UNIVERSITY OF CALIFORNIA, LOS ANGELES
Department of Economics
Economics 143 (Cameron) - Applied Regression
Analysis
Problem Set #7: Serially Correlated
Errors
Outline of Solutions
Solutions to the problems in this homework (which was not to be graded)
can be found in the corresponding Fall 1997 Problem Set #7 and its Solution Key. Other solution information can be
found in the key to the Fall 1997 final exam.
1. The following questions pertain to EXHIBIT 2. These are real data, and we will
explore a preliminary model to explain
the observed quarterly time-series variation in automobile loans at commercial
banks. The variables read by the program are defined as follows:
DATE = year and quarter in decimal form
(e.g. 1960.25=1960:1)
AUTOCRED = consumer installment credit
outstanding: automobiles, commercial banks (million $, end of month, not
seasonally adjusted) [CITIBASE variable CCIUAC; monthly data averaged for
each quarter (1960:1-1996:4)]
YP = gross national product, total [CITIBASE
variable GNP; quarterly (1960:1-1996:4)].
R = nominal interest rate, measured as
the rate on commercial paper, 6-mo (% per annum, not seasonally adjusted)
[CITIBASE variable FYCP; monthly data averaged for each quarter
(1960:1-1996:4)].
AUTOINV = inventories, business, retail
durables, motor vehicle dealers; billions [CITIBASE variable GLRDA; quarterly
(1960:1-1996:4)]
QTR1, QTR2, QTR3, QTR4 = set of quarterly
dummy variables, equal to one during each respective quarter, zero
otherwise.
a.) Which three variables in this data set
are the most highly correlated?
YP, AUTOCRED, AUTOINV
b.) According to REGRESSION
2A, approximately what
is the rate of change of outstanding automobile loans per year?
16160.
These are quarterly data, so the estimated
change per year is 4[4040.7], which is approximately 16160.
c.) Based solely on REGRESSION 2B, does multicollinearity
compromise our ability to discern the incremental effects on AUTOCRED of
changes in any of the individual explanatory variables? Explain.
DW = 0.31 (P-value=0.000) implies AR(1) errors at a
minimum. YP and AUTOINV are highly collinear, so their statistical significance
may be compromized in Regression 2B. However, it is still very high, with t-
ratios of 2.18 and 8.96, so despite the multicollinearity, we can still identify
the distinct contributions of each variable. However, this also assumes that the
OLS standard errors are valid--in fact, they are NOT. In reality, we do not know
the true standard errors yet.....
d.) Is there evidence of systematic "seasonal"
variations in the level of AUTOCRED, according to REGRESSION 2B? Explain.
This model, which ignores serial correlation in the
errors, suggests no seasonality, since the F-test of the joint significance
of the set of quarterly dummmy variables fails to reject that the
coefficients on QTR2, WTR3, WTR4 are jointly zero. [However, note that these
hypothesis tests are NOT valid, since the regression ignores serially correlated
errors in computing standard errors of coefficients.]
e.) What is the purpose of REGRESSION 2C? What
does it imply about the results obtained from REGRESSION 2B?
Regression 2C is intended to detect serial correlations
up to order=4 in
the initial OLS error terms. Despite the collinearity among the various lagged
errors, ELAG1 through ELAG3 are strongly significant. We need an autoregressive
error model. Point estimates are unbiased, but standard errors (and therefore t-
ratios) are wrong in Regression 2B.
f.) Is REGRESSION 2D
likely to be adequate to
correct the problems revealed by REGRESSION 2C?
Why or why not? Explain.
Regression 2D allows only for first-order serial
correlation in the data, whereas we have identified a more-commplex lag structure
in the errors in Regression 2C. These are quarterly data, so we might expect more
than just AR(1): et = rho*et-1 + epsilont.
Instead, try AR(4): et = rho1*et-1 +
rho2*et-2 + rho3*et-3 +
rho4*et-4 + epsilont.
g.) Suppose the REGRESSION
2E was your preferred
model. Does this specification suggest the presence of seasonal effects
in AUTOCRED? Which months tend to have the highest amount of outstanding
car loans? July, August, September (QTR3). Which
months tend to have the
lowest amount of outstanding car loans? January, February,
March (QTR1).
Two of the seasonal dummies are now strongly
statistically significant, and we are more inclined to believe this significance
since we have remedied much of the serially correlated error problem by using the
AUTO command.
h.) How do the implications of REGRESSION 2E differ
from those of REGRESSION 2B concerning the
effects on car loans of (a)
nominal interest rates, and (b) car dealer inventories? Explain.
The apparent effect of R changes from large and highly
significant to small and completely insignificant; same for the effects of
AUTOINV. Both effects remain positive, but our hopes for using these variables to
explain AUTOCRED have evaporated. It seems to be all just GNP and quarterly
effects!
i.) Compare the goodness-of-fit of REGRESSION 2B with that of REGRESSION 2E.
R2 in Regression 2B is 0.9848;
R2 in Regression 2E is 0.9994. Each model uses the same number of
regressors, so these are comparable, and these R2 values are corrected
for the original variables (recall the transformation of the data that is
necessary to get the AUTO parameter estimates). We would expect the predicted and
actual AUTOCRED seris to track pretty well in both cases, but almost perfectly for
Regression 2E.
