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  • 1. EC 603 ECONOMETRIC ASSIGNMENTSTable 21.1Macroeconomic Data, United States, 1970.1 to 1991.421.16. Using the data given in Table 21.1, obtain sample correlograms up to25 lags for the time series PCE, PDI, Profits, and Dividends. What generalpattern do you see? Intuitively, which one(s) of these time series seem to bestationary?PCE-1.00-0.500.000.501.000 5 10 15 20 25LagBartletts formula for MA(q) 95% confidence bands25 0.1599 0.0024 927.67 0.000024 0.1894 -0.0103 924.46 0.000023 0.2205 0.0016 920.02 0.000022 0.2526 -0.0093 914.1 0.000021 0.2864 -0.0346 906.44 0.000020 0.3212 -0.0356 896.74 0.000019 0.3553 -0.0184 884.73 0.000018 0.3889 -0.0344 870.24 0.000017 0.4227 -0.0183 853.13 0.000016 0.4560 -0.0330 833.2 0.000015 0.4895 -0.0144 810.33 0.000014 0.5226 -0.0225 784.33 0.000013 0.5561 -0.0209 755.1 0.000012 0.5897 -0.0071 722.45 0.000011 0.6234 -0.0018 686.21 0.000010 0.6580 -0.0179 646.23 0.00009 0.6936 -0.0101 602.28 0.00008 0.7295 -0.0232 554.04 0.00007 0.7659 -0.0187 501.36 0.00006 0.8021 -0.0514 444.01 0.00005 0.8383 -0.0301 381.87 0.00004 0.8725 -0.0408 314.81 0.00003 0.9059 -0.0193 243.04 0.00002 0.9378 -0.0379 166.58 0.00001 0.9696 0.9696 85.581 0.0000LAG 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 valuethree quarters ago is 0.9059. AC can be used to define the q in MA (q) only instationary seriesPAC shows that the correlation between the current value of PCE and its valuethree quarters ago is 0.0193 without the effect of the two previous lags. PACcan be used to define the p in AR (p) only in stationary seriesQ statistic tests the null hypothesis that all correlation up to lag 25 are equalto 0. This series show significant autocorrelation as shown in the Prob>Q valuewhich at any lag are less than 0.05, therefore rejecting the null that all lags arenot auto correlated.A graph of AC shows a slow decay in the trend, suggesting non-stationary asindicated also in the ac command.PDI-1.00-0.500.000.501.000 5 10 15 20 25LagBartletts formula for MA(q) 95% confidence bands.25 0.1849 0.0102 940.5 0.000024 0.2149 0.0146 936.21 0.000023 0.2471 -0.0761 930.49 0.000022 0.2819 0.0145 923.05 0.000021 0.3132 -0.0388 913.52 0.000020 0.3470 -0.0782 901.92 0.000019 0.3799 -0.0057 887.9 0.000018 0.4092 -0.0289 871.33 0.000017 0.4395 -0.0385 852.39 0.000016 0.4695 -0.0087 830.84 0.000015 0.4983 -0.0106 806.59 0.000014 0.5280 -0.0022 779.65 0.000013 0.5583 -0.0005 749.82 0.000012 0.5898 -0.0014 716.91 0.000011 0.6226 -0.0082 680.66 0.000010 0.6566 -0.0167 640.8 0.00009 0.6915 -0.0067 597.02 0.00008 0.7266 -0.0236 549.08 0.00007 0.7625 -0.0339 496.82 0.00006 0.7983 -0.0232 439.97 0.00005 0.8331 -0.0296 378.42 0.00004 0.8676 -0.0320 312.19 0.00003 0.9014 -0.0184 241.22 0.00002 0.9343 -0.0126 165.51 0.00001 0.9670 0.9670 85.124 0.0000LAG 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 valuethree quarters ago is 0.9014. AC can be used to define the q in MA (q) only instationary seriesPAC shows that the correlation between the current value of PCE and its valuethree quarters ago is 0.0184 without the effect of the two previous lags. PACcan be used to define the p in AR (p) only in stationary seriesQ statistic tests the null hypothesis that all correlation up to lag 25 is equal to0. This shows a significant autocorrelation as shown in the Prob>Q value whichat any lag are less than 0.05, therefore rejecting the null that all lags are notauto correlated. A graph of AC shows a slow decline in the trend, suggestingnon-stationary of PCE data as indicated also in the auto correlation (ac)command.Profits-1.00-0.500.000.501.000 5 10 15 20 25LagBartletts formula for MA(q) 95% confidence bands
