Tools and techniques for detrending time series data in macroeconomic analysis. Use when working with economic time series that need to be decomposed into trend and cyclical components. Covers HP filter, log transformations for growth series, and correlation analysis of business cycles.
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This skill provides guidance on decomposing economic time series into trend and cyclical components, a fundamental technique in business cycle analysis.
Economic time series like GDP, consumption, and investment contain both long-term trends and short-term fluctuations (business cycles). Separating these components is essential for:
The HP filter is the most widely used method for detrending macroeconomic data. It decomposes a time series into a trend component and a cyclical component.
Given a time series $y_t$, the HP filter finds the trend $\tau_t$ that minimizes:
$$\sum_{t=1}^{T}(y_t - \tau_t)^2 + \lambda \sum_{t=2}^{T-1}[(\tau_{t+1} - \tau_t) - (\tau_t - \tau_{t-1})]^2$$
Where:
Critical: The choice of λ depends on data frequency:
| Data Frequency | Recommended λ | Rationale |
|---|---|---|
| Annual | 100 | Standard for yearly data |
| Quarterly | 1600 | Hodrick-Prescott (1997) recommendation |
| Monthly | 14400 | Ravn-Uhlig (2002) adjustment |
Common mistake: Using λ=1600 (quarterly default) for annual data produces an overly smooth trend that misses important cyclical dynamics.
from statsmodels.tsa.filters.hp_filter import hpfilter
import numpy as np
# Apply HP filter
# Returns: (cyclical_component, trend_component)
cycle, trend = hpfilter(data, lamb=100) # For annual data
# For quarterly data
cycle_q, trend_q = hpfilter(quarterly_data, lamb=1600)
Important: The function parameter is lamb (not lambda, which is a Python keyword).
For most macroeconomic aggregates (GDP, consumption, investment), you should apply the natural logarithm before filtering:
import numpy as np
# Apply log transformation BEFORE HP filtering
log_series = np.log(real_series)
cycle, trend = hpfilter(log_series, lamb=100)
# The cycle now represents percentage deviations from trend
# e.g., cycle = 0.02 means 2% above trend
import pandas as pd
import numpy as np
from statsmodels.tsa.filters.hp_filter import hpfilter
# Load real (inflation-adjusted) data
real_consumption = pd.Series(...) # Real consumption expenditure
real_investment = pd.Series(...) # Real fixed investment
# Log transformation
ln_consumption = np.log(real_consumption)
ln_investment = np.log(real_investment)
# HP filter with λ=100 for annual data
cycle_c, trend_c = hpfilter(ln_consumption, lamb=100)
cycle_i, trend_i = hpfilter(ln_investment, lamb=100)
# Compute correlation of cyclical components
correlation = np.corrcoef(cycle_c, cycle_i)[0, 1]
print(f"Business cycle correlation: {correlation:.4f}")
Ensure these packages are installed:
pip install statsmodels pandas numpy
The HP filter is in statsmodels.tsa.filters.hp_filter.
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CLI to manage emails via IMAP/SMTP. Use `himalaya` to list, read, write, reply, forward, search, and organize emails from the terminal. Supports multiple accounts and message composition with MML (MIME Meta Language).