A Semi-Parametric Non-linear Neural Network Filter: Theory and Empirical Evidence

Abstract

In this work, we decompose a time series into trend and cycle by introducing a novel de-trending approach based on a family of semi-parametric artificial neural networks. Based on this powerful approach, we propose a relevant filter and show that the proposed trend specification is a global approximation to any arbitrary trend. Furthermore, we prove formally a famous claim by Kydland and Prescott (1981, 1997) that over long time periods, the average value of the cycles is zero. A simple procedure for the econometric estimation of the model is developed as a seven-step algorithm, which relies on standard techniques, where all relevant measures may be computed routinely. Next, using relevant DGPs, we compare and show by means of Monte Carlo simulations that our approach is superior to Hodrick–Prescott (HP) and Baxter and King (BK) regarding the generated distortionary effects and the ability to operate in various frequencies, including changes in volatility, amplitudes and phase. In fact, while keeping the structure of the model relatively simple, our approach is perfectly capable of addressing the case of stochastic trend, in the sense that the generated distortionary effects in the near unit root case are minimal and, by all means, considerably fewer than those generated by HP and BK. Application to EU15 business cycles clustering is presented and the empirical results are consistent with the rigorous theoretical framework developed in this work.

Appendix

Definition 1

(Trend set) Consider $g_{t_j } ,t\in T\subseteq {\mathbb {R}}^{+},j\in J\subseteq {\mathbb {R}}$ representing the trend of a time series $x_{t_j } \forall j\in J$, such that $g_{t_j } \in {\mathbb {R}}$, $\forall j\in J$. Without loss of generality, let $\mathop \bigcup \nolimits _{t_j } g_{t_j } =\{g_{t_j } :g_{t_j } is~the~trend~of~x_{t_j } \forall j\in J\}\subseteq {\mathbb {R}}$ be the trend set which is assumed to be closed and bounded.

Definition 2

(Time series as random variable) A time series model for the observed data $x_{t_j } ,j\in J$ is a specification of the joint distributions, or only the means and covariances, of a sequence of random variables $\left\{ {X_t } \right\} _{t\in T}$ of which {$x_{t_j } \}_{t\in T}$ is postulated to be a realization.

Definition 3

(Time series set) Consider $x_{t_j } ,j\in J$ an arbitrary macroeconomic time series, such that $x_{t_j } \in {\mathbb {R}}\forall t\in T\subseteq {\mathbb {R}}^{+}$. Without loss of generality, let $\bigcup _{j\in J} x_{t_j } \subseteq {\mathbb {R}}$ be the time series set.

Theorem 2

Proof

From Lemma 1 the trend set is compact. From Lemma 2 any function of the form $:\hbox {F}\left( \hbox {t} \right) \equiv \hbox {d}+\hbox {ct}+\mathop \sum \nolimits _{\mathrm{i=1}}^{\mathrm{N}} \hbox {a}_{\mathrm{i}} \upvarphi \left( {\hbox {b}_{\mathrm{i}} \hbox {t}} \right) ,\upvarphi _{\mathrm{i}} ,\hbox {b}_{\mathrm{i}} ,\hbox {d}\in {\mathbb {R}}, \hbox {c}\in {\mathbb {R}}-\left\{ 0 \right\} \}$ is non-constant, bounded and continuous. Then, from Theorem 1, the family: $\mathcal{F}=\{\hbox {F}\left( \hbox {t} \right) \in \hbox {C}\left( {\mathop \bigcup \nolimits _{j\in J} g_{t_i } } \right) :\hbox {F}\left( \hbox {t} \right) \equiv \hbox {d}+\hbox {ct}+\mathop \sum \nolimits _{\mathrm{i=1}}^{\mathrm{N}} \hbox {a}_{\mathrm{i}} {\upvarphi }\left( {\hbox {b}_{\mathrm{i}} \hbox {t}} \right) ,\upvarphi _{\mathrm{i}} ,\hbox {b}_{\mathrm{i}} ,\hbox {d},\hbox {c}\in {\mathbb {R}}\}$ is dense in $\hbox {C}\left( {\mathop \bigcup \nolimits _{j\in J} g_{t_j } } \right)$. $\square$

