Question 2 (a) A researcher wished to investigate the relationship between two time series, which we denote by x, and x2. The researcher estimated the following VAR(3), dx, = 40 + Ajdx-1 + A2dx:-2 + A 3dx1-3 + t, where dx = (dx 1t, dx 21)', dx = X: - X:-1 and & VWN(0,2). The researcher used the AIC to select the VAR order. i) What is the dimension of the matrix A ? ii) Write down the equation for dx 11. iii) Write out in full the matrix 2. (b) i) Derive the autoregressive polynomial associated with the VAR in (4) ii) How many roots does the autoregressive polynomial have? iii) What is the condition for the VAR to be stationary? (c) The researcher wished to test the null hypothesis that the VAR order is 2 against the alternative that it is 3. i) What are the restrictions on the coefficients in the equation for dx, implied by this null hypothesis? ii) When a sample of 200 observations was used to estimate the restricted and unrestricted models associated with the null hypothesis the following results were obtained: Lu = 181 LR = 180, where Ly and Lr denote the maximized value of the log-likelihood function for the unrestricted and restricted models respectively. Use this information to test the null hypothesis that the VAR order is 2 against the alternative that it is 3. In your answer identify the null and alternative hypotheses, the form and sampling distribution of the test statistic, it's sample and critical values and your test conclusion. (d) Assume that the VAR order is 2. i) What are the restrictions on the parameters of the model implied by the null hypothesis that dx2 does not Granger-cause dx? ii) When a sample of 200 observations was used to estimate the restricted and unrestricted models associated with the null hypothesis that dx2 does not Granger-cause dx | the following results were obtained: Lu = 190 LR = 185 Use this information to test the null hypothesis. In your answer identify the null and alternative hypotheses, the form and sampling distribution of your test statistic, it's sample and critical values and your test conclusion. iii) (e) When the researcher conducted some impulse response analysis she obtained the results reported below. Based on these results, what was the ordering of the variables in the VAR used by the researcher? Briefly explain. Impulse responses shock to dxi shock to dx 2 impact effect on dx 0.01 0.00 impact effect on dx2 0.21 0.03 Question 2 (a) A researcher wished to investigate the relationship between two time series, which we denote by x, and x2. The researcher estimated the following VAR(3), dx, = 40 + Ajdx-1 + A2dx:-2 + A 3dx1-3 + t, where dx = (dx 1t, dx 21)', dx = X: - X:-1 and & VWN(0,2). The researcher used the AIC to select the VAR order. i) What is the dimension of the matrix A ? ii) Write down the equation for dx 11. iii) Write out in full the matrix 2. (b) i) Derive the autoregressive polynomial associated with the VAR in (4) ii) How many roots does the autoregressive polynomial have? iii) What is the condition for the VAR to be stationary? (c) The researcher wished to test the null hypothesis that the VAR order is 2 against the alternative that it is 3. i) What are the restrictions on the coefficients in the equation for dx, implied by this null hypothesis? ii) When a sample of 200 observations was used to estimate the restricted and unrestricted models associated with the null hypothesis the following results were obtained: Lu = 181 LR = 180, where Ly and Lr denote the maximized value of the log-likelihood function for the unrestricted and restricted models respectively. Use this information to test the null hypothesis that the VAR order is 2 against the alternative that it is 3. In your answer identify the null and alternative hypotheses, the form and sampling distribution of the test statistic, it's sample and critical values and your test conclusion. (d) Assume that the VAR order is 2. i) What are the restrictions on the parameters of the model implied by the null hypothesis that dx2 does not Granger-cause dx? ii) When a sample of 200 observations was used to estimate the restricted and unrestricted models associated with the null hypothesis that dx2 does not Granger-cause dx | the following results were obtained: Lu = 190 LR = 185 Use this information to test the null hypothesis. In your answer identify the null and alternative hypotheses, the form and sampling distribution of your test statistic, it's sample and critical values and your test conclusion. iii) (e) When the researcher conducted some impulse response analysis she obtained the results reported below. Based on these results, what was the ordering of the variables in the VAR used by the researcher? Briefly explain. Impulse responses shock to dxi shock to dx 2 impact effect on dx 0.01 0.00 impact effect on dx2 0.21 0.03
