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I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA PANEL D ISCUSSION : T HE R OLE OF P OTENTIAL O UTPUT IN P OLICYMAKING James Bullard* Federal Reserve Bank of St. Louis 33rd Annual Economic Policy Conference St. Louis, MO October 17, 2008 Views expressed are those of the author and do not necessarily reflect official positions of the FOMC or the Federal Reserve System. S UMMARY I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA M Y DISCUSSION Describe ideas about “proper” detrending. Core idea requires explicit theory of both growth and fluctuations. Ambition: The data should then be detrended by the theoretical growth path. Question: How to get the growth path to look like the data? Answer: Simple growth model with occasional trend breaks and learning. Applications in RBC and NK models. J. Bullard and J. Duffy. “Learning and Structural Change in Macroeconomic Data.” J. Bullard and S. Eusepi. “Did the Great Inflation Occur Despite Policymaker Commitment to a Taylor Rule?” Policy: How Taylor rules can lead you astray. S UMMARY I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY M AIN IDEAS Equilibrium business cycle research: A wide class of models including RBC, NK. All based on the concept of a balanced growth path. Data as summarized by Perron (1989) and Hansen(2001) suggest breaks in trends. Nonstationary aspects of the data are difficult to reconcile with available models. I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA C URRENT P RACTICE Trend-cycle decomposition done mostly with atheoretic, statistical filters. See King and Rebelo (1999). The discipline implied by the balanced growth assumption is dropped. This is a mistake, but one that dominates the literature. S UMMARY I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY S OME WELL - KNOWN CRITICISMS Filters do not remove the same trend that the balanced growth path requires. Current practice does not respect the cointegration of the variables, the multivariate trend, that the model implies. Filtered trends imply changes in growth rates, and agents would want to react to these changes. The "business cycle facts" are not independent of the statistical filter employed. Estimation, e.g., Smets-Wouters does not address this issue. I NTRODUCTION C ORE IDEAS H OW E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA TO IMPROVE ON THIS ? The criticisms are correct in principle. They are quantitatively important. These issues cannot be resolved by alternative statistical filters, because those filters are atheoretic. Instead, use theory to tell us what the growth path should look like. S UMMARY I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA C ORE IDEAS "Model-consistent detrending." The trends removed from the data are exactly the same as the ones implied by the model. Allow agents to react to (rare) changes in trend growth rates. Respect the cointegration of the variables that the balanced growth path implies. The methodology has wide applicability. S UMMARY I NTRODUCTION C ORE IDEAS E NVIRONMENT F EATURES R ECURSIVE L EARNING A N APPLICATION TO US DATA OF THE ENVIRONMENT Simple equilibrium business cycle model with exogenous, stochastic growth. Replace rational expectations with learning via Evans and Honkapohja (2001). Verify expectational stability. Calibrate, allowing occasional trend breaks, inspired by Perron (1989). S UMMARY I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA M AIN FINDINGS A more satisfactory method of detrending. A large fraction of the observed variance of output relative to trend can be attributed to structural change. In the NK world, learning about a productivity slowdown can send inflation up by 300 b.p. S UMMARY I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY P REFERENCES As in Cooley and Prescott (1994), a representative household maximizes h i ∞ (1) Et ∑ βt η t ln Ct + θ ln 1 `ˆ t t=0 subject to Ct + It It = Kt+1 and ... Yt , (1 δ) Kt , (2) (3) I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY T ECHNOLOGY The production technology is Yt = ŝt Ktα Xt Nt `ˆ t 1 α , (4) Xt = γXt 1, X0 = 1, γ > 0. (5) Nt = ηNt 1, ρ ŝt 1 et , N0 = 1, η > 0. (6) ŝt = ŝ0 = 1, (7) I NTRODUCTION C ORE IDEAS E NVIRONMENT B ALANCED R ECURSIVE L EARNING A N APPLICATION TO US DATA GROWTH Aggregate output, consumption, investment, and capital will grow at gross rate γη along the balanced growth path. Define k̂t = XKt Nt t , ŷt = XYt Nt t , ĉt = XCt Nt t , and rewrite the first order conditions and constraints of the problem. The new, stationary system has a steady state ĉt , ŷt , k̂t , `ˆ t = c̄, ȳ, k̄, `¯ 8t. The steady state values depend upon the gross growth rates γ and η. S UMMARY I NTRODUCTION C ORE IDEAS K EY E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY RATIOS Capital-output ratio along a balanced