DSGE Model Forecasting

Transcription

1 University of Pennsylvania EABCN Training School May 1, 216

2 Introduction The use of DSGE models at central banks has triggered a strong interest in their forecast performance. The subsequent material draws from the article DSGE Model-Based Forecasting which is slated to appear as a chapter in the Handbook of Economic Forecasting: DSGE Model-Based Forecasting

3 Generating Forecasts with a DSGE Model DSGE Model = State Space Model Measurement Eq: y t = Ψ (θ) + Ψ 1(θ)t + Ψ 2(θ)s t State Transition Eq: s t = Φ 1(θ)s t 1 + Φ ɛ(θ)ɛ t Posterior distribution of DSGE model parameters: p(θ Y 1:T ) = p(y 1:T θ)p(θ), p(y 1:T ) = p(y 1:T θ)p(θ)dθ. p(y 1:T ) Objective of interest is predictive distribution: p(y T +1:T +H Y 1:T ) = p(y T +1:T +H θ, Y 1:T )p(θ Y 1:T )dθ. Use numerical methods to generate draws from predictive distribution.

4 Generating Draws from Predictive Distribution For j = 1 to n sim, select the j th draw from the posterior distribution p(θ Y 1:T ) and: 1 Use the Kalman filter to compute mean and variance of the distribution p(s T θ (j), Y 1:T ). Generate a draw s (j) T from this distribution. 2 A draw from S T +1:T +H (s T, θ, Y 1:T ) is obtained by generating a sequence of innovations ɛ (j) T +1:T +H. Then, starting from s(j) T, iterate the state transition equation with θ replaced by the draw θ (j) forward to obtain a sequence S (j) T +1:T +H : s (j) t = Φ 1 (θ (j) )s (j) t 1 + Φ ɛ(θ (j) )ɛ (j) t, t = T + 1,..., T + H. 3 Use the measurement equation to obtain Y (j) T +1:T +H : y (j) t = Ψ (θ (j) )+Ψ 1 (θ (j) )t+ψ 2 (θ (j) )s (j) t, t = T +1,..., T +H.

5 Remarks Algorithm generates n sim trajectories Y (j) T +1:T +H from the predictive distribution. Possible modification: execute Steps 2 and 3 m times for each j. Leads to m n sim draws from the predictive distribution. A point forecast ŷ T +h of y T +h : (i) specify loss function L(y T +h, ŷ T +h ); determine prediction that minimizes posterior expected loss: ŷ T +h T = argmin δ R n L(y T +h, δ)p(y T +h Y 1:T )dy T +h. y T +h Under the quadratic forecast error loss function L(y, δ) = tr[w (y δ) (y δ)], the optimal predictor is the posterior mean ŷ T +h T = y T +h p(y T +h Y 1:T )dy T +h 1 n sim y (j) T +h y T +h n, sim which can be approximated by a Monte Carlo average. j=1

6 DSGE Model Smets and Wouters (27) model, modified to absorb real time information as specified below. Observables: output, consumption, investment, real wage growth, hours worked, inflation, Federal Funds rate. Real time data, following Edge and Gürkaynak (21): Recursive out-of-sample forecasting. All estimation samples start in Blue Chip samples: forecast origins aligned with Blue Chip survey publication dates. We consider January, April, July, and October, ending April 211 Greenbook samples: forecast origins aligned with Greenbook dates. We consider March, June, September, and December, ending Sept. 24

7 DSGE Model versus Blue Chip ( ) Output Growth Inflation Interest Rates SW BC h = 1 is current quarter nowcast. Growth rates, inflation rates, interest rates are QoQ %

8 DSGE Model versus Greenbook ( ) Output Growth Inflation Interest Rates SW GB h = 1 is current quarter nowcast. Growth rates, inflation rates, interest rates are QoQ %

9 RMSEs Reported in the Literature Figure depicts RMSE ratios: DSGE (reported in various papers) / AR(2) (authors calculation).

10 Shock Decompositions In DSGE models the evolution of the endogenous variables is due to exogenous shocks. We can study the contribution of each shock to historical fluctuations as well as forecasts. This produces a story that goes along with the forecast.

