Nonstationary Time Series Data and Cointegration ECON 6002 Econometrics Memorial University of...

ECON 6002Econometrics Memorial University of Newfoundland

Adapted from Vera Tabakova’s notes

12.1 Stationary and Nonstationary Variables

12.2 Spurious Regressions

12.3 Unit Root Tests for Stationarity

12.4 Cointegration

12.5 Regression When There is No Cointegration

Figure 12.1(a) US economic time series

Yt-Y t-1On the right hand side

“Differenced series”

Fluctuates about a rising trend

Fluctuates about a zero mean

Figure 12.1(b) US economic time series

Yt-Y t-1On the right hand side

“Differenced series”

tE y

2var ty

cov , cov ,t t s t t s sy y y y

Stationary if:

1 , 1t t ty y v

Each realization of the process has a proportion rho of the previous one plus an error drawn from a distribution with mean zero and variance sigma squared

It can be generalised to a higher autocorrelation order

We just show AR(1)

1 , 1t t ty y v

1 0 1

22 1 2 0 1 2 0 1 2

21 2 0

( )

..... tt t t t

y y v

y y v y v v y v v

y v v v y

We can show that the constant mean of this series is zero

21 2[ ] [ .....] 0t t t tE y E v v v

1( ) ( )t t ty y v

1 , 1t t ty y v

We can also allow for a non-zero mean, by replacing yt with yt-mu

Which boils down to (using alpha = mu(1-rho))

( ) / (1 ) 1/ (1 0.7) 3.33tE y

1( ) ( ( 1)) , 1 t t ty t y t v

1t t ty y t v

Or we can allow for a AR(1) with a fluctuation around a linear trend (mu+delta times t)

The “de-trended” model , which is now stationary, behaves like an autoregressive model:

With alpha =(mu(1-rho)+rho times delta)And lambda = delta(1-rho)

Figure 12.2 (a) Time Series Models

Figure 12.2 (b) Time Series Models

Figure 12.2 (c) Time Series Models

1t t ty y v

1 0 1

2

2 1 2 0 1 2 01

1 01

( )

ss

t

t t t ss

y y v

y y v y v v y v

y y v y v

The first component is usually just zero, since it is so far in thepast that it has a negligible effect nowThe second one is the stochastictrend

A random walk is non-stationary, although the mean is constant:

0 1 2 0( ) ( ... )t tE y y E v v v y

21 2var( ) var( ... )t t vy v v v t

A random walk with a drift both wanders and trends:

1 0 1

2

2 1 2 0 1 2 01

1 01

( ) 2

ss

t

t t t ss

y y v

y y v y v v y v

y y v t y v

1t t ty y v

0 1 2 3 0( ) ( ... )t tE y t y E v v v v t y

21 2 3var( ) var( ... )t t vy v v v v t

1 1 1

2 1 2

:

:

t t t

t t t

rw y y v

rw x x v

21 217.818 0.842 , .70

( ) (40.837)t trw rw R

t

Both independent and artificially generated, but…

Figure 12.3 (a) Time Series of Two Random Walk Variables

Figure 12.3 (b) Scatter Plot of Two Random Walk Variables

Dickey-Fuller Test 1 (no constant and no trend)

1t t ty y v

1 1 1

1

1

1

t t t t t

t t t

t t

y y y y v

y y v

y v

Dickey-Fuller Test 1 (no constant and no trend)

0 0

1 1

: 1 : 0

: 1 : 0

H H

H H

Easier way to test the hypothesis about rho

Remember that the null is a unit root: nonstationarity!

Dickey-Fuller Test 2 (with constant but no trend)

1t t ty y v

Dickey-Fuller Test 3 (with constant and with trend)

1t t ty y t v

First step: plot the time series of the original observations on the

variable.

If the series appears to be wandering or fluctuating around a sample

average of zero, use Version 1

If the series appears to be wandering or fluctuating around a sample

average which is non-zero, use Version 2

If the series appears to be wandering or fluctuating around a linear

trend, use Version 3

An important extension of the Dickey-Fuller test allows for the

possibility that the error term is autocorrelated.

The unit root tests based on (12.6) and its variants (intercept excluded

or trend included) are referred to as augmented Dickey-Fuller tests.

11

m

t t s t s ts

y y a y v

1 1 2 2 2 3, ,t t t t t ty y y y y y

1 1

1 1

0.178 0.037 0.672

( ) ( 2.090)

0.285 0.056 0.315

( ) ( 1.976)

t t t

t t t

F F F

tau

B B B

tau

F = US Federal funds interest rate

B = 3-year bonds interest rate

In STATA:

use usa, cleargen date = q(1985q1) + _n - 1format %tq datetsset date

TESTING UNIT ROOTS “BY HAND”:* Augmented Dickey Fuller Regressionsregress D.F L1.F L1.D.Fregress D.B L1.B L1.D.B

In STATA:

TESTING UNIT ROOTS “BY HAND”:* Augmented Dickey Fuller Regressionsregress D.F L1.F L1.D.Fregress D.B L1.B L1.D.B

_cons .1778617 .1007511 1.77 0.082 -.0228016 .378525 LD. .6724777 .0853664 7.88 0.000 .5024559 .8424996 L1. -.0370668 .0177327 -2.09 0.040 -.0723847 -.001749 F D.F Coef. Std. Err. t P>|t| [95% Conf. Interval]

