1. EC 603 ECONOMETRIC ASSIGNMENTS
Table 21.1Macroeconomic Data, United States, 1970.1 to 1991.4
21.16. Using the data given in Table 21.1, obtain sample correlograms up to
25 lags for the time series PCE, PDI, Profits, and Dividends. What general
pattern do you see? Intuitively, which one(s) of these time series seem to be
stationary?
PCE
-1.00-0.50
0.000.501.00
0 5 10 15 20 25
Lag
Bartlett's formula for MA(q) 95% confidence bands
25 0.1599 0.0024 927.67 0.0000
24 0.1894 -0.0103 924.46 0.0000
23 0.2205 0.0016 920.02 0.0000
22 0.2526 -0.0093 914.1 0.0000
21 0.2864 -0.0346 906.44 0.0000
20 0.3212 -0.0356 896.74 0.0000
19 0.3553 -0.0184 884.73 0.0000
18 0.3889 -0.0344 870.24 0.0000
17 0.4227 -0.0183 853.13 0.0000
16 0.4560 -0.0330 833.2 0.0000
15 0.4895 -0.0144 810.33 0.0000
14 0.5226 -0.0225 784.33 0.0000
13 0.5561 -0.0209 755.1 0.0000
12 0.5897 -0.0071 722.45 0.0000
11 0.6234 -0.0018 686.21 0.0000
10 0.6580 -0.0179 646.23 0.0000
9 0.6936 -0.0101 602.28 0.0000
8 0.7295 -0.0232 554.04 0.0000
7 0.7659 -0.0187 501.36 0.0000
6 0.8021 -0.0514 444.01 0.0000
5 0.8383 -0.0301 381.87 0.0000
4 0.8725 -0.0408 314.81 0.0000
3 0.9059 -0.0193 243.04 0.0000
2 0.9378 -0.0379 166.58 0.0000
1 0.9696 0.9696 85.581 0.0000
LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]
-1 0 1 -1 0 1
. corrgram pce, lags(25) yw

2. AC shows that the correlation between the current value of PCE and its value
three quarters ago is 0.9059. AC can be used to define the q in MA (q) only in
stationary series
PAC shows that the correlation between the current value of PCE and its value
three quarters ago is 0.0193 without the effect of the two previous lags. PAC
can be used to define the p in AR (p) only in stationary series
Q statistic tests the null hypothesis that all correlation up to lag 25 are equal
to 0. This series show significant autocorrelation as shown in the Prob>Q value
which at any lag are less than 0.05, therefore rejecting the null that all lags are
not auto correlated.
A graph of AC shows a slow decay in the trend, suggesting non-stationary as
indicated also in the ac command.
PDI
-1.00-0.50
0.000.501.00
0 5 10 15 20 25
Lag
Bartlett's formula for MA(q) 95% confidence bands
.
25 0.1849 0.0102 940.5 0.0000
24 0.2149 0.0146 936.21 0.0000
23 0.2471 -0.0761 930.49 0.0000
22 0.2819 0.0145 923.05 0.0000
21 0.3132 -0.0388 913.52 0.0000
20 0.3470 -0.0782 901.92 0.0000
19 0.3799 -0.0057 887.9 0.0000
18 0.4092 -0.0289 871.33 0.0000
17 0.4395 -0.0385 852.39 0.0000
16 0.4695 -0.0087 830.84 0.0000
15 0.4983 -0.0106 806.59 0.0000
14 0.5280 -0.0022 779.65 0.0000
13 0.5583 -0.0005 749.82 0.0000
12 0.5898 -0.0014 716.91 0.0000
11 0.6226 -0.0082 680.66 0.0000
10 0.6566 -0.0167 640.8 0.0000
9 0.6915 -0.0067 597.02 0.0000
8 0.7266 -0.0236 549.08 0.0000
7 0.7625 -0.0339 496.82 0.0000
6 0.7983 -0.0232 439.97 0.0000
5 0.8331 -0.0296 378.42 0.0000
4 0.8676 -0.0320 312.19 0.0000
3 0.9014 -0.0184 241.22 0.0000
2 0.9343 -0.0126 165.51 0.0000
1 0.9670 0.9670 85.124 0.0000
LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]
-1 0 1 -1 0 1
. corrgram pdi, lags(25) yw