2. a) Read in the first portion of the data, check the descriptive statistics for things like the ranges of each variable.
|_smpl 1 115
|_read(int2.dat) obs p yp m1 m2 r
UNIT 88 IS NOW ASSIGNED TO: int2.dat
6 VARIABLES AND 115 OBSERVATIONS STARTING AT OBS 1
|_stat / pcor
NAME N MEAN ST. DEV VARIANCE MINIMUM MAXIMUM
OBS 115 7290.3 833.61 0.69490E+06 5901.0 8703.0
P 115 43.586 20.251 410.08 22.910 83.360
YP 115 1855.3 1290.9 0.16665E+07 499.00 4731.4
M1 115 188.80 56.699 3214.8 109.47 298.70
M2 115 1072.5 737.06 0.54326E+06 288.47 2795.1
R 115 6.9346 3.0969 9.5905 2.8600 16.213
CORRELATION MATRIX OF VARIABLES - 115 OBSERVATIONS
OBS 1.0000
P 0.95246 1.0000
YP 0.94975 0.99631 1.0000
M1 0.97198 0.91671 0.91567 1.0000
M2 0.94708 0.99172 0.99818 0.91324 1.0000
R 0.67879 0.63349 0.60464 0.65990 0.56556
OBS P YP M1 M2
There are some pretty high correlations here, so multicollinearity might be
suspected, at least to some degree.
|_* date might be more informative than the usual t variable:
|_genr date=1959+(time(-1)/4)
|_* annual inflation rate is four times quarterly inflation rate
|_* make sure it is expressed in percent, rather than decimal percent:
(Actually, just make sure the the way you express inflation MATCHES the way
the interest rate is expressed. Remember that changes of origin and/or changes of scale for a variable will affect parameter point estimates, but not the statistical significance of the coefficient on the variable.)
|_genr infl=( 4*(p-lag(p))/p )*100
..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
|_* any model using infl directly or indirectly loses the first observation
|_sample 2 115
|_* real interest rate is nominal rate less inflation
|_genr realr=r-infl
|_* express money stocks and gnp in real terms (1992 dollars)
Convert to constant 1992 dollars by dividing by the price index in each
period, and then multiplying by 100 (so the 1992 values are pretty much unchanged). This is because the price index is not 1.000 in 1992, but 100.0 in 1992.
|_* real m1
|_genr realm1=(m1/p)*100
|_* real m2
|_genr realm2=(m2/p)*100
|_* real gnp
|_genr realyp=(yp/p)*100
|_* check to make sure the numbers make sense
|_print r infl realr
R INFL REALR
3.603300 -0.3491925 3.952492
4.193300 0.6971678 3.496132
4.760000 1.562500 3.197500
4.686700 1.728608 2.958092
4.073300 1.549720 2.523580
3.373300 1.714531 1.658769
3.270000 1.366937 1.903063
3.013300 0.8525149 2.160785
2.860000 1.020408 1.839592
... bunch of output deleted
8.690000 4.222995 4.467005
7.910000 3.167710 4.742290
7.723300 2.588832 5.134468
7.700000 3.472135 4.227865
7.413300 1.803381 5.609919
6.543300 2.043359 4.499941
5.893300 3.066271 2.827029
5.726700 2.994232 2.732468
5.950000 3.068680 2.881320
6.846700 2.853343 3.993357
7.026700 3.119002 3.907698
|_plot r infl realr date / gnu line commfile=xnt.gnu datafile=xnt.dat
|_* try some simple regressions first
|_ols realr realyp
REQUIRED MEMORY IS PAR= 15 CURRENT PAR= 500
OLS ESTIMATION
114 OBSERVATIONS DEPENDENT VARIABLE = REALR
...NOTE..SAMPLE RANGE SET TO: 2, 115
R-SQUARE = 0.1185 R-SQUARE ADJUSTED = 0.1106
VARIANCE OF THE ESTIMATE-SIGMA**2 = 4.8319
STANDARD ERROR OF THE ESTIMATE-SIGMA = 2.1981
SUM OF SQUARED ERRORS-SSE= 541.17
MEAN OF DEPENDENT VARIABLE = 2.4722
LOG OF THE LIKELIHOOD FUNCTION = -250.538
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS F
REGRESSION 72.755 1. 72.755 15.057
ERROR 541.17 112. 4.8319 P-VALUE
TOTAL 613.92 113. 5.4329 0.000
ANALYSIS OF VARIANCE - FROM ZERO
SS DF MS F
REGRESSION 769.49 2. 384.74 79.626
ERROR 541.17 112. 4.8319 P-VALUE
TOTAL 1310.7 114. 11.497 0.000
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 112 DF P-VALUE CORR. COEFFICIENT AT MEANS
REALYP 0.80522E-03 0.2075E-03 3.880 0.000 0.344 0.3442 1.2487
CONSTANT -0.61472 0.8217 -0.7481 0.456-0.071 0.0000 -0.2487
|_ols realr realm2
REQUIRED MEMORY IS PAR= 15 CURRENT PAR= 500
OLS ESTIMATION
114 OBSERVATIONS DEPENDENT VARIABLE = REALR
...NOTE..SAMPLE RANGE SET TO: 2, 115
R-SQUARE = 0.0797 R-SQUARE ADJUSTED = 0.0715
VARIANCE OF THE ESTIMATE-SIGMA**2 = 5.0446
STANDARD ERROR OF THE ESTIMATE-SIGMA = 2.2460
SUM OF SQUARED ERRORS-SSE= 565.00
MEAN OF DEPENDENT VARIABLE = 2.4722
LOG OF THE LIKELIHOOD FUNCTION = -252.994
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS F
REGRESSION 48.925 1. 48.925 9.698
ERROR 565.00 112. 5.0446 P-VALUE
TOTAL 613.92 113. 5.4329 0.002
ANALYSIS OF VARIANCE - FROM ZERO
SS DF MS F
REGRESSION 745.66 2. 372.83 73.906
ERROR 565.00 112. 5.0446 P-VALUE
TOTAL 1310.7 114. 11.497 0.000
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 112 DF P-VALUE CORR. COEFFICIENT AT MEANS
REALM2 0.11918E-02 0.3827E-03 3.114 0.002 0.282 0.2823 1.0760
CONSTANT -0.18783 0.8797 -0.2135 0.831-0.020 0.0000 -0.0760
Result: each of the variables is individually statistically significant.
