t =
t-1 +
vt with -1 <
< 1 Economic time series do not adjust instantaneously to changes in the economic environment. One example of a dynamic model is the regression model with first-order autoregressive errors (an AR(1) error model). The equation errors have the form:
t =
t-1 +
vt with -1 <
< 1
where
(RHO) is the autoregressive parameter and vt
is another random error that is assumed to be zero mean,
homoskedastic and serially uncorrelated.
Test procedures for detecting the presence of AR(1) errors were discussed earlier in this guide. Users should be reminded that the appearance of autocorrelated errors may reflect misspecification in the structural part of the equation rather than a misspecified error structure.
The SHAZAM AUTO command is available for the
estimation of models with autoregressive errors. The general
command format is:
AUTO depvar indeps / options
|
where depvar is the dependent variable, indeps is a list of the explanatory variables and options is a list of desired options.
By default, SHAZAM assumes an AR(1) error model and implements
model estimation by the Cochrane-Orcutt method. An iterative estimation
procedure is used. The starting point is to obtain parameter estimates
by OLS. The OLS residuals ê are then used to obtain an estimate
of
from the regression:
êt =
êt-1 + vt
for t = 2, ..., N
The estimate of
is used to construct transformed observations
(the first observation is given a special
transformation) and parameter estimates are obtained by applying OLS to
the transformed model. A new estimate of
is computed and another round of parameter estimates is obtained.
The iterations stop when successive estimates of
differ by less than 0.001.
It is worthwhile noting that this is an example of a nonlinear least squares estimator that is derived by minimizing a sum of squared errors function. A specification of the objective function is given in Griffiths, Hill and Judge [1993, Equation (16.4.4), p. 529]. The first observation is included in the objective function by a special transformation. In general, the solution of nonlinear least squares problems requires the use of numerical optimisation algorithms. The Cochrane-Orcutt iterative method is an example of a solution algorithm.
Alternative estimation algorithms for the AR(1) error model are available.
A number of these are implemented in SHAZAM as options on the
AUTO command. The interested user can consult the
SHAZAM User's Reference Manual.
The SHAZAM AUTO command can also estimate models
with higher order autoregressive errors and models with moving
average errors.
After estimation with AR(1) errors it is useful to check if the
vt errors are serially uncorrelated.
The Durbin-Watson test is inappropriate because the transformed model
incorporates a lagged dependent variable.
A test that can be used is Durbin's h test.
SHAZAM reports Durbin's h test when the RSTAT option is
specified on the AUTO command.
This example, from Gujarati, examines the relationship between the
help-wanted index and the unemployment rate in the United States.
The data set contains quarterly data from
1962 to 1967. Gujarati chooses a log-log model for the analysis.
The SHAZAM command file (filename:
HWI.SHA)
that follows first estimates the model by OLS and tests for the
possibility of AR(1) errors. Cochrane-Orcutt iterative estimation is then
implemented.
SAMPLE 1 24 READ (HWI.txt) DATE HWI URATE * Transform to logarithms GENR LHWI=LOG(HWI) GENR LURATE=LOG(URATE) * OLS estimation - test for autocorrelated errors OLS LHWI LURATE / RSTAT DWPVALUE LOGLOG * * Cochrane-Orcutt iterative estimation AUTO LHWI LURATE / RSTAT LOGLOG STOP |
The SHAZAM output can be viewed. The OLS estimation results report:
|
SHAZAM reports the p-value for the Durbin-Watson test statistic
as .000672. This gives strong evidence for positive
serial correlation in the residuals. The right of the output reports
an estimate of the autoregressive parameter RHO as
0.54571. This value is less than 1 in absolute value and
so is in the acceptable region for stationarity.
Therefore, this model is a candidate for estimation with AR(1) errors.
The iterations in the Cochrane-Orcutt estimation procedure are shown below.
|
The starting point at iteration 1 with RHO=0 is OLS.
Iteration 2 uses the RHO estimate computed from the OLS
residuals (as reported on the OLS estimation output).
The difference in the RHO estimate from iteration 5 to 6
is 0.58044-0.57999=0.00045. This is less than
0.001 and so the iterations stop at iteration 6.
The final column (SSE) reports the sum of squared errors
(this refers to the v(t) error). This is the value
of the nonlinear least squares objective function that is being
minimized. Each iteration should show a smaller value for the
SSE. It can be noted that iterations 5 and 6 show
little improvement in the SSE.
The final estimate for RHO is 0.58044.
It is important to take careful note of this value. A result that
is not infrequently encountered is a RHO value near 1.
This indicates non-stationarity and other modelling approaches may need
to be investigated.
How successful was the model estimation procedure ? The
RSTAT option on the AUTO command
produces the following output after the display of the estimation
results.
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The RHO value reported at the extreme right
is now the estimate of the autoregressive parameter for the
vt errors.
(The previous RHO was the estimate of the autoregressive
parameter for the t errors.)
