The conintegrated VAR mode

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1. Getting Started..........................................................................................7
1.1 Using CATS................................................................................................................ 7
1.2 The Pre-Written Example Programs.................................................................... 7
1.3 Getting Started.......................................................................................................... 7
1.4 Reading Data.............................................................................................................. 8
1.5 The Data, Creating Series, and Data Transformations.................................... 9
1.6 Examining the data..................................................................................................11
2. Detrending and Graphing.......................................................................12
2.1 Detrending Data Using FILTER...........................................................................12
2.1.1 Detrending Using LINREG......................................................................12
2.2 Graphing the Results, Figure 2.4.........................................................................13
3. The Unrestricted Vector Autoregression Model.................................14
3.1 Moving Averages and First Differences.............................................................14
3.1.1 Moving Averages.......................................................................................14
3.1.2 First Differences........................................................................................14
3.2 Graphing Levels and Differences, Figures 3.3 and 3.4....................................14
3.3 Unrestricted VAR(2) Model.................................................................................15
3.4 VAR Residuals, Figure 3.5.....................................................................................16
3.5 Roots of the Companion Matrix, Table 3.1......................................................16
3.5.1 Graphing the Roots, Figure 3.6..............................................................17
3.6 VAR Estimation Results, Table 4.1......................................................................17
3.7 Computing F-Tests.................................................................................................18
3.8 The Vector Error Correction Model,Tables 4.2 through 4.4......................19
3.8.1 Using RATS Instructions..........................................................................19
4. Using CATS: Preliminary Analysis ........................................................20
4.1 Compiling and Executing CATS...........................................................................20
4.1.1 The CATS Wizard....................................................................................20
4.1.2 Using the @CATS Command...............................................................20
4.2 The VECM Estimation Results, Table 4.2..........................................................21
4.3 Graphing the Data in CATS, Figures 3.3 and 3.4............................................21
4.4 Graphing the Residuals, Figure 3.5......................................................................22
4.5 Roots of the Companion Matrix, Table 3.1, Figure 3.6.................................22
4.6 Residuals, Correlations and Specification Tests, Figures 4.1–4.6, Table 4.5.22
5. Cointegrated VAR Model........................................................................23
6. Deterministic Components....................................................................24
6.1 Figure 6.3..................................................................................................................24
6.2 Tables 6.2 and 6.3...................................................................................................24
6.2.1 Seasonal Autocorrelation/ARCH Tests—using the Preferences Dialog
Box.........................................................................................................25
6.2.2 Interpreting the Test Results..................................................................25
RATS Handbook for The Cointegrated VAR Model
4
7. Estimating the I(1) Model.......................................................................27
7.1 New Example File...................................................................................................27
7.2 Using SHIFT Rather Than EXO in CATS..........................................................27
7.3 Table 7.1....................................................................................................................27
7.4 Figures 7.1 through 7.5..........................................................................................28
8. Cointegration Rank..................................................................................29
8.1 Rank Test Statistics, Table 8.1.............................................................................29
9. Recursive Tests of Constancy.................................................................31
9.1 Setting the Rank of Pi.............................................................................................31
9.2 Recursive Log Likelihood, Figure 9.1..................................................................31
9.3 Recursive Trace Tests, Figure 9.2.......................................................................32
9.4 Recursive Eigenvalues and Fluctuation Tests, Figures 9.3-9.5......................32
9.5 Beta Constancy Tests, Figures 9.6 and 9.7.......................................................32
9.6 Changing the Estimation Sample, Figure 9.8.....................................................32
9.7 Another Estimation Sample, Figure 9.9..............................................................32
9.8 Prediction Errors, Figures 9.10-9.12...................................................................32
9.9 Backwards Recursive Tests, Figures 9.13-9.21................................................32
10. Testing Restrictions on Beta...................................................................34
10.1 Normalizing..............................................................................................................34
10.2 Two Restriction Formulations.............................................................................34
10.3 Some Simple Restrictions, Table 10.1................................................................34
10.3.1 Hypothesis H1 Using H*Phi....................................................................35
10.3.2 Hypotheses H1 and H2, Using R*Beta=0............................................36
10.3.3 Hypotheses H3 and H4............................................................................36
10.3.4 Hypotheses H5, a Joint Test...................................................................37
10.4 Hypothesis H6 and Table 10.2, Three Restrictions........................................37
10.5 Table 10.3, Unrestricted Constant Model........................................................37
