Question 4

Since both time series are stationary, we can use a standard linear regression model to examine the relationship between UK exports to China and China’s trade openness. The model will take the form of Y_t = a + bX_t + e_t, where Y_t is the UK exports to China at time t, X_t is China’s trade openness at time t, a is the intercept, b is the slope coefficient, and e_t is the error term at time t. The interpretation of the slope coefficient, b, is the change in UK exports to China associated with a one-unit increase in China’s trade openness, holding all other factors constant. A positive coefficient would indicate that as China becomes more open to trade, UK exports to China increase, while a negative coefficient would suggest the opposite. The intercept term, a, represents the value of UK exports to China when China’s trade openness is zero, which may or may not have practical significance depending on the context. It is important to note that while the stationary assumption is necessary for standard linear regression models, it is not sufficient to establish causality between the two variables. Further analysis, such as Granger causality testing or instrumental variable regression, may be necessary to establish causal relationships. Additionally, the model assumes a linear relationship between the two variables. Other functional forms may be more appropriate depending on the relationship between UK exports to China and China’s trade openness.

Question 9

Part (a)
Cointegration is a statistical property that describes the relationship between two or more non-stationary time series variables. In particular, it refers to a situation where two or more time series are integrated of the same order (i.e., have a unit root). Still, a stationary linear combination of these variables exists (i.e., it does not have a unit root). Cointegration implies that two or more variables may move together in the long run, despite their short-run fluctuations. This long-run relationship is not spurious but reflects a genuine economic link between the variables. For instance, consider the case of the Canadian GDP and the US GDP. Although both series may be non-stationary and exhibit high volatility over the short term, it is reasonable to assume that these two economies are connected in the long run. Cointegration analysis can help identify this relationship by testing whether a common stochastic trend is underlying the two series. Cointegration has important implications for econometric modeling and forecasting. In particular, it allows for developing models considering the long-run relationship between variables rather than simply analyzing their short-term behavior.

Part (b)

The output of the regression indicates that Canadian GDP (CAN) is being regressed against the US GDP (US) as the explanatory variable. The regression intercept is estimated to be -41.59938, representing the expected value of CAN when the US equals zero. However, this interpretation may not be meaningful in this case since the US GDP is never zero, and it is not plausible for it to be. The US GDP (US) coefficient is estimated to be 1.30094, indicating that a one-unit increase in US GDP is associated with a 1.30094 unit increase in Canadian GDP. This coefficient is statistically significant at the 1% level, as indicated by the t-value of 68.26 and the p-value of <2e-16. The regression output suggests that the US GDP is a significant determinant of the Canadian GDP, but it does not necessarily imply a causal relationship between the two variables. Other factors may affect the Canadian economy and explain the observed correlation with the US economy. For example, changes in commodity prices or exchange rates may affect both countries economies. Moreover, the regression does not provide any information about the direction or causality of the relationship between the two variables. The causality runs in the opposite direction, i.e., changes in the Canadian economy may also affect the US economy. Therefore, the regression should be interpreted as a descriptive tool rather than a causal inference.

Part (c)

The Engle-Granger test is a widely used econometric technique to determine whether two or more non-stationary time series variables are cointegrated. Cointegration means that there is a long-run equilibrium relationship between the variables. The test is a two-step procedure that first estimates a regression between the variables and then tests the residuals for stationarity using a unit root test such as the Augmented Dickey-Fuller (ADF) test.

In the first step of the Engle-Granger test, we estimate a regression model between the variables. In this case, we have used Canadian GDP (CAN) as the dependent variable and US GDP (US) as the explanatory variable. The regression output shows that the coefficient estimate for US GDP is positive and statistically significant at the 1% level, indicating that the two variables are positively related. However, this regression alone does not establish a long-run relationship between the variables. In the second step of the Engle-Granger test, we test the regression residuals for stationarity using the ADF test. The output provided shows the results of the ADF test. The test statistic of -2.2607 is less than the critical value of -1.95 at the 5% significance level, indicating that we reject the null hypothesis and conclude that the residuals are stationary. This suggests that the Canadian and US GDPs are cointegrated.

The two variables are related in the long run, and shocks to the US GDP impact the Canadian GDP and vice versa. However, this analysis alone needs to clarify the direction of causality. Further analysis is required to determine the direction of causality, such as Granger causality tests or vector error correction models. The Engle-Granger test provides strong evidence of cointegration between the Canadian and US GDP, implying that the two economies are closely linked in the long run. The regression analysis in part (b) may be useful for describing the long-run relationship between the two variables. Still, it is not sufficient to establish causality. The Engle-Granger test is useful for testing cointegration between non-stationary variables but has some limitations. One important limitation is that it assumes only one cointegrating relationship between the variables. If there are multiple cointegrating relationships, the test may not be reliable.