Tut9Sol - Tut 9 PDF

Preview

Summary

Tut 9...

Description

Tutorial 9 (Week 10) Unit Root Tests

Tutorial Assignment: Show that when β=1 that the series Yt is made up of its past shocks et, et-1 , … for the following process: Yt = β Yt-1 + et [Hint: Use recursive substitution to express Yt in terms of its past shocks et, et-1,… For an idea on how to solve this, refer to the lecture slide on unit root test.] Consider Yt = β Yt-1 + et (1) Yt-1 = β Yt-2 + et-1 (2) Substitute (2) into (1), this yields Yt = β ( β Yt-2 + et-1 ) + et = β2 Yt-2 + β et-1 + et Substitute Yt-2 with Yt-2 = β Yt-3 + et-2 which gives Yt = β2 ( β Yt-3 + et-2 ) + β et-1 + et = β3 Yt-3 + β2 et-2 + β et-1 + et If we do this substitution repeatedly on the RHS of the equation for the lag of Y, we get Yt = βt-1 Yt-(t-1) + βt-2 et-(t-2) +…+ β2 et-2 + β et-1 + et Note that Yt-(t-1)=Y 1 and from equation (1), we know that Y1 = β Y0 + e1 . Assume that Y0=0 so that Y1 = e1. Hence, we can re-write Yt as the sum of all past shocks: Yt = βt-1 e1 + βt-2 e2 +…+ β2 et-2 + β et-1 + et Objective: Conduct unit root testing on financial series. Determine the order of integration. Data sets: SIM_1.xls Question One The file SIM_1.xls contains 1000 simulated observations on the variables Y1 and Y2. i)

Plot the Y1 series against time. Does the process appear to be stationary? File/New/Workfile ... Undated or irregular 1

1 to 1000 File/Import/Text-Lotus-Excel ... by observation upper left data cell = A2 names for series = 2 Quick/Graph/Line Graph... y1 20

16

12

8

4

0 250

500

750

1000

Y1

The series plot appears to be trending upwards– the series appears to be nonstationary.

ii)

Use an ADF test to help determine whether the process has a unit root. Use a 5% level of significance. View/Unit Root Test ... Augmented Dickey-Fuller Trend and Intercept

Null Hypothesis: Y1 has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=21)

Augmented Dickey-Fuller test statistic Test critical values: 1% level 5% level 10% level *MacKinnon (1996) one-sided p-values.

2

t-Statistic

Prob.*

-33.94069 -3.967261 -3.414318 -3.129280

0.0000

Augmented Dickey-Fuller Test Equation Dependent Variable: D(Y1) Method: Least Squares Date: 19/10/04 Time: 08:12 Sample (adjusted): 2 1000 Included observations: 999 after adjustments Variable

Coefficient

Std. Error

t-Statistic

Prob.

Y1(-1) C @TREND(1)

-1.074402 5.220032 0.010978

0.031655 0.166032 0.000342

-33.94069 31.44001 32.09481

0.0000 0.0000 0.0000

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

0.536309 0.535378 1.004393 1004.769 -1420.396 1.996795

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

0.008474 1.473512 2.849642 2.864377 575.9917 0.000000

The ADF test statistic is less than the 5% critical values (or the p-value is less than 0.05) so we reject the null hypothesis and conclude there is no unit root. Thus, the trend that is evident in the series plot must be deterministic. This is supported by the ADF test equation where the trend term is statistically significant.

iii)

Estimate an appropriate ARMA model for this process.

The ADF test suggests the process doesn't have a unit root. The ADF estimating equation (chosen using the BIC criterion) includes a constant term and a time trend, but no augmentation terms: ∆yt = µ + λt + πyt-1 + εt which can be written as an AR(1) model with a constant and a time trend: (1 – θL)yt = µ + λt + εt The EViews commands for estimating this model: Quick/Generate Series ... t = @trend(0) Quick/Estimate Equation ... y1 c t y1(-1)

3

Dependent Variable: Y1 Method: Least Squares Date: 19/10/04 Time: 08:20 Sample (adjusted): 2 1000 Included observations: 999 after adjustments Variable

Coefficient

Std. Error

t-Statistic

Prob.

C T Y1(-1)

5.209055 0.010978 -0.074402

0.165769 0.000342 0.031655

31.42355 32.09481 -2.350372

0.0000 0.0000 0.0189

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

0.896226 0.896018 1.004393 1004.769 -1420.396 1.996795

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

9.967808 3.114758 2.849642 2.864377 4300.904 0.000000

All coefficients are statistically significant at the 5% level. However, the coefficient of yt-1 is not significantly different from zero at the 1% level. If we decide to omit yt-1 and estimate the model: yt = µ + λt + εt we obtain the results: Dependent Variable: Y1 Method: Least Squares Date: 19/10/04 Time: 08:22 Sample: 1 1000 Included observations: 1000 Variable

Coefficient

Std. Error

t-Statistic

Prob.

C T

4.848702 0.010217

0.063684 0.000110

76.13688 92.69948

0.0000 0.0000

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood

0.895946 0.895842 1.006177 1010.368 -1424.096

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic

4

9.962538 3.117656 2.852192 2.862008 8593.194

Durbin-Watson stat

2.144662

Prob(F-statistic)

0.000000

The true data generating process (DGP) for the Y1 process was model B with µ = 5 and λ = 0.01. Our estimated model is close to the true DGP. iv)

Repeat parts i) to iii) using the Y2 series. 50 0 -50 -100 -150 -200 -250 -300 250

500

750

1000

Y2

Like the Y1 series, the mean of the Y2 series appears to be changing over time – the series appears to be nonstationary. The EViews commands for the first step in the ADF test procedure: View/Unit Root Test… Augmented Dickey Fuller Trend and Intercept

Null Hypothesis: Y2 has a unit root Exogenous: Constant, Linear Trend Lag Length: 1 (Automatic based on SIC, MAXLAG=21)

Augmented Dickey-Fuller test statistic Test critical values: 1% level 5% level 10% level *MacKinnon (1996) one-sided p-values.