3. AC shows that the correlation between the current value of PDI and its value
three quarters ago is 0.9014. AC can be used to define the q in MA (q) only in
stationary series
PAC shows that the correlation between the current value of PCE and its value
three quarters ago is 0.0184 without the effect of the two previous lags. PAC
can be used to define the p in AR (p) only in stationary series
Q statistic tests the null hypothesis that all correlation up to lag 25 is equal to
0. This shows a significant autocorrelation as shown in the Prob>Q value which
at any lag are less than 0.05, therefore rejecting the null that all lags are not
auto correlated. A graph of AC shows a slow decline in the trend, suggesting
non-stationary of PCE data as indicated also in the auto correlation (ac)
command.
Profits
-1.00-0.50
0.000.501.00
0 5 10 15 20 25
Lag
Bartlett's formula for MA(q) 95% confidence bands

4. AC shows that the correlation between the current value of profits and its value
two quarters ago is 0.8968. AC can be used to define the q in MA (q) only in
stationary series
Q statistic tests the null hypothesis that all correlation up to lag 25 are equal
to 0. This series show significant autocorrelation as shown in the Prob>Q value
which at any lag are less than 0.05, therefore rejecting the null that all lags are
not auto correlated.
A graph of AC shows a slow decay in the trend and it reached a point there are
negative sign started at lag 21, suggesting non-stationary of profits data as
indicated also in the ac command in profits.
Dividends.
25 -0.0833 -0.0382 497.73 0.0000
24 -0.0679 -0.0049 496.86 0.0000
23 -0.0528 0.0810 496.29 0.0000
22 -0.0358 0.0187 495.95 0.0000
21 -0.0077 -0.0127 495.8 0.0000
20 0.0238 -0.1043 495.79 0.0000
19 0.0548 -0.0021 495.72 0.0000
18 0.0813 -0.0320 495.38 0.0000
17 0.1111 -0.0487 494.63 0.0000
16 0.1392 0.0415 493.25 0.0000
15 0.1680 -0.0254 491.12 0.0000
14 0.2062 0.0187 488.06 0.0000
13 0.2451 -0.0007 483.51 0.0000
12 0.2890 -0.0257 477.17 0.0000
11 0.3390 -0.0472 468.47 0.0000
10 0.3913 0.0045 456.65 0.0000
9 0.4427 0.1022 441.1 0.0000
8 0.5014 -0.1192 421.45 0.0000
7 0.5722 -0.0267 396.56 0.0000
6 0.6359 0.0229 364.55 0.0000
5 0.7014 -0.0612 325.49 0.0000
4 0.7718 -0.1231 278.55 0.0000
3 0.8384 -0.0310 222.39 0.0000
2 0.8968 -0.1455 156.9 0.0000
1 0.9539 0.9539 82.833 0.0000
LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]
-1 0 1 -1 0 1
. corrgram profits, lags(25) yw-1.00-0.50
0.000.501.00
0 5 10 15 20 25
Lag
Bartlett's formula for MA(q) 95% confidence bands

