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**Lecture 24 Univariate Time Series**

Economics 310 Lecture 24 Univariate Time Series

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**Concepts to be Discussed**

Time Series Stationarity Spurious regression Trends

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**Plot of Economic Levels Data**

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Plot of Rate Data

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**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.

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**Stationary Stochastic Process**

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**Test for Stationarity: Correlogram**

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Correlogram for PPI AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) PPIACO RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR. RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR. RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRR

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**Correlogram M1 AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) M1**

RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR. RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR. RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR. RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRR . RRRRRRRRRRRRRRRRRRRRRRRRRRRRR .

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**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

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**Testing Autocorrelation coefficients**

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**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

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**Unit Root Test for Stationarity**

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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).

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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.

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Dickey-Fuller Test

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**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 =

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**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 =

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**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 =

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**Trend Stationary vs Difference Stationary**

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Spurious Regression

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**Relate Price level to Money Supply**

Note: For this regression R-square= And DW = We have to fear a Spurious regression.

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Dickey-Fuller Test

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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.

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Cointegration

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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.

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**Test for Cointegration**

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**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 =

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**Error Correction Model**

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**Error Correction model: exchange rate & interest Rate**

Regression of exchange rate on interest rate Error Correction Model

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