Chapter 20
Serial Correlation
◼ Exercises
1. For the first-order autoregressive model, the autocorrelation is . Consider the first difference,
1,
t t t
v−
= −
which has Var[vt] = 2Var[t] − 2Cov[(t,
1t−
)] = 2u2[1/(1 − 2) − /(1 − 2)] = 2u2/(1 + )
and Cov[vt,
1t−
v−
] = 2Cov[t,
] − Var[t] − Cov[t,
] = u2[1/(1 − 2)][2 − 1 − 2] = u2[( − 1)/
For the moving average process, the first-order autocorrelation is Cov[(t,
1)] /
t−
Var[t] = −/(1 +
2. Derive the disturbance covariance matrix for the model yt = xt + t, t =
1t−
+ ut −
1.
t
u−
1t−
1t
1t
What
parameter is estimated by the regression of the ordinary least squares residuals on their lagged values?
Chapter 20 Serial Correlation 157
158 Greene • Econometric Analysis, Seventh Edition
To evaluate this expression, write
Therefore, the middle term is zero and the third is simply u2. Thus,
3. Since the regression contains a lagged dependent variable, we cannot use the Durbin-Watson statistic
4. It is commonly asserted that the Durbin-Watson statistic is only appropriate for testing for first-order
autoregressive disturbances. What combination of the coefficients of the model is estimated by the
Durbin-Watson statistic in each of the following cases: AR(1), AR(2), MA(1)? In each case, assume
that the regression model does not contain a lagged dependent variable. Comment on the impact on
your results of relaxing this assumption.
Chapter 20 Serial Correlation 159
Applications
1. Phillips Curve
–> date;1950.1$
–> peri;1950.1-2000.4$
+———————————————————————–+
| Ordinary least squares regression Weighting variable = none |
| Dep. var. = DP Mean= -.1926996283E-01, S.D.= 2.818214558 |
+———————————————————————–+
+———+————–+—————-+——–+———+———-+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+———————————————————————–+
| Ordinary least squares regression Weighting variable = none |
| Dep. var. = U Mean= -.3890391012E-01, S.D.= 2.799476915 |
| Model size: Observations = 199, Parameters = 3, Deg.Fr.= 196 |
160 Greene • Econometric Analysis, Seventh Edition
+———————————————+
| AR(1) Model: e(t) = rho * e(t-1) + u(t) |
| Initial value of rho = -.41378 |
| Maximum iterations = 100 |
+———————————————+
+———+————–+—————-+——–+———+———-+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+———+————–+—————-+——–+———+———-+
Constant .4704274598 .47671946 .987 .3237
2. Improved Phillips Curve Model
–> crea;newecon=dmy(1974.1,2000.4)$
–> regr;lhs=dp;rhs=one,unemp,newecon;plot$
+———————————————————————–+
| Ordinary least squares regression Weighting variable = none |
| Dep. var. = DP Mean= -.1926996283E-01, S.D.= 2.818214558 |
| Model size: Observations = 202, Parameters = 3, Deg.Fr.= 199 |
+———————————————————————–+
+———+————–+—————-+——–+———+———-+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+———+————–+—————-+——–+———+———-+
Constant .5507626279 .74399306 .740 .4600
3. GARCH Models
a. We used LIMDEP with the macroeconomics data in Table F5.1. The rate of inflation was computed
with all observations, then observations 6 to 204 were used to remove the missing data due to
lags. Least squares results were obtained first. The residuals were then computed and squared.
Using observations 15–204, we then computed a regression of the squared residual on a constant
and 8 lagged values. The chi-squared statistic with 8 degrees of freedom is 28.24. The critical
value from the table for 95% significance and 8 degrees of freedom is 15.51, so at this level of
significance, the hypothesis of no GARCH effects is rejected.
crea;pt=100*log(cpi_u/cpi_u[-1])$
crea;pt1=pt[-1];pt2=pt[-2];pt3=pt[-3];pt4=pt[-4]$
samp;6-204$
+———————————————————————–+
| Ordinary least squares regression Weighting variable = none |
| Dep. var. = PT Mean= .9589185961 , S.D.= .8318268241 |
+———————————————————————–+
+———+————–+—————-+——–+———+———-+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+———+————–+—————-+——–+———+———-+
Constant .1296044455 .67521735E-01 1.919 .0564
PT1 .2856136998 .69863942E-01 4.088 .0001 .97399582
samp;6-204$
crea;vt=et*et$
crea;ht=vt-8/36*vt[-1]-7/36*vt[-2]-6/36*vt[-3]-5/36*vt[-4]-4/36*vt[-5]-3/36*vt[-6]
-2/36*vt[-7]-1/36*vt[-8]$
samp;15-204$
162 Greene • Econometric Analysis, Seventh Edition
Once we have an estimate of 0 in hand, we then computed the set of variances according to the
ARCH(8) model, using the lagged squared residuals. Finally, we used these variance estimators
+———————————————————————–+
| Ordinary least squares regression Weighting variable = WT |
| Dep. var. = PT Mean= .8006997687 , S.D.= .6327877239 |
+———+————–+—————-+——–+———+———-+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+———+————–+—————-+——–+———+———-+
Constant .1468553158 .60127085E-01 2.442 .0155
PT1 .9760051110E-01 .88469908E-01 1.103 .2714 .77755556
The 8-period ARCH model produces quite a substantial change in the estimates. Once again, this
probably results from the restrictive assumption about the lag weights in the ARCH model. The
GARCH model follows.
+———————————————+
| GARCH MODEL |
| Maximum Likelihood Estimates |
| Model estimated: Jul 31, 2002 at 01:19:14PM.|
| Dependent variable PT |
| Weighting variable None |
+———————————————+
+———+————–+—————-+——–+———+———-+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+———+————–+—————-+——–+———+———-+
Regression parameters
Constant .1308478127 .61887183E-01 2.114 .0345
PT1 .1749239917 .70912277E-01 2.467 .0136 .98810078
Chapter 20 Serial Correlation 163
Lagged Variance Terms
Delta(1) .8556879884 .89322732E-01 9.580 .0000