## Содержание

- 2. Lecture Objectives Revisit the concept of non-stationary (unit root) process and its implications for analysis and
- 3. Outline Stationary and non-stationary variables Testing for unit roots Cointegration Testing for cointegration
- 4. Introduction Macro-econometric Forecasting and Analysis Many economic (macro/financial) variables exhibit trending behavior e.g., real GDP, real
- 5. Key Macro Series Appear to have trends Macro-econometric Forecasting and Analysis
- 6. Deterministic and Stochastic Trends in Data Two types of trends: deterministic or stochastic A Deterministic trend
- 7. Example: Processes with Trends Deterministic trend Stochastic trend
- 8. Stationary and non-stationary processes (1) Macro-econometric Forecasting and Analysis Consider the data generation process (DGP) If
- 9. Macro-econometric Forecasting and Analysis If model is said to be non-stationary and its associated (statistical) distribution
- 10. Reminder: Autoregressive AR(p) Process We shall check how shocks affect stationary and non-stationary variables, but first
- 11. Stochastic trends, autoregressive models and a unit root The condition for stationarity in an AR(p) model:
- 12. Consider a simple AR(1): yt = θyt-1 + νt, where θ takes any value for now
- 13. The Impact of Shocks for Stationary and Non-stationary Series (2) Representation at t=T: yT = θT+1y-1
- 14. Integration Macro-econometric Forecasting and Analysis Another way to write the stochastic trend model is: Thus the
- 15. Order of Integration: I(d) Macro-econometric Forecasting and Analysis In general, if yt is I(d) then: If
- 16. Problems due to Stochastic Trends (from a statistical perspective) Non-standard distribution of test statistics Spurious regression:
- 17. Figure 5: Distribution of OLS estimator for θ Macro-econometric Forecasting and Analysis
- 18. Testing For Unit Roots Macro-econometric Forecasting and Analysis Previous section suggests that I(1) variables need special
- 19. Testing for Unit Roots: Procedures Dickey Fuller Augmented Dickey Fuller Phillips Perron Kwiatkowski, Phillips, Schmidt and
- 20. Dickey Fuller Test Fuller (1976), Dickey and Fuller (1979) Example: consider a particular case of an
- 21. Dickey-Fuller Test (2) For the purpose of testing we reformulate the regression: Δyt = yt –
- 22. Dickey-Fuller Test (3) Important issue – shall deterministic components be included in the test model for
- 23. DF-Test (3): Deterministic Components are Known Say, we assume yt includes an intercept, but not a
- 24. If deterministic components are not included in the test, when they should be, then the test
- 25. The Augmented Dickey Fuller (ADF) Test The DF-test above is only valid if εt is a
- 26. The ADF-Test (2) Again, we have three choices: (1) include neither a constant nor a time
- 27. The ADF-Test: Lag Length Selection Three approaches are commonly used: Akaike Information Criterion (AIC) Schwarz-Bayesian Criterion
- 28. Dickey-Fuller (and ADF) Test: Criticism The power of the tests is low if the process is
- 29. The Phillips Perron (PP) test Rather popular in the analysis of financial time series The test
- 30. The PP test (2) Under the null hypothesis that ψ = 0, Zt statistic has the
- 31. The Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test The KPSS test is a stationarity test. The H0 is: yt ~I(0)
- 32. Testing for Higher Orders of Integration Just when we thought it is over... Consider: Δyt =
- 33. Working with Non-Stationary Variables Consider a regression model with two variables; there are 4 cases to
- 34. Cointegration Macro-econometric Forecasting and Analysis Important implication is that non-stationary time series can be rendered stationary
- 35. Cointegration Macro-econometric Forecasting and Analysis Three main implications of cointegration: Existence of cointegration implies a set
- 36. Cointegration Macro-econometric Forecasting and Analysis We will see that cointegrated systems (VECMs) are special VARS. Specifically,
- 37. Long-Run Equilibrium Relationships: Examples Macro-econometric Forecasting and Analysis Permanent Income Hypothesis (PIH) Postulates a long-run relationship
- 38. Term Structure Of Interest Rates Macro-econometric Forecasting and Analysis Models the relationship between the yields on
- 39. VECM Macro-econometric Forecasting and Analysis Cointegration postulates the existence of long-run equilibrium relationships between non-stationary variables
- 40. Bivariate VECMs Macro-econometric Forecasting and Analysis Consider a bivariate model containing two I(1) variables, say Assume
- 41. Phase Diagram: VECM Macro-econometric Forecasting and Analysis
- 42. Adjusting Back To Equilibrium Macro-econometric Forecasting and Analysis Suppose there is a positive shock in the
- 43. Adjustments Are Made by Y1,t Macro-econometric Forecasting and Analysis Long-run equilibrium is restored by y1,t decreasing
- 44. Adjustments Are Made by Y2,t Macro-econometric Forecasting and Analysis Long-run equilibrium is restored by y2,t increasing
- 45. Adjustments are made by both Y1,t and Y2,t Macro-econometric Forecasting and Analysis The previous two equations
- 46. VECM = Special VAR Macro-econometric Forecasting and Analysis A VECM is actually a special case of
