Ch21_TimeSeries1

By Tatre Jantarakolica

Time Series Econometrics 1. Nonstochastic Processes Random Walk without Drift: 1 1 1t t tY Y u−= + where: ut is white noise error term with mean = 0 and variance = σ2 Random Walk with Drift: 2 2 1t t tY Y uδ −= + +

EView’s Command

smpl 1 1 genr y1=0 genr y2=0 smpl 1 200 genr u=nrnd smpl 2 200 series y1=y1(-1)+u series y2=0.5+y2(-1)+u smpl 1 200 plot y1 y2

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2. Trend Stationary (TS) and Difference Stationary (DS) Pure random walk β1 = 0 , β2 = 0, β3 = 1

1 1 1t t tY Y u−= + Random walk with drift β1 ≠ 0 , β2 = 0, β3 = 1

2 1 2 1t t tY Y uβ −= + + Deterministic trend β1 ≠ 0 , β2 ≠ 0, β3 = 0

3 1 2t tY t uβ β= + + Random walk with drift and deterministic trend β1 ≠ 0 , β2 ≠ 0, β3 = 1

4 1 2 4 1t t tY t Y uβ β −= + + + Deterministic trend with stationary AR(1) component β1 ≠ 0 , β2 ≠ 0, β3 < 1

5 1 2 3 5 1t t tY t Y uβ β β −= + + +

EView’s Command

smpl 1 1 genr y1=0 genr y2=0 genr y3=0 genr y4=0 genr y5=0 smpl 1 200 genr u=nrnd genr t=@trend smpl 2 200 series y1=y1(-1)+u series y2=0.5+y2(-1)+u series y3=0.5+0.5*t+u series y4=0.5+0.5*t+y4(-1)+u series y5=0.5+0.5*t+0.9*y5(-1)+u smpl 1 200 genr yhat = 0.5+0.5*t

plot y1 y2 y3 yhat plot y1 y4 y5


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3. Spurious Regression 1t t tY Y u−= + 1t t tX X v−= + If Xt and Yt are uncorrelated I(1) processes, regression 1 2t t tY Xβ β ε= + + can lead to spurious problem.

EView’s Command

smpl 1 1 genr y=0 genr x=0 smpl 1 500 genr u=nrnd genr v=nrnd smpl 2 500 series y=y(-1)+u series x=x(-1)+v smpl 1 500 equation eq_21_7.ls y c x

Dependent Variable: Y Method: Least Squares Sample: 1 200 Included observations: 200

Variable Coefficient Std. Error t-Statistic Prob. C -4.282010 0.418138 -10.24067 0.0000 X 0.492647 0.043581 11.30409 0.0000

R-squared 0.392232 Mean dependent var -0.287086 Adjusted R-squared 0.389163 S.D. dependent var 4.043829 S.E. of regression 3.160498 Akaike info criterion 5.149286 Sum squared resid 1977.772 Schwarz criterion 5.182269 Log likelihood -512.9286 F-statistic 127.7824 Durbin-Watson stat 0.122278 Prob(F-statistic) 0.000000


4. Unit Root Test Example Autoregressive (Stationary) Series X1t = 0.05 + 0.95 X1t-1 + u1t Random Walk with Drift X2t = 0.05 + X2t-1 + u2t Explosive Series X3t = 0.05 + 1.05 X3t-1 + u3t

EView’s Command

smpl 1 1 genr x1 = 0 genr x2 = 0 genr x3 = 0 smpl 1 200 genr u1 = nrnd genr u2 = nrnd genr u3 = nrnd smpl 2 200 series x1 = 0.05+0.95*x1(-1)+u1 series x2 = 0.05+x2(-1)+u2 series x3 = 0.05+1.05*x3(-1)+u3 plot x1 plot x2 plot x3

Graphical Plot

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Autocorrelation Function (ACF) and Correlogram From series Windows (e.g. series X1), go to View and choose Correlogram … Then, choose number of Lags.


Result of Stationary Time Series

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Result of Nonstationary Time Series

Unit Root Test – ADF Test From series Windows, go to View and choose Unit Root Test…


From Unit Root Test Window, choose Test type (in this case, Augmented Dickey-Fuller), level of test or Test for unit root in (in this case, Level), equation form or Include in test equation (in this case, Trend and intercept), and Lag length (in this case, 1) or suggested by Schwartz Information Criterion by choose Automatic selection.