2. b)
|_* now try a multiple regression, checking for multicollinearity
|_ols realr realyp realm2 / resid=e predict=fitols exactdw auxrsqr
REQUIRED MEMORY IS PAR= 121 CURRENT PAR= 500
OLS ESTIMATION
114 OBSERVATIONS DEPENDENT VARIABLE = REALR
...NOTE..SAMPLE RANGE SET TO: 2, 115
DURBIN-WATSON STATISTIC = 0.54450
DURBIN-WATSON P-VALUE = 0.000000
The value should be near 2 if there is no serial correlation in the error terms. However, it is close to zero? How close? The p-value indicates that the area under the pdf for the DW test statistic, evaluated at 0.5445, is essentially zero. Thus we are way out in the left tail of the distribution, and must reject zero serial correlation in favor of positive serial correlation.
R-SQUARE OF REALYP ON OTHER INDEPENDENT VARIABLES = 0.9810
R-SQUARE OF REALM2 ON OTHER INDEPENDENT VARIABLES = 0.9810
R-SQUARE OF CONSTANT ON OTHER INDEPENDENT VARIABLES = 0.0000
Caution! There is a lot of multicollinearity here...
R-SQUARE = 0.2991 R-SQUARE ADJUSTED = 0.2865
VARIANCE OF THE ESTIMATE-SIGMA**2 = 3.8764
STANDARD ERROR OF THE ESTIMATE-SIGMA = 1.9689
SUM OF SQUARED ERRORS-SSE= 430.28
MEAN OF DEPENDENT VARIABLE = 2.4722
LOG OF THE LIKELIHOOD FUNCTION = -237.468
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS F
REGRESSION 183.64 2. 91.822 23.687
ERROR 430.28 111. 3.8764 P-VALUE
TOTAL 613.92 113. 5.4329 0.000
ANALYSIS OF VARIANCE - FROM ZERO
SS DF MS F
REGRESSION 880.37 3. 293.46 75.704
ERROR 430.28 111. 3.8764 P-VALUE
TOTAL 1310.7 114. 11.497 0.000
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 111 DF P-VALUE CORR. COEFFICIENT AT MEANS
REALYP 0.79391E-02 0.1347E-02 5.895 0.000 0.488 3.3941 12.3111
REALM2 -0.13001E-01 0.2431E-02 -5.348 0.000-0.453 -3.0793 -11.7369
CONSTANT 1.0527 0.7993 1.317 0.191 0.124 0.0000 0.4258
Despite the high multicollinearity, we still get sufficiently good resolution to be able to conclude that the coefficients on each variable are individually statistically significantly different from zero.
|_plot fitols realr date / gnu line commfile=xls.gnu datafile=xls.dat
2. c)
|_* quarterly data: check for first- and fourth-order autoregressive errors
|_* also called AR(1) and AR(4) errors
Just look at two lag-lengths for errors. Could do the intervening ones as well.
|_genr elag=lag(e)
|_genr elag4=lag(e,4)
|_* adjust sample (lose more observations by taking the fourth lags)
|_sample 3 115
|_plot e elag
113 OBSERVATIONS
*=E
M=MULTIPLE POINT
4.1053 | *
3.4737 | *
2.8421 | * *
2.2105 | * * ** * *
1.5789 | ***MM *M *
0.94737 | * * *MMMMM *
0.31579 | * MM*MMMM M M
-0.31579 | * ** M MM **
-0.94737 | * * ***MM
-1.5789 | * * * *
-2.2105 | * * M **
-2.8421 | * * **
-3.4737 | * M * * M *
-4.1053 | * M *
-4.7368 | *
-5.3684 | * strongly positively correlated
-6.0000 | *
________________________________________
-6.000 -3.000 0.000 3.000 6.000
ELAG
|_sample 6 115
|_plot e elag4
*=E
M=MULTIPLE POINT
4.1053 | *
3.4737 | *
2.8421 | * *
2.2105 | * * * M *
1.5789 | **MM * * *
0.94737 | * *M MM*M MM *
0.31579 | * MM**MMM M* *
-0.31579 | M * MM*M**
-0.94737 | * * M *M* **
-1.5789 | * * * *
-2.2105 | * ** M
-2.8421 | ** * *
-3.4737 | M* * M * *
-4.1053 | * ** *
-4.7368 | * also positively
-5.3684 | * correlated!
-6.0000 | *
________________________________________
-6.000 -3.000 0.000 3.000 6.000
ELAG4
If you are going to use a regression of E on ELAG1,...,ELAG4, you might want to use regression through the origin because the expected value of all errors should be zero. The fitted regression line should go through the means of all the data. Empirically, if you do not set the intercept to zero using the NOCONSTANT option, it will very likely be indistinguishable from zero. It is probably a moot point, therefore.
2. e) Now try a generalized least squares model that acknowledges AR(1) errors in the data. Recall that SHAZAM figures out an estimate of rho, then uses it to transform all the variables (including the intercept) by subtracting rho times the last-period value of the variable. Then the transformed data are used in a regression, and the resulting parameters (for data with well-behaved errors, are reported in the context of the original model. You almost don't need to know what is happening in the background.
This model assumes only first-order autoregressive errors.
