This site is an archived version of Voteview.com archived from University of Georgia on May 23, 2017. This point-in-time capture includes all files publicly linked on Voteview.com at that time. We provide access to this content as a service to ensure that past users of Voteview.com have access to historical files. This content will remain online until at least January 1st, 2018. UCLA provides no warranty or guarantee of access to these files.

45-734 PROBABILITY AND STATISTICS II (4th Mini AY1997-98)Note Set 13 Handouts

1. Moving Average Examples

1. The first example is data generated from an MA(2).

IDENT(5) Y1
```                      Correlogram of Y1
==============================================================
Date: 04/18/98   Time: 16:39
Sample: 1 100
Included observations: 99
==============================================================
Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob
==============================================================
. |**     |      . |**     |  1 0.293 0.293 8.7529 0.003
.*| .     |      **| .     |  2-0.183-0.294 12.192 0.002
.*| .     |      . |*.     |  3-0.084 0.085 12.929 0.005
.*| .     |      .*| .     |  4-0.069-0.144 13.425 0.009
.*| .     |      . | .     |  5-0.088-0.028 14.245 0.014
==============================================================
```
The Q-statistic is distributed as a c2 distribution with 5 degrees of freedom (5-0-0). Given that the p-value is 0.014 at lag 5, we reject the null hypothesis that Y1 is white noise.

First, try an MA(1) model:

LS Y1 C MA(1)
```============================================================
LS // Dependent Variable is Y1
Date: 04/18/98   Time: 17:07
Included observations: 99 after adjusting endpoints
Convergence achieved after 8 iterations
============================================================
Variable      CoefficienStd. Errort-Statistic  Prob.
============================================================
C          -0.142594   0.156414  -0.911646   0.3642
MA(1)         0.542028   0.085869   6.312245   0.0000
============================================================
R-squared            0.163421    Mean dependent var-0.140707
Adjusted R-squared   0.154796    S.D. dependent var 1.100216
S.E. of regression   1.011483    Akaike info criter 0.042831
Sum squared resid    99.24053    Schwarz criterion  0.095257
Log likelihood      -140.5950    F-statistic        18.94835
Durbin-Watson stat   2.174610    Prob(F-statistic)  0.000033
============================================================
Inverted MA Roots         -.54
============================================================
```
```                   Correlogram of Residuals
==============================================================
Date: 04/18/98   Time: 17:08
Sample: 2 100
Included observations: 99
Q-statistic probabilities adjusted for 1 ARMA term(s)
==============================================================
Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob
==============================================================
.*| .     |      .*| .     |  1-0.094-0.094 0.8922
.*| .     |      .*| .     |  2-0.133-0.143 2.7157 0.099
. | .     |      . | .     |  3 0.014-0.014 2.7354 0.255
.*| .     |      .*| .     |  4-0.072-0.094 3.2827 0.350
. | .     |      . | .     |  5 0.008-0.010 3.2890 0.511
==============================================================
```
The P-Value for the 2nd lag is below .1 so it is a good idea to check if adding an MA(2) term has an effect.

LS Y1 C MA(1) MA(2)
```============================================================
LS // Dependent Variable is Y1
Date: 04/18/98   Time: 17:11
Included observations: 99 after adjusting endpoints
Convergence achieved after 7 iterations
============================================================
Variable      CoefficienStd. Errort-Statistic  Prob.
============================================================
C          -0.142608   0.123090  -1.158568   0.2495
MA(1)         0.429426   0.100236   4.284151   0.0000
MA(2)        -0.204504   0.100171  -2.041551   0.0439
============================================================
R-squared            0.190999    Mean dependent var-0.140707
Adjusted R-squared   0.174144    S.D. dependent var 1.100216
S.E. of regression   0.999839    Akaike info criter 0.029512
Sum squared resid    95.96905    Schwarz criterion  0.108152
Log likelihood      -138.9358    F-statistic        11.33240
Durbin-Watson stat   1.967454    Prob(F-statistic)  0.000038
============================================================
Inverted MA Roots          .2      -.72
============================================================
```
The MA(2) coefficient is statistically significant and the correlogram of the residuals below indicates that the residuals are now white noise.
```                   Correlogram of Residuals
==============================================================
Date: 04/18/98   Time: 17:12
Sample: 2 100
Included observations: 99
Q-statistic probabilities adjusted for 2 ARMA term(s)
==============================================================
Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob
==============================================================
. | .     |      . | .     |  1 0.008 0.008 0.0073
. | .     |      . | .     |  2-0.009-0.009 0.0151
.*| .     |      .*| .     |  3-0.068-0.068 0.4946 0.482
.*| .     |      .*| .     |  4-0.059-0.058 0.8553 0.652
.*| .     |      .*| .     |  5-0.059-0.060 1.2260 0.747
==============================================================
```
2. In this second example, the series was generated from a MA(4) process.