Unit Root Test of Stationary Series (X1) – Level Test Null Hypothesis: X1 has a unit root Exogenous: Constant, Linear Trend Lag Length: 1 (Fixed)

t-Statistic Prob.* Augmented Dickey-Fuller test statistic -7.406746 0.0000 Test critical values: 1% level -4.005076

5% level -3.432682 10% level -3.140127

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

Augmented Dickey-Fuller Test Equation Dependent Variable: D(X1) Method: Least Squares Sample(adjusted): 3 200 Included observations: 198 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. X1(-1) -0.531530 0.071763 -7.406746 0.0000

D(X1(-1)) 0.060736 0.071894 0.844803 0.3993 C 0.095208 0.141188 0.674332 0.5009

@TREND(2) 0.000561 0.001227 0.456875 0.6483 R-squared 0.253346 Mean dependent var -0.002534 Adjusted R-squared 0.241800 S.D. dependent var 1.131346 S.E. of regression 0.985116 Akaike info criterion 2.827880 Sum squared resid 188.2679 Schwarz criterion 2.894310 Log likelihood -275.9601 F-statistic 21.94193 Durbin-Watson stat 1.990704 Prob(F-statistic) 0.000000


Unit Root Test of Nonstationary Series (X2) – Level Test Null Hypothesis: X2 has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=14)


5% level -3.432566 10% level -3.140059


Augmented Dickey-Fuller Test Equation Dependent Variable: D(X2) Method: Least Squares Sample: 2 200 Included observations: 199

Variable Coefficient Std. Error t-Statistic Prob. X2(-1) -0.056878 0.021507 -2.644656 0.0088

C 0.635091 0.213990 2.967855 0.0034 @TREND(2) 0.004893 0.002407 2.032962 0.0434

R-squared 0.035489 Mean dependent var 0.152559 Adjusted R-squared 0.025647 S.D. dependent var 1.011570 S.E. of regression 0.998513 Akaike info criterion 2.849862 Sum squared resid 195.4177 Schwarz criterion 2.899510 Log likelihood -280.5613 F-statistic 3.605911 Durbin-Watson stat 2.039938 Prob(F-statistic) 0.028980

Unit Root Test of I(1) Series (X2) – First Difference Test


Null Hypothesis: D(X2) has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=14)


5% level -3.432682 10% level -3.140127


Augmented Dickey-Fuller Test Equation Dependent Variable: D(X2,2) Method: Least Squares Sample(adjusted): 3 200 Included observations: 198 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. D(X2(-1)) -1.043168 0.071582 -14.57305 0.0000

C 0.215335 0.146020 1.474690 0.1419 @TREND(2) -0.000577 0.001266 -0.455705 0.6491


5. Cointegration Test Example: PCEt = β1 + β2PDIt + ut where PCEt and PDIt are both nonstationary and I(1) processes.

Then, if PCEt and PDIt are cointegrated series, ut = PCEt - β1 - β1PDIt must be stationary. Stationarity Test – Unit Root Test Result Null Hypothesis: PCE has a unit root Exogenous: Constant Lag Length: 0 (Automatic based on SIC, MAXLAG=11)


5% level -2.895109 10% level -2.584738



Null Hypothesis: D(PCE) has a unit root Exogenous: Constant Lag Length: 0 (Automatic based on SIC, MAXLAG=11)


5% level -2.895512 10% level -2.584952

*MacKinnon (1996) one-sided p-values. Null Hypothesis: PDI has a unit root Exogenous: Constant Lag Length: 0 (Automatic based on SIC, MAXLAG=11)


5% level -2.895109 10% level -2.584738

*MacKinnon (1996) one-sided p-values. Null Hypothesis: D(PDI) has a unit root Exogenous: Constant Lag Length: 0 (Automatic based on SIC, MAXLAG=11)


5% level -2.895512 10% level -2.584952

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

EView’s Command

equation eq1.ls pce c pdi genr uhat=resid

Dependent Variable: PCE Method: Least Squares Sample: 1970:1 1991:4 Included observations: 88

Variable Coefficient Std. Error t-Statistic Prob. C -171.4412 22.91725 -7.480880 0.0000