Theorem 3 (Linear time trend as degenerate form of NNF)

Proof

Let $x_{t_i } ,i\in I$ be a time series and let $\bar{m} \in \left\{ {1,..M} \right\}$. Then, $\exists \overline{\beta _{\bar{m}_i}} \in {\mathbb {R}}^{\bar{m}}$, $\overline{a_{\bar{m}_i}}\in {\mathbb {R}}^{\bar{m}}+ , \overline{a_{0 \bar{m}_i }}\in {\mathbb {R}}$ and $\overline{\delta _{\bar{m}_i }} \in {\mathbb {R}}$ such that the trend of the macroeconomic time series be given by the following expression:

$$\begin{aligned} g_t =\overline{a_{0 \bar{m}_i } }+\overline{\delta _{\bar{m}_i}}_ t+\mathop \sum \nolimits _{\mathrm{k=1}}^{\bar{m}} \overline{\alpha _k}\varphi \left( {\overline{\beta _k }t} \right) , \quad \forall i\in I \end{aligned}$$

Now, since I is a compact subset of ${\mathbb {R}}$ then it is closed and bounded and there exists $i_0 \in I$ such that $\overline{\beta _{\bar{m}_{i_0}}}=\hbox {max}\{\overline{\beta _{\bar{m}_i}}\in {\mathbb {R}}^{\bar{m}}$}. For, this $\overline{\beta _{\bar{m}_{i_0}}}$ we have that $\overline{\delta _{\bar{m}_{i_0}}} =\hbox {max}\left\{ {\overline{\delta _{\bar{m}_i}} ,\bar{m}_i \in \left\{ {1,\ldots M} \right\} } \right\}$, while the trend of this macroeconomic time series is given by the expression:

$$\begin{aligned} g_t =\overline{a_0} +\overline{\delta _{\bar{m} _{i_0 } } } t+\mathop \sum \nolimits _{\mathrm{k=1}}^{\bar{m}_{i_0}} \overline{\alpha _k } \varphi \left( {\overline{\beta _k }}t \right) , \quad i_0 \in I \end{aligned}$$

But, since: $\overline{\beta _{\bar{m}_{i_0}} }=\hbox {max}\left\{ {\overline{\beta _{\bar{m} _i}},\overline{\beta _{\bar{m}_i}}\in {\mathbb {R}}^{\bar{m}}} \right\}$, then: $\mathop \sum \nolimits _{\mathrm{k=1}}^{\bar{m}_{i_0}} \overline{\alpha _k } \varphi \left( {\overline{\beta _k }}t \right) =\mathop \sum \nolimits _{\mathrm{k=1}}^{\bar{m}_{i_0 } } \overline{\alpha _k }$ . In view of $\varphi$ being increasing monotonic and $\varphi :{\mathbb {R}}\rightarrow \left[ {0,1} \right]$, we have that: $g_t =\overline{a_0 }+ \overline{\delta _{\bar{m}_{i_0}}} t+\mathop \sum \nolimits _{\mathrm{k=1}}^{\bar{m}_{i_0}}\overline{\alpha _k }$.

Hence, the trend approximation is equal to the linear trend, i.e. $g_t =\gamma +\delta t$, where: $\gamma =\overline{a_0} +\mathop \sum \nolimits _{\mathrm{k=1}}^{\bar{m}_{i_0}} \overline{\alpha _k }$. $\square$

Theorem 4 (Mean value of the cycle is zero)

Proof

Let ${x_{{t_j}}}$, $j \in J$ be a time series whose cyclical component is given by the expression:

$$\begin{aligned} {c_{{t_j}}}={x_{{t_j}}}-{g_{{t_j}}}, \forall j \in J \hbox { and } t \in T. \end{aligned}$$

Let $\{{p_{{t_j}}}\}_{j \in J}$ be the respective probability measure assigned on each cyclical times series ${c_{{t_j}}} \forall t \in T$.

