanent changes in policy regimes we need structural models. 50

51 References Overview VARs: Stock, James H. and Mark W. Watson (2001). Vector Autoregressions, Journal of Economic Perspectives, 15(4): Structural Identification Söderlind, P. (2005). Lecture Notes in Empirical Macroeconomics, available online on Paul Söderlind s website. Kilian, L. (2013). Structural Vectorautoregressions, in Handbook of Research Methods and Applications, Chapter 22. Lütkepohl, H., and M. Krätzig (2004). Applied Time Series Econometrics, Cambridge University Press, Chapters 3 and 4. Enders, W. (2004). Applied Econometric Time Series, John Wiley & Sons, Ch. 5. Bayesian VARs Koop, G., and D. Korobilis (2010). Bayesian Multivariate Time Series Methods for Empirical Macroeconomics, available on Gary Koop s website Canova, F. (2007). Methods for Applied Macroeconomic Research, Princeton University Press, Chapter 10. Forecasting with Bayesian VARs Karlsson, S. (2013). Forecasting with Bayesian Vectorautoregressions, Handbook of Economic Forecasting, Vol. 2, Chapter 15. Bayesian VAR: Gibbs sampler (simulating the posterior distribution) Blake, A., and H. Mumtaz (2012). Applied Bayesian econometrics for central bankers, Technical Handbook No. 4, Center for Central Banking Studies, Bank of England. 51