- 47. VECM = Special VAR Macro-econometric Forecasting and Analysis
- 48. VECM = Special VAR Macro-econometric Forecasting and Analysis Obviously, we have a first order VAR with
- 49. Multivariate Methods: N > 2 Macro-econometric Forecasting and Analysis Can easily generalize the relationship between a
- 50. VAR with p lags > 1 Macro-econometric Forecasting and Analysis Allowing for p lags gives: where
- 51. Cointegration Macro-econometric Forecasting and Analysis If the vector time series yt is assumed to be I(1),
- 52. Granger Representation Theorem Macro-econometric Forecasting and Analysis Suppose yt, which can be I(1) or I(0), is
- 53. Examples: Rank of Long-Run Models Macro-econometric Forecasting and Analysis The form of for the two long-run
- 54. Key Implications of the GE Representation Theorem Macro-econometric Forecasting and Analysis The Granger-Engle theorem suggests the
- 55. Key Implications of the GE Representation Theorem Macro-econometric Forecasting and Analysis If , then the appropriate
- 56. Dealing With Deterministic Components Macro-econometric Forecasting and Analysis We can easily extend the base VECM to
- 57. Deterministic Components Macro-econometric Forecasting and Analysis Suppose we can decompose these parameters into their short-run and
- 58. Deterministic Components Macro-econometric Forecasting and Analysis The term represents the long-run relationship among the variables. The
- 59. Deterministic Components Macro-econometric Forecasting and Analysis The equation contains five important special cases summarized on the
- 60. Alternative Deterministic Structures Macro-econometric Forecasting and Analysis
- 61. Estimating VECM Models Macro-econometric Forecasting and Analysis If you are willing to assume that the error
- 62. Three Cases: Macro-econometric Forecasting and Analysis VECM is equivalent to the unconstrained VAR. No restrictions are
- 63. Reduced Rank (Cointegration) Case: FIML Macro-econometric Forecasting and Analysis If cannot be inverted (i.e., reduced rank
- 64. Reduced Rank Case: Johansen Estimator Macro-econometric Forecasting and Analysis We can also use the Johansen (1988)
- 65. Zero-Rank Case for Macro-econometric Forecasting and Analysis When , the VECM reduces to a VAR in
- 66. Identification Macro-econometric Forecasting and Analysis The Johansen procedure requires one to normalize the cointegrating vectors so
- 67. Identification: Triangular Restrictions Macro-econometric Forecasting and Analysis Suppose there are r long-run relationships. Identification can be
- 68. Triangular Restrictions Macro-econometric Forecasting and Analysis If there are N = 3 variables and r =
- 69. Structural Restrictions Macro-econometric Forecasting and Analysis Traditional identification methods can also be used with VECM’s, including
- 70. Open Economy Model Macro-econometric Forecasting and Analysis Assuming r = 2 long-run equations, the following restrictions
- 71. Cointegration Rank Macro-econometric Forecasting and Analysis So far we have taken the rank of the system
- 72. Cointegration Rank: Likelihood Ratio Test Macro-econometric Forecasting and Analysis Suppose we estimate the model assuming no
- 73. Cointegration Rank: Likelihood Ratio Test Macro-econometric Forecasting and Analysis Using the standard result for the likelihood
- 74. Cointegration Rank: Johansen Approach Macro-econometric Forecasting and Analysis A numerically equivalent approach was proposed by Johansen
- 75. Critical Values of the Likelihood Ratio Test Macro-econometric Forecasting and Analysis
- 76. Tests on the Cointegrating Vector (Long-Run Parameters) Macro-econometric Forecasting and Analysis Hypothesis tests on the cointegrating
- 77. Exogeneity Macro-econometric Forecasting and Analysis An important feature of a VECM is that all of the
- 78. Weak versus Strong Exogeneity Macro-econometric Forecasting and Analysis If the first channel does not exist, i.e.,
- 79. Example: Exogeneity Macro-econometric Forecasting and Analysis Consider the bi-variate term structure model with one cointegrating vector.
- 80. Impulse Response Functions Macro-econometric Forecasting and Analysis The dynamics of a VECM can be investigated using
- 81. Impulse Response Functions: VECM Macro-econometric Forecasting and Analysis This VECM can be expressed as a VAR
- 82. Appendices
- 83. Appendix A: Process moments, key results: AR(1) model with θ Macro-econometric Forecasting and Analysis Mean (first
- 84. Appendix A: Process moments, Simulation of an AR(1) model Macro-econometric Forecasting and Analysis Assume It follows
- 85. Appendix A: Process moments, key results: AR(1) model with θ = 1 Macro-econometric Forecasting and Analysis
- 86. Appendix A: Process moments, simulation of an I(1) Process Macro-econometric Forecasting and Analysis Notice that the
- 87. Appendix B: Enders Strategy Test H0: ψ=0 t-ratio test, 5% Crit. value is -3.45 Estimate Δyt
- 88. Enders Strategy was criticized for: triple- and double-testing for unit roots unrealistic outcomes: economic variables unlikely
- 89. Appendix B: Elder and Kennedy Strategy Test H0: ψ=0 t-ratio test, 5% Crit. value is -3.45
- 90. Nonstationary Asymptotics Macro-econometric Forecasting and Analysis
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