Now, provided that:$\sum \nolimits _{{\mathrm{t}\in {\mathrm{T}}}}({\hbox {x}_{{\hbox {t}_{\mathrm{j}}}}}-{\hbox {g}_{{\hbox {t}_{\mathrm{j}}}}}),{\hbox {p}_{{\hbox {t}_\mathrm{j}}}}$ converges absolutely, i.e. $\sum \nolimits _{t\in T}|({x_{{t_j}}}-{g_{{t_j}}}), {p_{{t_j}}}|< \infty$, the expected value of our cyclical component is given by the following expression:

$$\begin{aligned} E\left( {c_{t_j } } \right) =E\left( {(x_{t_j } -g_{t_j ,} )} \right) =\mathop \sum \nolimits _{t\in T} (x_{t_j } -g_{t_j ,} )p_{t_j } \forall j\in J, \quad t\in T \hbox { and } p_{t_j } \in \left[ {0,1} \right] \nonumber \\ \end{aligned}$$

(29)

But, since $\bigcup _{t_{i}}g_{t_{i}}$ is a dense subset on $\bigcup _{t_{j}}x_{t_{j}}$, then by the definition of density we have that $\forall \varepsilon >0$ and $\forall x_{t_{j} }, j \in J$ there exists $g_{t_{i}}$, $i \in I$ such that $\left| {x_{t_j } -g_{t_i }} \right| <\varepsilon \forall j\in J~and~ \forall i\in I$. Thus, $\forall \left\{ {\hbox {p}_{\mathrm{t}_{\mathrm{j}}}} \right\} _{\mathrm{j}\in \hbox {J}}$ in a relevant probability space, we have that: $|(x_{t_j }-g_{t_j })p_{t_j }|< \varepsilon p _{tj}< \varepsilon \, \forall j \in J$ and $\forall p_{t_j } \in \left[ {0,1} \right] .$

But, without loss of generality, for $\varepsilon _t =1-\frac{1}{2^{t}}$, $\forall t\in T$ we have that: $\sum \nolimits _{t \in T}({1}-\frac{1}{2^{t}})< \infty$.

Hence: $\mathop \sum \nolimits _{t\in T} \left| {(x_{t_j } -g_{t_j ,} )p_{t_j } } \right| <\infty$ (A.14) and, therefore, Eq. (29) defines the expected value of the cyclical component of the time series.

Thus:

$$\begin{aligned} \left( {c_{t_j } } \right) =E\left( {(x_{t_j } -g_{t_j ,} )} \right) =\mathop \sum \nolimits _{t\in T} (x_{t_j } -g_{t_j ,} )p_{t_j } \forall j\in J, t\in T \end{aligned}$$

(30)

But: $\mathop \sum \nolimits _{t\in T} (x_{t_j } -g_{t_j ,} )p_{t_j } =(x_{1_j } -g_{1_j ,} )p_{1_j } +\cdots +(x_{t_j } -g_{t_j ,} )p_{t_j } +\ldots$ and $(x_{1_j } -g_{1_j ,} )p_{1_j } +\cdots +(x_{t_j } -g_{t_j ,} )p_{t_j } <\varepsilon _1 p_{1_j } +\cdots +\varepsilon _T p_{T_j } +\ldots$, $\forall j\in J$, $t\in T$ because of the density of $\bigcup _{t_i } g_{t_i }$ on $\bigcup _{t_j } x_{t_j }$ which implies: $\left| {x_{t_j } -g_{t_i } } \right| <\varepsilon \forall j\in J~and~\forall i\in I$.