11 Draws from the Posterior Distribution of a Shock Decomposition For j = 1 to n sim, select the j th draw from the posterior p(θ Y 1:T ): 1 Use the simulation smoother to generate a draw S (j) :T distribution p(s (j) :T Y 1:T, θ (j) ). from the 2 For each structural shock i = 1,..., m (which is an element of the vector s t ): 1 Compute the sequence of shock innovations ɛ (j) i,1:t, for instance, by solving s (j) i,t = ρ(j) i s (j) i,t 1 + σ(j) i ɛ (j) i,t for ɛ(j) i,t. 2 Define a new sequence of innovations e 1:T (e t is of the same dimension as ɛ t) by setting the i th element e i,t = ɛ (j) i,t for t = 1,..., T and e i,t N(, σi 2 ) for t = T + 1,..., T + H. All other elements of e t, t = 1,..., T + H, are set equal to zero. 3 Starting from s = s (j), iterate the state transition equation forward using the innovations e 1:T +H to obtain the sequence S (j) 1:T +H. 4 Use the measurement equation to compute Ỹ (j) 1:T +H based on S (j) 1:T +H.

12 Shock Decomposition: Inflation π g b µ z λ f λ w r m * Notes: The shock decompositions for output growth (top) and inflation (bottom) are computed using model SWπ estimated on the last available vintage (May 211). The black and red lines represent the data and the forecasts, both in deviation from the steady state. The colored bars represent the contribution of each shock to the evolution of the variables.

13 Shock Decomposition: Output Growth Notes: The shock decompositions for output growth (top) and inflation (bottom) are computed using model SWπ estimated on the last available vintage (May 211). The black and red lines represent the data and the forecasts, both in deviation from the steady state. The colored bars represent the contribution of each shock to the evolution of the variables.

14 The Poverty of the Econometrician s Information Set Quality of forecasts is constrained by quality of model and quality of observations. Let s focus on the latter. Nowcasts: Professional forecasts beat DSGE forecasts in the short-run with respect to variables for which real time information is available. Potential solution? Augment the set of observables. We consider the following external information... Inflation Expectations External Nowcasts Interest Rate Expectations variables that may convey information about the state of the economy not contained in usual data set.

15 Incorporating Inflation Expectations High-inflation rates from lead to fairly large estimate of steady-state inflation rate (4 % annualized). = Upward bias in current inflation forecasts. Remedy: anchor target inflation rate using long-run inflation expectations. Augment measurement equations: [ ] πt O,4 1 4 = π + E t π t+k. 4 Modify policy rule: k=1 R t = ρ R R t 1 + (1 ρ R ) ( ψ 1 (π t π t ) +... Time-varying inflation target evolves according to: π t = ρ π π t 1 + σ π ɛ π,t.

16 With and Without Inflation Expectations Output Growth Inflation Interest Rates SW SW Loose SWπ Smets-Wouters: red Smets-Wouters with loose prior on π : salmon DSGE model with inflation expectations: magenta

17 Incorporating External Nowcasts (No Model Modification) Standard DSGE model forecasts ignore information from current quarter. Approach 1 (News): true Y T +1 = external info Z T + noise Approach 2 (Noise): external info Z T = true Y T +1 + noise Approaches are the same for hard conditioning: noise =. Under both approaches the forecaster essentially obtains information about the shocks ɛ T +1 as well as the state s T at forecast origin. The subsequent results are generated under Approach 2, using nowcasts for output growth, inflation, and interest rates.

18 Draws from the Predictive Distribution Conditional on External Nowcast (Noise Assumption) For j = 1 to n sim, select the j th draw from the posterior p(θ Y 1:T ): 1 Use the Kalman filter to compute mean and variance of the distribution p(s T θ (j), Y 1:T ). 2 In period T + 1 use z T +1 = y 1,T +1 η T +1, y 1,T +1 η T +1 as measurement equation for the nowcast z T +1 assuming η T +1 N(, σ 2 η). 3 Use the Kalman filter updating to compute p(s T +1 θ (j), Y 1:T, z T +1 ) and generate a draw s (j) T +1 from this distribution. 4 Draw a sequence of innovations ɛ (j) T +2:T +H and, starting from s(j) T +1, iterate the state transition equation forward to obtain the sequence S (j) T +2:T +H. 5 Use the measurement equation to compute Y (j) T +1:T +H based on S (j) T +1:T +H.

19 Draws from the Predictive Distribution Conditional on External Nowcast (News Assumption) For j = 1 to n sim, select the j th draw from the posterior p(θ Y 1:T ): 1 Use the Kalman filter to compute mean and variance of the distribution p(s T θ (j), Y 1:T ). 2 Generate a draw ỹ (j) 1,T +1 from the distribution p(ỹ 1,T +1 Y 1:T, z T +1 ) using y 1,T +1 = z T +1 + η T +1, z T +1 η T +1, η T +1 N(, σ 2 η). 3 Treating ỹ (j) 1,T +1 as observation for y 1,T +1 use the Kalman filter updating step to compute p(s T +1 θ (j), Y 1:T, ỹ (j) 1,T +1 draw s (j) T +1 from this distribution. 4 Draw a sequence of innovations ɛ (j) T +2:T +H ) and generate a and, starting from s(j) T +1, iterate the state transition equation forward to obtain the sequence S (j) T +2:T +H. 5 Use the measurement equation to obtain Y (j) T +1:T +H S (j) T +2:T +H. based on

