Question

Suppose that \(\left\\{y_{t}\right\\}\) and \(\left\\{z_{t}\right\\}\) are \(\mathrm{I}(1)\) series, but \(y_{t}-\beta z_{t}\) is \(\mathrm{I}(0)\) for some \(\beta \neq 0 .\) Show that for any \(\delta \neq \beta\) \(y_{t}-\delta z_{t}\) must be I(1).

Short answer

For any \(\delta \neq \beta\), \(y_t - \delta z_t\) is \(I(1)\).

Step-by-step solution

Understanding the Problem

We have two series, \(\{ y_t \}\) and \(\{ z_t \}\), both of which are integrated of order 1, denoted as \(I(1)\). This implies that both series are non-stationary, but their differences are stationary. We're given that the combination \(y_t - \beta z_t\) is stationary (\(I(0)\)). We need to show that any other linear combination, \(y_t - \delta z_t\), where \(\delta eq \beta\), is non-stationary (\(I(1)\)).

Using Definitions of Stationary and Non-stationary Series

A series that is \(I(1)\) can be made stationary by differencing once, while \(I(0)\) means the series is already stationary. Since \(y_t - \beta z_t\) is \(I(0)\), it fluctuates around a constant mean with constant variance, indicating a cointegration relationship between \(\{ y_t \}\) and \(\{ z_t \}\) with cointegrating vector \((1, -\beta)\).

Proof by Contradiction Setup

Assume for contradiction that there exists a \(\delta eq \beta\) such that \(y_t - \delta z_t\) is \(I(0)\). This would imply that this combination is also stationary, suggesting a different cointegration vector \((1, -\delta)\).

Analyzing the Cointegration Implications

If both \(y_t - \beta z_t\) and \(y_t - \delta z_t\) were \(I(0)\), this would mean two different stationary linear combinations exist, given by \(y_t = \beta z_t + u_t\) and \(y_t = \delta z_t + v_t\), where \(u_t\) and \(v_t\) are \(I(0)\). Subtract these equations: \((\beta - \delta)z_t = v_t - u_t\).

Deriving a Contradiction

Given that \(v_t - u_t\) is a stationary series and \((\beta - \delta)z_t\) would be \(I(0)\) when \(\beta eq \delta\), it implies \(z_t\) itself would have to be \(I(0)\) (since a non-zero constant multiplied by \(I(1)\) series can't be \(I(0)\)), contradicting the assumption that \(\{ z_t \}\) is \(I(1)\).

Conclude the Contradiction

Since \(z_t\) cannot be \(I(0)\), our initial assumption that \(y_t - \delta z_t\) is \(I(0)\) must be false for \(\delta eq \beta\). Hence, \(y_t - \delta z_t\) has to be \(I(1)\).

Key concepts

Integrated series

The concept of integrated series is central in time series analysis, particularly when addressing non-stationary processes. **Integrated series** refers to time series data that become stationary after differencing a certain number of times. The order of integration, denoted as \(I(d)\), indicates how many differences are needed to transform a non-stationary series into a stationary one.
For example, if a series is \(I(1)\), it means that it is not stationary (it may exhibit trends or random walks), but differencing it once will make it stationary. Such series are essential in econometrics because they often describe economic macro-variables like GDP or inflation over time.
In the given problem, both \(\{y_t\}\) and \(\{z_t\}\) are \(I(1)\) series. While individually they are non-stationary, the existence of a linear combination of the form \(y_t - \beta z_t\) being \(I(0)\) suggests cointegration—an important property that signifies a stable, long-term relationship between \(\{y_t\}\) and \(\{z_t\}\).
This property is fundamental, allowing statisticians and economists to make meaningful interpretations of the relationships underlying non-stationary data.

Stationary process

A **stationary process** is a key concept in time series analysis, where the statistical properties of the process (such as mean, variance, and autocorrelation) are constant over time. Stationarity is important because many statistical methods rely on the data being stationary. Stationary series do not show trends, have consistent variability, and their past values inform their future values in a stable manner.
In simple terms, a stationary series fluctuates around a constant mean and its fluctuations have limited variability. When a time series is \(I(0)\), it means that it is already stationary without needing any transformation, like differencing.
In our example, \(y_t - \beta z_t\) being \(I(0)\) illustrates that this specific combination is stable and fluctuates predictably around a fixed value, suggesting a cointegrating relationship. This implies a very special case where both series, despite being individually non-stationary, are bound together in a long-term relationship.

• The benefits of stationarity include the ability to model and forecast data accurately.
• It enables analysts to draw consistent and meaningful inferences from the data over time.

Recognizing stationarity or inducing it through transformations forms the foundation of effective time series analysis.

Contradiction proof

A **contradiction proof** is a logical method used to establish the validity or falsehood of a proposition by assuming the opposite and showing that this assumption leads to a contradiction. It is a powerful tool in mathematics and theoretical proofs.
In this context, we use contradiction to explore whether another linear combination \( y_t - \delta z_t \), where \( \delta eq \beta \), could also be \( I(0) \).
By assuming \( y_t - \delta z_t \) is \( I(0) \), we imply two different cointegrating vectors exist for the same series pairs. However, analyzing the implications reveals a contradiction: Subtracting the equations \( y_t = \beta z_t + u_t \) and \( y_t = \delta z_t + v_t \) gives \((\beta - \delta)z_t = v_t - u_t\). Here, \(v_t - u_t\) is \(I(0)\), leading to the conclusion that \(z_t\) must be \(I(0)\) for this to hold, contradicting its initial specification as \(I(1)\).
This contradiction proves that no other \( \delta eq \beta \) can result in \( y_t - \delta z_t \) being \( I(0) \), reinforcing the unique nature of the cointegrating relationship at \( \beta \).
The use of contradiction proofs showcases analytical rigor in establishing theoretical results and is significant in validating concepts like cointegration in econometric studies.