Chapter 21
Nonstationary Data
◼ Exercise
1. The autocorrelations are simple to obtain just by multiplying out
2
1
,,
t t t
v v v −
and so on. The autocovariances
are
22
1 2 2 2 2
1 , (1 ), ,0, ,+  +  − −  −
which provides the autocorrelations by division by the first of these.
The partial autocorrelations are messy, and can be obtained by the Yule-Walker equations. Alternatively
Applications
1. Macroeconomic Model
–> samp;1-204$
–> crea;c=log(realcons);y=log(realdpi)$
–> crea;c1=c[-1];c2=c[-2]$
–> samp;3-204$
–> regr;lhs=c;rhs=one,y,c1,c2$
+———————————————————————–+
| Ordinary least squares regression Weighting variable = none |
| Dep. var. = C Mean= 7.889033683 , S.D.= .5102401315 |
| Model size: Observations = 202, Parameters = 4, Deg.Fr.= 198 |
| Residuals: Sum of squares= .1519097328E-01, Std.Dev.= .00876 |
+———+————–+—————-+——–+———+———-+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+———+————–+—————-+——–+———+———-+
Constant .8165780259E-03 .10779352E-01 .076 .9397
Y .7869591065E-01 .29020268E-01 2.712 .0073 7.9998985
Chapter 21 Nonstationary Data 163
+———————————————————————–+
| Ordinary least squares regression Weighting variable = none |
| Dep. var. = E Mean= -.6947138134E-15, S.D.= .8693502258E-02 |
| Model size: Observations = 202, Parameters = 4, Deg.Fr.= 198 |
+———————————————————————–+
+———+————–+—————-+——–+———+———-+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+———+————–+—————-+——–+———+———-+
Constant .2437121418E-04 .57884755E-03 .042 .9665
Matrix Result has 2 rows and 2 columns.
1 2
2. ADF Test
To carry out the test, the rate of inflation is regressed on a constant, a time trend, the previous year’s value of
3. Reestimated Model in Example 10.4
–> samp;1–204$
–> crea;ddp1=infl[–1]–infl[–2]$
+———————————————————————–+
| Ordinary least squares regression Weighting variable = none |
+———————————————————————–+
+———+————–+—————-+——–+———+———-+
|Variable | Coefficient | Standard Error |t–ratio |P[|T|>t] | Mean of X|
+———+————--+—————-+——–+———+———-+
Constant 2.226039717 1.1342702 1.963 .0524
164 Greene • Econometric Analysis, Seventh Edition
–> wald;fn1=b_dp–1$
+———+————–+—————-+——–+———+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
–> samp;1-204$
–> crea;ct=realcons;yt=realgdp;gt=realgovt;rt=tbilrate$
–> crea;ct1=ct[-1];yt1=yt[-1]$
–> samp;2-204$
–> samp;1-204$
+———————————————————————–+
| Two stage least squares regression Weighting variable = none |
| Dep. var. = CT Mean= 3008.995074 , S.D.= 1456.900152 |
| Model size: Observations = 203, Parameters = 3, Deg.Fr.= 200 |
| Residuals: Sum of squares= 96595.67529 , Std.Dev.= 21.97677 |
+———————————————————————–+
+———+————–+—————-+——–+———+———-+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+———+————–+—————-+——–+———+———-+
Constant 6.666079115 8.6211817 .773 .4394
+———————————————————————–+
| Two stage least squares regression Weighting variable = none |
| Dep. var. = IT Mean= 654.5295567 , S.D.= 391.3705005 |
+———+————–+—————-+——–+———+———-+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+———+————–+—————-+——–+———+———-+
Constant -141.8297176 103.57113 -1.369 .1709
Time series identification for EC
Box-Pierce Statistic = 40.8498 Box-Ljung Statistic = 41.7842
Chapter 21 Nonstationary Data 165
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Lag | Autocorrelation Function |Box/Prc| Partial Autocorrelations X
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1 | .194*| |** | 7.65*| .194*| |** X
2 | .264*| |*** | 21.82*| .236*| |*** X
6 | .073 | |* | 39.52*| .018 | |* X
7 | .009 | |* | 39.53*| .003 | |* X
8 |-.078 | *| | 40.78*|-.109 | * | X
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Time series identification for EI
Box-Pierce Statistic = 27.4753 Box-Ljung Statistic = 28.3566
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Lag | Autocorrelation Function |Box/Prc| Partial Autocorrelations X
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1 | .244*| |*** | 12.13*| .244*| |*** X
2 | .143*| |** | 16.27*| .096 | |* X
3 | .037 | |* | 16.55*|-.019 | * | X