IDENT Y2
```                      Correlogram of Y2
==============================================================
Date: 04/18/98   Time: 17:16
Sample: 1 100
Included observations: 96
==============================================================
Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob
==============================================================
. |***    |      . |***    |  1 0.374 0.374 13.839 0.000
. |**     |      . |*.     |  2 0.210 0.081 18.243 0.000
. | .     |      .*| .     |  3-0.018-0.141 18.275 0.000
***| .     |     ***| .     |  4-0.402-0.442 34.800 0.000
.*| .     |      . |**     |  5-0.111 0.255 36.072 0.000
==============================================================
```
There are spikes at 1, 2, and 4 which indicates that we could start with the corresponding model:

LS Y2 C MA(1) MA(2) MA(4)
```============================================================
LS // Dependent Variable is Y2
Date: 04/18/98   Time: 17:18
Included observations: 96 after adjusting endpoints
Convergence achieved after 23 iterations
============================================================
Variable      CoefficienStd. Errort-Statistic  Prob.
============================================================
C          -0.247163   0.088734  -2.785441   0.0065
MA(1)         0.270512   0.014926   18.12332   0.0000
MA(2)         0.241476   0.023006   10.49631   0.0000
MA(4)        -0.741378   0.000426  -1741.355   0.0000
============================================================
R-squared            0.430208    Mean dependent var-0.249292
Adjusted R-squared   0.411628    S.D. dependent var 1.437408
S.E. of regression   1.102570    Akaike info criter 0.236061
Sum squared resid    111.8408    Schwarz criterion  0.342909
Log likelihood      -143.5490    F-statistic        23.15415
Durbin-Watson stat   1.583374    Prob(F-statistic)  0.000000
============================================================
Inverted MA Roots          .8  -.08 -.99  -.08+.99i      -.93
============================================================
```
Given the low P-Values at several of the lags, we reject the null hypothesis that we have white noise. Given the small spike at the 3rd lag -- the term we did not estimate -- this is what we should try next.
```                   Correlogram of Residuals
==============================================================
Date: 04/18/98   Time: 17:19
Sample: 5 100
Included observations: 96
Q-statistic probabilities adjusted for 3 ARMA term(s)
==============================================================
Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob
==============================================================
. |**     |      . |**     |  1 0.203 0.203 4.0771
. |*.     |      . |*.     |  2 0.126 0.088 5.6644
. |*.     |      . |*.     |  3 0.114 0.077 6.9852
. | .     |      . | .     |  4 0.026-0.020 7.0561 0.008
. | .     |      . | .     |  5-0.025-0.047 7.1193 0.028
.*| .     |      .*| .     |  6-0.116-0.119 8.5215 0.036
. | .     |      . | .     |  7-0.022 0.027 8.5709 0.073
. |*.     |      . |*.     |  8 0.098 0.137 9.6070 0.087
.*| .     |      .*| .     |  9-0.065-0.088 10.063 0.122
. | .     |      . | .     | 10 0.029 0.035 10.154 0.180
==============================================================
```
LS Y2 C MA(1) MA(2) MA(3) MA(4)
```============================================================
LS // Dependent Variable is Y2
Date: 04/18/98   Time: 17:23
Included observations: 96 after adjusting endpoints
Convergence achieved after 14 iterations
============================================================
Variable      CoefficienStd. Errort-Statistic  Prob.
============================================================
C          -0.241779   0.141800  -1.705070   0.0916
MA(1)         0.417588   0.033201   12.57753   0.0000
MA(2)         0.347918   0.024563   14.16449   0.0000
MA(3)         0.177436   0.034332   5.168260   0.0000
MA(4)        -0.657530   0.000420  -1564.053   0.0000
============================================================
R-squared            0.451218    Mean dependent var-0.249292
Adjusted R-squared   0.427095    S.D. dependent var 1.437408
S.E. of regression   1.087981    Akaike info criter 0.219325
Sum squared resid    107.7169    Schwarz criterion  0.352885
Log likelihood      -141.7457    F-statistic        18.70542
Durbin-Watson stat   1.895751    Prob(F-statistic)  0.000000
============================================================
Inverted MA Roots          .6  -.07+.99i  -.07 -.99      -.97
============================================================
```
All the MA terms are statistically significant and the P-Values of the correlogram indicate that we have white noise.
```                   Correlogram of Residuals
==============================================================
Date: 04/18/98   Time: 17:24
Sample: 5 100
Included observations: 96
Q-statistic probabilities adjusted for 4 ARMA term(s)
==============================================================
Autocorrelation Partial Correlation  AC     PAC  Q-Stat Prob
==============================================================
. | .     |      . | .     |  1 0.048 0.048 0.2327
. | .     |      . | .     |  2 0.049 0.047 0.4779
. | .     |      . | .     |  3 0.012 0.007 0.4920
. | .     |      . | .     |  4 0.001-0.002 0.4921
.*| .     |      .*| .     |  5-0.083-0.084 1.1997 0.273
.*| .     |      .*| .     |  6-0.131-0.125 2.9944 0.224
.*| .     |      . | .     |  7-0.066-0.049 3.4496 0.327
. |*.     |      . |*.     |  8 0.121 0.143 5.0195 0.285
.*| .     |      .*| .     |  9-0.092-0.096 5.9386 0.312
. | .     |      . | .     | 10 0.035 0.027 6.0755 0.415
==============================================================
```
To forecast this series issue the following commands (see Epple Notes XIII-13):

EXPAND 120

SMPL 1 120

Then go into the "Forecast" menu (click the button). In the forecast menu enter a name for the standard error series and click the plot button. You will see:

To plot the original series along with the forecast, issue the command:

PLOT Y2F Y2