http://www.nber.org/cycles.html

ECO 6120 : Economic fluctuations Understanding economic fluctuations: important empirical component in the analysis Great contributions in the 1970-1990 (Granger-Newbold, 1974, « spurious regressions»; Nelson and Plosser (1982) (Unit roots); Perron (1989) structural breaks. The notion of stationarity is fundamental (Spurious regressions) 1: in general for macro-econometrics and 2 : persistence of shocks

Dependent Variable: LRY_CA Method: Least Squares Date: 10/27/03 Time: 10:41 Sample(adjusted): 1961:1 2000:1 Included observations: 157 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C 4.248894 0.096100 44.21333 0.0000 LRY_US 1.029452 0.011317 90.96445 0.0000 R-squared 0.981612 Mean dependent var 12.98271 Adjusted R-squared 0.981494 S.D. dependent var 0.375136 S.E. of regression 0.051033 Akaike info criterion -3.100038 Sum squared resid 0.403674 Schwarz criterion -3.061105 Log likelihood 245.3530 F-statistic 8274.530 Durbin-Watson stat 0.024883 Prob(F-statistic) 0.000000

Consequently: 1) you should know if the series are stationary (using unit root tests) and 2) if non-stationary, you have to make the series stationary (first differencing, HP filter) or to use cointegration.

The traditional view (artificial series) Log (Y) Qt

Unit root tests Suppose that we want to test if ln y tends to revert toward its trend For this end, Nelson and Plosser (1982) propose to test: Where εt is a mean zero disturbance uncorrelated with the term between [] If b < 0, output tends to revert toward the trend if b = 0, this is not the case (fluctuations have a permanent component).

4.54 is rewritten as: There is however an important econometric complication (Dickey et Fuller, 1979), the OLS estimate of b is biased towards negative values if chocks are persistent. The usual t test cannot be used (see the detail discussion on page 203-205).

Augmented Dickey-Fuller test on lry Null Hypothesis: LRY has a unit root Exogenous: Constant, Linear Trend Lag Length: 1 (Automatic based on SIC, MAXLAG=13) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -2.492486 0.3315 Test critical values: 1% level -4.020396 5% level -3.440059 10% level -3.144465 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(LRY) Method: Least Squares Date: 10/23/03 Time: 14:51 Sample(adjusted): 1965:3 2002:4 Included observations: 150 after adjusting endpoints Variable Coefficient Std. Error t -Statistic Prob. LRY(-1) -0.033682 0.013514 -2.492486 0.0138 D(LRY(-1)) 0.316591 0.076800 4.122292 0.0001 C 0.435908 0.171816 2.537067 0.0122 @TREND(1965:1) 0.000239 0.000107 2.233035 0.0271

Augmented Dickey-Fuller test on Dlry Null Hypothesis: D(LRY) has a unit root Exogenous: Constant Lag Length: 0 (Automatic based on SIC, MAXLAG=13) t-Statistic Prob.* Augmented Dickey-Fuller test statistic -8.513613 0.0000 Test critical values: 1% level -3.474265 5% level -2.880722 10% level -2.577077 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(LRY,2) Method: Least Squares Date: 10/28/03 Time: 12:57 Sample(adjusted): 1965:3 2002:4

The Hodrick-Prescott (HP) filter (from EViews help) This is a smoothing method that is widely used among macroeconomists to obtain a smooth estimate of the long-term trend component of a series. The method was first used in a working paper (circulated in the early 1980's and published in 1997) by Hodrick and Prescott to analyze postwar U.S. business cycles. Technically, the Hodrick-Prescott (HP) filter is a two-sided linear filter that computes the smoothed series of by minimizing the variance of around , subject to a penalty that constrains the second difference of . That is, the HP filter chooses to minimize: Where λ is the smoothing parameter or the penalty parameter that controls the smoothness of the series . (100, 1600, 14400)

(From EViews 4.0 Help) To smooth the series using the Hodrick-Prescott filter, choose Procs/Hodrick-Prescott Filter.... First, provide a name for the smoothed series. EViews will suggest a name, but you can always enter a name of your choosing. Next, specify an integer value for the smoothing parameter, . The default values in EViews are set to be: 100 for annual data, 1,600 for quarterly, and 14,400 for monthly When you click OK, EViews displays a graph of the filtered series together with the original series.

The cyclical component is LRY-HPRY

Perron P. (1989): Unit roots can be explained by the presence of one structural break in a stationary series.