When using the Angle-Granger cointegration test, the primary condition is that the time series data must be non-stationary, meaning they have a unit root but are integrated of the same order (usually I(1)), meaning they need to be differenced once to achieve stationarity. The series should also exhibit a stable long-run equilibrium relationship, meaning that despite being individually non-stationary, a linear combination of the variables should be stationary (cointegrated). The test checks for this long-term equilibrium between the variables, requiring that the residuals from a cointegrating regression are stationary. Additionally, proper lag selection is necessary for ensuring valid results in the cointegration test.

The term "angle-granger cointegration" seems to be a typographical or conceptual error, as it does not correspond to any known statistical or econometric method. However, it is likely that you are referring to the **Engle-Granger cointegration** method, which is a well-established approach for testing and modeling cointegrated relationships between time series variables. Below, I outline the conditions and assumptions for using the **Engle-Granger cointegration** method:

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### **Conditions for Engle-Granger Cointegration**

1. **Non-Stationarity of Variables**:

- The individual time series being analyzed must be **non-stationary** (i.e., they have a unit root). This is typically confirmed using unit root tests such as the **Augmented Dickey-Fuller (ADF) test** or the **Phillips-Perron (PP) test**.

- The variables should be integrated of the same order (e.g., both are I(1), meaning they become stationary after first differencing).

2. **Linear Combination of Variables**:

- A linear combination of the non-stationary variables should result in a **stationary residual series**. This implies that the variables share a long-run equilibrium relationship.

3. **Residual Stationarity**:

- After estimating the cointegrating regression (e.g., \( y_t = \beta_0 + \beta_1 x_t + \epsilon_t \)), the residuals (\( \epsilon_t \)) must be tested for stationarity using a unit root test (e.g., ADF test on residuals).

- If the residuals are stationary, the variables are said to be cointegrated.

4. **No Serial Correlation in Residuals**:

- The residuals from the cointegrating regression should not exhibit significant serial correlation. If serial correlation is present, it may indicate misspecification of the model.

5. **Constant Cointegrating Relationship**:

- The cointegrating relationship should be stable over time. Structural breaks or changes in the relationship can invalidate the results.

6. **Appropriate Lag Selection**:

- When performing unit root tests or estimating the cointegrating regression, the appropriate number of lags should be selected to avoid misspecification.

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### **Steps in the Engle-Granger Cointegration Method**

1. **Test for Unit Roots**:

- Confirm that the individual time series are non-stationary (e.g., I(1)).

2. **Estimate the Cointegrating Regression**:

- Regress one variable on the other(s) to obtain the residuals.

3. **Test Residuals for Stationarity**:

- Perform a unit root test on the residuals to check if they are stationary.

4. **Interpret Results**:

- If the residuals are stationary, the variables are cointegrated, and you can proceed to estimate an Error Correction Model (ECM) to analyze the short-run dynamics.

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### **Limitations of the Engle-Granger Method**

- It is limited to testing cointegration between **two variables** at a time. For multiple variables, the **Johansen cointegration test** is more appropriate.

- The method assumes a linear cointegrating relationship, which may not hold in all cases.

- The results can be sensitive to the choice of the dependent variable in the cointegrating regression.