Henderson Filter

Motivation

• What is the kind of Henderson filter?

• What was the original idea of the filter?

• What is the set up of model?

Model setting

Let \(\{ X_t\}, t \in \mathbb{Z}\) be a time series process ,which is \(X_t = m_t + u_t\),where \(m_t\) is a nonlinear trend function.

Then we can apply a linear filter which leads to smooth trend estimates \(\{ Y_t\}, t \in \mathbb{Z}\).

So the Henderson filter is defined as

\[H=\sum_{i=-k}^{k}(\Delta^3\theta_i)^2\]

such that \(Y_t=\displaystyle\sum_{i=-k}^{k}(\Delta^3\theta_i)^2X_{t+i}\)

Parameter minimizing

The Henderson moving average of order 2k+1 will therefore be the solution by minimizing (OLS Estimation):\[\displaystyle\min_{\theta} \sum_{i} (\Delta^3 \theta_i)^2\]

with the restrictions[1]

\[ \sum_{i=-k}^{k} \theta_i = 1, \quad \sum_{i=-k}^{k} i \theta_i = 0, \quad \sum_{i=-k}^{k} i^2 \theta_i = 0 \]

Why those restrictions?

The Coefficients

Computation of the weights

The coefficients of these moving averages may also be calculated explicitly and, for an order 2k+1 average by positing n=k+2 we have:\[\theta_i = \frac{315[(n-1)^2 - i^2][n^2 - i^2][(n+1)^2 - i^2][3n^3 - 16 - 11i^2]}{8n(n^2-1)(4n^2-1)(4n^2-9)(4n^2-25)}\] [1]

Which is used in X-11 and how ?[2]

Henderson filters

• Widely used in the seasonal adjustment methods ,e.g X-11

• X-11:Filter based methods of seasonal adjustment.[3]

• Seasonal adjustment process is divided into two parts.

  • Initial Cleaning: clean the series from non-linearities,mainly outliers and calendar effects

  • Calculating Trends and Components by an enhanced version of the X-11 algorithm(Henderson filters)

• The length of the Henderson

  • Automatically selected between a 9-term, 13-term, or 23-term moving average based on the I/C ratio

13-point and 23-point for comparison

[1] 3.38589

[1] 13