5

t-Statistic

Prob.*

-1.957363 -3.967270 -3.414323 -3.129283

0.6233

The prob-value is greater than 0.05 so we fail to reject Ho and proceed to the second step in the ADF test procedure.

View/Unit Root Test… Augmented Dickey Fuller Intercept Null Hypothesis: Y2 has a unit root Exogenous: Constant Lag Length: 1 (Automatic based on SIC, MAXLAG=21)

Augmented Dickey-Fuller test statistic Test critical values: 1% level 5% level 10% level

t-Statistic

Prob.*

-1.565801 -3.436683 -2.864225 -2.568251

0.4998

*MacKinnon (1996) one-sided p-values.

Once again, the p-value is larger than 0.05, so we fail to reject the null hypothesis of a unit root and proceed to the last step of the ADF test procedure: View/Unit Root Test… Augmented Dickey Fuller None

Null Hypothesis: Y2 has a unit root Exogenous: None Lag Length: 1 (Automatic based on SIC, MAXLAG=21)

Augmented Dickey-Fuller test statistic Test critical values: 1% level 5% level 10% level *MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation Dependent Variable: D(Y2) Method: Least Squares Date: 19/10/04 Time: 08:28 Sample (adjusted): 3 1000 Included observations: 998 after adjustments

6

t-Statistic

Prob.*

0.092291 -2.567281 -1.941141 -1.616486

0.7118

Variable Y2(-1) D(Y2(-1)) R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood

Coefficient

Std. Error

t-Statistic

Prob.

2.39E-05 0.894291

0.000259 0.014393

0.092291 62.13404

0.9265 0.0000

0.795491 0.795286 1.016724 1029.594 -1431.653

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Durbin-Watson stat

-0.285250 2.247135 2.873051 2.882883 2.043699

Again, the prob-value is greater than 0.05 so we fail to reject H0 and conclude that the process has a unit root. Now that it has been concluded that Y2 has a unit root, first differences should be taken and the ADF test procedure should be repeated. Quick/Generate Series… dY2 = y2 – y2(-1) 8 6 4 2 0 -2 -4 -6 -8 250

500

750

1000

DY2

The series plot doesn't trend or meander too much – the first differenced series appears stationary. View/Unit Root Test… Augmented Dickey Fuller Trend and Intercept Null Hypothesis: DY2 has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=21)

Augmented Dickey-Fuller test statistic

7

t-Statistic

Prob.*

-7.505016

0.0000

Test critical values:

1% level 5% level 10% level

-3.967270 -3.414323 -3.129283

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation Dependent Variable: D(DY2) Method: Least Squares Date: 19/10/04 Time: 08:34 Sample (adjusted): 3 1000 Included observations: 998 after adjustments Variable

Coefficient

Std. Error

t-Statistic

Prob.

DY2(-1) C @TREND(1)

-0.107749 0.002005 -6.87E-05

0.014357 0.064512 0.000112

-7.505016 0.031075 -0.614287

0.0000 0.9752 0.5392

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

0.053622 0.051720 1.016542 1028.191 -1430.972 2.042256

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

-0.001844 1.043895 2.873692 2.888439 28.18860 0.000000

A p-value of 0.000 indicates that we should reject the null hypothesis and conclude that the series does not have a unit root. The trend term in the ADF test equation is statistically insignificant – the differenced series dY2 is stationary according to the ADF test. The ADF test equation indicates that dY2 should be modelled as an AR(1) process:

Dependent Variable: DY2 Method: Least Squares Date: 19/10/04 Time: 08:41 Sample (adjusted): 3 1000 Included observations: 998 after adjustments Variable

Coefficient

Std. Error

t-Statistic

Prob.

C DY2(-1)

-0.032252 0.892705

0.032423 0.014333

-0.994705 62.28160

0.3201 0.0000

R-squared Adjusted R-squared S.E. of regression

0.795692 0.795487 1.016224

Mean dependent var S.D. dependent var Akaike info criterion

8

-0.285250 2.247135 2.872067

Sum squared resid Log likelihood Durbin-Watson stat

1028.581 -1431.161 2.042412

Schwarz criterion F-statistic Prob(F-statistic)

2.881898 3878.997 0.000000

The coefficient of the constant is not significantly different from zero. If we estimate the model without an intercept we obtain: Dependent Variable: DY2 Method: Least Squares Date: 19/10/04 Time: 08:42 Sample (adjusted): 3 1000 Included observations: 998 after adjustments Variable

Coefficient

Std. Error

t-Statistic

Prob.

DY2(-1)

0.894492

0.014220

62.90216

0.0000

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood

0.795489 0.795489 1.016219 1029.603 -1431.657

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Durbin-Watson stat

-0.285250 2.247135 2.871056 2.875972 2.044045

The true data generating process for Y2 was the nonstationary AR(2) process: yt = 1.9yt-1 – 0.9yt-2 + εt with first-differences: ∆yt = 0.9∆yt-1 + εt Once again, our estimated model is close to the true DGP.

9...