Use INTQRT for this exercise. (i) In Example 18.7, we...

Use INTQRT for this exercise. (i) In Example $18.7,$ we estimated an error correction model for the holding yield on six-month T-bills, where one lag of the holding yield on three-month T-bills is the explanatory variable. We assumed that the cointegration parameter was one in the equation $h y 6_{t}=\alpha+\beta h y 3_{t-1}+u_{r}$ Now, add the lead change, $\Delta h y 3_{t}$ , the contemporaneous change, $\Delta h y 3_{t-1},$ and the lagged change, $\Delta h y 3_{t-2},$ of $h y 3_{t-1} .$ That is, estimate the equation $h y 6_{t}=\alpha+\beta h y 3_{t-1}+\phi_{0} \Delta h y 3_{t}+\phi_{1} \Delta h y 3_{t-1}+\rho_{1} \Delta h y 3_{t-2}+e_{t}$ and report the results in equation form. Test $\mathrm{H}_{0} : \beta=1$ against a two-sided alternative. Assume that the lead and lag are sufficient so that $\left\{\mathrm{hy} 3_{t-1}\right\}$ is strictly exogenous in this equation and do not worry about serial correlation. (ii) To the error correction model in $(18.39),$ add $\Delta h y 3_{t-2}$ and $\left(h y 6_{t-2}-h y 3_{t-3}\right)$ . Are these terms jointly significant? What do you conclude about the appropriate error correction model?

Use INTQRT for this exercise.
(i) In Example $18.7,$ we estimated an error correction model for the holding yield on six-month
T-bills, where one lag of the holding yield on three-month T-bills is the explanatory variable. We
assumed that the cointegration parameter was one in the equation $h y 6_{t}=\alpha+\beta h y 3_{t-1}+u_{r}$ Now, add the lead change, $\Delta h y 3_{t}$ , the contemporaneous change, $\Delta h y 3_{t-1},$ and the lagged change, $\Delta h y 3_{t-2},$ of $h y 3_{t-1} .$ That is, estimate the equation
$h y 6_{t}=\alpha+\beta h y 3_{t-1}+\phi_{0} \Delta h y 3_{t}+\phi_{1} \Delta h y 3_{t-1}+\rho_{1} \Delta h y 3_{t-2}+e_{t}$
and report the results in equation form. Test $\mathrm{H}_{0} : \beta=1$ against a two-sided alternative. Assume that the lead and lag are sufficient so that $\left\{\mathrm{hy} 3_{t-1}\right\}$ is strictly exogenous in this equation and do not worry about serial correlation.
(ii) To the error correction model in $(18.39),$ add $\Delta h y 3_{t-2}$ and $\left(h y 6_{t-2}-h y 3_{t-3}\right)$ . Are these terms jointly significant? What do you conclude about the appropriate error correction model?

Key Concepts

Joint Significance Test

A joint significance test is used to assess whether a set of additional explanatory variables, such as lagged or differenced terms, significantly improves the model’s explanatory power. In the context of an ECM, this test can determine if adding extra dynamics is necessary, informing the appropriate form of the error correction model for capturing both short- and long-term relationships.

Model Specification with Lead and Lag Variables

Model specification in time series analysis often involves including lead and lag variables to capture the dynamic adjustments and potential delayed effects in the relationship between variables. Including these variables helps in adequately modeling the temporal structure and potential asynchronous adjustments in the data, thereby improving the robustness of the estimation.

Exogeneity

Exogeneity is an assumption related to whether an explanatory variable is correlated with the error term in the model. In cointegration and ECM contexts, assuming a variable is strictly exogenous implies that it is determined outside of the system and, therefore, does not react to contemporaneous shocks, which ensures valid inference about the long-run relationship.

Cointegration

Cointegration refers to a statistical relationship between two or more non-stationary time series which move together over time such that a linear combination of them is stationary. This concept is crucial in assessing long-run equilibrium relationships, indicating that despite their individual trends, the series share a common stochastic drift or equilibrium path.

Error Correction Model

An error correction model (ECM) is used in econometrics to capture both the short?term dynamics and the long?term equilibrium relationship between cointegrated time series. The ECM corrects deviations from the long-run equilibrium by incorporating an adjustment term that represents the disequilibrium from the previous period, thereby linking short-run fluctuations with long-run trends.

Hypothesis Testing in Cointegration Context

In the cointegration framework, hypothesis testing often involves checking whether the estimated cointegration parameter equals a theoretical value, typically unity, or some other value derived from economic theory. This type of test is used to validate or refute theoretical restrictions on the long-run relationship and to determine the appropriateness of imposing certain error correction representations.

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