Economics 310

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 .
RRRRRRRRRRRRRRRRRRRRR
RRRRRRRRRRRR . 4 0.93 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 5
0.91 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 6 0.90 .
RRRRRRRRRRRRRRRRRRRRR
RRRRRRRRRR . 7 0.88 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 8
0.87 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 9 0.85 .
RRRRRRRRRRRRRRRRRRRRR
RRRRRRRRR . 10 0.84 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 11
0.83 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 12 0.81 .
RRRRRRRRRRRRRRRRRRRRR
RRRRRRRR . 13 0.80 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRR . 14
0.79 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRR . 15 0.78 .
RRRRRRRRRRRRRRRRRRRRR
RRRRRRR . 16 0.77 .
RRRRRRRRRRRRRRRRRRRRRRRRRRR . 17
0.76 .
RRRRRRRRRRRRRRRRRRRRRRRRRRR . 18 0.75 .
RRRRRRRRRRRRRRRRRRRRR
RRRRRR .
9
Correlogram M1
AUTOCORRELATION FUNCTION OF THE SERIES (1-B)
(1-B ) M1 1 0.99 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR 2
0.98 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR. 3 0.97 .
RRRRRRRRRRRRRRRRRRRRR
RRRRRRRRRRRRR. 4 0.96 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR. 5
0.95 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 6 0.94 .
RRRRRRRRRRRRRRRRRRRRR
RRRRRRRRRRRR . 7 0.93 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 8
0.92 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 9 0.91 .
RRRRRRRRRRRRRRRRRRRRR
RRRRRRRRRRR . 10 0.90 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 11
0.89 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 12 0.88 .
RRRRRRRRRRRRRRRRRRRRR
RRRRRRRRRR . 13 0.87 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 14
0.86 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 15 0.85 .
RRRRRRRRRRRRRRRRRRRRR
RRRRRRRRR . 16 0.84 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 17
0.83 .
RRRRRRRRRRRRRRRRRRRRRRRRRRRRR . 18 0.81 .
RRRRRRRRRRRRRRRRRRRRR
RRRRRRRR .
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 sd0.064, and 5 test cut off
  is 0.126
• For our rate data, sd0.059, 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
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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 -----------------------------
----------------------------------------------
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-square0.888162964 An
d 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
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Error Correction model exchange rate interest
Rate
Regression of exchange rate on interest rate
Error Correction Model