forked from burakbayramli/books
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathExpt1.txt
3298 lines (3113 loc) · 132 KB
/
Expt1.txt
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
Chapter 1. Introduction
/*==========================================================
Example 1.1. Keynes’s Consumption Function
*/==========================================================
?
? Read data
?
Read ; Nobs = 10 ; Nvar = 2 ; Names = C,X
; By Variables $
672.1 696.8 737.1 767.9 762.8
779.4 823.1 864.3 903.2 927.6
751.6 779.2 810.3 864.7 857.5
874.9 906.8 942.9 988.8 1015.7
?
? Plot the figure
?
Plot ; Lhs = X ; Rhs = C ; Regression Line $
/*==========================================================
Example 1.2. Income and Education - An Econometric Issue
*/==========================================================
?
? There are no computations in Example 1.2.
?
Chapter 2. Matrix Algebra
/*=========================================================
Section 2.9.2.
/*=========================================================
?
? The unconstrained solution requires computation
? of a - 2Ax = 0, or [x1 x2 x3]' = inv(2*A)a.
?
Matrix ; MA = [2,1,3 /
1,3,2 /
3,2,5 ]
; a = [5 / 4 / 2 ] $
Matrix ; list ; x = .5 * <MA> * a $
Calc ; List ; fn = a'x - qfr(x,MA) $
/*
Matrix X 1
+--------------
1| .1125000D+02
2| .1750000D+01
3| -.7250000D+01
FN = .24375000000000010D+02
The constrained solution requires solution of
[ -2A C' ] (x ) (-a)
[ C 0 ] (lambda) = ( 0)
C = [ 1 -1 1 ]
[ 1 1 1 ]
There are simpler ways to get this solution,
but the following is complete and explicit.
*/
Matrix ; C = [1, -1, 1 /
1, 1, 1] $
Matrix ; MTWOA = -2 * MA ; Minusa = -1 * a $
Matrix ; Zero = [0 / 0] ; Zero22 = [0,0/0,0]$
Matrix ; CT = C' $
Matrix ; D = [MTWOA , CT /
C , Zero22 ]$
Matrix ; q = [Minusa / zero ] $
Matrix ; XL = <D> * q $
/*
Note that the solution for x(2) is not identically zero because of rounding.
*/
Matrix ; List ; x = XL(1:3) ; lambda=XL(4:5) $
Calc ; List ; fn = a'x - qfr(x,MA) $
/*
Matrix X has 3 rows and 1 columns.
1
+--------------
1| .1500000D+01
2| .5551115D-15
3| -.1500000D+01
Matrix LAMBDA has 2 rows and 1 columns.
1
+--------------
1| -.5000000D+00
2| -.7500000D+01
FN = .22499999999999980D+01
Chapter 3. Probability and Distribution Theory
/*=========================================================
Example 3.1 Poisson Model for a Discrete Outcome.
No computations needed. To illustrate the distribution, try
Calc ; TBP( lambda) $ For example: Calc ; TBP(5) $ produces
/*=========================================================
Poisson distribution with lambda = 5.0000
[Probability of x occurrences, mean occurrences/pd = lambda.]
Mean = 5.00000, Standard deviation = 2.23607 x P(X=x) P(X<=x)
P(x) +---------------------------------------------------+ -- ------ -------
.200 | | 0 .00674 .00674
.192 + + 1 .03369 .04043
.184 | | 2 .08422 .12465
.176 | X X | 3 .14037 .26503
.168 | X X | 4 .17547 .44049
.160 | X X | 5 .17547 .61596
.152 | X X - | 6 .14622 .76218
.144 + x X X X + 7 .10444 .86663
.136 | X X X X | 8 .06528 .93191
.128 | X X X X | 9 .03627 .96817
.120 | X X X X | 10 .01813 .98630
.112 | X X X X - | 11 .00824 .99455
.104 | X X X X X | 12 .00343 .99798
.096 + X X X X X + 13 .00132 .99930
.088 | x X X X X X | 14 .00047 .99977
.080 | X X X X X X | 15 .00016 .99993
.072 | X X X X X X - | 16 .00005 .99998
.064 | X X X X X X X | 17 .00001 .99999
.056 | X X X X X X X | 18 .00000 1.00000
.048 + X X X X X X X + 19 .00000 1.00000
.040 | - X X X X X X X x | 20 .00000 1.00000
.032 | X X X X X X X X X | 21 .00000 1.00000
.024 | X X X X X X X X X - | 22 .00000 1.00000
.016 | X X X X X X X X X X - | 23 .00000 1.00000
.008 |X X X X X X X X X X X X x - - - - - - - | 24 .00000 1.00000
.000 ++---+---+---+---+---+---+---+---+---+---+---+---+--+ 25+ .00000 1.00000
0 2 4 6 8 10 12 14 16 18 20 22 24 x
*/
/*==================================================================
Example 3.2. Approximation to the Chi-Squared Distribution
Computes exact and approximate values of chi-squared probabilities.
