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**Time Series Analysis Definition of a Time Series process**

AR, MA, ARMA, ARIMA Vector Autoregression Impulse Response Forecasting

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**Four Components of a Time Series**

Trend Season Cycle Random (refer STAMP manual p.140)

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**Iterative Substitution in AR(1) Model**

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**AR(1) Time Series as a Function of Past Innovations (Impulses or Shocks)**

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**Time Dependent Variance**

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**Dicky-Fuller and Augmented Dicky-Fuller Tests**

Null hypotheses: There is unit root and time series in non-stationary =0 (1-)=0 Alternative hypothesis: There is no unit root and time series is stationary <0 (1-)<0 <1

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**Moving Average-MA Process**

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MA(2) Process

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**Autoregressive Process**

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ARMA(1,1) Process

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Co-integration

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**Error Correction Model**

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**Structure of a VAR Model**

. Simple Example

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**Impulse Response Analysis in a VAR Model**

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**Stamp Program for Time Series Analysis**

Estimation sample is (T = 116, n = 111). Log-Likelihood is (-2 LogL = ). Prediction error variance is Summary statistics ER Std.Error Normality H( 37) r( 1) r( 9) DW Q( 9, 6) Rs^ ER = Trend + Trigo seasonal + Expl vars + Irregular Eq 3 : Estimated coefficients of final state vector. Variable Coefficient R.m.s.e t-value Lvl [ ] Slp [ ] Sea_ [ ] Sea_ [ ] Sea_ [ ] Eq 3 : Estimated coefficients of explanatory variables. ER_ [ ] ER_ [ ] ER_ [ ] ER_ [ ] Eq 3 : Seasonal analysis (at end of period). Seasonal Chi^2( 3) test is [0.9818]. Seas Seas Seas Seas 4 Value

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**Forecasting of the Exchange Rate**

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References Burns, A and W. Michell, (1946), “Measuring Business Cycles” NBER, New York. Campbell J. Y. and R.J. Shiller (1987) Cointegration and Tests of Present Value Models, Journal of Political Economy, 95, 5, pp Cooley and Thomas F. and S.F. LeRoy (1985) Atheoretical Macroeconometrics, Journal of Monetary Economics, North Holland 16: Dickey D.A. and W.A. Fuller (1979) Distribution of the Estimator for Autoregressive Time Series with a Unit Root, Journal of the American Statistical Association, June. Dickey D.A. and W. A. Fuller (1981) Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root, Econometrics, 49:4 July, Engle R E and C.W.J. Granger (1987) Co-integration and Error Correction: Representation, Estimation and Testing. Econometrica, vol. 55, No. 2, pp Enders W. (1995) Applied Econometric Time Series, John Wiley and Sons Fair R.C.(1984) Specification, Estimation, and Analysis of Macroeconomic Models, Harvard. Garratt A., K. Lee, M.H. Pesaran and Y. Shin (2003) A Structural Cointegration VAR Approach to Macroeconometric Modelling, Economic Journal. Cooly Thomas F (1995) Frontiers of Business Cycle Research, Princeton. Doornik J.A and D.F. Hendry (2003) Econometric Modelling Using PCGive Volumes I, II and II, Timberlake Consultant Ltd, London. Hendry D.F. (1997) Dynamic Econometrics, Oxford University Press. Harris R. and R. Sollis (2003) Applied Time Series Modelling and Forecasting, John Willey. Holly S and M Weale Eds.(2000) Econometric Modelling: Techniques and Applications, pp.69-93, the Cambridge University Press. Johansen Soren (1988) Estimation and Hypothesis Testing of Cointegration Verctors in Gaussian Vector Autoregressive Models, Econometrica, 59:6, Johansen Soren (1988) Statistical Analysis of Cointegration Vectors, Journal of Economic Dynamics and Control , North Holland. Nelson C. R. and C. I. Plosser (1982) Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications, Journal of Monetary Economics. Pagan A. and M. Wickens (1989) A Survey of Some Recent Econometric Methods, Economic Journal, 99 pp Phillips P.C.B. (1987) Time Series Regression with an Unit Root, Econometrica, vol. 55, No. 2, Prescott, E.C. (1986), “Theory Ahead of Business Cycle Measurement,” Federal Reserve Bank of Minneapolis, Quarterly Review; Fall. Pindyck R.S and Robinfeld D.L. (1998) Econometric Models and Economic Forecasts, 4th edition, McGraw Hill. Quah, D.T., (1995), “Business Cycle Empirics: Calibration and Estimation,” The Economic Journal (November) Sims Christopher A (1980) Macroeconomics and Reality, 48:1 January, pp Wallis KF. (1989) Macroeconomic Forecasting: A Survey, Economic Journal, 99, March, pp Wallis Kenneth (1980) Econometric Implications of the Rational Expectations Hypothesis, Econometrics 48:1, pp,

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MULTIVARIATE TIME SERIES & FORECASTING 1. 2 : autocovariance function of the individual time series.

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