growth path k̄ = ȳ γ αβ β (1 δ) , (8) Consumption-output ratio along a balanced growth path c̄ γ = ȳ β (1 δ) γ αβ (γη β (1 δ ) 1 + δ) . (9) I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY L INEAR APPROXIMATION Need a linear system to apply Evans and Honkapohja (2001). Use logarithmic deviations from steady state. Define ! ! ˆt k̂t ` ĉt c̃t = ln , `˜ t = ln ¯ , (10) , k̃t = ln c̄ k̄ ` ỹt = ln ŷt ȳ , and s̃t = ln Rewrite system in terms of tilde variables. ŝt s̄ . (11) I NTRODUCTION C ORE IDEAS E NVIRONMENT M ORE ON LINEAR R ECURSIVE L EARNING A N APPLICATION TO US DATA APPROXIMATION The linearized system is still not satisfactory, because the log deviations involve c̄, ȳ, k̄, `¯ . Want the agents to learn the vector c̄, ȳ, k̄, `¯ when a change in growth occurs. Define ct = ln ĉt , kt = ln k̂t , yt = ln ŷt , `t = ln `ˆ t , and st = ln ŝt . Also define c = ln c̄, k = ln k̄, y = ln ȳ, ` = ln `¯ , and s = ln s̄ = 0. Rewrite the system in terms of these redefined variables; reduce system to three equations. S UMMARY I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING T HE SYSTEM UNDER A N APPLICATION TO US DATA S UMMARY RATIONAL EXPECTATIONS The system: ct kt st = B10 + B11 Et ct+1 + B12 Et kt+1 + B13 Et st+1 = D20 + D21 ct 1 + D22 kt 1 + D23 st 1 = ρst 1 + ϑt (12) (13) (14) The Bij and Dij are composites of fundamental parameters. Also, ϑt = ln et . I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA R ECURSIVE LEARNING Study this system under a recursive learning assumption. Assume agents have no specific knowledge of the economy. Endow them with a perceived law of motion which looks a lot like a VAR. S UMMARY I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY M ORE ON RECURSIVE LEARNING The system: ct kt st = B10 + B11 Et? ct+1 + B12 Et? kt+1 + B13 Et? st+1 + ∆t (15) = D20 + D21 ct 1 + D22 kt 1 + D23 st 1 (16) = ρst 1 + ϑt (17) The shock ∆t prevents perfect multicollinearity. It has standard deviation 1/1000th of et . I NTRODUCTION C ORE IDEAS E NVIRONMENT T HE PERCEIVED R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY LAW OF MOTION The agents use ct kt = a10 + a11 ct = a20 + a21 ct + a12 kt 1 + a22 kt 1 + a13 st 1 + a23 st 1 1, 1. (18) (19) This corresponds in form to the equilibrium law of motion for the economy. The agents are given equation (17). They could estimate ρ as well without materially changing the results. The presence of constant terms allows the agents to learn steady state values of variables. I NTRODUCTION C ORE IDEAS E NVIRONMENT T HE MAPPING R ECURSIVE L EARNING FROM PLM TO A N APPLICATION TO US DATA ALM Assume t 1 dating of expectations. Take expectations based on the PLM. Substitute to obtain the actual law of motion (ALM) S UMMARY I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING M ORE ON THE MAPPING A N APPLICATION TO US DATA FROM PLM TO S UMMARY ALM This implies ct = T10 + T11 ct 1 + T12 kt 1 + T13 st 1 + ∆t (20) where T10 = B10 + B11 [a10 + a11 a10 + a12 a20 ] + B12 [a20 + a21 a10 + a22 a20 ] , (21) h i T11 = B11 a211 + a12 a21 + B12 [a21 a11 + a22 a21 ] , (22) h i (23) T12 = B11 [a11 a12 + a12 a22 ] + B12 a21 a12 + a222 , T13 = B11 [a11 a13 + a12 a23 + a13 ρ] + h i B12 [a21 a13 + a22 a23 + a23 ρ] + B13 ρ2 . (24) I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING T HE SYSTEM UNDER Write 2 3 2 3 2 ct T11 T10 4kt 5 = 4D20 5 + 4D21 0 st 0 A N APPLICATION TO US DATA S UMMARY LEARNING T12 D22 0 3 32 ct 1 T13 D23 5 4kt 1 5 ρ st 1 2 1 0 + 40 0 0 0 32 3 0 ∆t 05 4 0 5 . 1 ϑt (25) I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY R ATIONAL EXPECTATIONS A stationary MSV solution solves T1i = a1i , for i = 0, 1, 2, 3, with all eigenvalues of the matrix 2 3 T11 T12 T13 4D21 D22 D23 5 0 0 ρ inside the unit circle. There is only one such solution for this model. (26) (27) I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY E XPECTATIONAL STABILITY Expectational stability is determined by the following matrix differential equation d (a) = T (a) dτ a, (28) where T = (T10 , T11 , T12 , T13 ) and a = ai,j with i = 1, 2 and j = 1, 2, 3, 4. A particular MSV solution is E-stable if the MSV fixed point of the differential equation (28) is locally asymptotically stable at that point. Calculated E-stability for this model and found that it holds at baseline parameter values (including the various values of γ and η used). A real time learning version can be implemented. I NTRODUCTION C ORE IDEAS E NVIRONMENT S TABILITY UNDER R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY CONSTANT GAIN LEARNING The system should be locally stable in the real time learning dynamics with gain of t 1 . With a constant gain, the system may depart the domain of attraction. But the constant gain also allows the agents to track the balanced growth path, should an underlying parameter change unexpectedly. The agents admit their model may be