Advanced Econometrics and Statistics

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1 Advanced Econometrics and Statistics Bernd Süssmuth IEW Institute for Empirical Research in Economics University of Leipzig November 11, 2010 Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

2 Outline II Macroeconometrics and Time Series Analysis II.1 Methods and Applications: Business Cyles II.1.1 Unobserved Components Model: Separating components 1 Visual Analysis 2 Detrending and di erencing bias 3 Using bandpass lters 4 Kydland-Prescott-type business cycle analysis II.1.2 Stochastic Processes: ARMA 1 Box-Jenkins in a nutshell 2 Simulating ARMAs Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

3 Visual Analysis Contents I 1 Unobserved Components Model (UCM) Visual Analysis Detrending and di erencing bias Using bandpass lters Kydland-Prescott-type business cycle analysis 2 Stochastic Processes: ARMA Box-Jenkins in a nutshell Simulating ARMAs Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

4 Visual Analysis Observed Components: National Income and Product Accounts IP (Private Investment) = II (Inventory Investment) + GFCF (Gross Fixed Capital Formation) CP (Private Consumption) CG (Governmental Consumption) PCA (Primary Current Account) = Ex (Exports) Im (Imports) = GDP (Gross Domestic Product) Note 1: In nominal terms series add up (in real terms not necessarily) Note 2: RD (Residual Demand) = CP + CG + PCA Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

5 Visual Analysis Additive UCM series = seasonal component + cyclical component + trend + rest bn $ US year Sales gures of US retailers in consecutive months Source: US Bureau of the Census Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

6 Visual Analysis Multiplicative UCM series = seasonal component cyclical component trend rest Visual indications: Volatlity increases with increasing trend Suggestive for transformation to linear model by taking logs ln series = ln seasonal component + ln cyclical component + ln trend + ln rest We abstract from the seasonal component in the following Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

7 Detrending and di erencing bias Contents I 1 Unobserved Components Model (UCM) Visual Analysis Detrending and di erencing bias Using bandpass lters Kydland-Prescott-type business cycle analysis 2 Stochastic Processes: ARMA Box-Jenkins in a nutshell Simulating ARMAs Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

8 Detrending and di erencing bias Why are rst log di erences so popular? U.R. (ADF) tests are biased towards stochastic trend model Most basic linear lter (given multiplicative UCM) First log di erences are a valid approximation for growth rates Proof of log 1 st di erences being 1 st order TSA of growth rates. Consider a Taylor series expansion and let f be a real valued function at a speci c point in time y t 1 : f (y t ) = f (y t 1 ) + f 0 (y t 1 ) (y t y t 1 ) +f 00 (y t 1 ) 1 2! (y t y t 1 ) 2 + f 000 (y t 1 ) 1 3! (y t y t 1 ) Assume second line! 0 and let f = ln y t f (y t ) f (y t 1 ) + 1 y t 1 (y t y t 1 ), ln y t ln y t 1 y t y t 1 y t 1 Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

9 Detrending and di erencing bias SIDE NOTE: Trigonometric functions Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

10 Detrending and di erencing bias Trigonometric functions (cont ed) Amplitude R j Phase φ j END OF SIDENOTE Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

11 Detrending and di erencing bias But: di erencing may distort contained dynamics In the sense that it may produce spurious cycles To see this, consider a stylized business cycle generated by 2π 2π y t = a 0 + a 1 t + b 1 cos 8 t + b 2 cos 4 t ) purely determinsitic model of double dip phenomenon ) two superimposed cycles with length of 8 and 4 periods, resp ly LogD lter is a 1 st order discrete approximation of a continuous time derivative: GR t Dy (t) y (t), where Dy (t) = d dt y (t) and GR t is denoting growth rate at time t. Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

12 Detrending and di erencing bias If we apply this formula to the double dip generator, we get GR t = a 1 2π 2π B 1 sin y t 8 t B 2 sin 4 t, where B 1 = b 12π 8y t ^ B 2 = b 22π 4y t. ) Obviously, the ratio of amplitudes of long vs. short cycle halved B 1 B 2 = b 1 2b 2 ) Short cycle is ampli ed by di erencing ) Results: less regularity, more volatility, and (possibly) spurious cycles What is to do to overcome these problems in the UCM framework? Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

13 Detrending and di erencing bias Why are rst log di erences so popular? U.R. (ADF) tests are biased towards stochastic trend model Most basic linear lter (given multiplicative UCM) First log di erences are a valid approximation for growth rates Proof of log 1 st di erences being 1 st order TSA of growth rates. Consider a Taylor series expansion and let f be a real valued function at a speci c point in time y t 1 : f (y t ) = f (y t 1 ) + f 0 (y t 1 ) (y t y t 1 ) +f 00 (y t 1 ) 1 2! (y t y t 1 ) 2 + f 000 (y t 1 ) 1 3! (y t y t 1 ) Assume second line! 0 and let f = ln y t f (y t ) f (y t 1 ) + 1 y t 1 (y t y t 1 ), ln y t ln y t 1 y t y t 1 y t 1 Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

