Business Cycles Theory

Transcription

1 Business Cycles Theory Identification of business cycles Jaromír Baxa 31st October, 2018

2 Plan of the talk 1. Brief summary of the frequency domain approach 2. Filtering and identification of business cycles 3. Recession dating procedures: NBER approach and alternatives. 4. Issues with forecasting of business cycles: Data and trend revisions 5. Leading indicators

3 Frequency domain: Spectral analysis Spectral analysis in econometrics: a method for finding dominant frequencies in time series and filtering Frequency ω: angular frequency measured in radians, (frequency at which cos(ωt) completes the cycle relative to cos(t) in a respective period of time T). Spectral representation of time series: Each (covariance stationary) time series yt can be expressed in terms of frequencies ωj as n y t =a0 + {α j cos( ω j t)+ β j sin ( ω j t )} j=1 yt shall be stationary according to definitions and mathematical theorems behind the theory (Fourier transformation), nevertheless one can in principle estimate spectra of any time series. Implication: All time series can be approached as a weighted sum of harmonic signals.

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5 Spectrum and Periodogram Consider spectral representation of time series n y t =a 0 + {α j cos( ω j t)+ β j sin ( ω j t )} j=1 Periodogram P = (sum of second powers of coefficients at cos and sin terms)*n/2. Estimating spectrum: smoothing periodogram by some window that determines smoothness of resulting spectrum. Example: Barlett window. For infinite samples, smoothed periodogram should converge to spectrum derived from the autocovariance function.

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8 Identification of business cycles using filters

9 Filtering Filter: operation that retains only some part of spectrum. Transfer function theorem: If a zero-mean stationary process Xt with spectral density fx(ω) is passed through a filter with frequency response function a(ω), then the spectral density of Yt is defined as: fy(ω)=fx(ω) a(ω) 2. Low-Pass filter: a(ω) 2=1 if ω ωc; 0 otherwise High-Pass filter: a(ω) 2=1 if ω ωc; 0 otherwise Band-Pass filter: a(ω) 2=1 if ωl ω ωh; 0 otherwise All operations with time series can be considered as filters. Filters can be used for identification of business cycles.

10 Moving Averages Linear filters: moving averages. Weights determine cyclical properties Moving average: MA4 = (x1 + x2 + x3+ x4)/4 Compositions: 4*MA4=(MA41 + MA42 + MA43+ MA44)/4 Result: 4*MA4 = (xt-3 + 2xt-2 + 3xt-1+ 4xt + 3xt+1 + 2xt+2 + xt+3)/16 Usually centered. Weighted moving average: WMA, weights not 1/k but change Examples: Spencer: 5*5*4*4 MA with the final MA being assigned the weights 3/4,+3/4,1,+3/4,-3/4.

11 Band-Pass Filter Band-pass filters and estimation of business cycles fluctuations between 6 and 32 quarters => cut all frequencies above π/3 and under π/16 ~ BPn(6,32) Annual data: BP(2,8) Popular type: Baxter-King (1999) Moving averages filter. Weights for BP12(6,32) and BP3(2,8): BP12(6,32): ; ; ; ; ; ; ; ; ; ; ; ; BP3(2,8): ; ; ;

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13 Hodrick-Prescott (1982) Filter Hodrick, R. J., & Prescott, E. C. (1997). Postwar US business cycles: an empirical investigation. Journal of Money, Credit, and Banking, Often used for removing business cycles, high pass nature Denoted as H-P(λ). Let y(t) be a log of time series that is a realization of a non-stationary stochastic process. τ t =min ( T T 1 2 ( y t τ t ) + λ [( τt +1 τt ) ( τt τt 1)]2 t=1 t =2 ) yt = log GNP, τ is trend component minimizing this formula and λ is apriori chosen parameter to derive trend, which students of business cycles would draw into the plot of GDP. (Kydland-Prescott, 1990)

14 Hodrick-Prescott Filter Which value of λ? Quarterly data: Annual: some authors suggest 6.5, others 10. Still, the choice of λ not without controversies: The value 1600 is known to generate trend that is procyclical, and Mehra (2004) recommend much larger values, i.e Important: To understand the implication of different λ s: If λ=0, then yt =τt. As λ increases up to infinity the trend becomes linear. H-P filter is very similar to high-pass filter with λ = 1600 it cuts off all frequencies with periods higher than 32 quarters. It does not remove seasonal fluctuations.