Total 17.5433842 78 .224915182 Root MSE = .35436 Adj R-squared = 0.4417 Residual 9.54348876 76 .12557222 R-squared = 0.4560 Model 7.99989546 2 3.99994773 Prob > F = 0.0000 F( 2, 76) = 31.85 Source SS df MS Number of obs = 79

. regress D.F L1.F L1.D.F

In STATA:

Augmented Dickey Fuller Regressions with built in functionsdfuller F, regress lags(1)dfuller B, regress lags(1)

Choice of lags if we want to allow For more than a AR(1) order

In STATA:

Augmented Dickey Fuller Regressions with built in functionsdfuller F, regress lags(1)


MacKinnon approximate p-value for Z(t) = 0.2484 Z(t) -2.090 -3.539 -2.907 -2.588 Statistic Value Value Value Test 1% Critical 5% Critical 10% Critical Interpolated Dickey-Fuller

Augmented Dickey-Fuller test for unit root Number of obs = 79

. dfuller F, regress lags(1)





In STATA:


Alternative: pperron uses Newey-West standard errors to account for serial correlation, whereas the augmented Dickey-Fuller test implemented in dfuller uses additional lags of the first-difference variable.

Also consider now using DFGLS (Elliot Rothenberg and Stock, 1996) to counteract problems of lack of power in small samples. It also has in STATA a lag selection procedure based on a sequential t test suggeste by Ng and Perron (1995)

In STATA:


Alternatives: use tests with stationarity as the nullKPSS (Kwiatowski, Phillips, Schmidt and Shin. 1992) which also has an automatic bandwidth selection tool or the Leybourne & McCabe test .

1t t t ty y y v

The first difference of the random walk is stationary

It is an example of a I(1) series (“integrated of order 1”First-differencing it would turn it into I(0) (stationary)

In general, the order of integration is the minimum number of times a series must be differenced to make it stationarity

1t t t ty y y v

10.340

( ) ( 4.007)

t tF F

tau

10.679

( ) ( 6.415)

t tB B

tau

So now we reject the Unit root after differencingonce:We have a I(1) series

In STATA:

ADF on differencesdfuller D.F, noconstant lags(0)dfuller D.B, noconstant lags(0)





Z(t) -4.007 -2.608 -1.950 -1.610 Statistic Value Value Value Test 1% Critical 5% Critical 10% Critical Interpolated Dickey-Fuller

Dickey-Fuller test for unit root Number of obs = 79

. dfuller D.F, noconstant lags(0)

1ˆ ˆt t te e v

ˆ1: t t tCase e y bx

2 1ˆ2 : t t tCase e y b x b

2 1ˆˆ3: t t tCase e y b x b t

Not the same as for dfuller, since the residuals are estimated errors no actual ones

Note: unfortunately STATA dfuller will not notice and give you erroneous critical values!

2ˆ 1.644 0.832 , 0.881

( ) (8.437) (24.147)t tB F R

t

1 1ˆ ˆ ˆ0.314 0.315

( ) ( 4.543)t t te e e

tau

Z(t) -4.543 -2.608 -1.950 -1.610 Statistic Value Value Value Test 1% Critical 5% Critical 10% Critical Interpolated Dickey-Fuller


. dfuller ehat, noconstant lags(1)

Check: These are wrong!

The null and alternative hypotheses in the test for cointegration are:

0

1

: the series are not cointegrated residuals are nonstationary

: the series are cointegrated residuals are stationary

H

H

12.5.1 First Difference Stationary

The variable yt is said to be a first difference stationary series.

Then we revert to the techniques we saw in Ch. 9

1t t ty y v

1t t t ty y y v

1 0 1 1t t t t ty y x x e

1t t ty y v

t ty v


Manipulating this one you can construct and Error Correction Modelto investigate the SR dynamics of the relationship between y and x

where

and

t ty t v

t ty t v


1 0 1 1t t t t ty t y x x e

1 1 2 0 1 1 1 1 2(1 ) ( )

1 1 2 0 1(1 ) ( )

To summarize:

If variables are stationary, or I(1) and cointegrated, we can estimate a regression relationship between the levels of those variables without fear of encountering a spurious regression.

Then we can use the lagged residuals from the cointegrating regression in an ECM model

This is the best case scenario, since if we had to first-differentiate the variables, we would be throwing away the long-run variation

Additionally, the cointegrated regression yields a “superconsistent” estimator in large samples

To summarize:

If the variables are I(1) and not cointegrated, we need to estimate a relationship in first differences, with or without the constant term.

If they are trend stationary, we can either de-trend the series first and then perform regression analysis with the stationary (de-trended) variables or, alternatively, estimate a regression relationship that includes a trend variable. The latter alternative is typically applied.

47Principles of Econometrics, 3rd Edition


Kit Baum has really good notes on these topics that can be used to learn also about extra STATA commands to handle the analysis:

http://fmwww.bc.edu/ec-c/s2003/821/ec821.sect05.nn1.pdf


For example, some of you should look at seasonal unit root analysis (command HEGY in STATA)

Panel unit roots would be here


Further issues


You may want to some time consider unit root tests that allow for structural Breaks

You can also take a look at the literature review in this working paper:

http://ideas.repec.org/p/wpa/wuwpot/0410002.html

Further issues


When you deal with more than 2 regressors you should consider the Johansen’s method to examine the cointegration relationships

Further issues

Nonstationary Time Series Data and Cointegration ECON 6002 Econometrics Memorial University of...

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