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“ ” “The alternative of a smoother parameter in the Hodrick-Prescott filter” Speaker: Miguel Ángel Ramírez Hernández

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Time Series: Retrospective, definition and components. The Hodrick-Prescott filter (1997) and criticism of the smoothing parameter (λ). Proposal, simulation and empirical evidence. References. 2 Miguel Ángel Ramírez Hernández

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“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 1 Jevons, W.S. (1884). Investigations in currency and finance. London: Macmillan and Co. page 4. 3 Miguel Ángel Ramírez Hernández

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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. 4 Miguel Ángel Ramírez Hernández

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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 in time series decomposition: trend-cycle. Harvey (1985) and Hodrick-Prescott (1997). Extensive use in RBC models. 5 Miguel Ángel Ramírez Hernández

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Time series definition Probability space: “y” is a random variable, real function defined in such that for every real number “a” Thus, for each “a”: Therefore, a random vector or vector of random variables of dimension K is a function “y" of in Euclidean space R k 6 Miguel Ángel Ramírez Hernández

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Distribution function “y” Assuming an index set Z that contains non-negative integers, discrete stochastic process is a real function: 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. 7 Miguel Ángel Ramírez Hernández

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Persons (1919) identified four components of the time series: 1) 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). 8 Miguel Ángel Ramírez Hernández

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Define a time series y t as the sum of a component of "growth" and a cyclical component : Optimization process particular: minimizing the variance of cyclical component and the variance of trend "second difference of serie". Smoothing parameter: λ Source: Authors' calculations based on Hodrick and Prescott (1997). 9 Miguel Ángel Ramírez Hernández

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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. 10 Miguel Ángel Ramírez Hernández

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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". Where: 11 Miguel Ángel Ramírez Hernández

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Matrix estimation In terms of Hodrick-Prescott (1997) If the smoothing parameter is non-negative, i.e. λ> 0, the breakdown of the series is obtained by minimizing the weighted sum of squares with respect to : Note: Stata incorporates hprescott command. 12 Miguel Ángel Ramírez Hernández

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The unique solution of the minimization is defined as: Where denotes a particular matrix, dimension and “I” denotes the identity matrix. 13 Miguel Ángel Ramírez Hernández

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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 14 Miguel Ángel Ramírez Hernández

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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. 15 Miguel Ángel Ramírez Hernández

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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). 16 Miguel Ángel Ramírez Hernández

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Norway: real effective exchange rate and HP filters with variations in the smoothing parameter, 1980-I to 2008-III. Source: Authors' calculations based on IMF and Central Bank of Norway (2013). Simulated data in Stata. Alternative proposal 17 Miguel Ángel Ramírez Hernández

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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. o Frickey, E. (1934), “The problem of secular trend”, Review of Economics and Statistics, 16, 199–206. o 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. o 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. o Harvey, A.C. (1985), “Trends and cycles in macroeconomic time series”, Journal of Business and Economic Statistics, 3, 216–227. o Hodrick, R.J. and Prescott, E.C. (1997). “Postwar US business cycles: an empirical investigation”, Journal of Money, Credit and Banking, 29, 1–16. o Hooker, R.H. (1901), “Correlation of the marriage rate with trade”, Journal of the Royal Statistical Society, 64, 485–503. o Kaldor, N. (1940), “A model of the trade cycle”, Economic Journal, 50, 78–92. o Kitchin, J. (1923), “Cycles and trends in economic factors”, Review of Economics and Statistics, 5, 10–16. o Kuznets, S. (1929), “Random events and cyclical oscillations”, Journal of the American Statistical Association, 24, 258–275. Continue… 18 Miguel Ángel Ramírez Hernández

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o Kydland, F.E. and Prescott, E.C. (1982), “Time to build and aggregate fluctuations”, Econometrica, 50, 1345–1370. o Jevons, W.S. (1884). Investigations in currency and finance. London: Macmillan and Co. page 4. o Long, J.B. and Plosser, C.I. (1983), “Real business cycles”, Journal of Political Economy, 91, 39–69. o Lucas, R.E. (1975), “An equilibrium model of the business cycle”, Journal of Political Economy, 83, 1113–1144. o Metzler, L.A. (1941), “The nature and stability of inventory cycles”, Review of Economics and Statistics, 23, 113–129. o Nordhaus, W.D. (1975), “The political business cycle”, Review of Economic Studies, 42, 169–190. o Persons, W.M. Indices of Business Conditions, Review of Economic Statistics (1919), pp. 5 – 107. o Samuelson, P.A. (1939), “Interactions between the multiplier analysis and the principle of the accelerator”, Review of Economics and Statistics, 21, 75–78. o Tinbergen, J. (1939b), Statistical Testing of Business-Cycle Theories, Volume 1I: Business Cycles in the United States of America, Geneva: League of Nations. o Tinbergen, J. (1940), “On a method of statistical business-cycle research. A reply”, Economic Journal, 50, 141–154. o Tinbergen, J. (1942), “Critical remarks on some business-cycle theories”, Econometrica, 10, 129–146. 19 Miguel Ángel Ramírez Hernández

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