15 λ=1 λ = 100 λ =

16 Dejong-Dave: Structural Macroeconometrics, p

17 Filtering: Comparing transfer functions of popular filters Source: Stock, J. - Watson, M. (1999): Business Cycle Fluctuations in U.S. Macroeconomic Time Series. In: Taylor, J., Woodford, M. (eds.): Handbook of Macroeconomics, Vol. 1. Elsevier.

18 Using output gaps in real-time: End-sample bias and revisions

19 Data Revisions Data source: OECD MEI Data Revisions Database.

20 Data source: OECD MEI Data Revisions Database; calculated gap is mine.

21 Using output gaps in real-time Evidence that the distinction between real-time and ex-post data matters for evaluation of monetary policy rules by Orphanides (2001); Orphanides and Van Norden (2002) provide analysis of revisions of output gap estimates. McCallum (2000) analyses the contribution of data revisions to errors in real-time output gap estimates and shows that even if data revisions were negligible, substantial differences might appear among real-time and ex-post gaps, which are measured with an often substantial delay of several years. This end-sample bias points to the fact that detrending works as a local filter, and as new data arrive, past potential output is affected. Later, it has been confirmed, this problem is not related to HP filters only, but to production function or other structural methods as well. See Kempkes (2014) and McMorrow et al. (2015). Furthermore, they suggest that prediction errors of output gaps are systematically correlated with business cycles so that cyclical component is underestimated in booms and rather overestimated in recessions.

22 Figure A.1: Revisions of underlying trend HP-filter, estimated till 2008Q4, 2009Q4, 2010Q4 (extrapolated); and 2015Q3 (dashed). Extrapolation of trends based on ARIMA(1,1,0) forecasts of trends, 95% confidence bands 14,1 14,05 Log of real GDP 14 13,95 13,9 13,85 13,8 13,75 13,7 13, Q1 2006Q1 2007Q1 2008Q1 2009Q1 2010Q1 2011Q1 2012Q1

23 Output gaps in real-time Alternative in real-time? Production function methods used by many institutions includig OECD, European Commission etc. prone to revisions of trend as much as HP filter does. J. Hamilton (2017): Why you should never use the Hodrick-Prescott filter proposes regression of yt on yt-8,, yt-11 (output third year ago) arguing that this filter Cancels out seasonal fluctuations Builds on the intuition that output gap is a difference between current value and value that would have been based on expectations two years ago

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25 Output gaps in real-time One-sided HP filter instead of two-sided version Can be calculated recursively for the first k observations, then for k+1 until k=n, and the values filtered at the end of each step are stored. No revisions of underlying trend but higher variability of the trend.

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30 Forecasting output gaps Implications: Proper assessment of cyclical position important for both fiscal and monetary policy, but their estimation or forecasting are extremely uneasy. Note that fiscal rules such as the Stability and Growth Pact and the new revised EU fiscal framework require balanced cyclically adjusted balances. Moreover: Revisions systematically correlated with the cycle. Expansions, revisions tend to increase the ex-post cyclical component Recessions, revisions tend to decrease the ex-post cyclical component In the Czech Republic, over the course of past decade, the revisions of the cyclical component are of the same magnitude as the cyclical component itself (BaxaPaulus, 2016). Masten and Gnip (Journal of Financial Stability, 2016) document that failure to recognize the appropriate position of the fiscal policy can make the EU surveillance inefficient.

31 Classical business cycles: Turning points

32 Classical business cycles Business cycles identified by filters are often symmetric and do not overlap with periods of ups and downs. Absolute declines and increases are considered as so called classical business cycles. This classical apprach originated in the 19th century. Typical description of the business cycles comes from Joseph Schumpeter: Expansion: increase in production and prices, low interest rates Crisis: stock market crash or a wave of bankruptcies Recession: drops in prices and output and increase of the (real) interest rate Recovery: when prices and incomes fall to correct imbalances, recovery occurs Burns and Mitchell in Measuring Business Cycles (1946): definition of business cycles later on established at NBER: Significant decline in economic activity spread across the economy, lasting more than a few months, normally visible in real GDP, real income, employment and industrial production. Burns and Mitchell shifted the focus from explanation of business cycles to its measurement and forecasting since they prefered the interpretation of fluctuations rather than periodic cycles.