*/==================================================================
Proc=apxchi(x,d)$
Calc;z=sqr(2*x)-sqr(2*d-1)
;list
;approx=Phi(z)
;exact=chi(x,d)$
Endproc
Exec;proc=apxchi(85,70)$
/*
APPROX = .89409039431135510D+00
EXACT = .89297135030469340D+00
*/
/*==================================================================
Example 3.3. Linear Transformation of Normal Variable
No computations.
*/==================================================================
/*==================================================================
Example 3.4. Linear Transformations
Linear transformation of normally distributed variable. No
computations.
*/==================================================================
/*==================================================================
Example 3.5. Regression in an Exponential Distribution
Conditional distribution for an exponential model. No computations.
*/==================================================================
/*==================================================================
Example 3.6. Poisson Regression
Poisson Regression. Linear conditional mean function. No computations.
*/==================================================================
/*==================================================================
Example 3.7. Conditional Variance in a Poisson Model
Poisson Regression. Conditional variance function. No computations.
*/==================================================================
/*==================================================================
Example 3.8. Uniform - Exponential Mixture Distibution.
No computations.
*/==================================================================
/*==================================================================
Example 3.9. Covariance in a Mixture Distibution.
No computations.
*/
/*==================================================================
Example 3.10. Decomposition of Variance.
No computations.
*/==================================================================
/*==================================================================
Example 3.11. Conditional Variance in a Poisson Regression.
Simple arithmetic computations.
*/==================================================================
/*==================================================================
Example 3.12. Analysis of Variance in a Poisson Model.
No computations.
*/==================================================================
Chapter 4. Statistical Inference
/*==================================================================
Example 4.1. Descriptive Statistics for a Random Sample
*/==================================================================
Read ; Nobs = 20 ; Nvar = 3 ; Names = I,Y,E$
1 20.5 12
2 31.5 16
3 47.7 18
4 26.2 16
5 44.0 12
6 8.28 12
7 30.8 16
8 17.2 12
9 19.9 10
10 9.96 12
11 55.8 16
12 25.2 20
13 29.0 12
14 85.5 16
15 15.1 10
16 28.5 18
17 21.4 16
18 17.7 20
19 6.42 12
20 84.9 16
Dstat ; Rhs = Y,E ; All ; Output = 3 $
Plot ; Lhs = E ; Rhs = Y $
Descriptive Statistics
All results based on nonmissing observations.
===============================================================================
Variable Mean Std.Dev. Minimum Maximum Cases
===============================================================================
Y 31.2780000 22.3758337 6.42000000 85.5000000 20
Skewness= 1.3343 Kurtosis= 3.8681
E 14.6000000 3.11870418 10.0000000 20.0000000 20
Skewness= .1649 Kurtosis= 1.7860
Matrix COV.MAT. has 2 rows and 2 columns.
Y E
+----------------------------
Y | .5006779D+03 .2359705D+02
E | .2359705D+02 .9726316D+01
Correlation Matrix for Listed Variables
Y E
Y 1.00000 .33815
E .33815 1.00000
/*==================================================================
Example 4.2. Sampling Distribution of a Sample Mean
Central limit theorem. Computes a draw from chi-squared 1 by squaring
a draw from standard normal. Averages 4 such draws. Repeats
1000 times, and plots a histogram of the draws.