PDI 0.967250 0.008069 119.8712 0.0000 R-squared 0.994051 Mean dependent var 2537.042 Adjusted R-squared 0.993981 S.D. dependent var 463.1134 S.E. of regression 35.92827 Akaike info criterion 10.02339 Sum squared resid 111012.3 Schwarz criterion 10.07969 Log likelihood -439.0292 F-statistic 14369.10 Durbin-Watson stat 0.531629 Prob(F-statistic) 0.000000


Cointegration Test by Testing Stationarity of Residual Null Hypothesis: UHAT has a unit root Exogenous: Constant Lag Length: 0 (Automatic based on SIC, MAXLAG=11)


5% level -2.895109 10% level -2.584738

*MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(UHAT) Method: Least Squares Sample(adjusted): 1970:2 1991:4 Included observations: 87 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. UHAT(-1) -0.275357 0.073267 -3.758250 0.0003

C -0.441396 2.615711 -0.168748 0.8664 R-squared 0.142492 Mean dependent var -0.405877 Adjusted R-squared 0.132404 S.D. dependent var 26.19315 S.E. of regression 24.39757 Akaike info criterion 9.249565 Sum squared resid 50595.53 Schwarz criterion 9.306252 Log likelihood -400.3561 F-statistic 14.12444 Durbin-Watson stat 2.278168 Prob(F-statistic) 0.000313

Error Correction Mechanism (ECM)

ΔPCEt = α0 + α1ΔPDIt + α2 ut + εt

EView’s Command equation eq2.ls d(pce) c d(pdi) uhat(-1)

Dependent Variable: D(PCE) Method: Least Squares Sample(adjusted): 1970:2 1991:4 Included observations: 87 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob. C 11.69183 2.195675 5.324936 0.0000

D(PDI) 0.290602 0.069660 4.171715 0.0001 UHAT(-1) -0.086706 0.054180 -1.600311 0.1133



6. Autoregressive Integrated Moving Average (ARIMA) Models Example MA(1) Y1t = 0.1 + u1t+ 0.5 u1t-1 AR(1) Y2t = 0.1 + 0.5 Y2t-1 + u2t

ARMA(1,1) Y3t = 0.1 + 0.5 Y3t-1 + u3t + 0.5 u3t-1 ARIMA(1,1,1) ΔY4t = 0.1 + 0.5 ΔY4t-1 + u4t + 0.5 u4t-1

EView’s Command

smpl 1 1 genr y1 = 0 genr y2 = 0 genr y3 = 0 genr y4 = 0 genr dy4 =0 smpl 1 200 genr u1 = nrnd genr u2 = nrnd genr u3 = nrnd genr u4 = nrnd smpl 2 200 series y1 = 0.1+ u1 + 0.5*u1(-1) series y2 = 0.1 + 0.5*y2(-1) + u2 series y3 = 0.1 + 0.5*y3(-1) + u3 + 0.5*u3(-1) series y4 = 0.1 + y4(-1) + u4 + 0.5*u4(-1) genr dy4 = y4 – y4(-1) series y4 = 0.1 + y4(-1) + 0.5*dy4(-1) + u4 + 0.5*u4(-1) plot y1 y2 y3 y4


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Correlogram of MA(1)


Correlogram of AR(1)


Correlogram of ARMA(1,1)


Correlogram of ARIMA(1,1,1)


Example: Finding ARIMA process of Price Index (CMI) – Box-Jenkins Methodology Step 1 & 2:

Identification and Estimation: The first step is to identify the order of ARIMA(p,d,q). To find the value of p, d, and q, we should firstly identify value of d by performing unit root test to find which order the series is integrated. Unit Root Test Find integrated order of the series using unit root test.

From EViews, Click the series, go to View, then choose unit root (follow the step of testing unit root). From this example, CMI series is I(1). Thus, d = 1 or the model is ARIMA(p,1,q).


