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“The alternative of a smoother parameter in the Hodrick-Prescott filter” Speaker: Miguel Ángel Ramírez Hernández Time Series: Retrospective, definition and components. The Hodrick-Prescott filter (1997) and criticism of the smoothing parameter (λ). Proposal, simulation and empirical evidence. References. Miguel Ángel Ramírez Hernández 2 “Every kind of periodic fluctuation, whether daily, weekly, monthly, quarterly, or yearly, must be detected and exhibited, not only as a subject of study in itself, but because we must ascertain and eliminate such periodic variations before we can correctly exhibit those which are irregular or non-periodic, and probably of more interest importance”1 William Stanley Jevons 1862 1Jevons, W.S. (1884). Investigations in currency and finance. London: Macmillan and Co. page 4. Miguel Ángel Ramírez Hernández 3 In the half nineteenth century: W. S. Jevons (1862) pioneered the analysis of time series. Early twentieth century Hooker (1901): Concept trend. First half of the twentieth century Study of business cycles: Kitchin (1923) and Frickey (1934). Formalizing cyclic models: Kuznets (1929); Frisch (1933); Samuelson (1939); Kaldor (1940); Metzler (1941) and Tinbergen (1939.1940, 1942). 1950-1960: period of "steady state" in the level intellectual of cycles. Miguel Ángel Ramírez Hernández 4 1970-1980: early distinction of economic cycles, growth cycles and political-economic cycles, Nordhaus (1975). Simultaneously, Lucas (1975) with notable differences: General Economic Cycle (GEC). From GEC were raised derivations with rational expectation. Development of "RBC“ models. Kydland-Prescott (1982): prototype RBC Long and Plosser (1983). 1980-1990: special interest decomposition: trend-cycle. in time series Harvey (1985) and Hodrick-Prescott (1997). Extensive use in RBC models. Miguel Ángel Ramírez Hernández 5 Time series definition Probability space: (, , P ) “y” is a random variable, real function defined in that for every real number “a” Thus, for each “a”: such Aa w | y(w) a F : R 0,1 F (a) P( Aa ) Therefore, a random vector or vector of random variables of dimension K is a function “y" of in Euclidean space Rk y w y1 w,..., yk w´ a (a1 ,...,ak )´ R k Aa w | y1 ( w) a1 ,..., yk | ( w) ck Miguel Ángel Ramírez Hernández 6 F : R k 0,1 F (a) P( Aa ) Distribution function “y” Assuming an index set Z that contains non-negative integers, discrete stochastic process is a real function: t Z, y (t , w) y:Z R Generally, the random variable corresponding to "t" is denoted as {yt}. Finally, a time series is defined as an underlying stochastic process embodiment and whose order performs to equidistant frequency. Miguel Ángel Ramírez Hernández 7 1) Persons (1919) identified four components of the time series: A long-term development (trend). 2) A cyclical component with periods longer than t +1 (cycle). 3) A component that contains fluctuations up and down within a year (seasonal cycle / seasonal). 4) A component excluding movements in 1), 2) and 3). (residual / irregular/random element). Miguel Ángel Ramírez Hernández 8 Define a time series yt as the sum of a component of "growth" and a cyclical component : yt gt ct for t 1,2,3,..T Optimization process particular: minimizing the variance of cyclical component and the variance of trend "second difference of serie". 2 T T 2 min c g g ( g g ) t t t 1 t 1 t 2 g t Tt 1 t 1 t 1 Smoothing parameter: λ 12 2 2 Source: Authors' calculations based on Hodrick and Prescott (1997). Miguel Ángel Ramírez Hernández 9 The evidence and relevance of λ smoothing factors that suggest the authors to annual data is 100 and 1600 for quarterly data. However, the parameter λ has a number of inconsistencies outlined mainly by Cogley and Nason (1995); Guay, ST-Amant (2005). Reviews: 1. The trend and cycle components present deviations prominent when the estimator λ is not consistent. 2. Real Cycles Spurious, derivatives of overidentifying in the order of time series. 3. Variance and trend cycle series are particular; assume a priori smoothing factors can disturb inferences, for example, the unemployment rate and the natural rate estimates by the HP filter. Miguel Ángel Ramírez Hernández 10 i. Proposal: Promptly identify the order of integration of the trend and proceed to use the corresponding variance. ii. Consider and weight the factor λ by a coefficient inverse of "angular frequency". 12 af 2 2 Where: 1 af T Miguel Ángel Ramírez Hernández 11 Matrix estimation In terms of Hodrick-Prescott (1997) yt ( y1 , y2 ,..., yT ), yt RT g t RT yt gt ct c RT If the smoothing parameter is non-negative, i.e. λ> 0, the breakdown of the series yt is obtained by minimizing the weighted sum of squares with respect to gt : 2 yg g R 2 2 T Note: Stata incorporates hprescott command. Miguel Ángel Ramírez Hernández 12 The unique solution of the minimization is defined as: g , y I Z´Z y 1 Where Z denotes a particular matrix, dimension T 2 T and “I” denotes the identity matrix. Z T 2T 1 2 1 0 1 2 : ... ... ... 