M9902: The Quarterly Projection Model František Brázdik Macroeconomic Forecasting Division frantisek.brazdik@cnb.cz Czech National Bank November 2011 Czech National Bank QPM 1 / 73 Outline 1 Trend and cycles 2 Structure of the Quarterly Projection Model 3 Parameters setup 4 Properties of the Model Czech National Bank QPM 2 / 73 Trend and cycles Outline 1 Trend and cycles 2 Structure of the Quarterly Projection Model 3 Parameters setup 4 Properties of the Model Czech National Bank QPM 3 / 73 Trend and cycles Time series analysis Analysis of time series data is based on smoothing past data in order to separate the underlying pattern in the data series from randomness. The underlying pattern then can be projected into the future and used as the forecast. The underlying pattern can also be broken down into sub patterns to identify the component factors that influence each of the values in a series: decomposition Decomposition methods: identify separate components of the basic underlying pattern that tend to characterize economics and business series. Czech National Bank QPM 4 / 73 Trend and cycles In search for trends Czech National Bank QPM 5 / 73 Trend and cycles Decomposition Techniques Goal: separation of data into several unobservable components, generally in an additive or multiplicative form. Components: trend, seasonal pattern, cycle, and residual or irregular pattern Seasonal component: the periodic fluctuations of constant length Trend-cycle component: long term changes in the level of series Czech National Bank QPM 6 / 73 Trend and cycles Detrending methods Detrending Trend Component: The tendency of a variable to grow over time, either positively or negatively. Basic forces in trend: population change, price change, technological change, productivity change, product life cycles The long term movements or trend in a series can be described by a straight line or a smooth curve. The long-term trend is estimated from the seasonally adjusted data for the variable of interest Interpretation: Long run equilibrium: trends Cyclical fluctuations: gaps Czech National Bank QPM 7 / 73 Trend and cycles Detrending methods Trend analysis Assume seasonally adjusted data Trend-Cycle decomposition: Series = Trend + Cycle + Noise No general-automatic techniques for detrending Simple techniques: Smoothing Moving average: The average eliminate some higher frequency noise in the data, and leaves a smooth trend-cycle component. What order to use? Simple centered moving average: can be defined for any odd order. A moving average of order k, is defined as the average consisting of an observation and the m = (k-1)/2 points on either side. Centered moving average: take the simple centered moving average, assign weights and create weighted average Advanced techniques of detrending: Fitting a polynomial Using a structural model Czech National Bank QPM 8 / 73 Trend and cycles Detrending methods Detrending techniques overview I Watson detrending: greater business cycle persistence; trend component follows a random walk with drift and cyclical component is a stationary finite order AR process. Harvey-Clark detrending: local linear trend model Hodrick-Prescott filter: univariate method Kalman filter: multivariate method, structural method Bandpass filter: not widely used, frequency domain analysis Czech National Bank QPM 9 / 73 Trend and cycles Detrending methods Detrending techniques overview II Detrending comparison: US GDP gap Czech National Bank QPM 10 / 73 QPM structure Outline 1 Trend and cycles 2 Structure of the Quarterly Projection Model 3 Parameters setup 4 Properties of the Model Czech National Bank QPM 11 / 73 QPM structure Motivation for QPM Separate econometric methods: Inconsistencies Short experience with FPAS: Forecasting and Policy Analysis System State: Insufficient data and experience Participation of other departments Communication of results The further step on the way to complex structural models: DSGE Research tool Czech National Bank QPM 12 / 73 QPM structure Features of QPM Reflects inflation targeting regime: In December 1997: after an exchange rate crisis CNB adopted a series of end-year inflation targets Regime proved very effective in combating inflation and anchoring Evolution toward a more transparent inflation targeting regime where monetary policy is anchored by a medium-term perspective Change to point inflation target: Inflation target band The