Durbin's h statistic is computed to be 0.31712. This
is less than the 5% critical value from the standard normal distribution
of 1.96 and so the null hypothesis of no serial correlation is not
rejected. However, it can be noted that only 24 observations were
available and Durbin's h test may not be reliable in small samples.
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This example is from Gujarati [1995, Section 12.7, pp. 433+].
|_SAMPLE 1 24
|_READ (HWI.txt) DATE HWI URATE
UNIT 88 IS NOW ASSIGNED TO: HWI.txt
3 VARIABLES AND 24 OBSERVATIONS STARTING AT OBS 1
|_* Transform to logarithms
|_GENR LHWI=LOG(HWI)
|_GENR LURATE=LOG(URATE)
|_* OLS estimation - test for autocorrelated errors
|_OLS LHWI LURATE / RSTAT DWPVALUE LOGLOG
OLS ESTIMATION
24 OBSERVATIONS DEPENDENT VARIABLE = LHWI
...NOTE..SAMPLE RANGE SET TO: 1, 24
DURBIN-WATSON STATISTIC = 0.91077
DURBIN-WATSON POSITIVE AUTOCORRELATION TEST P-VALUE = 0.000672
NEGATIVE AUTOCORRELATION TEST P-VALUE = 0.999329
R-SQUARE = .9550 R-SQUARE ADJUSTED = .9530
VARIANCE OF THE ESTIMATE-SIGMA**2 = .33773E-02
STANDARD ERROR OF THE ESTIMATE-SIGMA = .58115E-01
SUM OF SQUARED ERRORS-SSE= .74302E-01
MEAN OF DEPENDENT VARIABLE = 4.9226
LOG OF THE LIKELIHOOD FUNCTION(IF DEPVAR LOG) = -82.8653
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 22 DF P-VALUE CORR. COEFFICIENT AT MEANS
LURATE -1.5375 .7114E-01 -21.61 .000 -.977 -.9772 -1.5375
CONSTANT 7.3084 .1110 65.82 .000 .997 .0000 7.3084
DURBIN-WATSON = .9108 VON NEUMANN RATIO = .9504 RHO = .54571
RESIDUAL SUM = .00000 RESIDUAL VARIANCE = .33773E-02
SUM OF ABSOLUTE ERRORS= 1.1162
R-SQUARE BETWEEN OBSERVED AND PREDICTED = .9550
R-SQUARE BETWEEN ANTILOGS OBSERVED AND PREDICTED = .9563
RUNS TEST: 4 RUNS, 13 POS, 0 ZERO, 11 NEG NORMAL STATISTIC = -3.7492
|_*
|_* Cochrane-Orcutt iterative estimation
|_AUTO LHWI LURATE / RSTAT LOGLOG
DEPENDENT VARIABLE = LHWI
..NOTE..R-SQUARE,ANOVA,RESIDUALS DONE ON ORIGINAL VARS
LEAST SQUARES ESTIMATION 24 OBSERVATIONS
BY COCHRANE-ORCUTT TYPE PROCEDURE WITH CONVERGENCE = .00100
ITERATION RHO LOG L.F. SSE
1 .00000 -82.8653 .74302E-01
2 .54571 -78.8009 .52180E-01
3 .57223 -78.8066 .52111E-01
4 .57836 -78.8108 .52106E-01
5 .57999 -78.8120 .52106E-01
6 .58044 -78.8124 .52106E-01
LOG L.F. = -78.8124 AT RHO = .58044
ASYMPTOTIC ASYMPTOTIC ASYMPTOTIC
ESTIMATE VARIANCE ST.ERROR T-RATIO
RHO .58044 .02763 .16622 3.49203
R-SQUARE = .9685 R-SQUARE ADJUSTED = .9670
VARIANCE OF THE ESTIMATE-SIGMA**2 = .23684E-02
STANDARD ERROR OF THE ESTIMATE-SIGMA = .48667E-01
SUM OF SQUARED ERRORS-SSE= .52106E-01
MEAN OF DEPENDENT VARIABLE = 4.9226
LOG OF THE LIKELIHOOD FUNCTION(IF DEPVAR LOG) = -78.8124
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 22 DF P-VALUE CORR. COEFFICIENT AT MEANS
LURATE -1.4712 .1251 -11.76 .000 -.929 -.9351 -1.4712
CONSTANT 7.2077 .1955 36.87 .000 .992 .0000 7.2077
DURBIN-WATSON = 1.8594 VON NEUMANN RATIO = 1.9402 RHO = .03757
RESIDUAL SUM = .84449E-02 RESIDUAL VARIANCE = .23717E-02
SUM OF ABSOLUTE ERRORS= .90632
R-SQUARE BETWEEN OBSERVED AND PREDICTED = .9684
R-SQUARE BETWEEN ANTILOGS OBSERVED AND PREDICTED = .9710
RUNS TEST: 12 RUNS, 11 POS, 0 ZERO, 13 NEG NORMAL STATISTIC = -.3854
DURBIN H STATISTIC (ASYMPTOTIC NORMAL) = .31712
MODIFIED FOR AUTO ORDER=1
|_STOP
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