33 Classical Cycles

34 Classical Cycles and Growth Cycles Recessions vs. growth recessions Expansions and recessions: refer to absolute declines in output and other measures. Alternative is to examine cyclical fluctuations in economic time series that are deviations from their long-run trends: growth cycles Classical cycles imply shorter duration of recessions and slower recoveries relatively to growth rates in contractionary phases Stock-Watson (1999): Classical cycles tend to have recessions that are considerably shorter than expansions because of underlying trend growth, growth recessions and expansions have approximately the same duration. The study of growth cycles has advantages and disadvantages relative to classical cycles. On the one hand, separation of the trend and cyclical component is inconsistent with some modern macroeconomic models, in which productivity shocks (for example) determine both long-run economic growth and the fluctuations around that growth trend. From this perspective, the trend-cycle dichotomy is only justified if the factors determining long-run growth and those determining cyclical fluctuations are largely distinct (...) and in fact some economies which have had very high growth rates, such as postwar Japan, exhibit growth cycles but have few absolute declines and thus have few classical business cycles.

35 Business Cycles Measurement Chronology of business cycles, duration and other measures of fluctuations. Identification problem: economic series are noisy and especially time series gathered at the monthly frequency such as industrial production contain many ups and downs => necessary to determine, which change of sign of the growth rates is associated with cycle. To establish a turning point (peak or trough) necessary to show that: Decline in the economic activity is general, hence it does not occur just in one time series Peak is followed by trough (peaks and troughs alternate) At the NBER, peaks and troughs are determined based on a voting procedure. The Committee applies its judgment based on many time series and broad definition of recession (significant decline in economic activity spreads across the economy not necessarily ahown by a fall in GDP) and it has no fixed rule to determine whether a contraction is only a short interruption of an expansion, or an expansion is only a short interruption of a contraction. Hence, it is based on human judgment that allows to take into account various factors and links that are difficult to model and inclued in a formal quantitative analysis.

36 NBER Approach

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38 Simplified Rules and Formalized Procedures A. Okun (1962): recession iff 2 subsequent quarters of negative GDP growth. Does not much the NBER chronology fully, concerned on GDP only recession, for example, did not exhibit two quarters of negative growth. Bry-Boschan, 1971 sequential application of moving averages and determining of turning points based on several simple rules (peaks and troughs should alternate, peak shall be higher than previous one etc.). Quarterly version BBQ. Toolboxes for various software packages exist. Simplified approach by sequential application of logical rules: 1. Identification of potential of peaks and troughs Peak: y t 2 y t 1 < y t > y t +2 y t +1 Trough: y t 2 y t 1 > y t < y t +2 y t+1 2. Manual check whether peaks and troughs alternate, if not deletion of higher troughs and lower peaks. Example of nonalternating troughs => lower trough accepted ( T 2 ). P T1 T2

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42 Business Cycles Stylized Facts 2 approaches: gaps obtained via detrending or via the turning point analysis. Zarnowitz (1982, ch. 2) on classical business cycles stylized facts: The term "business cycle," is a misnomer insofar as no unique periodicities are involved, but its wide acceptance reflects the recognition of important regularities of long standing. The observed fluctuations vary greatly in amplitude and scope as well as duration, yet they also have much in common. First, they are national, indeed often international, in scope, showing up in a multitude of processes, not just in total output, employment, and unemployment. Second, they are persistent-lasting, as a rule several years, that is, long enough to permit the development of cumulative movements in the downward as well as upward direction. For all their differences, business cycles consist of patterns of recurrent, serially correlated and cross-correlated movements in many economic (and even other) activities. In the U.S., the individual business cycles show considerable variability over time, however most of the variability is caused by some rare outliers: Average expansion 3 years, contraction 1 year, full cycle about 4 years. SD's 3-4 quarters. Turning points analysis might help to assess about relevancy of detrending methods: countries with high growth rates (Japan till 80's, transition countries): many growth recessions but recessions as such are rare.