character of the regime was further enhanced by publication of unconditional forecasts Linked to quarterly data Small open-economy gap model Czech National Bank QPM 13 / 73 QPM structure Model of trends and cycle Two separate blocks: Long run equilibrium trends Cyclical fluctuations - gaps These blocks are separable Czech National Bank QPM 14 / 73 QPM structure QPM trends Long Run Trends First step: filter trend series History - estimated by a simple statistical model (Kalman filter) and expert judgement Forecast - exogenous (expert judgement), respecting steady state properties of QPM Important equilibrium values: Real output growth Real wage growth Real exchange rate appreciation Real interest rate Stationarity is required: growth rates in focus Monetary decisions have small impact on long term real trends Czech National Bank QPM 15 / 73 QPM structure QPM cycles Cyclical Part of QPM Description of the position of the Czech economy Monetary policy characteristics: Inflation targeting regime Forward looking policy Focus on deviations from the target −→ reaction to expected inflation a year ahead Floating exchange rate - endogenous Description of behavior economic agents includes forward looking components Price frictions: Wage stickiness Final price stickiness Expectation stickiness Czech National Bank QPM 16 / 73 QPM structure QPM scheme Scheme of model Czech National Bank QPM 17 / 73 QPM structure QPM scheme Real Economy I IS curve (Aggregate demand): Output: function of lagged output, the real interest rate, the real exchange rate and foreign demand Includes impact of a change in interest rates with longer maturity on aggregate demand and take into account expectations about yield-curve on the dynamic properties of the model Real impact of monetary policy in a sticky-price model of a small open economy Marginal costs: cost of producing additional unit of a good Czech National Bank QPM 18 / 73 QPM structure QPM scheme Real Economy II Real Marginal Costs Gap: Approximation of inflationary pressures from the real economy. Marginal costs consist of the costs arising from the increasing volume of production (the "output gap") and wage costs (the "real wage gap"). A positive real marginal cost gap implies an inflationary effect of the real economy mct = λyt + wrt Output Gap: Czech National Bank QPM 19 / 73 QPM structure QPM scheme Real Economy III Standard economic theory: higher real interest rate reduce aggregate demand by increasing the reward to saving Output gap: responds negatively to the difference between the real interest rate and its equilibrium value Open economy: the exchange rate matters Currency appreciation will, all else equal, make domestic goods more expensive in foreign markets and reduce demand for domestic goods abroad; cheaper imports may displace domestic goods yt = α1yt−1 − rmcit−1 + α2yf t + εy t rmcit = β1 β3rct + β4r4t + (1 − β3 − β4) r4 f t + β2zt Real Wage Gap: Czech National Bank QPM 20 / 73 QPM structure QPM scheme Real Economy IV Introduced in January 2007 Wage costs are above their equilibrium level, they have an inflationary effect The effect of a deviation of the current level of the average real wage from its equilibrium level, which in the long run rises at the same rate as equilibrium real output (non-accelerating inflation real output) wrt = wrt−1 + wt 4 − πt 4 − wrt 4 + εwr t Czech National Bank QPM 21 / 73 QPM structure QPM scheme Phillips Curves I Price Inflation: Phillips curve has been modified for a small open economy Blocks for various goods Import price effects Wage setters derive their nominal wage demand real consumer wage x for fuel, food, or adjusted excl. fuel inflation Administered prices are exogenous in baseline Czech National Bank QPM 22 / 73 QPM structure QPM scheme Phillips Curves II πx t = γx 1 π4Mx t + 4zx t + γx 2 Eπ4t + 4zx t − 4zt + 1 − γx 1 − γx 2 πx t−1 + γx 3 mct + επx t Wage Inflation: wt = δ1Ew4t + (1 − δ1)wt−1 − δ2 wrt − δ3yt + εw t Czech National Bank QPM 23 / 73 QPM structure QPM scheme Expectations I Price Inflation Expectations: Expected inflation: a weighted combination of a backward-looking and a forward-looking component (the expected value of overall CPI inflation over the next four quarters) Overall CPI: an explicit link between changes in administered and energy prices and pressures on the rate of inflation for market prices Eπ4t = λ1πt+1 + 1 − λ1 πt−1 + εE4 t Wage Inflation