I have many time series data at work. Some are annual, some are quarterly, and some are monthly. I am exploring Digital Signal Processing (DSP) as a complementary approach to ARIMA and other methods. As a first step I am taking Hodrick-Prescott filter (HP Filter) for trend removing and forecasting. I am using R for my work.
How do I use HP filter for forecasting? What are the drawbacks that I should be aware of? I read many positive and negative reviews of HP Filter. I would like to get a concrete suggestions.
Best Answer
Short answer: Don't use the HP Smoother for forecasting.
Long answer: For a time series $\{y_t\}_{t=1}^T$, the HP Smoother computes trend and cycle components $\{\tau_t\}_{t=1}^T$ and $\{c_t\}_{t=1}^T$ from the solution to the optimization problem: $$ \min_{\{\tau_t\}_{t=1}^T} \left(\sum_{t=1}^T(y_t - \tau_t)^2 + \lambda \sum_{t=3}^{T} (\Delta^2 \tau_t) ^2 \right) $$ where $\Delta$ is the differencing operator and $\lambda$ a tuning parameter. Say that you computed trend and cycle components with the HP Filter and now you want to forecast $y_{T+1}$, the next observation out of sample. We have $y_{T+1} = c_{T+1} + \tau_{T+1}$, so you need to forecast both trend and cycle components. The cycle component can be forecasted through a dynamic model that you presumably fitted to $\{c_t\}_{t=1}^T$. But how do you forecast the trend? There is no obvious way of extrapolating the trend component that was computed from minimizing the criterion. The only way that I have seen the HP filter used in "forecasting" was to ex post apply the HP filter to a larger dataset $\{y_t\}_{t=1}^{T+1}$, then extract $\tau_{T+1}$ and $c_{T+1}$. This is however not a forecast since it requires data up to $T+1$. Besides, the in-sample trend and cycle components $\{\tau_t\}_{t=1}^T$ and $\{c_t\}_{t=1}^T$ that you get from applying the HP Smoother in $T+1$ are not the same as you would get when applying it in $T$, so this iterative application of the Smoother really does not make sense.
I also recommend this source.