  • 4. AC shows that the correlation between the current value of profits and its valuetwo quarters ago is 0.8968. AC can be used to define the q in MA (q) only instationary seriesQ statistic tests the null hypothesis that all correlation up to lag 25 are equalto 0. This series show significant autocorrelation as shown in the Prob>Q valuewhich at any lag are less than 0.05, therefore rejecting the null that all lags arenot auto correlated.A graph of AC shows a slow decay in the trend and it reached a point there arenegative sign started at lag 21, suggesting non-stationary of profits data asindicated also in the ac command in profits.Dividends.25 -0.0833 -0.0382 497.73 0.000024 -0.0679 -0.0049 496.86 0.000023 -0.0528 0.0810 496.29 0.000022 -0.0358 0.0187 495.95 0.000021 -0.0077 -0.0127 495.8 0.000020 0.0238 -0.1043 495.79 0.000019 0.0548 -0.0021 495.72 0.000018 0.0813 -0.0320 495.38 0.000017 0.1111 -0.0487 494.63 0.000016 0.1392 0.0415 493.25 0.000015 0.1680 -0.0254 491.12 0.000014 0.2062 0.0187 488.06 0.000013 0.2451 -0.0007 483.51 0.000012 0.2890 -0.0257 477.17 0.000011 0.3390 -0.0472 468.47 0.000010 0.3913 0.0045 456.65 0.00009 0.4427 0.1022 441.1 0.00008 0.5014 -0.1192 421.45 0.00007 0.5722 -0.0267 396.56 0.00006 0.6359 0.0229 364.55 0.00005 0.7014 -0.0612 325.49 0.00004 0.7718 -0.1231 278.55 0.00003 0.8384 -0.0310 222.39 0.00002 0.8968 -0.1455 156.9 0.00001 0.9539 0.9539 82.833 0.0000LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]-1 0 1 -1 0 1. corrgram profits, lags(25) yw-1.00-0.500.000.501.000 5 10 15 20 25LagBartletts formula for MA(q) 95% confidence bands
  • 5. AC shows that the correlation between the current value of devidends and itsvalue three quarters ago is 0.9088. AC can be used to define the q in MA (q)only in stationary seriesQ statistic tests the null hypothesis that all correlation up to lags 25 are equalto 0. This series show significant autocorrelation as shown in the Prob>Qvalue, 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 ofdevidends data as indicated also in the ac command.Conclusion.Thus it seems that the GDP time series is non-stationary. The correlograms ofthe other U.S. economic time series shown above indicates a similar pattern,leading to the conclusion that all these time series are non-stationary; theymay 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 outif these series contain a unit root. If a unit root exists, how would youcharacterize 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 stationarity25 0.1757 0.0427 973.83 0.000024 0.2079 0.0155 969.94 0.000023 0.2442 -0.0253 964.6 0.000022 0.2831 -0.0599 957.33 0.000021 0.3219 -0.0809 947.71 0.000020 0.3586 -0.0759 935.46 0.000019 0.3918 -0.0441 920.49 0.000018 0.4220 -0.0218 902.87 0.000017 0.4509 -0.0008 882.72 0.000016 0.4799 0.0037 860.04 0.000015 0.5102 0.0008 834.7 0.000014 0.5421 -0.0085 806.46 0.000013 0.5752 -0.0126 775.01 0.000012 0.6091 -0.0251 740.07 0.000011 0.6434 -0.0270 701.4 0.000010 0.6775 -0.0274 658.82 0.00009 0.7111 -0.0164 612.21 0.00008 0.7442 -0.0351 561.52 0.00007 0.7774 -0.0182 506.7 0.00006 0.8097 -0.0046 447.61 0.00005 0.8420 -0.0048 384.29 0.00004 0.8751 -0.0432 316.64 0.00003 0.9088 -0.0348 244.44 0.00002 0.9408 -0.0627 167.49 0.00001 0.9718 0.9718 85.968 0.0000LAG 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 nullhypothesis that the time series under consideration has a unit root; that is, itis nonstationary. The alternative hypothesis is that the time series isstationary.PDI._cons 181.1675 121.5525 1.49 0.140 -60.51135 422.8463L1. -.0412101 .0312141 -1.32 0.190 -.1032722 .020852gdpD.gdp Coef. Std. Err. t P>|t| [95% Conf. Interval]MacKinnon approximate p-value for Z(t) = 0.6199Z(t) -1.320 -3.528 -2.900 -2.585Statistic Value Value