5. AC shows that the correlation between the current value of devidends and its
value three quarters ago is 0.9088. AC can be used to define the q in MA (q)
only in stationary series
Q statistic tests the null hypothesis that all correlation up to lags 25 are equal
to 0. This series show significant autocorrelation as shown in the Prob>Q
value, therefore rejecting the null that all lags are not auto correlated.
A graph of AC shows a slow decay in the trend, suggesting non-stationary of
devidends data as indicated also in the ac command.
Conclusion.
Thus it seems that the GDP time series is non-stationary. The correlograms of
the other U.S. economic time series shown above indicates a similar pattern,
leading to the conclusion that all these time series are non-stationary; they
may be non-stationary in mean or variance or both.
21.17. For each of the time series of exercise 21.16, use the DF test to find out
if these series contain a unit root. If a unit root exists, how would you
characterize such a time series?
The Dickey-Fuller test is one of the most commonly use tests for stationarity,
simply a test for unit root. A unit root test is a test for stationarity
25 0.1757 0.0427 973.83 0.0000
24 0.2079 0.0155 969.94 0.0000
23 0.2442 -0.0253 964.6 0.0000
22 0.2831 -0.0599 957.33 0.0000
21 0.3219 -0.0809 947.71 0.0000
20 0.3586 -0.0759 935.46 0.0000
19 0.3918 -0.0441 920.49 0.0000
18 0.4220 -0.0218 902.87 0.0000
17 0.4509 -0.0008 882.72 0.0000
16 0.4799 0.0037 860.04 0.0000
15 0.5102 0.0008 834.7 0.0000
14 0.5421 -0.0085 806.46 0.0000
13 0.5752 -0.0126 775.01 0.0000
12 0.6091 -0.0251 740.07 0.0000
11 0.6434 -0.0270 701.4 0.0000
10 0.6775 -0.0274 658.82 0.0000
9 0.7111 -0.0164 612.21 0.0000
8 0.7442 -0.0351 561.52 0.0000
7 0.7774 -0.0182 506.7 0.0000
6 0.8097 -0.0046 447.61 0.0000
5 0.8420 -0.0048 384.29 0.0000
4 0.8751 -0.0432 316.64 0.0000
3 0.9088 -0.0348 244.44 0.0000
2 0.9408 -0.0627 167.49 0.0000
1 0.9718 0.9718 85.968 0.0000
LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]
-1 0 1 -1 0 1
. corrgram devidents, lags(25) yw

6. nonstationarity of time series data. Unit root tests are based on the null
hypothesis that the time series under consideration has a unit root; that is, it
is nonstationary. The alternative hypothesis is that the time series is
stationary.
PDI
.
_cons 181.1675 121.5525 1.49 0.140 -60.51135 422.8463
L1. -.0412101 .0312141 -1.32 0.190 -.1032722 .020852
gdp
D.gdp Coef. Std. Err. t P>|t| [95% Conf. Interval]
MacKinnon approximate p-value for Z(t) = 0.6199
Z(t) -1.320 -3.528 -2.900 -2.585
Statistic Value Value Value
Test 1% Critical 5% Critical 10% Critical
Interpolated Dickey-Fuller
Dickey-Fuller test for unit root Number of obs = 87
. dfuller gdp, regress lags(0)
_cons 125.1689 113.5706 1.10 0.274 -100.7184 351.0563
LD. -.4321718 .0990718 -4.36 0.000 -.6292216 -.2351219
L1. -.0238446 .0291793 -0.82 0.416 -.0818811 .034192
gdp
D.gdp Coef. Std. Err. t P>|t| [95% Conf. Interval]
MacKinnon approximate p-value for Z(t) = 0.8140
Z(t) -0.817 -3.530 -2.901 -2.586
Statistic Value Value Value
Test 1% Critical 5% Critical 10% Critical
Interpolated Dickey-Fuller
Augmented Dickey-Fuller test for unit root Number of obs = 86
. dfuller gdp, regress lags(1)

7. .
_cons 29.86419 18.07331 1.65 0.102 -6.070392 65.79877
L1. -.0042874 .0063841 -0.67 0.504 -.0169808 .0084059
pdi
D.pdi Coef. Std. Err. t P>|t| [95% Conf. Interval]
MacKinnon approximate p-value for Z(t) = 0.8540
Z(t) -0.672 -3.528 -2.900 -2.585
Statistic Value Value Value
Test 1% Critical 5% Critical 10% Critical
Interpolated Dickey-Fuller
Dickey-Fuller test for unit root Number of obs = 87
. dfuller pdi, regress lags(0)
_cons 29.73848 18.74492 1.59 0.116 -7.544422 67.02138
LD. -.0503178 .1093677 -0.46 0.647 -.2678458 .1672103
L1. -.0039528 .0065606 -0.60 0.548 -.0170017 .009096
pdi
D.pdi Coef. Std. Err. t P>|t| [95% Conf. Interval]
MacKinnon approximate p-value for Z(t) = 0.8704
Z(t) -0.603 -3.530 -2.901 -2.586
Statistic Value Value Value
Test 1% Critical 5% Critical 10% Critical
Interpolated Dickey-Fuller
Augmented Dickey-Fuller test for unit root Number of obs = 86
. dfuller pdi, regress lags(1)