|_sample 2 115
|_auto realr realyp realm2 / predict=fitauto rstat
REQUIRED MEMORY IS PAR= 23 CURRENT PAR= 500
DEPENDENT VARIABLE = REALR
..NOTE..R-SQUARE,ANOVA,RESIDUALS DONE ON ORIGINAL VARS
LEAST SQUARES ESTIMATION 114 OBSERVATIONS
BY COCHRANE-ORCUTT TYPE PROCEDURE WITH CONVERGENCE = 0.00100
ITERATION RHO LOG L.F. SSE
1 0.00000 -237.468 430.28
2 0.72552 -195.053 203.11
3 0.72622 -195.053 203.11
LOG L.F. = -195.053 AT RHO = 0.72622
ASYMPTOTIC ASYMPTOTIC ASYMPTOTIC
ESTIMATE VARIANCE ST.ERROR T-RATIO
RHO 0.72622 0.00415 0.06439 11.27916
R-SQUARE = 0.6692 R-SQUARE ADJUSTED = 0.6632
VARIANCE OF THE ESTIMATE-SIGMA**2 = 1.8298
STANDARD ERROR OF THE ESTIMATE-SIGMA = 1.3527
SUM OF SQUARED ERRORS-SSE= 203.11
MEAN OF DEPENDENT VARIABLE = 2.4722
LOG OF THE LIKELIHOOD FUNCTION = -195.053
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS
REGRESSION 410.82 2. 205.41
ERROR 203.11 111. 1.8298
TOTAL 613.92 113. 5.4329
ANALYSIS OF VARIANCE - FROM ZERO
SS DF MS
REGRESSION 1107.5 3. 369.18
ERROR 203.11 111. 1.8298
TOTAL 1310.7 114. 11.497
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 111 DF P-VALUE CORR. COEFFICIENT AT MEANS
REALYP 0.73525E-02 0.2253E-02 3.264 0.001 0.296 3.1433 11.4014
REALM2 -0.11836E-01 0.4014E-02 -2.949 0.004-0.270 -2.8034 -10.6851
CONSTANT 0.76829 1.786 0.4301 0.668 0.041 0.0000 0.3108
The transformed model no longer suffers from serially correlated errors.
|_plot fitauto realr date / gnu line commfile=xuto.gnu datafile=xuto.dat
DURBIN-WATSON = 2.1598 VON NEUMANN RATIO = 2.1789 RHO = -0.08769
RESIDUAL SUM = -1.2958 RESIDUAL VARIANCE = 1.8449
SUM OF ABSOLUTE ERRORS= 115.67
R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.6668
RUNS TEST: 64 RUNS, 59 POS, 0 ZERO, 55 NEG NORMAL STATISTIC = 1.1435
DURBIN H STATISTIC (ASYMPTOTIC NORMAL) = -1.2893
MODIFIED FOR AUTO ORDER=1
Explore a model that allows for second-order autocorrelation in the error terms.
|_auto realr realyp realm2 / order=2 predict=fitauto2 rstat
REQUIRED MEMORY IS PAR= 24 CURRENT PAR= 500
DEPENDENT VARIABLE = REALR
..NOTE..R-SQUARE,ANOVA,RESIDUALS DONE ON ORIGINAL VARS
LEAST SQUARES SECOND-ORDER AUTOCORRELATION
BY COCHRANE-ORCUTT TYPE PROCEDURE WITH CONVERGENCE =0.001000
114 OBSERVATIONS
ITERATION RHO1 RHO2 SSE SSE/N LOG.L.F.
1 0.00000 0.00000 430.27888 3.7743761 -237.46839
2 0.63173 0.12833 199.84788 1.7530516 -194.14548
3 0.63122 0.12952 199.84494 1.7530257 -194.14558
4 0.63119 0.12958 199.84485 1.7530250 -194.14559
Only the first-order rho term is individually statistically significant.
ASYMPTOTIC ASYMPTOTIC ASYMPTOTIC
ESTIMATE VARIANCE ST.ERROR T-RATIO AUTOCORRELATION
RHO1 0.63119 0.00862 0.09287 6.79660 0.72516
RHO2 0.12958 0.00862 0.09287 1.39529 0.58729
COVARIANCE -0.00625
REAL ROOTS - AUTOREGRESSIVE PROCESS DISPLAYS DAMPED EXPONENTIAL BEHAVIOUR
R-SQUARE = 0.6745 R-SQUARE ADJUSTED = 0.6686
VARIANCE OF THE ESTIMATE-SIGMA**2 = 1.8004
STANDARD ERROR OF THE ESTIMATE-SIGMA = 1.3418
SUM OF SQUARED ERRORS-SSE= 199.84
MEAN OF DEPENDENT VARIABLE = 2.4722
LOG OF THE LIKELIHOOD FUNCTION = -194.146
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS
REGRESSION 414.08 2. 207.04
ERROR 199.84 111. 1.8004
TOTAL 613.92 113. 5.4329
ANALYSIS OF VARIANCE - FROM ZERO
SS DF MS
REGRESSION 70.546 3. 23.515
ERROR 199.84 111. 1.8004
TOTAL 270.39 114. 2.3718
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 111 DF P-VALUE CORR. COEFFICIENT AT MEANS
REALYP 0.79708E-02 0.2239E-02 3.560 0.001 0.320 3.4077 12.3603
REALM2 -0.12872E-01 0.3967E-02 -3.245 0.002-0.294 -3.0490 -11.6211
CONSTANT 0.73880 1.953 0.3782 0.706 0.036 0.0000 0.2988
DURBIN-WATSON = 2.0569 VON NEUMANN RATIO = 2.0751 RHO = -0.03607
RESIDUAL SUM = -1.6903 RESIDUAL VARIANCE = 1.8151
SUM OF ABSOLUTE ERRORS= 114.99
R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.6723
