BUS606: Operations and Supply Chain Management

The concept of trend is used in economics and other sciences to represent long-term smooth variations. The causes of these variations are often associated with structural phenomena such as population growth, technological progress, capital accumulation, new practices of business and economic organization. For most economic time series, the trends evolve smoothly and gradually, whether in a deterministic or stochastic manner. When there is sudden change of level and/or slope this is referred to as a structural change. It should be noticed however that series at a higher levels of aggregation are less susceptible to structural changes. For example, a technological change is more likely to produce a structural change for some firms than for the whole industry.

The identification and estimation of the secular or long-term trend have posed serious challenges to statisticians. The problem is not of statistical or mathematical character but originates from the fact that the trend is a latent (nonobservable) component and its definition as a long-term smooth movement is statistically vague. The concept of long-period is relative, since a trend estimated for a given series may turn out to be just a long business cycle as more years of data become available. To avoid this problem statisticians have used two simple solutions. One is to estimate the trend and the business cycles jointly, calling it the trend-cycle. The other solution is to estimate the trend over the whole series, and to refer to it as the longest non-periodic variation.

It should be kept in mind that many systems of time series are redefined every fifteen years or so in order to maintain relevance. Hence, the concept of longterm trend loses importance. For example, in Canada, the system of Retail and Wholesale Trade series was redefined in 1989 to adopt the 1980 Standard Industrial Classification (SIC), and again in 2003 to conform to the North American Industrial Classification System (NAICS), following the North American Free Trade Agreement. The following examples illustrate the necessity of such reclassifications. The 1970 Standard Industrial Classification (SIC) considered computers as business machines, e.g. cash registers, desk calculators. The 1980 SIC rectified the situation by creating a class for computers and other goods and services. The last few decades witnessed the birth of new industries involved in photonics (lasers), bio-engineering, nano-technology, electronic commerce. In the process, new professions emerged, and Classification systems had to keep up with these new realities.

There is a large number of deterministic and stochastic models which have been proposed for trend estimation.

Deterministic models are based on the assumption that the trend can be well approximated by mathematical functions of time such as polynomials of low degree, cubic splines, logistic functions, Gompertz curves, modified exponentials. Stochastic trends models assume that the trend can be better modelled by differences of low order together with autoregressive and moving average terms. Stochastic trend models are appropriate when the trend is assumed to follow a nonstationary stochastic process where the non-stationarity is modelled with finite differences of low order.

A typical stochastic trend model often used in structural time series modelling, is the so-called random walk with constant drift. In the classical notation this model is

$\mu_{t}=\mu_{t-1}+\beta+\xi_{t}, t=1,2, \ldots n ; \xi_{t}-N\left(0, \sigma_{\xi}^{2}\right)$                     (8.a)

$\Delta \mu_{t}=\beta+\xi_{t}$, where $\mu_{t}$ denotes the trend, $\beta$ a constant drift and $\left\{\xi_{t}\right\}$ is a normal white noise process. Solving the difference equation (2.15a) and assuming $\xi_{0}=0$ , we obtain

$\mu_{t}=\beta t+\Delta^{-1} \xi_{t}=\beta t+\Sigma_{j=0}^{\infty} \xi_{t-j}, t=1, \ldots, n$                         (8.b)

which show that a random walk with constant drift consists of a linear deterministic trend plus a non-stationary infinite moving average.

Another type of stochastic trend belongs to the ARIMA $(p, d, q)$ class, where $p$ is the order of the autoregressive polynomial, $q$ is the order of the moving average polynomial and $d$ the order of the finite difference operator $\Delta=(1-B)$. The backshift operator $B$ is such that $B^{n} z_{t} \equiv z_{t-n}$. The ARIMA $(p, d, q)$ model is written as

$\phi_{p}(B)(1-B)^{d} z_{t}=\theta_{q}(B) a_{t}, a_{t} \sim N\left(0, \sigma_{a}^{2}\right)$                              (9)

where $z_{t}$ now denotes the trend, $\phi_{p}(B)$ the autoregressive polynomial in $B$ of order $p, \quad \theta_{q}(B)$ stands for the moving average polynomial in $B$ of order $q$, and $\left\{a_{t}\right\}$ denotes the innovations assumed to follow a normal white noise process. For example, with $p=1$, $d=2$, $q=0$, model (9) becomes

$\left(1-\phi_{1} B\right)(1-B)^{2} z_{t}=a_{t}$                                                           (10)

where $z_{t}$ now denotes the trend, $\phi_{p}(B)$ the autoregressive polynomial in $B$ of order $p, \quad \theta_{q}(B)$ stands for the moving average polynomial in $B$ of order $q$, and{a_{t}\right\}\) denotes the innovations assumed to follow a normal white noise process. For example, with $p=1$, $d=2$, $q=0$,, model (9) becomes

$\left(1-\phi_{1} B\right)(1-B)^{2} z_{t}=a_{t}$                                                          (11)

which means that after applying first order differences twice, the transformed series can be modelled by an autoregressive process of order one.

The Hodrick & Prescott filter follows the cubic smoothing spline approach. The framework used in Hodr& Prescott is that a given time series $X$ is the sum of a growth component $T$ and a cyclical component $C$ such that $X = T + C$. The measure of the smoothness of the trend T is the sum of the squares of its second order difference. The $C$ are deviations from $T$ and the conceptual framework is that over long time periods, their average is near zero.

The Hodrick-Prescott (HP) filter was not developed to be appropriate, much less optimal, for specific time series generating processes. Rather, apart from the choice of the smoothing parameter λ, the same filter is intended to be applied to all series. Nevertheless, the smoother that results can be viewed in terms of optimal signal extraction literature pioneered by Wiener and Whittle  and extended by Bell to incorporate integrated time series generating processes. King & Rebelo and Ehglen  analyzed the HP filter in this framework, motivating it as a generalization of the exponential smoothing filter. On the other hand, Kaiser& Maravall showed that under certain restriction the HP filter can be well approximated by an Integrated Moving Average model of order 2, whereas Harvey & Jaeger interpreted the HP filter in terms of structural time series models.