5Stationary Stochastic Process Stochastic Random ProcessRealizationA 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.
8Correlogram for PPIAUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) PPIACORRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR.RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR.RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
9Correlogram M1 AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) M1 RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR.RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR.RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR.RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRR .RRRRRRRRRRRRRRRRRRRRRRRRRRRRR .
10Testing autocorrelation coefficients If data is white noise, the sample autocorrelation coefficient is normally distributed with mean zero and variance ~ 1/nFor our levels data sd=0.064, and 5% test cut off is 0.126For our rate data, sd=0.059, and 5% test cut off is 0.115
12Ljung-Box test for PPI SERIES (1-B) (1-B ) PPIACO NET NUMBER OF OBSERVATIONS = 242MEAN= VARIANCE= STANDARD DEV.=LAGS AUTOCORRELATIONS STD ERRMODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS (CHI-SQUARE)LAG Q DF P-VALUE LAG Q DF P-VALUE
14Results of our testIf 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).
15Testing for unit rootIn 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.
17Dickey-Fuller Test for our level data-PPI |_coint ppiaco m1 employ...NOTE..SAMPLE RANGE SET TO: 1, 242...NOTE..TEST LAG ORDER AUTOMATICALLY SETTOTAL NUMBER OF OBSERVATIONS = 242VARIABLE : PPIACODICKEY-FULLER TESTS - NO.LAGS = NO.OBS = 227NULL TEST ASY. CRITICALHYPOTHESIS STATISTIC VALUE 10%CONSTANT, NO TRENDA(1)=0 T-TESTA(0)=A(1)=AIC =SC =CONSTANT, TRENDA(1)=0 T-TESTA(0)=A(1)=A(2)=A(1)=A(2)=AIC =SC =
18Dickey-Fuller Test for our level data-M1 VARIABLE : M1DICKEY-FULLER TESTS - NO.LAGS = NO.OBS = 229NULL TEST ASY. CRITICALHYPOTHESIS STATISTIC VALUE 10%CONSTANT, NO TRENDA(1)=0 T-TESTA(0)=A(1)=AIC =SC =CONSTANT, TRENDA(1)=0 T-TESTA(0)=A(1)=A(2)=A(1)=A(2)=AIC =SC =
19Dickey-Fuller on First Difference-PPI VARIABLE : DIFFPPIDICKEY-FULLER TESTS - NO.LAGS = NO.OBS = 226NULL TEST ASY. CRITICALHYPOTHESIS STATISTIC VALUE 10%CONSTANT, NO TRENDA(1)=0 T-TESTA(0)=A(1)=AIC =SC =CONSTANT, TRENDA(1)=0 T-TESTA(0)=A(1)=A(2)=A(1)=A(2)=AIC =SC =
24CointegrationWe 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.
26CointegrationWe 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.