RUNS TEST: 64 RUNS, 61 POS, 0 ZERO, 53 NEG NORMAL STATISTIC = 1.1876
|_plot fitauto2 realr date / gnu line commfile=xut2.gnu datafile=xut2.dat
|_* since these are quarterly data, suspect AR(4) errors
|_auto realr realyp realm2 / order=4 predict=fitauto4 rstat
DEPENDENT VARIABLE = REALR
..NOTE..R-SQUARE,ANOVA,RESIDUALS DONE ON ORIGINAL VARS
REQUIRED MEMORY IS PAR= 34 CURRENT PAR= 500
AUTOREGRESSIVE ERROR MODEL, ORDER= 4
ITERATION 0 ESTIMATES AND ERROR SUM OF SQUARES
0.79391E-02 -0.13001E-01 1.0527 0.00000 0.00000
0.00000 0.00000 430.28
ITERATION 1 ESTIMATES AND ERROR SUM OF SQUARES
0.93068E-02 -0.15413E-01 1.2151 -0.53817 0.42543E-01
-0.22177 -0.15955 181.19
ITERATION 2 ESTIMATES AND ERROR SUM OF SQUARES
0.84431E-02 -0.13949E-01 1.6742 -0.54521 0.48646E-01
-0.21823 -0.14101 179.72
ITERATION 3 ESTIMATES AND ERROR SUM OF SQUARES
0.84102E-02 -0.13897E-01 1.8152 -0.55751 0.44179E-01
-0.22204 -0.13885 179.59
ITERATION 4 ESTIMATES AND ERROR SUM OF SQUARES
0.83281E-02 -0.13763E-01 1.8478 -0.55614 0.45216E-01
-0.22207 -0.13976 179.59
ITERATION 5 ESTIMATES AND ERROR SUM OF SQUARES
0.83311E-02 -0.13765E-01 1.8450 -0.55620 0.45120E-01
-0.22230 -0.14049 179.59
ITERATION 6 ESTIMATES AND ERROR SUM OF SQUARES
0.83243E-02 -0.13754E-01 1.8492 -0.55623 0.45161E-01
-0.22230 -0.14042 179.59
RESIDUAL CORRELOGRAM
LM-TEST FOR HJ:RHO (J)=0,STATISTIC IS CHI-SQUARE(1)
LAG RHO STD ERR T-STAT LM-STAT
1 -0.0063 0.0937 -0.0674 0.2343
2 -0.0027 0.0937 -0.0283 0.0075
3 0.0048 0.0937 0.0515 0.0223
4 -0.0534 0.0937 -0.5705 0.6323
5 0.0426 0.0937 0.4551 0.2463
6 -0.1100 0.0937 -1.1740 1.5028
7 0.0201 0.0937 0.2149 0.0510
8 0.0307 0.0937 0.3282 0.1199
9 0.1317 0.0937 1.4058 2.1576
10 -0.0280 0.0937 -0.2986 0.0992
11 0.0299 0.0937 0.3191 0.1123
12 -0.1026 0.0937 -1.0958 1.3388
13 0.0415 0.0937 0.4432 0.2196
14 0.0787 0.0937 0.8407 0.7863
15 0.1864 0.0937 1.9902 4.5010
CHISQUARE WITH 15 D.F. IS 10.305
With dependence of the current error on the last FOUR errors, only the first and third lags are individually statistically significant.
ASYMPTOTIC
ESTIMATE VARIANCE ST.ERROR T-RATIO
RHO 1 0.55623 0.00867 0.09313 5.97248
RHO 2 -0.04516 0.01130 0.10629 -0.42486
RHO 3 0.22230 0.01095 0.10464 2.12437
RHO 4 0.14042 0.00880 0.09379 1.49726
R-SQUARE = 0.7075 R-SQUARE ADJUSTED = 0.7022
VARIANCE OF THE ESTIMATE-SIGMA**2 = 1.6179
STANDARD ERROR OF THE ESTIMATE-SIGMA = 1.2720
SUM OF SQUARED ERRORS-SSE= 179.59
MEAN OF DEPENDENT VARIABLE = 2.4722
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 111 DF P-VALUE CORR. COEFFICIENT AT MEANS
REALYP 0.83243E-02 0.2111E-02 3.944 0.000 0.351 3.5588 12.9085
REALM2 -0.13754E-01 0.3505E-02 -3.924 0.000-0.349 -3.2579 -12.4172
CONSTANT 1.8492 1.960 0.9432 0.348 0.089 0.0000 0.7480
DURBIN-WATSON = 2.0074 VON NEUMANN RATIO = 2.0252 RHO = -0.00631
RESIDUAL SUM = -6.8596 RESIDUAL VARIANCE = 1.6179
SUM OF ABSOLUTE ERRORS= 106.88
R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.7097
RUNS TEST: 66 RUNS, 53 POS, 0 ZERO, 61 NEG NORMAL STATISTIC = 1.5658
|_plot fitauto4 realr date / gnu line commfile=xut4.gnu datafile=xut4.dat
NOW look at the full sample, up to the second quarter of 1997.
Commentary not yet added; output only. NOTE: there is some decidedly different "action" in the longer time-series of data.
|_smpl 1 154
|_read(int2.dat) obs p yp m1 m2 r
UNIT 88 IS NOW ASSIGNED TO: int2.dat
6 VARIABLES AND 154 OBSERVATIONS STARTING AT OBS 1
|_stat / pcor
NAME N MEAN ST. DEV VARIANCE MINIMUM MAXIMUM
OBS 154 7777.8 1115.2 0.12437E+07 5901.0 9702.0
P 154 57.727 30.278 916.77 22.910 112.10
YP 154 2998.5 2307.8 0.53261E+07 499.00 8012.4
M1 154 225.50 84.035 7061.8 109.47 408.30
M2 154 1655.8 1197.3 0.14334E+07 288.47 3908.8
R 154 6.6940 2.8511 8.1289 2.8600 16.213
CORRELATION MATRIX OF VARIABLES - 154 OBSERVATIONS
OBS 1.0000
