1. Lecture 24 Univariate Time Series
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
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9. Correlogram M1 AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) M1
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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= VARIANCE= STANDARD DEV.=
LAGS AUTOCORRELATIONS STD ERR
MODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS (CHI-SQUARE)
LAG Q DF P-VALUE LAG Q DF P-VALUE

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 = NO.OBS = 227
NULL TEST ASY. CRITICAL
HYPOTHESIS STATISTIC VALUE 10%
CONSTANT, NO TREND
A(1)=0 T-TEST
A(0)=A(1)=
AIC =
SC =
CONSTANT, TREND
A(1)=0 T-TEST
A(0)=A(1)=A(2)=
A(1)=A(2)=
AIC =
SC =

18. Dickey-Fuller Test for our level data-M1
VARIABLE : M1
DICKEY-FULLER TESTS - NO.LAGS = NO.OBS = 229
NULL TEST ASY. CRITICAL
HYPOTHESIS STATISTIC VALUE 10%
CONSTANT, NO TREND
A(1)=0 T-TEST
A(0)=A(1)=
AIC =
SC =
CONSTANT, TREND
A(1)=0 T-TEST
A(0)=A(1)=A(2)=
A(1)=A(2)=
AIC =
SC =

19. Dickey-Fuller on First Difference-PPI
VARIABLE : DIFFPPI
DICKEY-FULLER TESTS - NO.LAGS = NO.OBS = 226
NULL TEST ASY. CRITICAL
HYPOTHESIS STATISTIC VALUE 10%
CONSTANT, NO TREND
A(1)=0 T-TEST
A(0)=A(1)=
AIC =
SC =
CONSTANT, TREND
A(1)=0 T-TEST
A(0)=A(1)=A(2)=
A(1)=A(2)=
AIC =
SC =

20. Trend Stationary vs Difference Stationary

21. Spurious Regression

22. Relate Price level to Money Supply
Note: For this regression R-square=
And DW =
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 = DURBIN-WATSON = E-01
DICKEY-FULLER TESTS ON RESIDUALS - NO.LAGS = 14 M = 2
TEST ASY. CRITICAL
STATISTIC VALUE 10%
NO CONSTANT, NO TREND
T-TEST
AIC =
SC =

29. Error Correction Model

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