Null Hypothesis: CMI has a unit root Exogenous: Constant, Linear Trend Lag Length: 1 (Automatic based on SIC, MAXLAG=12)


5% level -3.455376 10% level -3.153438


Null Hypothesis: D(CMI) has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=12)


5% level -3.455376 10% level -3.153438


Identify p and q Next step is to identify order p and q of the model. We should first specify the

maximum lags length to limit the combination of specification of the models to be tested (for example, in this case, we set maximum lags length = 2). Then, the combinations of the models to be tested include ARIMA(0,1,1), ARIMA(1,1,0), ARIMA(1,1,1), ARIMA(1,1,2), ARIMA(2,1,0), ARIMA(2,1,1), and ARIMA(2,1,2). To identify the optimal order of the lags number of the ARIMA model, Schwarz Information Criteria (SIC) and Akaike Information Criteria (AIC) can be used by choose the model with the lowest value of SIC of AIC (in this case, we use SIC as our choosing criteria).

From EViews, click Estimate model, specify the order of ARIMA model. For example, if the model to be estimated is ARIMA(2,1,2), the command is:

D(CMI) C AR(1) AR(2) MA(1) MA(2) If the model is ARIMA(1,1,1), the command is: D(CMI) C AR(1) MA(1) If the model is ARIMA(1,0,0), the command is: CMI C AR(1)

For example, if we estimate ARIMA(2,1,2), we can specify the command as follows:


Then, click OK. Dependent Variable: D(CMI) Method: Least Squares Sample (adjusted): 1997M04 2005M06 Included observations: 99 after adjustments Convergence achieved after 18 iterations Backcast: 1997M02 1997M03

Variable Coefficient Std. Error t-Statistic Prob.

C 0.360691 0.222549 1.620722 0.1084 AR(1) -0.317276 0.684607 -0.463443 0.6441 AR(2) 0.213323 0.216329 0.986106 0.3266 MA(1) 0.805658 0.687702 1.171521 0.2444 MA(2) 0.024205 0.395562 0.061191 0.9513


Inverted AR Roots .33 -.65 Inverted MA Roots -.03 -.77


After estimate all combination of the models, then, compare SIC (or AIC) indices to identify the optimal lags length (in this case, ARIMA(2,1,1) provide the lowest lags length).

ARIMA 0,1,1 1,1,0 1,1,1 1,1,2 2,1,0 2,1,1 2,1,2

AIC 3.2428 3.3317 3.2534 3.2432 3.3237 3.2411 3.2836

SIC 3.3204 3.4098 3.3576 3.3735 3.4285 3.3722 3.4409

Then, the selected model is ARIMA(2,1,1) Dependent Variable: D(CMI) Method: Least Squares Sample (adjusted): 1997M04 2005M06 Included observations: 99 after adjustments Convergence achieved after 17 iterations Backcast: 1997M03


C 0.407365 0.239446 1.701278 0.0922 DUM* -5.901357 0.530451 -11.12517 0.0000 AR(1) -0.200610 0.107485 -1.866398 0.0651 AR(2) 0.213423 0.109349 1.951762 0.0539 MA(1) 0.966275 0.022858 42.27297 0.0000


Inverted AR Roots .37 -.57 Inverted MA Roots -.97

*In this example, since there is price war in the market in February, 2002. To eliminate this

effect, we use dummy variable by setting DUM=1 in February, 2002 and DUM=0 in all others.


Step 3: Diagnostic Checking: Since residual terms of ARIMA model are assumed to

be white-noise, we can diagnostically check the estimated model by testing the property of the estimated residual of the estimated model – using Correlogram. In this example, correlogram result shows that residual terms of the model are white-noise process.

Step 4: Forecasting: The last step is forecasting process. To ensure that the estimated model provides accurate prediction, forecasting error index, such as, Root Mean Squares Error (RMSE), or Theil’s Inequality Coefficient, is used. In EViews’ estimated result Windows, from menu bar click Forecast.


In Forecast Window, fills in Series to forecast (in this case, CMI), names the

forecasted series (in this case, cmif), choose the forecasting Method (in this case, Dynamic forecast), and Forecast sample (in this case, 1997m01 2005m05).

