Economics 310

Lecture 24Univariate Time Series

Concepts to be Discussed Time Series Stationarity Spurious regression Trends

Plot of Economic Levels Data

PPI, M1, Employment

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Plot of Rate DataExchange Rate & Interest Rate

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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.

Stationary Stochastic Process

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Correlogram for PPIAUTOCORRELATION 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 .

Correlogram M1AUTOCORRELATION 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 .

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

Testing Autocorrelation coefficients

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Ljung-Box test for PPISERIES (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

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).

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.

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 = 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 ---------------------------------------------------------------------------

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 ---------------------------------------------------------------------------

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 ---------------------------------------------------------------------------

Trend Stationary vs Difference Stationary

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Relate Price level to Money Supply

Coefficients Standard Error t StatIntercept 78.34396361 0.809216059 96.81464M1 0.041353791 0.000947232 43.65753

Note: For this regression R-square=0.888162964

And DW = 0.028682

We have to fear a Spurious regression.

Dickey-Fuller Test

Coefficients Standard Error t StatIntercept 3.9116703 1.139598437 3.432499Trend 0.0050264 0.002010218 2.500442ppilag -0.0387279 0.012275819 -3.15481

Coefficients Standard Error t StatIntercept 2.017339156 1.912609208 1.054758trend -0.040007924 0.017782745 -2.24982m1lag 0.007145816 0.004785534 1.493212

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.

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.

Test for Cointegration

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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 ---------------------------------------------------------------------------

Error Correction Model

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Error Correction model: exchange rate & interest Rate

Coefficients Standard Error t StatIntercept 84.4016519 1.934688878 43.62544BA6M 1.82542313 0.24291891 7.514537

Regression of exchange rate on interest rate

Error Correction Model

Coefficients Standard Error t StatIntercept -0.0369947 0.095387016 -0.38784diffintr 0.7803093 0.153370941 5.087726Residuals -0.0153463 0.007787867 -1.97054