8. .
_cons 17.96975 11.32464 1.59 0.116 -4.554495 40.49399
LD. .1812105 .1083336 1.67 0.098 -.0342607 .3966817
L1. -.0015963 .0043604 -0.37 0.715 -.0102691 .0070764
pce
D.pce Coef. Std. Err. t P>|t| [95% Conf. Interval]
MacKinnon approximate p-value for Z(t) = 0.9156
Z(t) -0.366 -3.530 -2.901 -2.586
Statistic Value Value Value
Test 1% Critical 5% Critical 10% Critical
Interpolated Dickey-Fuller
Augmented Dickey-Fuller test for unit root Number of obs = 86
. dfuller pce, regress lags(1)
_cons 19.16936 11.10396 1.73 0.088 -2.908295 41.24701
L1. -.0008961 .0043215 -0.21 0.836 -.0094885 .0076963
pce
D.pce Coef. Std. Err. t P>|t| [95% Conf. Interval]
MacKinnon approximate p-value for Z(t) = 0.9376
Z(t) -0.207 -3.528 -2.900 -2.585
Statistic Value Value Value
Test 1% Critical 5% Critical 10% Critical
Interpolated Dickey-Fuller
Dickey-Fuller test for unit root Number of obs = 87
. dfuller pce, regress lags(0)

9. .
_cons 5.633873 2.867579 1.96 0.053 -.0696273 11.33737
LD. .2758812 .1041843 2.65 0.010 .0686629 .4830995
L1. -.0334463 .0205288 -1.63 0.107 -.0742773 .0073847
profits
D.profits Coef. Std. Err. t P>|t| [95% Conf. Interval]
MacKinnon approximate p-value for Z(t) = 0.4679
Z(t) -1.629 -3.530 -2.901 -2.586
Statistic Value Value Value
Test 1% Critical 5% Critical 10% Critical
Interpolated Dickey-Fuller
Augmented Dickey-Fuller test for unit root Number of obs = 86
. dfuller profits, regress lags(1)
_cons 5.450201 2.883648 1.89 0.062 -.2832636 11.18367
L1. -.028913 .0207324 -1.39 0.167 -.0701346 .0123085
profits
D.profits Coef. Std. Err. t P>|t| [95% Conf. Interval]
MacKinnon approximate p-value for Z(t) = 0.5849
Z(t) -1.395 -3.528 -2.900 -2.585
Statistic Value Value Value
Test 1% Critical 5% Critical 10% Critical
Interpolated Dickey-Fuller
Dickey-Fuller test for unit root Number of obs = 87
. dfuller profits, regress lags(0)

10. By concluding, therefore that, on the basis of data analysis, the Dickey–Fuller
test, for the quarterly periods of 1970 to 1991, the U.S. GDP time series,
Personal Disposable Income (PDI), Personal Consumption Expenditure (PCE),
Profits and Devidends were nonstationary; that means it contained a unit root.
.
_cons .3726058 .2910401 1.28 0.204 -.2062613 .9514729
LD. .7006808 .080416 8.71 0.000 .5407365 .8606251
L1. .0004995 .0038417 0.13 0.897 -.0071416 .0081405
devidents
D.devidents Coef. Std. Err. t P>|t| [95% Conf. Interval]
MacKinnon approximate p-value for Z(t) = 0.9681
Z(t) 0.130 -3.530 -2.901 -2.586
Statistic Value Value Value
Test 1% Critical 5% Critical 10% Critical
Interpolated Dickey-Fuller
Augmented Dickey-Fuller test for unit root Number of obs = 86
. dfuller devidents, regress lags(1)
_cons .5979509 .3919752 1.53 0.131 -.1814009 1.377303
L1. .01042 .0050228 2.07 0.041 .0004333 .0204067
devidents
D.devidents Coef. Std. Err. t P>|t| [95% Conf. Interval]
MacKinnon approximate p-value for Z(t) = 0.9988
Z(t) 2.075 -3.528 -2.900 -2.585
Statistic Value Value Value
Test 1% Critical 5% Critical 10% Critical
Interpolated Dickey-Fuller
Dickey-Fuller test for unit root Number of obs = 87
. dfuller devidents, regress lags(0)