2. (a) Describe a necessary and a sufficient condition for a long-run relationsh
ID: 1143849 • Letter: 2
Question
2. (a) Describe a necessary and a sufficient condition for a long-run relationship to exist between two variables X and Y.5 (b) Suppose you want to test for a cointegrating (long-run) relationship between persona consumption expenditure (PCE) and personal disposable income for the United States over the period 1970q1-1990q4. Employing Dickey-Fuller (1979) and Engle-Granger (te you have tested for a unit root in PCE and PDI, variables for the United States over the period 1970q1-1990q4. The results you and a long-run relationship between these have obtained are reported as follows: 5+5 Table 1: Null hypothesis that the underlying variables have a unit root PCE APCE PDI APDI Augmented Dickey-Fuller Test -0.2074 -7.6151 -0.6716 9,6362 Statistic 2.8955 Table 2: Null hypothesis is that there is a cointegration (long-run) relationship between PCE Critics valuesforADFtestasreported in Mackinnon(1996)forthe5%significance level is and PDI Dependent Variable: PCE Method: Fully Modified Least Squares (FMOLS) Variable Coefficient Std. Erro t-Statistic Prob. 0.0000 40.22536 4.533455 0.0000 2545.508 458.8979 108809.4 PDI 0.014122 68.74311 -182.3598 R-squared Adjusted R-squared S.E. of regression Long-run variance 0.993992 Mean dependent var 0.993921 S.D. dependent var 35.77864 Sum squared resid 3821.590 Table 3: Cointegration Test- Engle-Granger Engle-Granger tau-statistic Engle-Granger z-statistic Value -3.779071 -23.95216 Prob.+ 0.0197 0.0160 MacKinnon (1996) p-values. (0) Based on the results of the ADF test statistic, explain the order of integration of the economic time-series PCE and PDI (ii) Evaluate and interpret results based Engle-Granger cointegration tests, tau-statistic and z- statisticExplanation / Answer
Answer:
2 a) The necessary and sufficient conditions for a long-run relationship between two variables are:
b) i) The null hypothesis of the ADF statistics is rejected for I(1) series at 5 percent. The statistics reports -7.62 and -9.36 respectively for PCE and PDI. The critical value based on McKinnon (1996) is 2.90. Hence according to rule of thumb if the statistics vaklue is less than the critical value, we reject the null. Whereas for level the ADF statistics is -0.207 and -0.67, which is greater than the I(1) statistics. According to the ADF rule, the greater the value in negative the higher the chance for a series to be stationary. Hence both the series PCE and PDI are stationary at I(1).
ii) The Table 2 above present the FMOLS result for Engle- Granger cointegration test. The dependent variable for the test is PCE. The coefficient value of PDI is 0/97. This means that PCE is cointegrated to PDI of the form (1, -0.97). The adjusted R square of the model is 0.99 which means that the model has 99 percent of goodness of fit.
The cointegration table result in Table 3 presnts that tau and z statistic. Both the statistic p-value is below 5 percent. Hence the null hyopthesis that there exisits a no cointegration relationship between the two variable is rejected. To note that tau statistic is t test of Engle Granger cointegration and z -statistics is normalised autocorrelation coefficient.
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