10.5.1 Table 10.3, Hypotheses H7, H8, H9, Stationarity.............................38
10.5.2 Table 10.3, Hypotheses H10, H11, and H12......................................40
10.5.3 Table 10.3, Hypotheses H13 through H16.........................................40
10.5.4 Table 10.3, Hypotheses H17 through H20.........................................42
10.5.5 Table 10.3, Hypotheses H21 through H24.........................................43
10.5.6 Table 10.3, Hypotheses H25 through H29.........................................44
11. Testing Restrictions on a...............................................................................45
11.1 Testing for Weak Exogeneity...............................................................................45
11.1.1 Tests of Long-run Weak Exogeneity....................................................45
11.2 Models with Weakly Exogenous Variables.......................................................46
11.3 Testing a Known Vector in a.....................................................................46
12. Identification of the Long-run Structure...............................................47
12.1 Just-identifying Restrictions, Table 12.1.............................................................47
12.1.1 Example One..............................................................................................47
12.1.2 Example Two..............................................................................................47
RATS Handbook for The Cointegrated VAR Model
5
12.2 Over-identifying Restrictions, Table 12.3 ........................................................48
12.2.1 Example Three...........................................................................................48
12.2.2 Example Four.............................................................................................48
12.3 Lack of Identification, Table 12.4.........................................................................49
12.3.1 Example Five...............................................................................................49
12.3.2 Example Six.................................................................................................49
12.4 Recursive tests of a and b, Figures 12.1–12.7.................................................50
13. Identification of the Short-Run Structure.............................................51
13.1 The Data...................................................................................................................51
13.2 Creating Error Correction Terms and First Differences..............................51
13.3 Estimating the VECM Model, and F-Tests.........................................................52
13.4 Restrictions on the Short-Run Model, Tables 13.1 and 13.2........................52
13.5 The VAR in Triangular Form, Tables 13.3 and 13.4.......................................53
13.6 Tables 13.5-13.9......................................................................................................54
14. Identification of Common Trends..........................................................55
14.6.1 The Unrestricted MA Representation.................................................55
14.6.2 The MA Representation Subject to Restrictions on a and b........56
14.6.3 Remaining Sections of Chapter 14........................................................57
15. Identification of a Structural MA Model................................................58
15.1 Structural Models and Variable Order...............................................................58
15.2 Structural MA Model Illustration.......................................................................58
15.3 Are the Labels Credible?.......................................................................................61
15.3.1 Generating Table 15.1—SVAR 1 Model..............................................61
15.3.2 Generating Table 15.1—SVAR 2 Model..............................................61
15.3.3 Generating Table 15.1—VAR Model....................................................62
16. Analyzing I(2) Data with the I(1) Model................................................63
16.1 Stochastic and Deterministic Trends.................................................................63
16.1.1 Figure 16.1..................................................................................................63
16.1.2 Figure 16.2..................................................................................................63
16.2 I(2) Symptoms in I(1) Models...............................................................................64
16.2.1 Dummy Variables......................................................................................64
16.2.2 Characteristic Roots.................................................................................64
16.2.3 Graphing Cointegration Relations.........................................................65
16.3 Evaluating the Nominal-to-Real Transformation.............................................65
16.3.1 Testing Long-Run Price Homogeneity..................................................65
17. Estimating the I(2) Model.......................................................................69
17.1 Estimating the I(2) Model and Determining Rank...........................................69
17.2 Setting the Reduced Ranks...................................................................................69
17.3 Graphing Cointegrating Relations.......................................................................69
17.4 Identifying Restrictions..........................................................................................69
RATS Handbook for The Cointegrated VAR Model
6
18. Testing Hypotheses on the I(2) Model..................................................71
18.1 Testing Price Homogeneity..................................................................................71
18.1.1 Long-run Price Homogeneity.................................................................71
18.2 Medium-run Price Homogeneity.........................................................................71
18.3 Assessing I(1) Results Within the I(2) Model...................................................72
18.3.1 Restrictions of the I(1) Model, Tables 18.2 and 18.3........................72
18.3.2 Restrictions on Beta, Table 18.4 ..........................................................72
18.3.3 Data Consistent Long-Run Structure, Table 18.5.............................73
19. Specific-to-General and General-to-Specific........................................74
19.1 Figures 19.2 and 19.3.............................................................................................74
19.2 Figure 19.4................................................................................................................75
20. Wage, Price and Unemployment Dynamics.........................................76
20.1 The Data...................................................................................................................76




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