7.5.4 Equivalence with ARMA Models

7.5.4 Equivalence with ARMA Models

Exponential smoothing models have been shown to be equivalent to a restricted class of ARMA models (Abraham and Ledolter, 1983). Suppose the data have been differenced to obtain a covariance stationary process. Then we know from the Wold representation theorem that there exists a linear MA model for the data. Since exponential smoothing rules are also linear, it seems reasonable to expect the two representations to be related.

An example is the unobserved components model in (7.47). Consider the change in $Y_t,\mathrm{i}.\mathrm{e}.$ ,

$$\begin{array}{c}\Delta Y_t\equiv (1-L)Y_t=(1-L){\mu }_t+(1-L){\epsilon }_t\\ \\ =v_t+{\epsilon }_t-{\epsilon }_{t-1}.\end{array}$$

To find the equivalent MA representation, consider the autocorrelation function for $\Delta Y_t$ :

$$\gamma (0)=E[\Delta Y_t{]}^2=E[(v_t+{\epsilon }_t-{\epsilon }_{t-1}{)}^2]={\sigma }_v^2+2{\sigma }_{\epsilon }^2,$$

$$\begin{array}{r}\gamma (1)=E\left[\Delta Y_t\Delta Y_{t-1}\right]=E\left[(v_t+{\epsilon }_t-{\epsilon }_{t-1})(v_{t-1}+{\epsilon }_t-{\epsilon }_{t-1})\right]=-{\sigma }_s^2,\end{array}$$

$$\gamma (k)=0\mathrm{for}k\ge 2.$$

Next consider an ARIMA(0,1,1) model for $Y_t$

$$(1-L)y_t={\eta }_t+\theta {\eta }_{t-1}\equiv {\eta }_t-\tilde{\theta }{\eta }_{t-1},$$

where $\tilde{\theta }=-\theta$ . The variance and covariances of $\Delta Y_t$ are given by

$$\begin{array}{rl}& \gamma (0)={\sigma }_{\eta }^2(1+{\theta }^2),\\ & \\ & \gamma (1)=-\tilde{\theta }{\sigma }_{\eta }^2,\\ & \gamma (k)=0\mathrm{for}k\ge 2.\end{array}$$

Hence if ${\sigma }_{\eta }^2(1+{\tilde{\theta }}^2)=({\sigma }_v^2+2{\sigma }_{\epsilon }^2)$ and ${\sigma }_{\eta }^2\tilde{\theta }={\sigma }_{\epsilon }^2$ , the two representations will be identical.13 To see this, write the ARIMA(0,1,1) model in (7.52) as an infinite AR recursion:

$${\eta }_t=\sum _{j=0}^{\infty }{\tilde{\theta }}^j(y_{t-j}-y_{t-j-1})$$

$$=y_t-(1-\tilde{\theta })\sum _{j=1}^{\infty }{\tilde{\theta }}^{j-1}{y}_{t-j}.$$

Rearranging, we can express the forecast as an infinite sum of the past:

$$f_{t+1|t}=(1-\tilde{\theta })\sum _{j=1}^{\infty }{\tilde{\theta }}^{j-1}{y}_{t+1-j}.$$

Noting that we can write

$$f_{t+1|t}=(1-\tilde{\theta })[y_t+\tilde{\theta }{y}_{t-1}+{\tilde{\theta }}^2{y}_{t-2}+\cdot \cdot \cdot ],$$

$$\tilde{\theta }{f}_{t|t-1}=(1-\tilde{\theta })[\tilde{\theta }{y}_{t-1}+{\tilde{\theta }}^2{y}_{t-2}+\cdot \cdot \cdot ],$$

it follows that

$$f_{t+1|t}-\tilde{\theta }{f}_{t|t-1}=(1-\tilde{\theta })y_t,$$

or, after rearranging,

$$f_{t+1|t}=f_{t|t-1}+(1-\tilde{\theta })(y_t-f_{t|t-1}),$$

which is the same as the exponential smoothing model with $\delta =\tilde{\theta }$ . Hence, the choice of the smoothing parameter, δ, is directly related to the MA coefficient, ˜θ , which is suggestive of a method for estimating the smoothing coefficient.

7.5.5 Extensions of the Exponential Smoother

The ARIMA, unobserved component, and exponential smoothing methods described above are equivalent in the sense that they model time variation in the mean $y_t={\mu }_t+{\epsilon }_t$ and attempt to predict ${\mu }_{t+1}$ given information at time t. This turns out to be equivalent to parameterizing the mean as is done in the unobserved components model. Alternatively, we could construct a more complicated model for the forecast. Suppose such a model is

$$y_t={\mu }_t+{\beta }_t^{\prime }{x}_t+{\epsilon }_t,\text{\hspace{1em}(7.53)}$$

for a known $p\cdot$ -dimensional vector of predictors, $x_t.$ and a $p\cdot$ -dimensional vector of time-varying coefficients, ${\beta }_t$ . This setup generalizes the earlier methods. Most filtering methods set $x_t$ to be some nonstochastic process. Examples include

a. double exponential smoothing models, $x_t=t;$

b. triple exponential smoothing models, $x_t=(t,t^2/2{)}^{\prime };$

c. trigonometric models, $x_t=(\sin (2\pi t/S),\cos (2\pi t/S){)}^{\prime }$ , where S is the number of observations in a cycle $(S=4$ for quarterly data, $S=12$ for monthly data, etc.).

In each case, discounted least squares can be used to obtain estimates for ${\mu }_t$ and ${\beta }_t$ and hence generate the forecast $f_{t+1|t}={\mu }_{t+1|t}+{\beta }_{t+1|t}(t+1)$ . The updating equations that use the discount factor δ for the coefficients are

$${\mu }_{t+1|t}=(1-{\delta }^2)\sum _{j=0}^t{\delta }^j{y}_{t-j}-(1-\delta {)}^2\sum _{j=0}^t{\delta }^{j}j{y}_{t-j},$$

$${\beta }_{t+1|t}=(1-\delta {)}^2\sum _{j=0}^t{\delta }^j{y}_{t-j}-{\delta }^{-1}(1-\delta {)}^3\sum _{j=0}^t{\delta }^{j}j{y}_{t-j},\text{\hspace{1em}(7.54)}$$

see, e.g., Abraham and Ledolter (1983).

7.6 CONCLUSION

Univariate ARIMA models have long been the workhorse in applied forecasting. ARIMA models tend to work best for time series with a clearly defined and stable persistent component such as the unemployment rate or the rate of inflation. However, they do not work so well for series such as stock returns for which the evidence of a sizable persistent component in the conditional mean is much weaker.

Provided that the lag length is chosen in a reasonable way, ARMA forecasts are often difficult to outperform empirically for many macroeconomic variables that contain a persistent component. Alternatives such as exponential smoothing and unobserved components models have been proposed and are used extensively to model and predict time series with strong seasonal components such as company sales or corporate dividends.

The success of ARIMA models owes much to their simplicity of use; these models are easy to estimate and impose minimal requirements on the information set used by the forecaster. We consider the main limitations to these models—the restriction to linear models and the use of a univariate information set (the past history of the variable being predicted)—in the following two chapters.