43 Business Cycles Stylized Facts High advantage of stylized facts derived from classic cycles: not sensitive on detrending methods. That detrending matters for stylized facts of the growth cycles is extensively discussed in Fabio Canova (1998): Detrending and Business Cycles Facts, Journal of Monetary Economics. The basic result of his paper is that analysis of the growth cycles can hardly help with selection of the true business cycles models. For example, the RBC model assumes that business cycles are caused by the real factors and that movement in monetary aggregates do not lead the cycle. See Kydland and Prescott 1990 wrote an article on that called Real Cycles and Monetary Myth based on analysis of first and second moments of detrended aggregated series using the HP filter. Their results not widely accepted: large narrative evidence on the fact that swings in monetary policy cause swings in real activity such as during the Great recession (M. Friedman, A. Schwartz: A Monetary History of the United States, 1963) or during the Volcker disinflation in the early 1980's. Second implication of Canova's findings pointed out by Craig Burnside (JME, 1998) is that if detrending matters, it is also hard to derive universal business cycle stylized facts both for one country over time but especially for more countries. Widely discussed. Generally, the prevailing view is that whereas business cycles have relatively uniform characteristics in developed countries, in developing countries the variety of business cycles characteristics increases (Rand Tarp: Business Cycles in Developing Countries Are They Different? World, Development, 2002). Related to discussion on whether transition or permanent shocks are more frequent: Aguiar Gopinath: Emerging Market Business Cycles: The Cycle Is the Trend, Journal of Political Economy, 2007/1).

44 European Perspective CEPR, Euro Area Business Cycle Dating Committee, PEAK 1974q3 1980q1 1992q1 2008q1 2011q3 TROUGH 1975q1 1982q3 1993q3 2009q2 2013Q1 Economic Cycle Research Institute (ECRI)

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46 Forecasting turning points Forecasting output gaps or the time series as such: ARIMA or any other methods. However, forecasting of turning points nontrivial. Generally, two options available: Forecasting turning points sequence 1,0,-1: feasible, but not very convincing results (Zellner, A. et al. "Forecasting turning points in international output growth rates using Bayesian exponentially weighted autoregression, time-varying parameter, and pooling techniques." Journal of Econometrics (1991); papers by Harding and Pagan, 2008 and newer): Hardly more than 50% turning points correctly estimated. Finding indicators that turn before the turning point occur (so called leading indicators). Caveats: Data on many series available with lag and are revised (next 2 slides: revision of the Czech GDP and HP output gap): for example in practice one needs to forecast not only the future inflation or GDP, but also the current one (nowcasting). Leading indicators need to fulfill two criteria: they need to move before the indicator of interest or at least before the value of the indicator of interest is known and second, they should not signalize turning points when no turning point occurs.

47 Leading indicators

48 Leading indicators Economic indicators can be classified into three categories according to their usual timing in relation to the business cycle: leading indicators, lagging indicators, and coincident indicators. Leading indicators are indicators that usually change before the economy as a whole changes. They are therefore useful as short-term predictors of the economy. Examples: Average weekly hours (manufacturing), Average weekly jobless claims for unemployment insurance, Manufacturers' new orders, Building permits for new private housing units, Stock market index, Money Supply, Interest rate spread, Consumer expectations... Usually one composite indicator derived (usually using factor analysis) to an composite leading indicator. Reference: CLI by the OECD Coincident: number of employees, industrial production, personal income, manufacturing and trade sales (usually available before the GDP numbers) + various indicators claimed to capture the economic conditions at higher frequency) Lagging: inventories investment, labour costs, duration of unemployment, ratio of consumer credits relatively to income...

49 ADS Business Conditions Index Aruoba-Diebold-Scotti business conditions index Example of index that is designed to track real business conditions at high frequency. Its underlying (seasonally adjusted) economic indicators (weekly initial jobless claims; monthly payroll employment, industrial production, personal income less transfer payments, manufacturing and trade sales; and quarterly real GDP) blend high- and lowfrequency information and stock and flow data. Both the ADS index and this web page are updated as data on the index's underlying components are released. The average value of the ADS index is zero. Progressively bigger positive values indicate progressively better-than-average conditions, whereas progressively more negative values indicate progressively worse-than-average conditions. The ADS index may be used to compare business conditions at different times. A value of -3.0, for example, would indicate business conditions significantly worse than at any time in either the or the 2001 recession, during which the ADS index never dropped below Published at Philladelphia FED Aruoba, S.B., Diebold, F.X. and Scotti, C. (2009), "Real-Time Measurement of Business Conditions," Journal of Business and Economic Statistics 27:4 (October 2009), pp