misspecified. How would the system respond to any small enough parameter change? I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US A N APPLICATION TO US DATA S UMMARY DATA Model is too simple to match directly with data. But it is also a well-known benchmark model. So it is possible to assess how important the detrending issue is for determining the nature of the business cycle as well as for the performance of the model relative to the data. I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA D ATA Quarterly U.S. data 1948Q1 to 2002Q1. Concern that the aggregates add up. Subtract real government purchases and farm business product from real GDP to get nonfarm private sector output. Using nonfarm private sector hours from the establishment survey. S UMMARY I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY M ORE ON THE DATA Nonfarm private sector productivity created from these. Consumption defined as personal consumption expenditures for nondurable goods and services, plus net exports of services, less farm business product. Investment defined as gross private domestic investment plus personal consumption expenditures on consumer durables, plus net exports of goods. Series essentially add up despite chain-weighting. Allocated discrepencies using the consumption-output ratio for that year. I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA A STANDARD CALIBRATION Cooley and Prescott (1994). β = .987, θ = 1.78, α = .4, ρ = .95, σe = .007. Growth rates of productivity and labor input: allow those to change. Gain chosen informally at g = .00025; does not seem to impact results importantly. S UMMARY I NTRODUCTION C ORE IDEAS B REAKS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA ALONG THE BALANCED GROWTH PATH Productivity slowdown: Hansen (2001), Perron (1989), Bai, Lumsdaine, and Stock (1998). Only allow trend breaks where clear econometric evidence is available? S UMMARY I NTRODUCTION C ORE IDEAS H OW E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY TO CHOOSE BREAK DATES One approach: conformity between measured productivity and labor input, in the model and in the data. 1 2 3 Choose break dates and growth rates. Compare implied trends in measured productivity and labor input to data. If discrepencies exist, return to 1, otherwise terminate at a fixed point. Use a search process (genetic algorithm) to implement this process. I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA O PTIMAL T REND B REAKS Initial annual growth rate, percent Break date Mid-sample annual growth rate, percent Break date Ending annual growth rate, percent N (t) 1.20 1961, Q2 1.91 1.91 X (t) 2.47 1973, Q3 1.21 1993, Q3 1.86 TABLE : Optimal break dates and growth rates for fundamental factors driving growth in the model, based on a search of possible dates and growth rates. These choices produce measured productivity and hours series that conform best to the US data. S UMMARY I NTRODUCTION C ORE IDEAS E NVIRONMENT D EFINING A R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY TREND A trend is the economy’s path if only low frequency shocks occur. Turn the noise on the business cycle shock down, multiplying σe by 1/1000. What happens in the economy where the only meaningful shocks are those to productivity growth and hours growth? Normalization: initially on a balanced growth path. I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY A RTIFICIAL ECONOMIES Confirm that estimated coefficients are initially close to RE values. Collect an additional 217 quarters of data, with trends breaking as described above. Detrend the data using the same (multivariate) trend that is used for the actual data. Collect statistics over a large number of simulations. I NTRODUCTION C ORE IDEAS E NVIRONMENT B USINESS R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY CYCLE STATISTICS TABLE 3. B USINESS C YCLE S TATISTICS Relative Contemporaneous Volatility Volatility Correlations Data Model Data Model Data Model Output 3.25 3.50 1.00 1.00 1.00 1.00 Consumption 3.40 2.16 1.05 0.62 0.60 0.75 Investment 14.80 8.86 4.57 2.53 0.65 0.92 Hours 2.62 1.54 0.81 0.44 0.65 0.80 Productivity 2.52 2.44 0.77 0.70 0.61 0.92 TABLE : Business cycle statistics, model-consistent detrending. I NTRODUCTION C ORE IDEAS E NVIRONMENT NK A PPLICATION R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY I NTRODUCTION C ORE IDEAS E NVIRONMENT R ECURSIVE L EARNING A N APPLICATION TO US DATA S UMMARY Provides some microfoundations for current, atheoretical practices. Structural change accounts for a large fraction of observed variability of output. Learning provides the “glue” that holds the various balanced growth paths together. Adjustment following a trend change is relatively rapid. Learning about structural change could have large effects on policy. S UMMARY
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