Question

Suppose the process \(\left\\{\left(x_{t}, y_{t}\right): t=0,1,2, \ldots\right\\}\) satisfies the equations $$y_{t}=\beta x_{t}+u_{t}$$ and $$\Delta x_{t}=\gamma \Delta x_{t-1}+v_{t}$$ where \(\mathrm{E}\left(u_{t} | I_{t-1}\right)=\mathrm{E}\left(v_{t} | I_{t-1}\right)=0, I_{t-1}\) contains information on \(x\) and \(y\) dated at time \(t-1\) and earlier, \(\beta \neq 0,\) and \(|\gamma|<1\left[\text { so that } x_{t}, \text { and therefore } y_{t}, \text { is } I(1)\right] .\) Show that these two equations imply an error correction model of the form $$\Delta y_{t}=\gamma_{1} \Delta x_{t-1}+\delta\left(y_{t-1}-\beta x_{t-1}\right)+e_{t}$$ where \(\gamma_{1}=\beta \gamma, \delta=-1,\) and \(e_{t}=u_{t}+\beta v_{t}\). (Hint: First subtract \(y_{t-1}\) from both sides of the first equation. Then, add and subtract \(\beta x_{t-1}\) from the right-hand side and rearrange. Finally, use the second equation to get the error correction model that contains \(\Delta x_{t-1}\).)

Short answer

The given equations imply the error correction model: \(\Delta y_{t} = \gamma_{1} \Delta x_{t-1} + \delta (y_{t-1} - \beta x_{t-1}) + e_{t}\), with \(\gamma_{1} = \beta \gamma\), \(\delta = -1\), and \(e_{t} = u_{t} + \beta v_{t}\).

Step-by-step solution

Express Difference Equation

Start by subtracting \(y_{t-1}\) from both sides of the equation \(y_{t} = \beta x_{t} + u_{t}\). This gives us:\[y_{t} - y_{t-1} = \beta x_{t} + u_{t} - y_{t-1}\]This expression represents the change in \(y\), or \(\Delta y_{t}\).

Add and Subtract \(\beta x_{t-1}\)

Next, add and subtract \(\beta x_{t-1}\) from the right side of the equation:\[\Delta y_{t} = \beta x_{t} + u_{t} - \beta x_{t-1} + \beta x_{t-1} - y_{t-1}\]Rearrange terms:\[\Delta y_{t} = \beta (x_{t} - x_{t-1}) + (u_{t} - (y_{t-1} - \beta x_{t-1}))\]

Introduce \(\Delta x_{t}\)

Recognize that \(x_{t} - x_{t-1}\) is \(\Delta x_{t}\):\[\Delta y_{t} = \beta \Delta x_{t} + (u_{t} - (y_{t-1} - \beta x_{t-1}))\]This separates the equation into a term involving the change in \(x\) and an error correction term.

Substitute \(\Delta x_{t}\) Using Second Equation

From \(\Delta x_{t} = \gamma \Delta x_{t-1} + v_{t}\), substitute into the equation:\[\Delta y_{t} = \beta (\gamma \Delta x_{t-1} + v_{t}) + (u_{t} - (y_{t-1} - \beta x_{t-1}))\]This yields:\[\Delta y_{t} = \beta \gamma \Delta x_{t-1} + \beta v_{t} + u_{t} - (y_{t-1} - \beta x_{t-1})\]

Introduce Error Correction and Constants

Now, recognize the error correction term as \((y_{t-1} - \beta x_{t-1})\):\[\Delta y_{t} = \beta \gamma \Delta x_{t-1} + \beta v_{t} + u_{t} - (y_{t-1} - \beta x_{t-1})\]If we set \(\gamma_{1} = \beta \gamma\), \(\delta = -1\), and \(e_{t} = u_{t} + \beta v_{t}\), the equation becomes:\[\Delta y_{t} = \gamma_{1} \Delta x_{t-1} + \delta(y_{t-1} - \beta x_{t-1}) + e_{t}\]This matches the error correction model form.

Key concepts

Error Correction Model

An error correction model (ECM) is a tool used in econometrics to describe the long-term relationship between time series variables, while accounting for short-term deviations. It's handy when working with data that is non-stationary but have a cointegrated relationship. In our case, the original exercise provides an equation that transforms into an ECM format.
This model helps in understanding how variables return to equilibrium after a short-term shock. For example, if two economic indicators are known to move together, but one suddenly changes due to an external factor, the ECM shows how quickly they will return to their long-term equilibrium relationship.

• ECMs are essential when dealing with economic data because such data are often subject to short-term fluctuations.
• They efficiently combine short-term dynamics with long-term equilibrium conditions.
• The key component in ECM is the error correction term, representing the speed at which a dependent variable returns to equilibrium after a change in other variables.

In our solution, the ECM derived helps to show how changes in time series are adjusted through the equation \(\Delta y_{t}=\gamma_{1} \Delta x_{t-1}+\delta\left(y_{t-1}-\beta x_{t-1}\right)+e_{t}\) representing the relationship properly.

Difference Equations

Difference equations are a widely used mathematical tool in econometrics and time series analysis to model the discrete change of variables over time. In the original exercise, the equations for \(\Delta x_t\) and \(\Delta y_t\) are insightful examples of difference equations. They indicate changes in variables from one period to the next.

• Difference equations are essential for modeling time-dependent processes using discrete intervals.
• They help provide a clear understanding of the dynamics by which variables evolve.
• Econometric models often employ difference equations to capture trends, cycles, and other temporal effects.

An understanding of difference equations allows analysts to derive insights related to economic dynamics. For instance, in our model, solving the difference equation for \(\Delta x_t\) relates directly to changes in \(x_t\) and helps in forming the foundation for developing the error correction model.

Stationary Processes

Stationary processes are fundamental in time series analysis, referring to a time series whose statistical properties like mean and variance do not change over time. This concept is vital for model reliability and predictability.

• A stationary process should have constant mean and variance over time, and its autocorrelation should not change.
• Stationarity is crucial for ensuring that the models provide accurate predictions and reflections on trends.
• In our example, despite \(x_t\) being \(I(1)\), which suggests non-stationarity, the aim of the error correction model is to help manage and account for such non-stationary variables.

A non-stationary time series can lead to spurious results if not correctly handled. Understanding and transforming non-stationary data into a stationary form ensures powerful and reliable econometric analysis.

Time Series Analysis

Time series analysis is a statistical method employed to study datasets that extend across time intervals. It's used to identify patterns, trends, and cycles, which are pivotal for forecasting and understanding temporal dynamics of economic data.

• This analysis can disentangle different components such as seasonality, trend, and noise, providing a comprehensive understanding of time-dependent data.
• Time series analysis is employed to develop forecasting models to predict future observations based on past data.
• In applying time series concepts to the original exercise, it underpins the derivation of the error correction model aiming to enhance understanding of long-term equilibrium relationships amidst short-term changes.

While working with time series data, recognizing the integration order and potential cointegration between variables is crucial. From the analysis, we derive insights into how variables can adjust dynamically over time, making informed economic and business decisions.