Introduction

In the realm of time series analysis, understanding the trend component is crucial for making accurate forecasts and informed decisions. A trend represents the general direction or tendency of a time series over an extended period. In this chapter, we will explore how to calculate the estimated trend value and utilize it to create interval forecasts for a non-stationary time series.

Accurate forecasting often requires separating the underlying trend from the noise in a time series. The trend captures the long-term movement in the data, which can be obscured by short-term fluctuations and irregularities. Estimating the trend allows us to make informed predictions about the future behavior of the time series.

Calculating the Estimated Trend Value

To calculate the estimated trend value, we employ various techniques such as linear regression, moving averages, or exponential smoothing. One common method is using linear regression to fit a trend line to the data points. The equation of the trend line provides an estimate of the trend component. For instance, consider a simple linear trend model:

$$Y_t=a+b t+e_t$$

Where:

• $Y_t$ is the value of the time series at time $t$.
• $a$ is the intercept of the trend line.
• $b$ is the slope of the trend line, indicating the rate of change over time.
• $e_t$ represents the error term at time $t$, capturing the noise or randomness.

By estimating $a$ and $b$ through regression analysis, we can calculate the trend value for any given point in the time series.

Forming an Interval Forecast

Once we have the estimated trend value, we can use it to create interval forecasts. Interval forecasts provide a range within which we expect the future values of the time series to lie. These intervals account for the uncertainty inherent in forecasting.

To form an interval forecast, we take into account both the trend component and the variability around the trend. One approach is to combine the estimated trend with a measure of variability, such as the standard deviation of the residuals from the trend line. This gives us a prediction interval that reflects the potential range of future values.

The interval forecast can be expressed as: Forecast $\pm z \times$ Standard Deviation of Residuals. Here, $z$ is the critical value from the standard normal distribution corresponding to the desired level of confidence.

Example: Consider a retail sales dataset that exhibits an increasing linear trend. By calculating the trend using linear regression and calculating the standard deviation of the residuals, we can create a 95% confidence interval forecast for future sales. This interval accounts for both the upward trend and the potential variability around it.