*/==================================================================
Sample ; 1-1000$
Create ; Means = (1/4)*(
(Rnn(0,1))^2+(Rnn(0,1))^2
+(Rnn(0,1))^2+(Rnn(0,1))^2)$
Histogram ; Rhs=means ; int=30$ (30 Bars)
Histogram for MEANS NOBS= 1000, Too low: 0, Too high: 0
Bin Lower limit Upper limit Frequency Cumulative Frequency
========================================================================
0 .026 .175 46 ( .0460) 46( .0460)
1 .175 .324 75 ( .0750) 121( .1210)
2 .324 .473 103 ( .1030) 224( .2240)
3 .473 .623 113 ( .1130) 337( .3370)
4 .623 .772 112 ( .1120) 449( .4490)
5 .772 .921 100 ( .1000) 549( .5490)
6 .921 1.070 77 ( .0770) 626( .6260)
7 1.070 1.219 66 ( .0660) 692( .6920)
8 1.219 1.368 58 ( .0580) 750( .7500)
9 1.368 1.517 47 ( .0470) 797( .7970)
10 1.517 1.667 39 ( .0390) 836( .8360)
11 1.667 1.816 31 ( .0310) 867( .8670)
12 1.816 1.965 28 ( .0280) 895( .8950)
13 1.965 2.114 25 ( .0250) 920( .9200)
14 2.114 2.263 13 ( .0130) 933( .9330)
15 2.263 2.412 17 ( .0170) 950( .9500)
16 2.412 2.562 10 ( .0100) 960( .9600)
17 2.562 2.711 13 ( .0130) 973( .9730)
18 2.711 2.860 3 ( .0030) 976( .9760)
19 2.860 3.009 5 ( .0050) 981( .9810)
20 3.009 3.158 3 ( .0030) 984( .9840)
21 3.158 3.307 4 ( .0040) 988( .9880)
22 3.307 3.457 4 ( .0040) 992( .9920)
23 3.457 3.606 2 ( .0020) 994( .9940)
24 3.606 3.755 3 ( .0030) 997( .9970)
25 3.755 3.904 1 ( .0010) 998( .9980)
26 3.904 4.053 1 ( .0010) 999( .9990)
27 4.053 4.202 0 ( .0000) 999( .9990)
28 4.202 4.352 0 ( .0000) 999( .9990)
29 4.352 4.501 1 ( .0010) 1000(1.0000)
/*==================================================================
Example 4.3. Sampling Distribution of the Sample Minimum.
No computations.
*/==================================================================
/*==================================================================
Example 4.4. Mean Squared Error of the Sample Variance.
No computations.
*/==================================================================
/*==================================================================
Example 4.5. Quadratic Loss Function.
No computations.
*/==================================================================
/*==================================================================
Example 4.6. Likelihood Function for the Exponential Distribution.
No computations.
*/==================================================================
/*===================================================================
Example 4.7. Likelihood Function for the Normal Distribution.
No computations.
*/==================================================================
/*==================================================================
Example 4.8. Variance Bound for the Poisson Distribution.
No computations.
*/==================================================================
/*==================================================================
Example 4.9. Information Matrix for the Normal Distribution.
No computations.
*/==================================================================
/*==================================================================
Example 4.10. Convergence of the Sample Minimum in Exponential
Sampling. No computations.
*/==================================================================
/*==================================================================
Example 4.11. Estimating a Function of the Mean
No computations.
*/==================================================================
/*==================================================================
Example 4.12. Probability Limit of a Function of x-bar and s-squared
No computations.
*/==================================================================
/*==================================================================
Example 4.13. Limiting Distribution of t(n-1)
The following plots the distribution of t for 2, 10, 40,and infinite
degrees of freedom.
*/==================================================================
Sample ; 1-101$ Plot and connect 100 segments
Create ; t=trn(-4,.08)$ Values -4 to +4 in steps of .08
?
? This procedure obtains the value of the density over a
? grid of values contained in variable t, and puts then
? in a variable passed as fn.
?
Proc = tdensity(fn,t,d)$
Create;fn=Gma((d+1)/2)/Gma(d/2) / sqr(d*pi) *
(1+t*t/d)^(-(d+1)/2)$
Endproc$
?
? Compute for 2, 10, 40, infinity. (Last is N(0,1).)
Execute ; Proc=tdensity(t2,t,2)$
Execute ; Proc=tdensity(t10,t,10)$
Execute ; Proc=tdensity(t40,t,40)$
Create ; tinf=N01(t)$
?