20 Timing Forecast End of Est. External Forecast Origin Sample T Nowcast T + 1 h = 1 h = 2 Apr 92 91:Q4 92:Q1 from Apr 92 BC 92:Q1 92:Q2 Jul 92 92:Q1 92:Q2 from Jul 92 BC 92:Q2 92:Q3 Oct 92 92:Q2 92:Q3 from Oct 92 BC 92:Q3 92:Q4 Jan 93 92:Q3 92:Q4 from Jan 93 BC 92:Q4 93:Q1

21 With and Without Blue Chip Nowcasts Output Growth Inflation Interest Rates SWπ SWπ now BC DSGE model with inflation expectations: magenta DSGE model with inflation expectations and nowcasts: salmon

22 With and Without Blue Chip Nowcasts Consumption Growth Investment Growth Real Wage Growth DSGE model with inflation expectations: magenta DSGE model with inflation expectations and nowcasts: salmon

23 Incorporating Interest Rate Expectations We are utilizing multi-step interest-rate forecasts from Blue Chip. BC interest-rate forecasts are treated as observations of agents expectations in the model. Particularly useful for forecasting when interest rates are near ZLB. SW model allows for a serially correlated monetary policy disturbances r m t = ρ r mr m t 1 + σ r mɛm t. Augment rt m by anticipated policy shocks that capture future expected deviations from the systematic part of the monetary policy rule: K rt m = ρ r mrt 1 m + σ r mɛ m t + σ r m,kɛ m k,t k. k=1 Policy shocks ɛ m k,t k, k = 1,..., K, are known to agents at time t k, but affect the policy rule with a k period delay in period t.

24 Timing Forecast End of Est. External Interest Rate Exp Forecast Origin Sample T Nowcast T + 1 RT e +2 T +1,..., Re T +5 T +1 h = 1 Apr 92 91:Q4 92:Q1 from Apr 92 BC 92:Q2-93:Q1 92:Q1 Jul 92 92:Q1 92:Q2 from Jul 92 BC 92:Q3-93:Q2 92:Q2 Oct 92 92:Q2 92:Q3 from Oct 92 BC 92:Q4-93:Q3 92:Q3 Jan 93 92:Q3 92:Q4 from Jan 93 BC 93:Q1-93:Q4 92:Q4

25 With and Without Interest Rate Expectations Output Growth Inflation Interest Rates SWπ now SWπ R now DSGE model with inflation expectations and nowcasts: salmon DSGE model with inflation expectations and nowcasts and interest rate expectations: turquoise

26 With and Without Interest Rate Expectations Consumption Growth Investment Growth Real Wage Growth DSGE model with inflation expectations and nowcasts: salmon DSGE model with inflation expectations and nowcasts and interest rate expectations: turquoise

27 Main Findings Re Usefulness of External Information Positive: Mixed: A time-varying inflation target combined with long-run inflation expectations worked well in a number of models. Using external nowcasts improves DSGE model forecasts in the short-run. For some series improvement carries over to medium-run. Interest-rate expectations in conjunction with anticipated policy results improve interest rate forecasts in particular near ZLB. Anticipated policy shocks tend to trigger large output and inflation movements which lead to forecast deterioration.

28 Forecasts Conditional on Interest Rate Paths Central banks are often interest in forecasts conditional on an interest rate path. In DSGE model, we can control the interest rate path with unanticipated shocks; anticipated shocks.

29 The Effects of Monetary Policy Shocks Euler Equation y t = E[y t+1 ] (R t E[π t+1 ]) Phillips curve π t = βe[π t+1 ] + κy t Monetary policy rule with unanticipated and anticipated monetary policy shocks: R t = 1 β π t + ɛ R t + K k=1 ɛ R k,t k

30 Impulse Responses to Anticipated Shock Output: y t+h ɛ R K,t = ψ 1+K h, h =,..., K where ψ = (1 + κ/β) 1. Inflation: ( K 1 h = κψ ψ K h + β K h + ψ K h π t+h ɛ R K,t where β/ψ = β + κ. i=1 ( ) ) i β, h =,..., K, ψ

31 The Effects of Policy Anticipation Unanticipated policy shocks Interest Rates Output Growth Inflation.18.5 Percent Percent Percent Six-periods ahead anticipated policy shocks Interest Rates Output Growth Inflation.1.1 Percent Percent Percent

32 The Effects of Policy Anticipation... and Forecasts Conditional on an FFR Path 1.4 Interest Rates Output Growth Inflation