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Hi, my name is Dimas, I am a data enthusiast. At this moment I am writing several chapters related to Big Data, Macroprudential effect on the economy, and a couple of economic and IT research. If you are interested in collaborating, please write your mail to [email protected]

Thanks for stopping by. All the information here is curated from the most inspirational article on the site.

The Hodrick-Prescott filter separates a time series into growth and cyclical components with

\( y_t = g_t + c_t \)

yt=gt+ct

where yt is a time series, gt is the growth component of yt, and ct is the cyclical component of yt for t=1,...,T.

The objective function for the Hodrick-Prescott filter has the form

Tt=1c2t+λT−1t=2((gt+1−gt)−(gt−gt−1))2

with a smoothing parameter λ. The programming problem is to minimize the objective over all g1,...,gT.

The conceptual basis for this programming problem is that the first sum minimizes the difference between the data and its growth component (which is the cyclical component) and the second sum minimizes the second-order difference of the growth component, which is analogous to minimization of the second derivative of the growth component.

Note that this filter is equivalent to a cubic spline smoother.

Using data similar to the data found in Hodrick and Prescott [1], plot the cyclical component of GNP. This result should coincide with the results in the paper. However, since the GNP data here and in the paper are both adjusted for seasonal variations with conversion from nominal to real values, differences can be expected due to differences in the sources for the pair of adjustments. Note that our data comes from the St. Louis Federal Reserve FRED database [2], which was downloaded with the Datafeed Toolbox™.

We run the filter for different smoothing parameters λ = 400, 1600, 6400, and ∞. The infinite smoothing parameter just detrends the data.

The following code generates Figure 1 from Hodrick and Prescott [1].

The blue line is the cyclical component with smoothing parameter 1600 and the red line is the difference with respect to the detrended cyclical component. The difference is smooth enough to suggest that the choice of smoothing parameter is appropriate.

We will now reconstruct Table 1 from Hodrick and Prescott [1]. With the cyclical components, we compute standard deviations, autocorrelations for lags 1 to 10, and perform a Dickey-Fuller unit root test to assess non-stationarity. As in the article, we see that as lambda increases, standard deviations increase, autocorrelations increase over longer lags, and the unit root hypothesis is rejected for all but the detrended case. Together, these results imply that any of the cyclical series with finite smoothing is effectively stationary.

[1] Hodrick, Robert J, and Edward C. Prescott. "Postwar U.S. Business Cycles: An Empirical Investigation." *Journal of Money, Credit, and Banking*. Vol. 29, No. 1, 1997, pp. 1–16.

[2] U.S. Federal Reserve Economic Data (FRED), Federal Reserve Bank of St. Louis, `https://fred.stlouisfed.org/`

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