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Economics 310

Economics 310. Lecture 24 Univariate Time Series. Concepts to be Discussed. Time Series Stationarity Spurious regression Trends. Plot of Economic Levels Data. Plot of Rate Data. Stationary Stochastic Process. Stochastic Random Process Realization

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Economics 310

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  1. Economics 310 Lecture 24 Univariate Time Series

  2. Concepts to be Discussed • Time Series • Stationarity • Spurious regression • Trends

  3. Plot of Economic Levels Data

  4. Plot of Rate Data

  5. Stationary Stochastic Process • Stochastic Random Process • Realization • A Stochastic process is said to be stationary if its mean and variance are constant over time and the value of covariance between two time periods depends only on the distance or lag between the two time periods and not on the actual time at which the covariance is computed. • A time series is not stationary in the sense just define if conditions are violated. It is called a nonstationary time series.

  6. Stationary Stochastic Process

  7. Test for Stationarity: Correlogram

  8. Correlogram for PPI AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) PPIACO 1 0.98 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR. 2 0.96 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR. 3 0.95 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 4 0.93 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 5 0.91 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 6 0.90 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 7 0.88 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 8 0.87 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 9 0.85 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 10 0.84 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 11 0.83 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 12 0.81 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 13 0.80 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRR . 14 0.79 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRR . 15 0.78 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRR . 16 0.77 . + RRRRRRRRRRRRRRRRRRRRRRRRRRR . 17 0.76 . + RRRRRRRRRRRRRRRRRRRRRRRRRRR . 18 0.75 . + RRRRRRRRRRRRRRRRRRRRRRRRRRR .

  9. Correlogram M1 AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) M1 1 0.99 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR 2 0.98 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR. 3 0.97 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR. 4 0.96 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR. 5 0.95 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 6 0.94 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 7 0.93 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 8 0.92 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 9 0.91 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 10 0.90 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 11 0.89 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 12 0.88 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 13 0.87 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 14 0.86 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 15 0.85 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 16 0.84 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 17 0.83 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 18 0.81 . + RRRRRRRRRRRRRRRRRRRRRRRRRRRRR .

  10. Testing autocorrelation coefficients • If data is white noise, the sample autocorrelation coefficient is normally distributed with mean zero and variance ~ 1/n • For our levels data sd=0.064, and 5% test cut off is 0.126 • For our rate data, sd=0.059, and 5% test cut off is 0.115

  11. Testing Autocorrelation coefficients

  12. Ljung-Box test for PPI SERIES (1-B) (1-B ) PPIACO NET NUMBER OF OBSERVATIONS = 242 MEAN= 112.01 VARIANCE= 129.36 STANDARD DEV.= 11.374 LAGS AUTOCORRELATIONS STD ERR 1 -12 0.98 0.96 0.95 0.93 0.91 0.90 0.88 0.87 0.85 0.84 0.83 0.81 0.06 13 -18 0.80 0.79 0.78 0.77 0.76 0.75 0.29 MODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS (CHI-SQUARE) LAG Q DF P-VALUE LAG Q DF P-VALUE 1 236.25 1 .000 10 2060.22 10 .000 2 465.31 2 .000 11 2234.89 11 .000 3 687.34 3 .000 12 2405.13 12 .000 4 902.18 4 .000 13 2571.38 13 .000 5 1110.17 5 .000 14 2733.79 14 .000 6 1311.32 6 .000 15 2892.87 15 .000 7 1506.47 7 .000 16 3048.83 16 .000 8 1696.30 8 .000 17 3201.65 17 .000 9 1880.78 9 .000 18 3351.37 18 .000

  13. Unit Root Test for Stationarity

  14. Results of our test • If a time series is differenced once and the differenced series is stationary, we say that the original (random walk) is integrated of order 1, and is denoted I(1). • If the original series has to be differenced twice before it is stationary, we say it is integrated of order 2, I(2).

  15. Testing for unit root • In testing for a unit root, we can not use the traditional t values for the test. • We used revised critical values provided by Dickey and Fuller. • We call the test the Dickey-Fuller test for unit roots.