P 0.97844 1.0000
YP 0.96366 0.99135 1.0000
M1 0.96764 0.95163 0.95484 1.0000
M2 0.97120 0.99526 0.99442 0.94357 1.0000
R 0.27728 0.19827 0.11614 0.19576 0.13778
1.0000
OBS P YP M1 M2
R
|_* date might be more informative than the usual t variable:
|_genr date=1959+(time(-1)/4)
|_* annual inflation rate is four times quarterly inflation rate
|_* make sure it is expressed in percent, rather than decimal percent:
|_genr infl=( 4*(p-lag(p))/p )*100
..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
|_* any model using infl directly or indirectly loses the first observation
|_sample 2 154
|_* real interest rate is nominal rate less inflation
|_genr realr=r-infl
|_* express money stocks and gnp in real terms (1992 dollars)
|_* real m1
|_genr realm1=(m1/p)*100
|_* real m2
|_genr realm2=(m2/p)*100
|_* real gnp
|_genr realyp=(yp/p)*100
|_plot r infl realr date / gnu line commfile=int.gnu datafile=int.dat
|_* try some simple regressions first
|_ols realr realyp
REQUIRED MEMORY IS PAR= 20 CURRENT PAR= 500
OLS ESTIMATION
153 OBSERVATIONS DEPENDENT VARIABLE = REALR
...NOTE..SAMPLE RANGE SET TO: 2, 154
R-SQUARE = 0.0740 R-SQUARE ADJUSTED = 0.0678
VARIANCE OF THE ESTIMATE-SIGMA**2 = 4.2275
STANDARD ERROR OF THE ESTIMATE-SIGMA = 2.0561
SUM OF SQUARED ERRORS-SSE= 638.35
MEAN OF DEPENDENT VARIABLE = 2.5963
LOG OF THE LIKELIHOOD FUNCTION = -326.374
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS F
REGRESSION 50.982 1. 50.982 12.060
ERROR 638.35 151. 4.2275 P-VALUE
TOTAL 689.34 152. 4.5351 0.001
ANALYSIS OF VARIANCE - FROM ZERO
SS DF MS F
REGRESSION 1082.3 2. 541.17 128.012
ERROR 638.35 151. 4.2275 P-VALUE
TOTAL 1720.7 153. 11.246 0.000
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 151 DF P-VALUE CORR. COEFFICIENT AT MEANS
REALYP 0.40844E-03 0.1176E-03 3.473 0.001 0.272 0.2720 0.7050
CONSTANT 0.76582 0.5527 1.386 0.168 0.112 0.0000 0.2950
|_ols realr realm2
REQUIRED MEMORY IS PAR= 20 CURRENT PAR= 500
OLS ESTIMATION
153 OBSERVATIONS DEPENDENT VARIABLE = REALR
...NOTE..SAMPLE RANGE SET TO: 2, 154
R-SQUARE = 0.0658 R-SQUARE ADJUSTED = 0.0596
VARIANCE OF THE ESTIMATE-SIGMA**2 = 4.2649
STANDARD ERROR OF THE ESTIMATE-SIGMA = 2.0652
SUM OF SQUARED ERRORS-SSE= 644.00
MEAN OF DEPENDENT VARIABLE = 2.5963
LOG OF THE LIKELIHOOD FUNCTION = -327.048
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS F
REGRESSION 45.336 1. 45.336 10.630
ERROR 644.00 151. 4.2649 P-VALUE
TOTAL 689.34 152. 4.5351 0.001
ANALYSIS OF VARIANCE - FROM ZERO
SS DF MS F
REGRESSION 1076.7 2. 538.35 126.228
ERROR 644.00 151. 4.2649 P-VALUE
TOTAL 1720.7 153. 11.246 0.000
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 151 DF P-VALUE CORR. COEFFICIENT AT MEANS
REALM2 0.78304E-03 0.2402E-03 3.260 0.001 0.256 0.2565 0.7626
CONSTANT 0.61627 0.6298 0.9784 0.329 0.079 0.0000 0.2374
|_* now try a multiple regression, checking for multicollinearity
|_ols realr realyp realm2 / resid=e predict=fitols exactdw auxrsqr
REQUIRED MEMORY IS PAR= 208 CURRENT PAR= 500
OLS ESTIMATION
153 OBSERVATIONS DEPENDENT VARIABLE = REALR
...NOTE..SAMPLE RANGE SET TO: 2, 154
DURBIN-WATSON STATISTIC = 0.43242
DURBIN-WATSON P-VALUE = 0.000000
R-SQUARE OF REALYP ON OTHER INDEPENDENT VARIABLES = 0.9619
R-SQUARE OF REALM2 ON OTHER INDEPENDENT VARIABLES = 0.9619
R-SQUARE OF CONSTANT ON OTHER INDEPENDENT VARIABLES = 0.0000
R-SQUARE = 0.0767 R-SQUARE ADJUSTED = 0.0644
VARIANCE OF THE ESTIMATE-SIGMA**2 = 4.2430
STANDARD ERROR OF THE ESTIMATE-SIGMA = 2.0599
SUM OF SQUARED ERRORS-SSE= 636.45
MEAN OF DEPENDENT VARIABLE = 2.5963
LOG OF THE LIKELIHOOD FUNCTION = -326.146
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS F
REGRESSION 52.887 2. 26.443 6.232
ERROR 636.45 150. 4.2430 P-VALUE
TOTAL 689.34 152. 4.5351 0.003
ANALYSIS OF VARIANCE - FROM ZERO
SS DF MS F
REGRESSION 1084.2 3. 361.42 85.180