However:$\left| {x_{t_j } -g_{t_i } } \right|<\varepsilon _t \Leftrightarrow -\varepsilon _t<x_{t_j } -g_{t_i } <\varepsilon _t$, $\forall j\in J$,$\forall t\in T$ and $\forall i\in I$

$$\begin{aligned} -\mathop \sum \nolimits _{t\in T} \varepsilon _t p_{t_j } \le \mathop \sum \nolimits _{t\in T} \left( {x_{t_j } -g_{t_j } } \right) p_{t_j } \le \mathop \sum \nolimits _{t\in T} \varepsilon _t p_{t_j } \forall j\in J~and~ \forall p_{t_j } \in \left[ {0,1} \right] \end{aligned}$$

Now, without loss of generality, for $\varepsilon _t =1-\frac{1}{2^{t}}$,$\forall t\in T$ we have that:

$$\begin{aligned}&-\mathop \sum \nolimits _{t\in T} \left( {1-\frac{1}{2^{t}}} \right) p_{t_j } \le \mathop \sum \nolimits _{t\in T} \left( {x_{t_j } -g_{t_j } } \right) p_{t_j }\nonumber \\&\quad \le \mathop \sum \nolimits _{t\in T} \left( {1-\frac{1}{2^{t}}} \right) p_{t_j } \forall j\in J~and~ \forall p_{t_j } \in \left[ {0,1} \right] \end{aligned}$$

(31)

But, given that $1-\frac{1}{2^{t}}>0$ and $p_{tj}\in [0,1]$, we have that:

$$\begin{aligned}&\mathop \sum \nolimits _{t\in T} \left( {1-\frac{1}{2^{t}}} \right) p_{t_j } \le \mathop \sum \nolimits _{t\in T} \left( {1-\frac{1}{2^{t}}} \right) \mathop \sum \nolimits _{t\in T} p_{t_j }\nonumber \\&\quad = \mathop \sum \nolimits _{t\in T} \left( {1-\frac{1}{2^{t}}} \right) \rightarrow 0 \hbox { since }: \mathop \sum \nolimits _{t\in T} p_{t_j } =1 \end{aligned}$$

(32)

Similarly: $-\mathop \sum \nolimits _{t\in T} \left( {1-\frac{1}{2^{t}}} \right) p_{t_j } \rightarrow 0$

Hence, given that: $-\mathop \sum \nolimits _{t\in T} \varepsilon _t p_{t_j } \le \mathop \sum \nolimits _{t\in T} \left( {x_{t_j } -g_{t_j } } \right) p_{t_j } \le \mathop \sum \nolimits _{t\in T} \varepsilon _t p_{t_j } \forall j\in J$, $\forall p_{t_j } \in \left[ {0,1} \right]$ and that: $-\mathop \sum \nolimits _{t\in T} \left( {1-\frac{1}{2^{t}}} \right) p_{t_j } \rightarrow 0$ and $\mathop \sum \nolimits _{t\in T} \left( {1-\frac{1}{2^{t}}} \right) p_{t_k } \rightarrow 0$ we get:

$$\begin{aligned} E\left( {c_{t_j } } \right) =0, \forall j\in J,t\in T \hbox { and } \forall p_{t_j } \in \left[ {0,1} \right] \end{aligned}$$

(33)

$\square$

Theorem 5 (Mean value of the NNF cycle is zero)

Proof

Based on Theorem 2, $\mathcal{F}=\{F\left( t \right) \in C\left( X \right) :F\left( t \right) \equiv d+ct+\mathop \sum \nolimits _{i=1}^N a_i \varphi \, \left( {\beta _i t} \right) ,\alpha _i ,\beta _i ,d,c\in {\mathbb {R}}\}$ is dense in $\bigcup _{t_i } g_{t_i }$. Hence, in view of Theorem 4, we have that: $\left( {c_{t_j } } \right) =0\forall j\in J$. $\square$