Cointegration and Error Correction Exercise Class, Econometrics II

Transcription

1 u n i v e r s i t y o f c o p e n h a g e n Faculty of Social Sciences Cointegration and Error Correction Exercise Class, Econometrics II Department of Economics March 19, 2017 Slide 1/39

2 Todays plan! 1 #3.4 Engle-Granger Analysis for Danish Consumption Slide 2/39

3 #Grapical Inspection of Consumption 1 log of Consumption, c, seems to be trending. The alternative hypothesis will be trend stationary. Slide 3/39

4 #Grapical Inspection of Disposable Income 1 log of disposable income, y, seems to be trending. The alternative hypothesis will be trend stationary. Slide 4/39

5 #Grapical Inspection of Wealth 1 log of disposable income, w, seems to be trending. The alternative hypothesis will be trend stationary. Slide 5/39

6 #Grapical Inspection of ARBLOS 1 log of ARBLOS does not have a trend. The alternative hypothesis will be stationary around a constant mean. Slide 6/39

7 #3.4.1 Augmented Dickey-Fuller Critical Values 1 Reproduced from Lecture Note 5: Slide 7/39

8 #3.4.1 ADF Test. Consumption 1 We include a trend since the alternative is trend stationarity. 2 I will start out with ve lags and I will ignore the misspeciction tests. This is simply out of convenience here! 1 Then I will remove the insignicant lags on by one. Slide 8/39

9 #3.4.1 ADF Test. Consumption 1 We end up with a model with one lag. 2 Critical value is so we cannot reject the null of a unit root. Slide 9/39

10 #3.4.1 ADF Test. Disposable Income 1 We include a trend since the alternative is trend stationarity. 2 I will start out with ve lags and I will ignore the misspeciction tests. This is simply out of convenience here! 1 Then I will remove the insignicant lags on by one. Slide 10/39

11 #3.4.1 ADF Test. Disposable Income 1 We end up witha model without and lagged dierences. 2 Critical value is so we can reject the null of a unit root. 1 Y t is trend stationary. 3 This seems strange. We would expect Y t and C t to cointegrate given economic theory. Slide 11/39

12 #3.4.1 ADF Test. Wealth 1 We include a trend since the alternative is trend stationarity. 2 I will start out with ve lags and I will ignore the misspeciction tests. This is simply out of convenience here! 1 Then I will remove the insignicant lags on by one. Slide 12/39

13 #3.4.1 ADF Test. Wealth 1 Critical value is so we cannot reject the null of a unit root. Slide 13/39

14 #3.4.1 ADF Test. ARBLOS 1 We only include a constant since the alternative is stationary around a constant mean. 2 I will start out with ve lags and I will ignore the misspeciction tests. This is simply out of convenience here! 1 Then I will remove the insignicant lags on by one. Slide 14/39

15 #3.4.1 ADF Test. ARBLOS 1 Critical value is so we can reject the null of a unit root. 1 Was also indicated by the ACF for ARBLOS. Slide 15/39

16 #3.4.1 Automatic ADF-test. PcGive 1 Remember from last class that we can also get the test statistics automatically from PcGive. 2 PcGive => Category: Other models => Model class: Descriptive Statistics using PcGive => Formulate. 3 Pick variable(s) of interest in (log-)levels => Unit-root test settings. Slide 16/39

17 #3.4.2 Engle-Granger Two-Step Approach 1 We estimate the static regression: C t = β0 + β1y t + β2w t + u t 2 We will include, Y t in the regression even though it seemed to be trend stationary. Slide 17/39

18 #3.4.2 Engle-Granger Two-Step Approach 1 There is a less than one to one increase in consumption given an increase in disposable income and wealth. 1 Consumption-income ratio not constant in steady-state! 2 Remember that we cannot say anything about the signicance of our estimated parameters. 1 Distribution depends upon unknown parameters and is non-normal. 2 Hence we cannot test hypothesis on our parameters. Slide 18/39

19 #3.4.2 Engle-Granger Two-Step Approach 1 Consistency of our parameter estimates holds even if our model is misspecied. 1 This holds as long as the misspecication is related to stationary terms. 2 The stochastic trends will dominate as T and any misspecication related to the stationary terms will dissapear. 2 Our estimates are super consistent. 1 Rate of convergence is T 2. 2 However they might be biased is small samples. Slide 19/39

20 #3.4.3 Engle-Granger Residual Based Test 1 If the variables cointegrate then the linear combination u t = β x t = C t β0 β1y t β2w t is stationary! 2 Testing for no-cointegration corresponds to testing for a unit root in u t. 3 This is a standard ADF-test on our estimated residuals, û t. 1 Remember to leave out the constant since the residuals are mean zero. 2 I will start out with 4 lags and remove insignicant lags. Slide 20/39

21 #3.4.3 Engle-Granger Residual Based Test 1 We end up with a well-specied model. Slide 21/39

22 #3.4.3 Critical values. E-G Residual Based Test 1 Note that the critical values depend upon the no. of estimated parameters in the static regression. 1 We have two estimated parameters (two I (1) variables, Y t and W t) and a constant. 2 The test statistic is The 5% critical value is We cannot reject the null of a unit root 2 This means that we cannot reject the null of no-cointegration! Slide 22/39

23 #3.4.3 Engle-Granger Two-Step Approach 1 We can try to check if it would make a dierence to add a trend to the static model! 2 We run the regression: C t = β0 + β1t + β2y t + β3w t + u t. Slide 23/39

24 #3.4.3 Engle-Granger Residual Based Test 1 Then use an ADF-test on our estimated residuals. 2 We still cannot reject the null of no-cointegration. 1 Now the critical value is Notice that the test statistic is almost identical to that where we had no trend. Slide 24/39

