Understanding the HP Filter

The HP Filter has two objectives, with the importance of each objective denoted by the user given value of lambda:

Objective 1: minimize the $\tau_t$ in the term in the square brackets such that we minimize the changes in the estimated growth rate over time.

Objective 2: We want to bring the $\tau_t$ to be as close as possible to $y_t$ to minimize the first sum in the equation.

What I am failing to understand is why these acts of minimization will help us discover what the cyclical component of GDP is? Is there a fundamental gap in my knowledge?

• I don't think the purpose of the filter is to discover a cyclical component. – Michael Greinecker Apr 5 '18 at 11:47
• It's to separate the trend and cyclical components, no? – Joseph Apr 13 '18 at 6:49

The function $\tau$ is the 'line of best fit', the growth trend. The 'residual' $y-\tau$ is the output gap. So, $\tau$ detrends $y$ over time, to give the detrended business cycle $y-\tau$. $\quad y-\tau$ fluctuates around 0—it is positive when the output gap is positive and the economy is in a boom, and negative when the economy is in recession.
The first term makes the line of best fit follow the points in a more curvy way than just a straight line; the second term makes the line of best fit straighter (the higher $\lambda$ is).