ERROR 636.45 150. 4.2430 P-VALUE
TOTAL 1720.7 153. 11.246 0.000
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 150 DF P-VALUE CORR. COEFFICIENT AT MEANS
REALYP 0.80497E-03 0.6034E-03 1.334 0.184 0.108 0.5360 1.3895
REALM2 -0.82199E-03 0.1227E-02 -0.6700 0.504-0.055 -0.2692 -0.8006
CONSTANT 1.0672 0.7134 1.496 0.137 0.121 0.0000 0.4110
|_* quarterly data: check for first- and fourth-order autoregressive errors
|_* also called AR(1) and AR(4) errors
|_genr elag=lag(e)
|_genr elag4=lag(e,4)
|_* adjust sample
|_sample 3 154
|_plot e elag
REQUIRED MEMORY IS PAR= 20 CURRENT PAR= 500
FOR MAXIMUM EFFICIENCY USE AT LEAST PAR= 23
152 OBSERVATIONS
*=E
M=MULTIPLE POINT
6.0000 |
5.3684 | M
4.7368 | ** * *
4.1053 |
3.4737 | * *
2.8421 | * *
2.2105 | * * * * ***
1.5789 | * * MMM * **
0.94737 | * **MMMM *
0.31579 | * **M*M **
-0.31579 | * *M*MM*MMMM M
-0.94737 | ** * MM* *
-1.5789 | * * *MM **M * * *
-2.2105 | *M* ** *
-2.8421 | * * * M ***M *
-3.4737 | * *
-4.1053 | ** * * * *
-4.7368 | * *
-5.3684 | *
-6.0000 |
________________________________________
-6.000 -3.000 0.000 3.000 6.000
ELAG
|_sample 6 154
|_plot e elag4
REQUIRED MEMORY IS PAR= 20 CURRENT PAR= 500
FOR MAXIMUM EFFICIENCY USE AT LEAST PAR= 23
149 OBSERVATIONS
*=E
M=MULTIPLE POINT
6.0000 |
5.3684 | * *
4.7368 | * M *
4.1053 |
3.4737 | * *
2.8421 | * *
2.2105 | M * *** *
1.5789 | M M M* M* ** *
0.94737 | * MMM * *
0.31579 | ** MMM*****
-0.31579 | ** M*MMMMMMMMMM* *
-0.94737 | * ** ** **MM* *
-1.5789 | ** M *M *MM * M
-2.2105 | * * * ** * *
-2.8421 | * M* * M*M*
-3.4737 | * *
-4.1053 | * * * * * *
-4.7368 | * *
-5.3684 | *
-6.0000 |
________________________________________
-6.000 -3.000 0.000 3.000 6.000
ELAG4
|_sample 2 154
|_auto realr realyp realm2 / predict=fitauto rstat
REQUIRED MEMORY IS PAR= 30 CURRENT PAR= 500
DEPENDENT VARIABLE = REALR
..NOTE..R-SQUARE,ANOVA,RESIDUALS DONE ON ORIGINAL VARS
LEAST SQUARES ESTIMATION 153 OBSERVATIONS
BY COCHRANE-ORCUTT TYPE PROCEDURE WITH CONVERGENCE = 0.00100
ITERATION RHO LOG L.F. SSE
1 0.00000 -326.146 636.45
2 0.78026 -253.405 244.43
3 0.79138 -253.385 244.29
4 0.79252 -253.385 244.28
5 0.79265 -253.386 244.28
LOG L.F. = -253.386 AT RHO = 0.79265
ASYMPTOTIC ASYMPTOTIC ASYMPTOTIC
ESTIMATE VARIANCE ST.ERROR T-RATIO
RHO 0.79265 0.00243 0.04929 16.08126
R-SQUARE = 0.6456 R-SQUARE ADJUSTED = 0.6409
VARIANCE OF THE ESTIMATE-SIGMA**2 = 1.6285
STANDARD ERROR OF THE ESTIMATE-SIGMA = 1.2761
SUM OF SQUARED ERRORS-SSE= 244.28
MEAN OF DEPENDENT VARIABLE = 2.5963
LOG OF THE LIKELIHOOD FUNCTION = -253.386
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS
REGRESSION 445.05 2. 222.53
ERROR 244.28 150. 1.6285
TOTAL 689.34 152. 4.5351
ANALYSIS OF VARIANCE - FROM ZERO
SS DF MS
REGRESSION 1476.4 3. 492.14
ERROR 244.28 150. 1.6285
TOTAL 1720.7 153. 11.246
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 150 DF P-VALUE CORR. COEFFICIENT AT MEANS
REALYP 0.22393E-02 0.1313E-02 1.705 0.090 0.138 1.4910 3.8655
REALM2 -0.37458E-02 0.2689E-02 -1.393 0.166-0.113 -1.2268 -3.6482
CONSTANT 2.0538 1.870 1.098 0.274 0.089 0.0000 0.7910
DURBIN-WATSON = 2.1515 VON NEUMANN RATIO = 2.1657 RHO = -0.08236
RESIDUAL SUM = -1.3090 RESIDUAL VARIANCE = 1.6400
SUM OF ABSOLUTE ERRORS= 143.60
R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.6432
RUNS TEST: 81 RUNS, 76 POS, 0 ZERO, 77 NEG NORMAL STATISTIC = 0.5683
DURBIN H STATISTIC (ASYMPTOTIC NORMAL) = -1.2853
MODIFIED FOR AUTO ORDER=1
|_auto realr realyp realm2 / order=2 predict=fitauto2 rstat
REQUIRED MEMORY IS PAR= 31 CURRENT PAR= 500
DEPENDENT VARIABLE = REALR
..NOTE..R-SQUARE,ANOVA,RESIDUALS DONE ON ORIGINAL VARS
LEAST SQUARES SECOND-ORDER AUTOCORRELATION
BY COCHRANE-ORCUTT TYPE PROCEDURE WITH CONVERGENCE =0.001000
153 OBSERVATIONS
ITERATION RHO1 RHO2 SSE SSE/N LOG.L.F.