  16. Dickey-Fuller Test

  17. Dickey-Fuller Test for our level data-PPI |_coint ppiaco m1 employ ...NOTE..SAMPLE RANGE SET TO: 1, 242 ...NOTE..TEST LAG ORDER AUTOMATICALLY SET TOTAL NUMBER OF OBSERVATIONS = 242 VARIABLE : PPIACO DICKEY-FULLER TESTS - NO.LAGS = 14 NO.OBS = 227 NULL TEST ASY. CRITICAL HYPOTHESIS STATISTIC VALUE 10% --------------------------------------------------------------------------- CONSTANT, NO TREND A(1)=0 T-TEST -0.46372 -2.57 A(0)=A(1)=0 2.5444 3.78 AIC = -1.298 SC = -1.057 --------------------------------------------------------------------------- CONSTANT, TREND A(1)=0 T-TEST -2.7258 -3.13 A(0)=A(1)=A(2)=0 4.1554 4.03 A(1)=A(2)=0 3.7243 5.34 AIC = -1.323 SC = -1.067 ---------------------------------------------------------------------------

  18. Dickey-Fuller Test for our level data-M1 VARIABLE : M1 DICKEY-FULLER TESTS - NO.LAGS = 12 NO.OBS = 229 NULL TEST ASY. CRITICAL HYPOTHESIS STATISTIC VALUE 10% --------------------------------------------------------------------------- CONSTANT, NO TREND A(1)=0 T-TEST -1.5324 -2.57 A(0)=A(1)=0 1.8752 3.78 AIC = 2.678 SC = 2.888 --------------------------------------------------------------------------- CONSTANT, TREND A(1)=0 T-TEST -1.9984 -3.13 A(0)=A(1)=A(2)=0 2.2216 4.03 A(1)=A(2)=0 2.6252 5.34 AIC = 2.673 SC = 2.898 ---------------------------------------------------------------------------

  19. Dickey-Fuller on First Difference-PPI VARIABLE : DIFFPPI DICKEY-FULLER TESTS - NO.LAGS = 14 NO.OBS = 226 NULL TEST ASY. CRITICAL HYPOTHESIS STATISTIC VALUE 10% --------------------------------------------------------------------------- CONSTANT, NO TREND A(1)=0 T-TEST -4.2399 -2.57 A(0)=A(1)=0 8.9971 3.78 AIC = -1.299 SC = -1.057 --------------------------------------------------------------------------- CONSTANT, TREND A(1)=0 T-TEST -4.0255 -3.13 A(0)=A(1)=A(2)=0 5.9875 4.03 A(1)=A(2)=0 8.9725 5.34 AIC = -1.291 SC = -1.033 ---------------------------------------------------------------------------

  20. Trend Stationary vs Difference Stationary

  21. Spurious Regression

  22. Relate Price level to Money Supply Note: For this regression R-square=0.888162964 And DW = 0.028682 We have to fear a Spurious regression.

  23. Dickey-Fuller Test

  24. Cointegration • We can have two variables trending upward in a stochastic fashion, they seem to be trending together. The movement resembles two dancing partners, each following a random walk, whose random walks seem to be unison. • Synchrony is intuitively the idea behind cointegrated time series.

  25. Cointegration

  26. Cointegration • We need to check the residuals from our regression to see if they are I(0). • If the residuals are I(0) or stationary, the traditional regression methodology (including t and f tests) that we have learned so far is applicable to data involving time series.

  27. Test for Cointegration

  28. Cointegrating regression: PPI and M1 OINTEGRATING REGRESSION - CONSTANT, NO TREND NO.OBS = 242 REGRESSAND : PPIACO R-SQUARE = 0.8882 DURBIN-WATSON = 0.2868E-01 DICKEY-FULLER TESTS ON RESIDUALS - NO.LAGS = 14 M = 2 TEST ASY. CRITICAL STATISTIC VALUE 10% --------------------------------------------------------------------------- NO CONSTANT, NO TREND T-TEST -2.4007 -3.04 AIC = -1.200 SC = -0.974 ---------------------------------------------------------------------------

  29. Error Correction Model

  30. Error Correction model: exchange rate & interest Rate Regression of exchange rate on interest rate Error Correction Model

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