0 ... ... Miguel Ángel Ramírez Hernández ... ... 1 ... ... ... 1 2 0 0 ... 1 13 Simulation in Stata Step 1: Define the matrix Z. mkmat … … … …, matrix(Z) mat list Z Step 2: Estimate the transpose of Z. matrix Z´=Z’ mat list Z’ Step 3: Define the identity matrix I. mkmat … … …, matrix(identity) mat list identity Step 4: Multiply the transpose matrix Z by matrix Z. matrix Z´Z=Z’*Z Step 5: Estimate the smoothing parameter λ and multiply this scalar by the result of the matrix obtained in Step 4. matrix lambdaZ´Z= λ *Z´Z Miguel Ángel Ramírez Hernández 14 Step 6: The result obtained in step 5, add the identity matrix. matrix lambdaZ´Z+I=identity+lambdaZ´Z mat list lambdaZ´Z+I Step 7: Estimate inverse of the resulting matrix in step 6. matrix inversalambdaZ´Z+I=invsym(lambdaZ´Z+I) Step 8: Enter the time series mkmat serie matlist Step 9: Finally multiply the vector of the time series estimated by the matrix in step 7. Outcome: Trend of the time series. Note: The cycle component is obtained by subtracting the trend component in original time series. Miguel Ángel Ramírez Hernández 15 I extracted the real effective exchange rate of Norway (REER) for a quarterly period 1980-I to 2008-III. Objective: To analyze the changes in the balance of goods and non-factor services. (REER-proxy). Miguel Ángel Ramírez Hernández 16 4.65 Norway: real effective exchange rate and HP filters with variations in the smoothing parameter, 1980-I to 2008-III. Serie LTCR lambda=1600 lambda=13.08159669 lambda=239.4300893 4.5 4.55 4.6 Alternative proposal 1980-1 1990-1 2000-1 2010-1 Source: Authors' calculations based on IMF and Central Bank of Norway (2013). Simulated data in Stata. Miguel Ángel Ramírez Hernández 17 o o o o o o o o o o Cogley, T. and Nason, J.M. (1995), “Effects of the Hodrick–Prescott filter on trend and difference stationary time series. Implications for business cycle research”, Journal of Economic Dynamics and Control, 19, 253–278. Frickey, E. (1934), “The problem of secular trend”, Review of Economics and Statistics, 16, 199–206. Frisch, R. (1933), “Propagation problems and impulse problems in dynamic economics”, in Economic Essays in Honour of Gustav Cassel, London: George Allen & Unwin, 171–205. Guay, A. y St.-Amant, P. (2005). “Do the Hodrick-Prescott and Baxter-King Filters Provide a Good Approximation of BusinessCycles. Annals of Economics and Statistics / Annales d”Économie et de Statistique, 77,133-155. Harvey, A.C. (1985), “Trends and cycles in macroeconomic time series”, Journal of Business and Economic Statistics, 3, 216–227. Hodrick, R.J. and Prescott, E.C. (1997). “Postwar US business cycles: an empirical investigation”, Journal of Money, Credit and Banking, 29, 1–16. Hooker, R.H. (1901), “Correlation of the marriage rate with trade”, Journal of the Royal Statistical Society, 64, 485–503. Kaldor, N. (1940), “A model of the trade cycle”, Economic Journal, 50, 78–92. Kitchin, J. (1923), “Cycles and trends in economic factors”, Review of Economics and Statistics, 5, 10–16. Kuznets, S. (1929), “Random events and cyclical oscillations”, Journal of the American Statistical Association, 24, 258–275. Continue… Miguel Ángel Ramírez Hernández 18 o o o o o o o o o o o Kydland, F.E. and Prescott, E.C. (1982), “Time to build and aggregate fluctuations”, Econometrica, 50, 1345–1370. Jevons, W.S. (1884). Investigations in currency and finance. London: Macmillan and Co. page 4. Long, J.B. and Plosser, C.I. (1983), “Real business cycles”, Journal of Political Economy, 91, 39–69. Lucas, R.E. (1975), “An equilibrium model of the business cycle”, Journal of Political Economy, 83, 1113–1144. Metzler, L.A. (1941), “The nature and stability of inventory cycles”, Review of Economics and Statistics, 23, 113–129. Nordhaus, W.D. (1975), “The political business cycle”, Review of Economic Studies, 42, 169–190. Persons, W.M. Indices of Business Conditions, Review of Economic Statistics (1919), pp. 5 – 107. Samuelson, P.A. (1939), “Interactions between the multiplier analysis and the principle of the accelerator”, Review of Economics and Statistics, 21, 75–78. Tinbergen, J. (1939b), Statistical Testing of Business-Cycle Theories, Volume 1I: Business Cycles in the United States of America, Geneva: League of Nations. Tinbergen, J. (1940), “On a method of statistical business-cycle research. A reply”, Economic Journal, 50, 141–154. Tinbergen, J. (1942), “Critical remarks on some business-cycle theories”, Econometrica, 10, 129–146. Miguel Ángel Ramírez Hernández 19