## 16. Predictions with Time-Series Data

*Time series data* involves measurements that are ordered over time, in which the sequence of observations is important. Most procedures for data analysis cannot be used for this data, as these procedures are based on the assumption that the errors are independent. Thus, different forms of analysis are needed.

The main goal of analysing time-series data is to make predictions. An important assumption here is that the relations between variables remain constant.

## 16.1. Time-Series Components

Most time-series have the following four components:

*Trend component*: Values grow or decrease steadily over long periods of time.*Seasonality component*: An oscillatory patterns that is specific per season (quarter year) repeats itself.*Cyclical component*: And oscillatory or cyclical pattern that is not related to seasonal behaviour.*Irregular component*: No pattern is regular enough to only exist through these predictable trends; each series of data will also have irregular components (similar to the random error term).

Analysis of time-series data involves constructing a formal model in which most of these components are explicitly or implicitly present, in order to describe the behaviour of the data series. In building this model the series components can either be regarded as being fixed over time, or as steadily evolving over time.

## 16.2. Moving Averages

*Moving averages *are the basis for many practical adjustment procedures. It can be used to remove the irregular component or smooth seasonal component:

*Removing the irregular component*: This is done by replacing each observation with the average of itself and its neighbours. The theory is that this will decrease the effect of the irregular component on each data point.*Smoothing the seasonal component*: This is done by producing four-period moving averages in such a manner that the seasonal values become one single seasonal moving average. This does mean that the values have shifted in time (in comparison to the original series), but this can be corrected by centring the averages. The specific procedure always depends on the amount of stability the pattern is assumed to have, and whether seasonality is thought to be additive or multiplicative (in the latter case: use logarithms).

If there is an assumption of a stable seasonal pattern a further seasonal-adjustment approach can be used: the*seasonal index method*. Here the original series is expressed as a percentage of the centred 4-point moving average series.

Additionally moving averages are very suitable for detecting cyclical components and/or trends.

## 16.3. Predictions using smoothing

There are a various prediction methods, and the choice you make should always depend on the resources, the objectives, and the available data.

*Simple exponential smoothing* is a more basic prediction method that is appropriate when the series is non-seasonal and has no consistent trends. It predicts future values on the basis of an estimate of the current level of the time series. This estimate is comprised of a weighted average of current and past values, where most weight is given to the most recent observations (with decreasing weight the older the observation is).

The smoothed series is then *^x*, with *^x _{t} = (1 – α)^x_{t-1} + αx_{t}*. Where

*t*signifies t the moment in the time series, and α is the

*smoothing constant*. The smoothing constant is a value between 0 and 1 and is different per situation. It is possible to rely on experience or judgment to choose this value, or to try several different values and see which is more successful.

The *Holt-Winters exponential smoothing procedure* is a more advanced prediction method that allows for trend. It functions just like the simple exponential smoothing procedure, but with the added variable for the trend estimate *T _{t-1}*.

An extension of this method also allows for seasonality. This is done by using a set of recursive estimates from the time-series. For this a level factor (α), a trend factor (β) and a multiplicative seasonal factor (γ) are used.

## 16.4. Predictions using Auto-Regression

The procedure of *autoregressive models* uses the available time-series data to estimate the parameters of a model of the process that could have generated the time series. This is based on *autocorrelation*, correlation patterns between adjacent periods. The model that is formed by this is: *x _{t} = γ +φ_{1}x_{t-1} + ε_{t}*. Where

*γ*and

*φ*are fixed parameters. The parameter γ allows for rhe mean of the series

_{1}*x*to be other than 0. The random variables

_{t}*ε*have a mean of 0, fixed parameters and are not correlated with each other.

_{t}This is called a *first-order autoregressive model*. It is possible to extend this model by making the current value of the series dependent on the two most recent observations, this is then called a *second-order autoregressive model*.

## 16.5. The Box-Jenkins approach

It is good to briefly mention the *Box-Jenkins approach* to predictions in time-series data. In this procedure one (1) defines a broad class of models for predictions, and then (2) develop a methodology for picking a suitable model on the basis of the characteristics of the available data. This has three general stages:

- Selecting a specific model that might be appropriate, based on summary statistics.
- Estimated the unknown coefficients in this model.
- Applying checks to determine whether the model adequately represents the available data.

This approach is useful due to its flexibility.

A general model class that can be used here is that of *autoregressive integrated moving average models* (*ARIMA models*).

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