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51 Economic Sentiment Indicator (ESI) Total Economy, Euro area 120,0 110,0 100,0 90,0 80,0 70,0 60,0 I.85 I.86 I.87 I.88 I.89 I.90 I.91 I.92 I.93 I.94 I.95 I.96 I.97 I.98 I.99 I.00 I.01 I.02 I.03 I.04 I.05 I.06 I.07 I.08 I.09 I.10 I.11 I.12 I.13 I.14 I.15 I.16 I.17 I.18 European equivalent: Economic Sentiment Indicator. Maintained at the European Commission. Available here: economic-databases/business-and-consumer-surveys/download-busi ness-and-consumer-survey-data/time-series_en

52 Main Points Filters, transfer function, periodogram and spectrum. Problems with filters and alternatives Business cycles: growth vs. classical cycles. NBER procedure (and similar one used in Europe or in other countries) take into account many series and are not based on formal statistical procedure, but on voting of members of a committee. BBQ and Zellner Procedure Leading and Coincident indicators Data revisions.

53 Problem Set Periodogram of time series is derived as a discrete Fourier transformation of the autocovariance function. Use the formula for the DFT from the lecture notes by Lukáš Vácha (p. 4), the autocovariance function is the x t. Now consider the white noise process. Think how its autocovariance function looks like. Then, derive the spectrum analytically. Estimate spectrum of time series you ve generated in your last problem set. How seasonality affected the spectrum? Use R-function spec.pgram or any other. Estimate output gap using the Hodrick-Prescott filter. Use either time series of real GDP or industrial production of any country you want to, but the data shall cover at least the first quarter of this year. Then, reestimate the output gap for the first quarter of 2016 with the data only up to the first quarter of Describe the difference. Utilize the R-package mfilter. Readings: The NBER report on last business cycle turning point/last CEPR report. What was - based on one of the texts - the main reason for the committee to say the recession is already over? Link to NBER report Link to CEPR statement Have a look at OECD composite leading indicator. Check especially the EU, the U.S. and country of your origin. Which phase of the cycle the economy seems to be right now and what are the prospects for the near future? Link to OECD CLI: and Business Cycles Stylized facts: Download time series for the real GDP and two of the following series: Consumption, Investment, Government expenditures, Current account deficit, Industrial production, Construction, Wages, Employment, Unemployment, Inflation and Interest rates. Convert selected series into quarterly frequency. All shall be in real-terms and seasonally adjusted. Estimate the turning points using the simplified procedure based on logical rules and asses which variables have synchronized turning points, which lag or lead the cycle.

54 Appendix

55 Seasonal Adjustment Methods If time series are not seasonally adjusted from the source, there are various possibilities how to seasonally adjust them: Building a model that explicitly account for seasonal fluctuations inside its structure, e.g. SARIMA or more advanced methods based on state space decomposition. Seasonally adjust the time series using automatic filtering methods These methods are prefered if the first option is not available or not feasible, because seasonal terms are usually identified better when estimated jointly with the model of interest. Furthermore, the automatic methods are designed in order to work sufficiently good on many time series, but in some specific cases they may keep the seasonal pattern unfiltered.

56 Seasonal Adjustment Methods X11 (Census) and X12 (Tramo/Seats) Based on sequential application of filtering methods and estimation of ratio-to-moving average. Three main steps: 1. Estimate the trend by a moving average 2. Remove the trend leaving a seasonal and irregular component 3. Estimate the seasonal component using moving averages Seasonality cannot be identified until the trend is known and an estimate of the trend cannot be found until the series has been seasonally adjusted. Hence an iterative approach is needed.

57 The X11 method On the website of Australian Bureau of Statistics nice detailed description ( 8404/c890aa8e ca256ce10018c9d8!opendocument). How the method proceeds Step 1: Estimation of trend using moving average (centered, 5 or 13 terms). Preliminary trend line estimated, only very long periodic fluctuations are passed. Result: seasonal + irregular component Step 2: Preliminary estimation of seasonal pattern. Weighted moving average (3*3MA, weights might differ in various implementations, hence in future steps are skipped) on the difference between original series and trend from the step 1. Result: Seasonal Component Step 3: First estimation of seasonal adjusted series Seasonal component from step 2 is adjusted. Step 4: Better estimation of trend on seasonally adjusted series (Henderson) Step 5: Final estimation of seasonal component (as in step 2) and seasonally adjusted series (follows step 3).