1 0.00000 0.00000 636.44898 4.1597972 -326.14577
2 0.71908 0.07770 241.85314 1.5807395 -252.60053
3 0.70870 0.11063 241.25585 1.5768356 -252.45344
4 0.70813 0.11729 241.20476 1.5765017 -252.45068
5 0.70804 0.11907 241.19735 1.5764533 -252.45219
6 0.70802 0.11956 241.19578 1.5764430 -252.45279
ASYMPTOTIC ASYMPTOTIC ASYMPTOTIC
ESTIMATE VARIANCE ST.ERROR T-RATIO AUTOCORRELATION
RHO1 0.70802 0.00644 0.08027 8.82104 0.80417
RHO2 0.11956 0.00644 0.08027 1.48962 0.68894
COVARIANCE -0.00518
REAL ROOTS - AUTOREGRESSIVE PROCESS DISPLAYS DAMPED EXPONENTIAL BEHAVIOUR
R-SQUARE = 0.6501 R-SQUARE ADJUSTED = 0.6454
VARIANCE OF THE ESTIMATE-SIGMA**2 = 1.6080
STANDARD ERROR OF THE ESTIMATE-SIGMA = 1.2681
SUM OF SQUARED ERRORS-SSE= 241.20
MEAN OF DEPENDENT VARIABLE = 2.5963
LOG OF THE LIKELIHOOD FUNCTION = -252.453
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS
REGRESSION 448.14 2. 224.07
ERROR 241.20 150. 1.6080
TOTAL 689.34 152. 4.5351
ANALYSIS OF VARIANCE - FROM ZERO
SS DF MS
REGRESSION 43.842 3. 14.614
ERROR 241.20 150. 1.6080
TOTAL 285.04 153. 1.8630
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 150 DF P-VALUE CORR. COEFFICIENT AT MEANS
REALYP 0.29758E-02 0.1372E-02 2.169 0.032 0.174 1.9814 5.1368
REALM2 -0.52583E-02 0.2812E-02 -1.870 0.063-0.151 -1.7222 -5.1214
CONSTANT 2.5554 2.119 1.206 0.230 0.098 0.0000 0.9842
DURBIN-WATSON = 2.0606 VON NEUMANN RATIO = 2.0742 RHO = -0.03525
RESIDUAL SUM = -1.4427 RESIDUAL VARIANCE = 1.6168
SUM OF ABSOLUTE ERRORS= 142.92
R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.6483
RUNS TEST: 81 RUNS, 77 POS, 0 ZERO, 76 NEG NORMAL STATISTIC = 0.5683
|_* since these are quarterly data, suspect AR(4) errors
|_auto realr realyp realm2 / order=4 predict=fitauto4 rstat
DEPENDENT VARIABLE = REALR
..NOTE..R-SQUARE,ANOVA,RESIDUALS DONE ON ORIGINAL VARS
REQUIRED MEMORY IS PAR= 45 CURRENT PAR= 500
AUTOREGRESSIVE ERROR MODEL, ORDER= 4
ITERATION 0 ESTIMATES AND ERROR SUM OF SQUARES
0.80497E-03 -0.82199E-03 1.0672 0.00000 0.00000
0.00000 0.00000 636.45
ITERATION 1 ESTIMATES AND ERROR SUM OF SQUARES
0.14028E-02 -0.20424E-02 1.4733 -0.66218 0.97346E-01
-0.20255 -0.11043 225.45
ITERATION 2 ESTIMATES AND ERROR SUM OF SQUARES
0.43965E-02 -0.85009E-02 4.3594 -0.63423 0.65321E-01
-0.21590 -0.13879 216.17
ITERATION 3 ESTIMATES AND ERROR SUM OF SQUARES
0.44606E-02 -0.85885E-02 4.3486 -0.62442 0.58877E-01
-0.22699 -0.12799 216.09
ITERATION 4 ESTIMATES AND ERROR SUM OF SQUARES
0.44875E-02 -0.86544E-02 4.3783 -0.62417 0.58353E-01
-0.22753 -0.12824 216.09
ITERATION 5 ESTIMATES AND ERROR SUM OF SQUARES
0.44909E-02 -0.86591E-02 4.3768 -0.62402 0.58255E-01
-0.22765 -0.12811 216.09
RESIDUAL CORRELOGRAM
LM-TEST FOR HJ:RHO (J)=0,STATISTIC IS CHI-SQUARE(1)
LAG RHO STD ERR T-STAT LM-STAT
1 0.0009 0.0808 0.0106 0.0070
2 0.0090 0.0808 0.1107 0.1143
3 0.0077 0.0808 0.0954 0.0754
4 -0.0184 0.0808 -0.2274 0.0849
5 0.0105 0.0808 0.1295 0.0200
6 -0.0802 0.0808 -0.9920 1.0730
7 0.0037 0.0808 0.0459 0.0023
8 0.0946 0.0808 1.1697 1.4996
9 0.0845 0.0808 1.0458 1.2127
10 0.0268 0.0808 0.3312 0.1199
11 0.0624 0.0808 0.7718 0.6545
12 -0.0335 0.0808 -0.4143 0.1887
13 0.0461 0.0808 0.5700 0.3589
14 0.0837 0.0808 1.0357 1.1725
15 0.1494 0.0808 1.8477 3.8069
CHISQUARE WITH 15 D.F. IS 9.227
ASYMPTOTIC
ESTIMATE VARIANCE ST.ERROR T-RATIO
RHO 1 0.62402 0.00646 0.08040 7.76117
RHO 2 -0.05826 0.00886 0.09412 -0.61893
RHO 3 0.22765 0.00873 0.09342 2.43686
RHO 4 0.12811 0.00662 0.08137 1.57438
R-SQUARE = 0.6865 R-SQUARE ADJUSTED = 0.6823
VARIANCE OF THE ESTIMATE-SIGMA**2 = 1.4406
STANDARD ERROR OF THE ESTIMATE-SIGMA = 1.2003
SUM OF SQUARED ERRORS-SSE= 216.09
MEAN OF DEPENDENT VARIABLE = 2.5963
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 150 DF P-VALUE CORR. COEFFICIENT AT MEANS
REALYP 0.44909E-02 0.1418E-02 3.167 0.002 0.250 2.9902 7.7522
REALM2 -0.86591E-02 0.2828E-02 -3.062 0.003-0.243 -2.8360 -8.4336
CONSTANT 4.3768 1.893 2.312 0.022 0.185 0.0000 1.6858
DURBIN-WATSON = 1.9948 VON NEUMANN RATIO = 2.0079 RHO = 0.00086
RESIDUAL SUM = -4.2515 RESIDUAL VARIANCE = 1.4406
SUM OF ABSOLUTE ERRORS= 132.78
R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.6867
RUNS TEST: 89 RUNS, 70 POS, 0 ZERO, 83 NEG NORMAL STATISTIC = 1.9695
|_plot fitols realr date / gnu line commfile=ols.gnu datafile=ols.dat
|_plot fitauto realr date / gnu line commfile=auto.gnu datafile=auto.dat
|_plot fitauto2 realr date / gnu line commfile=aut2.gnu datafile=aut2.dat
|_plot fitauto4 realr date / gnu line commfile=aut4.gnu datafile=aut4.dat
Updated: 6:51 PM 12/7/98; Prepared by: Trudy Ann Cameron; Site Index