Question

Let \(\left\\{y_{t}\right\\}\) be an \(\mathrm{I}(1)\) sequence. Suppose that \(\hat{g}_{n}\) is the one-step-ahead forecast of \(\Delta y_{n+1}\) and let \(\hat{f}_{n}=\hat{g}_{n}+y_{n}\) be the one-step-ahead forecast of \(y_{n+1} .\) Explain why the forecast errors for forecasting \(\Delta y_{n+1}\) and \(y_{n+1}\) are identical.

Short answer

The forecast errors for \( \Delta y_{n+1} \) and \( y_{n+1} \) are identical because both simplify to \( y_{n+1} - \hat{g}_n - y_n \).

Step-by-step solution

Understand the components

To solve this, let's first understand the components given. We have a series \( \{ y_t \} \) which is \( \mathrm{I}(1) \), meaning it is integrated of order 1. This implies that the series \( \Delta y_t = y_t - y_{t-1} \) is stationary. The term \( \hat{g}_n \) is the one-step-ahead forecast of \( \Delta y_{n+1} \), and \( \hat{f}_n = \hat{g}_n + y_n \) is the one-step-ahead forecast of \( y_{n+1} \).

Forecast error of \( \Delta y_{n+1} \)

The forecast error, \( e_n^{\Delta} \), for \( \Delta y_{n+1} \) is given by the actual value minus the forecasted value:\[e_n^{\Delta} = \Delta y_{n+1} - \hat{g}_n.\]Substitute for \( \Delta y_{n+1} \):\[e_n^{\Delta} = (y_{n+1} - y_n) - \hat{g}_n.\]

Forecast error of \( y_{n+1} \)

Similarly, the forecast error, \( e_n^y \), for \( y_{n+1} \) is the actual value minus the forecasted value:\[e_n^y = y_{n+1} - \hat{f}_n.\]Given \( \hat{f}_n = \hat{g}_n + y_n \), substitute \( \hat{f}_n \):\[e_n^y = y_{n+1} - (\hat{g}_n + y_n).\]Simplifying gives:\[e_n^y = y_{n+1} - \hat{g}_n - y_n.\]

Compare forecast errors

We now compare the two expressions for the forecast errors:1. For \( \Delta y_{n+1} \):\[e_n^{\Delta} = y_{n+1} - \hat{g}_n - y_n.\]2. For \( y_{n+1} \):\[e_n^y = y_{n+1} - \hat{g}_n - y_n.\]Both expressions are identical, hence the forecast errors are the same.

Key concepts

Time Series Analysis

Time series analysis is a statistical technique used to model and analyze sequences of data points collected over time. It is particularly useful when studying patterns such as trends, cycles, and seasonal variations.
Time series data is often used in fields like economics, finance, and meteorology.In the context of the original exercise, we are dealing with a time series represented by \( \{y_t\} \). An important characteristic of this series is that it is integrated of order 1, or \( I(1) \). This implies that the actual data is non-stationary, meaning its statistical properties, like mean and variance, change over time.
However, when the difference between consecutive data points, \( \Delta y_t = y_t - y_{t-1} \), is taken, the resulting series is stationary.Key concepts in time series analysis include:

• **Stationarity**: A stationary series has a constant mean, variance, and autocorrelation over time.
• **Integration**: Refers to the order, \( d \), necessary to transform a non-stationary series into a stationary one through differencing.
• **Autocorrelation**: Measures how a time series is correlated with its past values.

Understanding these concepts helps in building robust statistical models for time-dependent data, which is crucial for forecasting.

Forecasting

Forecasting is the process of making predictions about future values based on observed data. In time series analysis, forecasting involves the development of models that capture the underlying patterns and use them to predict future events.
In the exercise, the aim is to forecast \( y_{n+1} \), the next value in the time series, and \( \Delta y_{n+1} \), its difference with the current value. The one-step-ahead forecasts, \( \hat{g}_n \) for \( \Delta y_{n+1} \) and \( \hat{f}_n = \hat{g}_n + y_n \) for \( y_{n+1} \), incorporate information at time \( n \).Forecasting involves:

• **Choosing a Model**: Selecting an appropriate model such as ARIMA that fits the time series data.
• **Estimating Parameters**: Using historical data to estimate the parameters for the chosen model.
• **Calculating Forecasts**: Deriving forecast values based on the model and parameters.
• **Evaluating Accuracy**: Comparing forecasted values against actual outcomes using approaches like measuring forecast errors.

In the provided exercise, both forecast errors \( e_n^{\Delta} \) and \( e_n^y \) are calculated and found to be the same. This shows that using the same forecast base \( \hat{g}_n \) leads to identical errors for both the change and level of the series at the next time step.

Integrated Processes

Integrated processes denote a class of time series models where the current value is derived from the values at previous time points. They are key in transforming non-stationary time series into stationary ones by differencing.A time series is said to be integrated of order \( d \), noted as \( I(d) \), if differencing the series \( d \) times results in a stationary series.
The series \( \{y_t\} \) in the exercise is \( I(1) \), indicating it needs one such differencing to attain stationarity.Characteristics of integrated processes include:

• **Random Walk**: An \( I(1) \) process often resembles a random walk, where successive differences are stationary.
• **Stochastic Trend**: Non-stationary components that persist over time and can be removed by differencing.
• **Long Memory**: Situations where past values significantly influence future values for an extended period.

Understanding integrated processes is crucial because they allow for identifying and modeling time series that exhibit long-term structures. This aids in robust forecasting, as these processes can properly account for the persistence and changes in the underlying trends of the data.