The Hodrick-Prescott Filter: The Standard Approach
The HP filter is the most widely used method in macroeconomics. It finds a smooth trend by minimizing both the fit to the data and the variability of the trend.
Basic Example with Quarterly Data
Let’s start with quarterly GDP construction data:
# Apply HP filter to quarterly data
gdp_hp <- gdp_construction |>
augment_trends(
value_col = "index",
methods = "hp"
)
# View results
head(gdp_hp)
#> # A tibble: 6 × 3
#> date index trend_hp
#> <date> <dbl> <dbl>
#> 1 1995-01-01 100 101.
#> 2 1995-04-01 100 101.
#> 3 1995-07-01 100 102.
#> 4 1995-10-01 100 103.
#> 5 1996-01-01 97.8 103.
#> 6 1996-04-01 101. 104.Now visualize the trend and the cyclical component (gap):
# Calculate the cycle (deviation from trend)
gdp_hp <- gdp_hp |>
mutate(cycle = index - trend_hp)
# Plot 1: Trend vs Original
p1 <- gdp_hp |>
select(date, index, trend_hp) |>
pivot_longer(cols = c(index, trend_hp), names_to = "series") |>
mutate(series = ifelse(series == "index", "Original", "HP Trend")) |>
ggplot(aes(x = date, y = value, color = series)) +
geom_line(linewidth = 0.9) +
labs(
title = "GDP Construction: Original vs HP Trend",
subtitle = "Quarterly data with standard HP filter (λ=1600)",
x = NULL,
y = "Index",
color = NULL
) +
theme_minimal() +
theme(legend.position = "bottom")
# Plot 2: Cyclical component
p2 <- gdp_hp |>
ggplot(aes(x = date, y = cycle)) +
geom_hline(yintercept = 0, linetype = "dashed", color = "gray50") +
geom_line(linewidth = 0.9, color = "#0072B2") +
labs(
title = "Cyclical Component (Output Gap)",
subtitle = "Deviations from HP trend",
x = "Date",
y = "Gap (index points)",
color = NULL
) +
theme_minimal()
print(p1)
The cyclical component shows periods when the construction sector was above trend (booms) and below trend (recessions).
The Lambda Parameter
The HP filter has one key parameter: lambda (λ), which controls smoothness:
- Higher λ: Smoother trend, captures only long-term movements
- Lower λ: More flexible trend, follows data more closely
Standard values: - Quarterly data: λ = 1600 (Hodrick-Prescott’s original recommendation) - Monthly data: λ = 14400 (adjusted for frequency) - Annual data: λ = 100
Let’s see the effect of different lambda values:
# Test different lambda values on quarterly data
lambdas <- c(400, 1600, 6400)
gdp_lambdas <- gdp_construction
for (lambda in lambdas) {
temp <- gdp_construction |>
augment_trends(
value_col = "index",
methods = "hp",
smoothing = lambda
) |>
select(trend_hp)
names(temp) <- paste0("hp_", lambda)
gdp_lambdas <- bind_cols(gdp_lambdas, temp)
}
# Plot
gdp_lambdas |>
select(date, index, starts_with("hp_")) |>
pivot_longer(
cols = c(index, starts_with("hp_")),
names_to = "method",
values_to = "value"
) |>
mutate(
method = case_when(
method == "index" ~ "Original",
method == "hp_400" ~ "λ = 400 (flexible)",
method == "hp_1600" ~ "λ = 1600 (standard)",
method == "hp_6400" ~ "λ = 6400 (very smooth)"
)
) |>
ggplot(aes(x = date, y = value, color = method)) +
geom_line(linewidth = 0.8) +
labs(
title = "HP Filter with Different Lambda Values",
subtitle = "Quarterly GDP construction data",
x = "Date",
y = "Index",
color = NULL
) +
theme_minimal() +
theme(legend.position = "bottom")
Recommendation: Use the standard values (1600 for quarterly, 14400 for monthly) unless you have a specific reason to change them.
HP Filter on Monthly Data
The HP filter works exactly the same way on monthly data, but uses a different default lambda:
# Apply HP filter to monthly IBC-Br data
ibcbr_hp <- ibcbr |>
slice_tail(n = 120) |> # Last 10 years
augment_trends(
value_col = "index",
methods = "hp"
) |>
mutate(cycle = index - trend_hp)
# Plot trend and cycle
p1 <- ibcbr_hp |>
select(date, index, trend_hp) |>
pivot_longer(cols = c(index, trend_hp), names_to = "series") |>
mutate(series = ifelse(series == "index", "Original", "HP Trend")) |>
ggplot(aes(x = date, y = value, color = series)) +
geom_line(linewidth = 0.8) +
labs(
title = "IBC-Br Economic Activity: HP Trend",
subtitle = "Monthly data with λ=14400",
x = NULL,
y = "Index",
color = NULL
) +
theme_minimal() +
theme(legend.position = "bottom")
p2 <- ibcbr_hp |>
ggplot(aes(x = date, y = cycle)) +
geom_hline(yintercept = 0, linetype = "dashed", color = "gray50") +
geom_line(linewidth = 0.8, color = "#0072B2") +
geom_ribbon(aes(ymin = pmin(cycle, 0), ymax = 0), alpha = 0.3, fill = "#D55E00") +
geom_ribbon(aes(ymin = 0, ymax = pmax(cycle, 0)), alpha = 0.3, fill = "#009E73") +
labs(
title = "Business Cycle Component",
subtitle = "Green = above trend (expansion), Red = below trend (contraction)",
x = "Date",
y = "Gap (index points)"
) +
theme_minimal()
print(p1)
The cyclical component clearly shows the economic contraction during the pandemic and subsequent recovery.