ValueTest 1% Critical 5% Critical 10% CriticalInterpolated Dickey-FullerDickey-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 351.0563LD. -.4321718 .0990718 -4.36 0.000 -.6292216 -.2351219L1. -.0238446 .0291793 -0.82 0.416 -.0818811 .034192gdpD.gdp Coef. Std. Err. t P>|t| [95% Conf. Interval]MacKinnon approximate p-value for Z(t) = 0.8140Z(t) -0.817 -3.530 -2.901 -2.586Statistic Value Value ValueTest 1% Critical 5% Critical 10% CriticalInterpolated Dickey-FullerAugmented 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 65.79877L1. -.0042874 .0063841 -0.67 0.504 -.0169808 .0084059pdiD.pdi Coef. Std. Err. t P>|t| [95% Conf. Interval]MacKinnon approximate p-value for Z(t) = 0.8540Z(t) -0.672 -3.528 -2.900 -2.585Statistic Value Value ValueTest 1% Critical 5% Critical 10% CriticalInterpolated Dickey-FullerDickey-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 67.02138LD. -.0503178 .1093677 -0.46 0.647 -.2678458 .1672103L1. -.0039528 .0065606 -0.60 0.548 -.0170017 .009096pdiD.pdi Coef. Std. Err. t P>|t| [95% Conf. Interval]MacKinnon approximate p-value for Z(t) = 0.8704Z(t) -0.603 -3.530 -2.901 -2.586Statistic Value Value ValueTest 1% Critical 5% Critical 10% CriticalInterpolated Dickey-FullerAugmented 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 40.49399LD. .1812105 .1083336 1.67 0.098 -.0342607 .3966817L1. -.0015963 .0043604 -0.37 0.715 -.0102691 .0070764pceD.pce Coef. Std. Err. t P>|t| [95% Conf. Interval]MacKinnon approximate p-value for Z(t) = 0.9156Z(t) -0.366 -3.530 -2.901 -2.586Statistic Value Value ValueTest 1% Critical 5% Critical 10% CriticalInterpolated Dickey-FullerAugmented 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 41.24701L1. -.0008961 .0043215 -0.21 0.836 -.0094885 .0076963pceD.pce Coef. Std. Err. t P>|t| [95% Conf. Interval]MacKinnon approximate p-value for Z(t) = 0.9376Z(t) -0.207 -3.528 -2.900 -2.585Statistic Value Value ValueTest 1% Critical 5% Critical 10% CriticalInterpolated Dickey-FullerDickey-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 11.33737LD. .2758812 .1041843 2.65 0.010 .0686629 .4830995L1. -.0334463 .0205288 -1.63 0.107 -.0742773 .0073847profitsD.profits Coef. Std. Err. t P>|t| [95% Conf. Interval]MacKinnon approximate p-value for Z(t) = 0.4679Z(t) -1.629 -3.530 -2.901 -2.586Statistic Value Value ValueTest 1% Critical 5% Critical 10% CriticalInterpolated Dickey-FullerAugmented 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 11.18367L1. -.028913 .0207324 -1.39 0.167 -.0701346 .0123085profitsD.profits Coef. Std. Err. t P>|t| [95% Conf. Interval]MacKinnon approximate p-value for Z(t) = 0.5849Z(t) -1.395 -3.528 -2.900 -2.585Statistic Value Value ValueTest 1% Critical 5% Critical 10% CriticalInterpolated Dickey-FullerDickey-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–Fullertest, 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 .9514729LD. .7006808 .080416 8.71 0.000 .5407365 .8606251L1. .0004995 .0038417 0.13 0.897 -.0071416 .0081405devidentsD.devidents Coef. Std. Err. t P>|t| [95% Conf. Interval]MacKinnon approximate p-value for Z(t) = 0.9681Z(t) 0.130 -3.530 -2.901 -2.586Statistic Value Value ValueTest 1% Critical 5% Critical 10% CriticalInterpolated Dickey-FullerAugmented Dickey-Fuller test for unit root Number of obs = 86. dfuller devidents, regress lags(1)_cons .5979509 .3919752 1.53 0.131 -.1814009 1.377303L1. .01042 .0050228 2.07 0.041 .0004333 .0204067devidentsD.devidents Coef. Std. Err. t P>|t| [95% Conf. Interval]MacKinnon approximate p-value for Z(t) = 0.9988Z(t) 2.075 -3.528 -2.900 -2.585Statistic Value Value ValueTest 1% Critical 5% Critical 10% CriticalInterpolated Dickey-FullerDickey-Fuller test for unit root Number of obs = 87. dfuller devidents, regress lags(0)