Expectations: Ew4t = λ2wt+1 + 1 − λ2 wt−1 + εEw4 t Czech National Bank QPM 24 / 73 QPM structure QPM scheme Uncovered interest rate parity Nominal Exchange Rate: UIP condition: arbitrage condition; international investors will equalize effective rates of return on investments in different currencies, allowing for any country-specific risk premiums foreign investor expecting a depreciation (appreciation) of the koruna will demand a higher (lower) return from Czech assets Moving average form st = φst+1 + (1 − φ) st−1 + 2 Etπ 4 − Etπf 4 + 2 zt + it 4 − if t 4 − premt + εs t Czech National Bank QPM 25 / 73 QPM structure QPM scheme Reaction Function Nominal Interest Rate: Forward-looking reaction function CPI inflation expecte to be above the target rate: central bank push up the short-term Excess demand: the central bank increases short-term interest rate Long-term level for rates and some additional dynamic structure Interest rate inertia: interest rate smoothing it = ψit−1 + (1 − ψ) ineutral t + Πt + εi t ineutral t = rt + π4t+4 + εi t Πt = κ1 π4t+4 − π4target t+4 + κ2yt Czech National Bank QPM 26 / 73 Parameters Outline 1 Trend and cycles 2 Structure of the Quarterly Projection Model 3 Parameters setup 4 Properties of the Model Czech National Bank QPM 27 / 73 Parameters Calibration vs. Estimation QPM is calibrated, partially estimated Problems in estimation: Short data sample Structural changes in economy Changes of monetary policy regime It is impossible to estimate some parameters: identification problems Czech National Bank QPM 28 / 73 Parameters Calibration of QPM Parameters setup: Restrictions on parameters originating from economic theory Parameters are set to mach the properties of data Responses to structural shocks Parameters checks: Reactions to shocks Residuals In-sample simulations Curve-fitting estimates Czech National Bank QPM 29 / 73 Model Properties Outline 1 Trend and cycles 2 Structure of the Quarterly Projection Model 3 Parameters setup 4 Properties of the Model Czech National Bank QPM 30 / 73 Model Properties Price shock I Positive shock to the output gap Upward pressure on inflation Currency depreciation Central bank increases interest rate Cumulative effect on output is very close to zero: feature of linear models; Offsetting of excess supply to counteract the effects of shocks that create excess demand Czech National Bank QPM 31 / 73 Model Properties Price shock II Czech National Bank QPM 32 / 73 Model Properties Aggregate demand shock I Positive shock to the output gap Upward pressure on inflation Currency depreciation Central bank increases interest rate Cumulative effect on output is very close to zero: feature of linear models; Offsetting of excess supply to counteract the effects of shocks that create excess demand Czech National Bank QPM 33 / 73 Model Properties Aggregate demand shock II Czech National Bank QPM 34 / 73 Model Properties Exchange rate shock I Depreciation acts to increase aggregate demand, opening a positive output gap Czech National Bank QPM 35 / 73 Model Properties Exchange rate shock II Czech National Bank QPM 36 / 73 Model Properties Inflation target change I Lower the target rate of inflation by one percentage point To achieve disinflation: raise the short rate Appreciation: Import prices fall The combined effect of the import price decline and the excess supply gap works to gradually pull down the rate of inflation Note: purely nominal shock, and since the model is super-neutral, there is no change to any real equilibrium in this shock, including the real exchange rate. The nominal exchange rate changes, of course, with the cumulative Cumulative effects on output and employment Sacrifice ratio: a cumulative loss of output vs. lower inflation by a percentage point Czech National Bank QPM 37 / 73 Model Properties Inflation target change II Czech National Bank QPM 38 / 73 Model Properties Data fitting Residuals I Conflict between estimated parameters and calibrated The parameters have to be chosen so as to give reasonable model behavior Examined how well the model performs over the historical sample Identify systematic biases Czech National Bank QPM 39 / 73 Model Properties Data fitting Residuals II Czech National Bank QPM 40 / 73 Model Properties Data fitting In-Sample Simulations Czech National Bank QPM 41 / 73 Model Properties Data fitting Modeling tools Implementation in Matlab IRIS by Jaromír Beneš Czech