Correlograms and Time-Series Stationarity

This study material discusses correlograms and time-series stationarity. It includes plots of autocorrelations of oil prices between 1883 and 1970, as well as the results of the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test and the Augmented Dickey-Fuller test. The material concludes that the price of oil between 1883 and 1970 is stationary.

Correlograms and Time-Series Stationarity
Question 2
Part (a)
Based on the two correlogram plots below, we can tell that the time series data is stationary. The two plots are
generated in STATA using different command and the objective is to demonstrate the overall trend in the
autocorrelations of Oil prices between 1883 and 1970. Plot 1 is provided below while plot 2 is given in the
appendix.
-0.50 0.00 0.50 1.00
Autocorrelations of Priceofoil
0 10 20 30 40
Lag
Bartlett's formula for MA(q) 95% confidence bands
Plot 1: Correlogram Generate for Price of Oil between 1883 and 1970
Part (b)
The Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test is a Unit test for stationarity. The null hypothesis is that
the time series data is stationary. From the results indicated below (based on four augmentation term) we can
conclude that the data is stationary. We will not reject the null hypothesis can state the Price of oil between
1883 and 1970 is stationary.

11 .0466
10 .0442
9 .0429
8 .0426
7 .0432
6 .0448
5 .0478
4 .0527
3 .0601
2 .0709
1 .0924
0 .155
Lag order Test statistic
10%: 0.119 5% : 0.146 2.5%: 0.176 1% : 0.216
Critical values for H0: Priceofoil is trend stationary
Autocovariances weighted by Bartlett kernel
Maxlag = 11 chosen by Schwert criterion
KPSS test for Priceofoil
Part (c)
The Augmented Dickey-Fuller test is a Unit root test for non-stationarity. The null hypothesis is that the time
series data is non-stationary, from the results indicated below the p-value is less than 0.05; as such, we will
reject the null hypothesis and conclude that the price of oil between 1883 and 1970 is stationary.
MacKinnon approximate p-value for Z(t) = 0.0053
Z(t) -3.625 -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 Priceofoil, lags(0)
Part (d)

Based on the results of Plot 3 and 4, we can state that first order difference for Price of Oil between 1883 and
1970 are stationary. With regard to order of integration, we can conclude that Price of Oil betwee 1883 and
1970 is intregate on both order 0 and 1 i.e. P(0) and P(1).
-0.40 -0.20 0.00 0.20 0.40
Autocorrelations of FDpriceofoil
0 10 20 30 40
Lag
Bartlett's formula for MA(q) 95% confidence bands
Plot 3: Correlogram for First Order Difference of PriceofOil
Appendix
Plot 2: Correlogram with figures Oil Pricesofoil

.
40 -0.0355 -0.0720 205.89 0.0000
39 -0.0216 0.1094 205.69 0.0000
38 -0.0056 0.0112 205.61 0.0000
37 0.0286 0.0488 205.61 0.0000
36 0.0065 0.1701 205.48 0.0000
35 0.0008 0.0625 205.47 0.0000
34 -0.0197 0.0848 205.47 0.0000
33 -0.0184 -0.0411 205.42 0.0000
32 0.0204 0.0090 205.37 0.0000
31 0.0987 0.1217 205.31 0.0000
30 0.1043 0.1889 203.96 0.0000
29 0.0559 0.1177 202.47 0.0000
28 0.0458 0.0933 202.05 0.0000
27 0.0778 -0.0107 201.77 0.0000
26 0.1445 -0.0276 200.99 0.0000
25 0.1894 -0.0235 198.32 0.0000
24 0.1879 0.0174 193.81 0.0000
23 0.1925 0.1136 189.44 0.0000
22 0.2040 -0.1468 184.92 0.0000
21 0.1891 0.1402 179.93 0.0000
20 0.1105 -0.0819 175.7 0.0000
19 0.0775 -0.0480 174.28 0.0000
18 0.0757 0.0192 173.59 0.0000
17 -0.0204 0.1085 172.94 0.0000
16 -0.1281 0.0920 172.89 0.0000
15 -0.2491 0.0032 171.09 0.0000
14 -0.2371 0.0730 164.36 0.0000
13 -0.2739 -0.0201 158.34 0.0000
12 -0.3153 0.2020 150.42 0.0000
11 -0.3984 -0.1087 140.06 0.0000
10 -0.3052 -0.1382 123.73 0.0000
9 -0.2119 -0.1572 114.27 0.0000
8 -0.0939 -0.1038 109.77 0.0000
7 -0.0220 -0.1620 108.9 0.0000
6 0.1371 -0.0324 108.85 0.0000
5 0.2700 -0.2647 107.04 0.0000
4 0.4001 0.2544 100.08 0.0000
3 0.3778 0.0667 84.987 0.0000
2 0.5041 -0.0497 71.686 0.0000
1 0.7283 0.7309 48.282 0.0000
LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]
-1 0 1 -1 0 1
Plot 4: Correlogram with figures for FDpriceofoil

40 -0.0297 -0.1999 59.813 0.0227
39 0.0116 0.1014 59.667 0.0182
38 -0.0548 -0.0750 59.645 0.0140
37 0.1308 0.0326 59.172 0.0118
36 -0.0363 -0.0168 56.522 0.0160
35 0.0369 -0.1425 56.322 0.0126
34 -0.0380 -0.0517 56.119 0.0099
33 -0.0594 -0.0824 55.908 0.0076
32 -0.0840 0.0231 55.401 0.0063
31 0.1302 -0.0016 54.409 0.0058
30 0.1071 -0.1171 52.064 0.0075
29 -0.0727 -0.1978 50.504 0.0080
28 -0.0788 -0.1364 49.799 0.0068
27 -0.0674 -0.1200 48.983 0.0060
26 0.0374 -0.0224 48.396 0.0049
25 0.0864 -0.0077 48.219 0.0035
24 -0.0072 -0.0069 47.287 0.0031
23 -0.0016 -0.0445 47.28 0.0021
22 0.0423 -0.1475 47.28 0.0013
21 0.1141 0.1040 47.067 0.0009
20 -0.0872 -0.1661 45.539 0.0009
19 -0.0596 0.0434 44.661 0.0008
18 0.1739 0.0176 44.256 0.0005
17 0.0231 -0.0464 40.862 0.0010
16 0.0341 -0.1404 40.803 0.0006
15 -0.2445 -0.1417 40.676 0.0004
14 0.0831 -0.0665 34.247 0.0019
13 0.0036 -0.1447 33.514 0.0014
12 0.0894 -0.0643 33.513 0.0008
11 -0.3214 -0.3428 32.688 0.0006
10 0.0027 -0.0892 22.162 0.0143
9 -0.0483 -0.0426 22.161 0.0084
8 0.0796 0.0082 21.929 0.0050
7 -0.1641 -0.0212 21.308 0.0033
6 0.0487 0.0563 18.703 0.0047
5 0.0075 -0.0620 18.476 0.0024
4 0.2829 0.1895 18.471 0.0010
3 -0.2818 -0.3338 11.007 0.0117
2 -0.1783 -0.1883 3.6895 0.1581
1 -0.0939 -0.0939 .79373 0.3730
LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]
-1 0 1 -1 0 1