14 Using bandpass lters Contents I 1 Unobserved Components Model (UCM) Visual Analysis Detrending and di erencing bias Using bandpass lters Kydland-Prescott-type business cycle analysis 2 Stochastic Processes: ARMA Box-Jenkins in a nutshell Simulating ARMAs Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

15 Using bandpass lters Preliminary thoughts Bandpass lters like the one of Baxter/King are widely used today... Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

16 Using bandpass lters Preliminary thoughts Bandpass lters are widely used today... Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

17 Using bandpass lters Ideal bandpass lter β (ω) = ( 1 if ω1 ω ω 2 0 else Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

18 Using bandpass lters SIDE NOTE Angular, ordinary and Nyquist frequency Note: ω j are angular (not ordinary) frequencies Angular frequency ω [radian], ordinary frequency f [time units] period length (T ) = 2π ω = 2π 2πf, where ω = 2πf, f = ω 2π Frequency is depicted on horizontal axis either [0, π] or [0, 0.5] Why? B/c highest possible freq is Nyquist freq: 2 periods length END OF SIDE NOTE Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

19 Using bandpass lters Baxter-King lters (band: 6 to 32 quarters) vs. HP [black colored line] Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

20 Using bandpass lters Bandpass lter work even if series follows a stochastic TM Most popular BP lter: Baxter-King lter, Christiano-Fitzgerald lter Alternative: double HP lter I I I lter the series using HP(0.52) ^ HP(677.13) take di erence of trend estimates How does this work? The former de nes a low-pass lter dampening the uctuations with a period smaller than 5 quarters (1.25 years), whereas the latter de nes a low-pass lter cutting o the uctuations with a period smaller than 8 years. The resulting component retains to a given extent the uctuations with a period between 5 quarters and 14 8 years, and in this respect produces estimates of the cycle that are comparable to the BK cycle although slightly noisier, without su ering from unavailability of the end of sample estimates Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

21 Kydland-Prescott-type business cycle analysis Contents I 1 Unobserved Components Model (UCM) Visual Analysis Detrending and di erencing bias Using bandpass lters Kydland-Prescott-type business cycle analysis 2 Stochastic Processes: ARMA Box-Jenkins in a nutshell Simulating ARMAs Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

22 Kydland-Prescott-type business cycle analysis Two more examples... Süssmuth and Woitek (2004), Business Cycles and Comovement in Mediterranean Economies. A National and Areawide Perspective, Emerging Markets Finance & Trade 40, 7 27 Süssmuth (2004), A note on death penalty executions and business cycles in U.S. federal states: Is there any nexus?, Economics Bulletin 11, 1-9 Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

23 Stochastic Processes: ARMA Box-Jenkins in a nutshell Contents I 1 Unobserved Components Model (UCM) Visual Analysis Detrending and di erencing bias Using bandpass lters Kydland-Prescott-type business cycle analysis 2 Stochastic Processes: ARMA Box-Jenkins in a nutshell Simulating ARMAs Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

24 Stochastic Processes: ARMA Box-Jenkins in a nutshell Central steps of the Box-Jenkins strategy: Step 1 Transform the data in such a way that they can be described by a covariance stationary process. Step 2 Parsimoniously specify a low order ARMA process Step 3 Estimate the parameters of the process Step 4 Check whether estimated process ts the data well Step 5 If Step 4 is OK, forecast; else return to Step 2 Note, habitually the Box-Jenkins approach also is concerned with stochastic trends, i.e. ARIMA models. Step 4 is routinely done with the help of the ACF.The order of ARMA models is chosen on the base of information criteria (AIC, BIC). Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

25 Stochastic Processes: ARMA Box-Jenkins in a nutshell ACF/PACF indicate which model is a good starting point ARMA(p, q) % & ACF decays (exp or sine-wave) starting at lag q PACF decays (exp or sine-wave) starting at lag p Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

26 Stochastic Processes: ARMA Simulating ARMAs Contents I 1 Unobserved Components Model (UCM) Visual Analysis Detrending and di erencing bias Using bandpass lters Kydland-Prescott-type business cycle analysis 2 Stochastic Processes: ARMA Box-Jenkins in a nutshell Simulating ARMAs Bernd Süssmuth (University of Leipzig) Advanced Econometrics November 11, / 26

Long-run trend, Business Cycle & Short-run shocks in real GDP

MPRA Munich Personal RePEc Archive Long-run trend, Business Cycle & Short-run shocks in real GDP Muhammad Farooq Arby State Bank of Pakistan September 2001 Online at http://mpra.ub.uni-muenchen.de/4929/