National Bank QPM 42 / 73 Model Properties Data fitting Czech National Bank QPM 43 / 73 Appendix Filters Univariate filtering I Hodrick-Prescott filter: optimally extracts a trend which is stochastic but moves smoothly over time and is uncorrelated with the cyclical component Mathematics of HP filter: Decomposition: yt = τt + ct Solve: min T t=1(yt − τt)2 + λ ∗ T−1 t=2 [(τt+1 − τt) − (τt − τt−1)]2 λ = 100 ∗ (number of periods in a year)2 Assumption that the trend is smooth is imposed by assuming that the sum of squares of the second differences of τt is small Sensitivity of the trend to short-term fluctuations is achieved by modifying a multiplier λ Czech National Bank QPM 44 / 73 Appendix Filters Univariate filtering II Drawbacks: One-time permanent shock, split growth rates present: Filter identifies non-existing shifts in the trend It pushes noise in data to Normal distribution Misleading predictive outcome: Analysis is purely historical and static Czech National Bank QPM 45 / 73 Appendix Filters Univariate filtering III Trend: Czech National Bank QPM 46 / 73 Appendix Filters Univariate filtering IV Gap: Czech National Bank QPM 47 / 73 Appendix Kalman filter Kalman filter I Separate the cyclical component of a time series from raw data Can handle more series and exploit relations between them Kalman filter is a powerful tool for: Estimation Prediction Smoothing Kalman filter: Online estimation procedure States are estimated, when the new observations are coming in Kalman smoother: Off-line estimation procedure The state estimation of is not only based on all previous observations, but also on all later observations Czech National Bank QPM 48 / 73 Appendix Kalman filter Kalman filter II F is the state transition model B is the control-input model H is the observation model w is the process noise z is the measurement v is the measurement error u is the exogenous control Czech National Bank QPM 49 / 73 Appendix Kalman filter Kalman filter structure Czech National Bank QPM 50 / 73 Appendix Simple filtering model Description of variables Measurement variables: ∆EU LGDP, EU LGDPGAP EXPERT State variables: ∆EU LGDP EQ, MU, EU LGDPGAP Exogenous-variables: EU RMCIGAP Shocks: ν’s Coefficients: a1, a2, a3 and µSS Variance: σ1, σ2, σ3, σ4 Remark: In the following slides the filtering is actually smoothing Czech National Bank QPM 51 / 73 Appendix Simple filtering model Description of model Measurement equations: ∆EU LGDP = ∆EU LGDP EQ + + 4 ∗ (EU LGDPGAP − EU LGDPGAP{−1}) EU LGDPGAP = EU LGDPGAP EXPERT + σ4 ∗ ν4 State equations: ∆EU LGDP EQ = µ + σ1 ∗ ν1 µ = (1 − a3) ∗ µSS + a3 ∗ µ{−1} + σ3 ∗ ν3 EU LGDPGAP = a1 ∗ EU LGDPGAP{−1} + + a2 ∗ EU RMCIGAP{−1} + σ2 ∗ ν2 Czech National Bank QPM 52 / 73 Appendix Filtering results Filtering results: EU Eq. trajectories Czech National Bank QPM 53 / 73 Appendix Filtering results Filtering results: EU Gap estimate Czech National Bank QPM 54 / 73 Appendix Filtering results Filtering results: Removing volatility Czech National Bank QPM 55 / 73 Appendix Filtering results Model setting: Changes in volatility of gap σ2 Czech National Bank QPM 56 / 73 Appendix Complex model Filtering domestic variables First step: Decompose real variables: trend and cycle Simple model for: Real interest rate, Real exchange rate, Exchange risk premium Second step: Utilize measurement of inflation and wage growth Fit simple backward-looking Phillips curves: relation between inflation and output gap Fit IS curve: relation between output gap and gaps in real interest and exchange rate Decompose: domestic output, real wage, unemployment Czech National Bank QPM 57 / 73 Appendix Complex model Filtering results: Domestic Eq. trajectory Czech National Bank QPM 58 / 73 Appendix Complex model Filtering results: Domestic output gap Czech National Bank QPM 59 / 73 Appendix Expert judgement Description: Second step model Measurement variables: DOT LGDP, DOT UNR, PIE CORE, PIE W , DOT LWR, LWR GAP EXPERT, LGDP GAP EXPERT, UNR GAP EXPERT State variables: DOT LGDP EQ, MU, LGDP GAP, DOT UNR EQ, UNR GAP, PIE CORE S, PIE W S, DOT LWR EQ, LWR GAP Exogenous-variables: RRC GAP, RR4 GAP, EU RR4 GAP, LZ GAP, EU LGDP GAP, PIE M XENERGY 4, DOT LZ CORE EQ4, DOT LZ EQ4, E0 CORE4, E0 PIE W 4, DOT LWR PRIOR, E0 PIE4 Shocks: νs Variance: σs Czech National Bank QPM 60 / 73 Appendix Model details Model I Measurement equations: DOT LGDP = DOT LGDP EQ + 4 ∗ (LGDP GAP − LGDP GAP{−1}) DOT