Question

(a) Assume that the two series \(y_{t}\) and \(x_{t}\) are \(I(1)\) and assume that both \(y_{t}-\beta_{1} x_{t}\) and \(y_{t}-\beta_{2} x_{t}\) are \(I(0)\). Show that this implies that \(\beta_{1}=\beta_{2}\), i.e. there can be only one unique cointegrating parameter. (b) Explain intuitively why the Durbin-Watson statistic in a regression of the \(I(1)\) variables \(y_{t}\) upon \(x_{t}\) is informative about the question of cointegration between \(y_{t}\) and \(x_{t}\). (c) Explain what is meant by 'superconsistency'. (d) Consider three \(I\) (1) variables \(y_{t}, x_{t}\) and \(z_{t}\). Assume that \(y_{t}\) and \(x_{t}\) are cointegrated, and that \(x_{t}\) and \(z_{t}\) are cointegrated. Does this imply that \(y_{t}\) and \(z_{t}\) are also cointegrated? Why (not)?

Short answer

(a) \(\beta_{1} = \beta_{2}\). (b) Durbin-Watson statistic checks for residual autocorrelation. (c) 'Superconsistency' is faster convergence to the true parameter. (d) No, \(y_t\) and \(z_t\) may not be cointegrated.

Step-by-step solution

Understand the definitions and implications

Recall that a series is said to be integrated of order one, denoted as \(I(1)\), if it becomes stationary after taking the first difference. Similarly, a series is \(I(0)\) if it is stationary without differencing.

Analyze the condition for cointegration with parameters \(\beta_1\) and \(\beta_2\)

Given: \(y_{t} - \beta_{1} x_{t}\) and \(y_{t} - \beta_{2} x_{t}\) are both \(I(0)\). Since both expressions are stationary, the difference between them must also be stationary. Therefore: \[ (y_{t} - \beta_{1} x_{t}) - (y_{t} - \beta_{2} x_{t}) \rightarrow (\beta_{2}x_{t} - \beta_{1}x_{t}) = x_{t}(\beta_{2} - \beta_{1}) \]As \( x_{t} \) is \( I(1) \), for the above expression to be \( I(0) \), it must be that \(\beta_{2} - \beta_{1} = 0\), thus \(\beta_{1} = \beta_{2}\).

Explain the Durbin-Watson Statistic

The Durbin-Watson statistic measures the amount of autocorrelation in the residuals of a regression analysis. When regressing \(I(1)\) variables, significant residual autocorrelation would indicate that the series are not cointegrated. Absence of autocorrelation (Durbin-Watson statistic near 2) would suggest potential cointegration.

Define 'Superconsistency'

In the context of cointegration, 'superconsistency' refers to the property that the estimator of the cointegration parameter converges to the true value at a faster rate than the typical \( \frac{1}{\sqrt{n}} \) rate found in ordinary least squares regression. This occurs because the integration properties of the series provide additional information that strengthens the estimate.

Analyze cointegration relationship among three variables

Even if \( y_{t} \) is cointegrated with \( x_{t} \) and \( x_{t} \) is cointegrated with \( z_{t} \), it does not necessarily imply that \( y_{t} \) is cointegrated with \( z_{t} \). Cointegration is a specific pairwise property that must be individually tested between each pair of series.

Key concepts

I(1) Series

An I(1) series, or an integrated order 1 series, becomes stationary only after the first difference. To put it simply, you need to subtract the previous value from the current value to achieve stationarity. For example, if you have a time series data set representing weekly sales figures, the differences between consecutive weeks would form an I(0) series if the original series was I(1). This property is crucial in econometrics as it helps in determining the appropriate methods for time series analysis.

I(0) Series

An I(0) series is already stationary, meaning its statistical properties like mean and variance are constant over time. This type of series doesn’t trend or exhibit long-term growth and decay patterns. Hence, there's no need for any differencing. For example, the series of daily temperature deviations from a long-term average would typically be I(0). Understanding whether your data is I(1) or I(0) helps in choosing the right modeling techniques, ensuring accurate results in analysis.

Durbin-Watson Statistic

The Durbin-Watson statistic is used to detect the presence of autocorrelation in the residuals from a regression analysis. Autocorrelation occurs when residuals (errors) are not random but follow a pattern. This statistic ranges from 0 to 4;

• Values around 2 indicate no autocorrelation.
• Values approaching 0 suggest positive autocorrelation.
• Values nearing 4 indicate negative autocorrelation.

If you regress two I(1) variables and find significant residual autocorrelation, it means the variables are not cointegrated. A Durbin-Watson statistic close to 2 could signal that the series might be cointegrated. This is essential for validating the statistical relationship between time series.

Superconsistency

Superconsistency is a term used in the context of cointegration. It refers to the phenomenon where the estimator of the cointegration parameter converges to the true value much faster than the usual rate of convergence found in ordinary least squares (OLS) regression. Normally, OLS estimators converge at a rate of \(\frac{1}{\sqrt{n}}\), where \(n\) is the sample size. However, for cointegrated systems, the rate is typically \(\frac{1}{n}\), yielding better estimates with smaller sample sizes. This occurs because the system's structure gives additional information, reducing estimation errors significantly. This property is particularly useful in economic models dealing with non-stationary series.

Autocorrelation in Residuals

Autocorrelation in the residuals of a regression analysis indicates that the residuals are not random. Instead, they follow a predictable pattern. This is problematic because most regression models assume that residuals are uncorrelated. If there is autocorrelation in the residuals, it suggests that the model might be mispecified or missing key variables. Detecting autocorrelation helps in refining the model for better accuracy.

There are important methods to detect autocorrelation:

• Durbin-Watson statistic
• Autocorrelation function (ACF)
• Partial autocorrelation function (PACF)

Addressing autocorrelation can involve adding lagged variables, using differencing methods, or switching to more complex models like ARIMA (AutoRegressive Integrated Moving Average).