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Forecast: CMIFActual: CMIForecast sample: 1997M01 2005M05Adjusted sample: 1997M04 2005M05Included observations: 98

Root Mean Squared Error 7.026952Mean Absolute Error 5.278022Mean Abs. Percent Error 4.837119Theil Inequality Coefficient 0.033602 Bias Proportion 0.413649 Variance Proportion 0.026065 Covariance Proportion 0.560286


7. Generalized Autoregressive Conditional Heteroscedastic (GARCH) Models

Example

0 1 2 3t tSET IBR GOLDB USDSβ β β β ε= + + + + where: SETt = Return on Stock Exchange of Thailand (SET). IBRt = Inter-bank rate. GOLDBt = Gold price. USDSt = Exchange rate ($US/฿Baht). εt = Residual which has GARCH(p,q) process:

2 2 2 2 2 2 20 1 1 2 2 1 1 2 2t t t p t p t t q t qσ α δ σ δ σ δ σ α ε α ε α ε− − − − − −= + + + + + + + +

Testing GARCH Effect Firstly, estimate the model using OLS without GARCH(p,q) process.

EView’s Command

equation modelols.ls set c ibr goldb usds Then, from EViews’ estimated results Windows, go to Views, choose Residual Tests, and select ARCH LM Test..


Then, in Lag Specification Window, specify the lag number to be tested in Lags to include: (in this case, Lags to include: = 1).

ARCH Test:

F-statistic 134.4074 Prob. F(1,115) 0.000000 Obs*R-squared 63.05211 Prob. Chi-Square(1) 0.000000

Test Equation: Dependent Variable: RESID^2 Method: Least Squares Sample (adjusted): 12/03/2003 5/13/2004 Included observations: 117 after adjustments


C 180.7126 74.34768 2.430642 0.0166 RESID^2(-1) 0.733750 0.063290 11.59342 0.0000

R-squared 0.538907 Mean dependent var 674.5738 Adjusted R-squared 0.534897 S.D. dependent var 966.4475 S.E. of regression 659.1020 Akaike info criterion 15.83658 Sum squared resid 49957771 Schwarz criterion 15.88380 Log likelihood -924.4399 F-statistic 134.4074 Durbin-Watson stat 1.992110 Prob(F-statistic) 0.000000

In this case, since p-value of the ARCH effect test (F-statistic or Chi-Square (Obs*R-squared = (117*0.538907) = 63.05211)) is less than level of significance 0.05, thus, null hypothesis that there is no ARCH effect is rejected, thus, there exists significant ARCH effect in this model with 0.05 significant level.


Identify Order (p,q) and Estimation The next step is to identify order of GARCH(p,q) by estimating GARCH models in several orders and choose the model with the lowest AIC or SIC. Estimate GARCH(p,q) using MLE

From EViews Equation Estimation Window, go to Estimation settings/Method, choose ARCH – Autoregressive Conditional Heteroskedasticity.


From GARCH Equation Estimation Window, specify order of GARCH(p,q) – p in GARCH box, and q in ARCH box (in this case, we use GARCH(1,1)).

From estimated results of GARCH models with different orders, choose the model with the lowest AIC or SIC (in this case, the most appropriated model is GARCH(0,1) or ARCH(1) model).

GARCH 0,1 1,1 1,2 2,1 2,2

AIC 8.8701 9.0228 8.9792 8.9430 8.9438

SIC 9.0110 9.1872 9.1670 9.1308 9.1551

The estimated result can be shown as follows:


Dependent Variable: SET Method: ML - ARCH (Marquardt) - Normal distribution Sample (adjusted): 12/02/2003 5/13/2004 Included observations: 118 after adjustments Convergence achieved after 234 iterations Variance backcast: ON GARCH = C(5) + C(6)*RESID(-1)^2

Coefficient Std. Error z-Statistic Prob.

C 3180.374 198.1774 16.04812 0.0000 IBR -64.52450 41.88617 -1.540473 0.1234

GOLDB 0.058916 0.013876 4.245984 0.0000 USDS -72.51075 2.641941 -27.44602 0.0000

Variance Equation

C 93.71101 38.35904 2.442997 0.0146 RESID(-1)^2 0.933997 0.327187 2.854630 0.0043


Estimate Variance (Volatility) of Dependent Variable from Estimated Model

Variance or volatility of dependent variable (in this case, return on SET) can be estimated from the estimated result σ2

t = 93.711 + 0.934ε2t-1

From EViews estimated result Windows menu bar, go to Proc, choose Make GARCH Variance Series...


Then, in Make GARCH Variance Window, define name of GARCH series in Conditional Variance: box (in this case, garch01), and click OK.

The estimated variance series is then estimated and keep as garch01.

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Ch21_TimeSeries1

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