UNR = DOT UNR EQ − 4 ∗ (UNR GAP − UNR GAP{−1}) PIE CORE = PIE CORE S PIE W = PIE W S DOT LWR = DOT LWR EQ + 4 ∗ (LWR GAP − LWR GAP{−1}) LWR GAP = LWR GAP EXPERT + std w3 ∗ ν LWR GAP EXPERT LGDP GAP = LGDP GAP EXPERT + std w1 ∗ ν LGDP GAP EXPERT UNR GAP = UNR GAP EXPERT + std w2 ∗ ν UNR GAP EXPERT Czech National Bank QPM 61 / 73 Appendix Model details Model II State equations: DOT LGDP EQ = MU{−1} + a1 ∗ DOT UNR EQ + std v1 ∗ ν DOT LGDP EQ LGDP GAP = LGDP GAP C01 ∗ LGDP GAP{−1} − RMCI GAP C02 ∗ (b2 ∗ RRC GAP{−1} + b3 ∗ RR4 GAP{−1} + b4 ∗ EU RR4 GAP{−1}) −RMCI GAP C01 ∗ LZ GAP{−1} + LGDP GAP C02 ∗ EU LGDP GAP + std v2 ∗ ν LGDP GAP MU = (1 − a3) ∗ MU SS + a3 ∗ MU{−1} + std v3 ∗ ν MU DOT UNR EQ = std v4 ∗ ν DOT UNR EQ UNR GAP = UNR GAP C01 ∗ UNR GAP{−1} + UNR GAP C02 ∗ LGDP GAP + std v5 ∗ ν UNR GAP PIE CORE S = PIE CORE C01 ∗ (PIE M XENERGY 4 + DOT LZ CORE EQ4) + PIE CORE C02 ∗ (PIE CORE C05 ∗ E0 CORE4 + (1 − PIE CORE C05) ∗ E0 PIE4) + (1 − PIE CORE C01 − PIE CORE C02) ∗ PIE CORE S{−1} + RMC GAP C01 ∗ PIE CORE C03 ∗ LGDP GAP + PIE CORE C03 ∗ LWR GAP + std v6 ∗ ν PIE CORE PIE W S = PIE W C01 ∗ E0 PIE W 4 + (1 − PIE W C01) ∗ PIE W S{−1} + PIE W C02 ∗ (LWR GAP − PIE W C03 ∗ LGDP GAP) + std v7 ∗ ν PIE W DOT LWR EQ = DOT LGDP EQ + DOT LWR PRIOR + std v8 ∗ ν DOT LWR EQ LWR GAP = f 1 ∗ LWR GAP{−1} + std v9 ∗ ν LWR GAP Fixing: unemployment gap in 1999Czech National Bank QPM 62 / 73 Appendix Expert judgement simulations Filtering results: Expert judgement Czech National Bank QPM 63 / 73 Appendix Expert judgement simulations Filtering results: Expert judgement Czech National Bank QPM 64 / 73 Appendix Advanced filtering Advanced filtering Criticism of simple models: lack of reference to unemployment J. Galí,F. Smets and R. Wouters (2011): Address this issue in an extended model Conclusion: Model-based output gap resembles conventional measures of the cyclical component of log GDP. Comparison of a variety of statistical detrending methods HP filter, band-pass filter, quadratic detrending, and the Congressional Budget Office’s measure Czech National Bank QPM 65 / 73 Appendix Advanced filtering Advanced filtering Czech National Bank QPM 66 / 73 Appendix Advanced filtering In search for future trends Czech National Bank QPM 67 / 73 Appendix Advanced filtering List of Variables I a gap of the variable a a trend (equilibrium) value of the variable a af variable a for the foreign country εa residual in the equation for the variable a mc real marginal costs y real output rw real wage rmci real monetary condition index r4 real 1Y interbank rate r real 3M interbank rate rc real rate of newly-issued bank loans z real exchange rate Czech National Bank QPM 68 / 73 Appendix Advanced filtering List of Variables II π4target inflation target (y-o-y) π price inflation (q-o-q) π4 price inflation (y-o-y) w wage inflation (q-o-q) w4 wage inflation (y-o-y) π4M imported inflation (y-o-y) s nominal exchange rate prem risk premium i nominal short-term interest rate ineutral policy neutral short-term interest rate α, β, γ, δ, φ, ψ, κ, λ parameters Czech National Bank QPM 69 / 73 Appendix Literature For Further Reading I Cbo’S Method For Estimating Potential Output: An Update, http://www.cbo.gov/doc.cfm?index=3020&type=0 Jordi Galí and Frank Smets and Rafael Wouters Unemployment In An Estimated New Keynesian Model, National Bureau Of Economic Research,vol. 17084, 2011 Peter K. Clark The Cyclical Component of U.S. Economic Activity, The Quarterly Journal of Economics,vol. 102,1987 Czech National Bank QPM 70 / 73 Appendix Literature For Further Reading II Rudolph E. Kalman A New Approach to Linear Filtering and Prediction Problems Transactions of the ASME–Journal of Basic Engineering, vol. 82, Series D, 1960 Greg Welch and Gary Bishop An introduction to the Kalman filter. University of North Carolina, July, 2006; 2000. Harvey, Andrew C, 1985 Trends and Cycles in Macroeconomic Time Series Journal of Business and Economic Statistics, Vol. 3 p. 216 Czech National Bank QPM 71 / 73 Appendix Literature For Further Reading III Watson, Mark M, 1986 Univariate Detrending Methods with Stochastic Trends Journal of Monetary Economics, Vol. 18, p. 49 Athanasios Orphanides and Simon van Norden, 2002 The Unreliability of Output-Gap Estimates in Real Time The Review of Economics and Statistics, Vol. 84, Num. 4 Czech National Bank QPM 72 / 73 Appendix Literature Additional one ... Czech National Bank QPM 73 / 73