25 #3.4.4 Engle-Granger Residual Based Test 1 We will now dene a new variable: ecm t = û t = C t ˆβ0 ˆβ1Y t ˆβ2W t. 1 We will assume that our variables cointegrate anyways. 2 There are some large deviations as seen in the time series graphs while the ACF indicates that there is some persistence left. 1 The ACF goes exponentially towards zero but lags at a longer horizon become signicant. Hence, not strong support fo cointegration! Slide 25/39

26 #3.4.5 Engle-Granger ECM model 1 We will estimate now the following error correction models (ECM). 1 C t = α0 + α1 C t 1 + α 2 Y t + α3 Y t 1 + α 4 W t + α5 W t 1 + α6arblos t + α7ecm t 1 + ut. 2 Y t = α0 + α1 Y t 1 + α 2 C t + α3 C t 1 + α 4 W t + α5 W t 1 + α6arblos t + α7ecm t 1 + ut. 3 W t = α0 + α1 W t 1 + α 2 Y t + α3 Y t 1 + α 4 C t + α5 C t 1 + α6arblos t + α7ecm t 1 + ut. 2 ecm t 1 is the deviation from equilibrium. 3 α7, the coecient on ecm t 1 determines the speed of adjustment back to equilibrium. 4 We need α7 to be negative and statistically signicant in the rst equation for C t to error correct! 1 We need α7 to be positive and statistically signicant in the second equation for Y t to error correct! 2 We need α7 to be positive and statistically signicant in the third equation for W t to error correct! 3 You can see this by looking at the error correction term: ecm t = û t = C t ˆβ0 ˆβ1Y t ˆβ2W t. 5 Remember that all parameters are stationary (given cointegration) and standard inference applies! Slide 26/39

27 #3.4.5 Single equation ECM for C t 1 Estimated error correction model (ECM) for C t. 1 Again, I am disregarding the misspecication tests (we should use heterosc. robust s.e.'s!) 2 It seems like consumption error corrects. 1 The coecient indicates that around 30% of deviations from equilibrium are removed each quarter. Slide 27/39

28 #3.4.5 Single equation ECM for Y t 1 Estimated error correction model (ECM) for Y t. 2 It does not seem like disposable income error corrects. Slide 28/39

29 #3.4.5 Single equation ECM for W t 1 Estimated error correction model (ECM) for W t. 2 It does not seem like wealth error corrects. Slide 29/39

30 #3.4.6 Single equation ECM for C t 1 We will now concentrate on the error correction model (ECM) for C t and remove insignicant terms. 1 Remember that all our variables are stationary and regular inference applies to all our parameters. Slide 30/39

31 #3.4.6 Single equation ECM for C t 1 I will try to get a well-specied model before I remove insignicant terms. 2 I will ignore the RESET23 test and focus on the other misspecication tests. 1 The model suers from non-normality which is not that important here since we are using OLS. Asymptotic normality is given by the CLT. 2 The model suers from heteroscedasticity which means our s.e.'s are estimated incorrectly. 3 Do a graphical inspection of the residuals. Slide 31/39

32 #3.4.6 Single equation ECM for C t 1 We seem to have a problem with some large outliers. 1 Especially in 1975:4, 1977:3 and 1977:4. Slide 32/39

33 #3.4.7 Single equation ECM for C t 1 Solution? 1 Use dummy variables in 1975:4, 1977:3 and 1977:4. 2 This should be known from PS#1. 2 Adding a dummy for a certain observation means that this observation is ignored under estimation. 3 Most times we should have a good reason to include the dummies. In our case the outliers are associated with transitory policy shocks so this will be our justication for using these three dummy variables. 4 We will use the algebra le dummy.alg which we also used in PS#1. 5 Note that the dummies will take the value of one for the specied time period and the value zero for the rest of the observations. Slide 33/39

34 #3.4.7 Single equation ECM for C t 1 Now it seems like our model is well-specied. 1 I ignore the borderline signicant RESET23 test and proceed by removing insignicant terms one at the time. Slide 34/39

35 #3.4.7 Single equation ECM for C t 1 Our nal model seems well specied but disposable income seems to be insignicant. 1 I ignore the borderline signicant RESET23 test. Slide 35/39

36 #3.4.7 Recursive Estimation of Parameter Estimates 1 Recursive estimation means that we will estimate all our coecients for all sample lenghts up to T. 2 We start with a base sample: 1973 : 2 + N, then we add on observation to get a sample: 1973 : 2 + N + 1, then 1973 : 2 + N + 2. This is done recursively until all observations are included in the sample, i.e. we reach 2003:2. 3 Note that when estimating a model we assume that all parameters are constant and therefore should our estimated coecients not uctuate too much. Slide 36/39

37 #3.4.7 Recursive Estimation of Parameter Estimates 1 PcGive => Single Equation Dynamic Modelling using PcGive => Formulate => Speciy model => OK => OLS => OK=> Tick the box: Recursive Estimation. 2 Set initialization to two times the amount of coecients (16 in our case). Slide 37/39

38 #3.4.7 Recursive Estimation of Parameter Estimates 1 After estimating: Test => Recursive Graphics => Beta coecient +/- 2 SE. 2 Our estimates should be relatively stable. 3 Our estimates should lie within our condence bands. 1 Furthermore, our condence bands should narrow as our sample increases for each t. Slide 38/39

39 #3.4.7 Recursive Estimation of Parameter Estimates 1 Our parameters seem relatively stable over time dc_1 +/-2SE Constant +/-2SE dw +/-2SE ARBLOS +/-2SE ecm_1 +/-2SE DUM754 +/-2SE DUM773 +/-2SE Slide 39/ DUM774 +/-2SE