? Now plot the four densities in the same figure.
Plot ; lhs=t ;rhs=t2,t10,t40,tinf ; Fill intervals
; Yaxis=Density
; Title=t Densities with Different Degrees of Freedom$
/*==================================================================
Example 4.14. Asymptotic Distribution of the Mean of an Exponential
Sample.
We generate the plot in the text for the mean of a sample of 16 from
exponential with parameter theta=1. Note, N and Theta are reserved
names in LIMDEP, so we use other names
*/==================================================================
Sample ; 1-101$
Calc ; thet = 1 ; nobs=16 ; df=2*nobs $
?
? Density of the chi-squared variable with 2n degrees of freedom
? Computed for t = 0 to 50.
Create ; t = trn(0,.5)
; Exact = .5^(df/2)/Gma(df/2) * Exp(-.5*t)*(t^(df/2-1)) $
?
? For the simple linear transformation from t to Xbar=theta/2n*t,
? just scale the variable down and the density up.
?
Create ; Xbar_n = t*thet/df
; Exact = Df/thet * Exact $
?
? Asymptotic distribution is normal with mean theta, and
? standard deviation theta/sqr(n).
?
Create ; Asymp = 1/Sqr(thet/nobs) *
N01((Xbar_n-thet)/Sqr(Thet/Nobs))$
?
? Now, just plot two densities.
?
Plot ; Lhs = Xbar_n
; Rhs = Exact,Asymp
; Fill;Endpoints=0,1.75
; Yaxis=Density
; Title=Asymptotic and Exact Distribution$
/*==================================================================
Example 4.15. Asymptotic Inefficiency of the Median in Normal
Sampling
No computations.
*/==================================================================
/*==================================================================
Example 4.16. Asymptotic Distribution for a Function of Two Estimates
No computations.
*/==================================================================
/*==================================================================
Example 4.17. Asymptotic Moments of the Sample Variance
No computations.
*/==================================================================
/*==================================================================
Example 4.18. Poisson Likelihood Function.
No computations.
*/==================================================================
/*==================================================================
Example 4.19. Likelihood for the Normal Distribution.
No computations.
*/==================================================================
/*==================================================================
Example 4.20. Multivariate Normal Mean Vector
No computations.
*/==================================================================
/*==================================================================
Example 4.21. Information Matrix for a Multivariate Normal Distribution
No computations.
*/==================================================================
/*==================================================================
Example 4.22. Variance Estimators for an MLE
*/==================================================================
Read ; Nobs=20 ; Nvar=3 ; Names=
I, Y, X $
1 20.5 12
2 31.5 16
3 47.7 18
4 26.2 16
5 44.0 12
6 8.28 12
7 30.8 16
8 17.2 12
9 19.9 10
10 9.96 12
11 55.8 16
12 25.2 20
13 29.0 12
14 85.5 16
15 15.1 10
16 28.5 18
17 21.4 16
18 17.7 20
19 6.42 12
20 84.9 16
?
? 1. Compute Maximum Likelihood Estimator
? Results that will follow are shown.
?
Maximize ; Fcn = -log(beta+e)-y/(beta+e)
; Labels = beta
; Start = 0 $
/*=================================================================
Note that the standard error shown below is the square root of the
BHHH estimate which we compute below. This is the one that LIMDEP
uses.
+---------------------------------------------+
| User Defined Optimization |
| Maximum Likelihood Estimates |
| Dependent variable Function |
| Weighting variable ONE |
| Number of observations 20 |
| Iterations completed 2 |
| Log likelihood function -88.43626 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
BETA 15.60272720 10.025547 1.556 .1196
*/=================================================================
?
? Compute variance estimators. Compute as a set of observations
? sum, then take reciprocals of sums.
Create ; Hessian = 1/(beta+e)^2 - 2*y/(beta+e)^3
; EHessian = -1/(beta+e)^2
; BHHH = (-1/(beta+e) + y/(beta+e)^2)^2$
Calc;List; V1 = -1/Sum(Hessian)
; V2 = -1/Sum(EHessian)
; V3 = 1/Sum(BHHH)$
/*=================================================================
V1 = .46163366852196030D+02
V2 = .44254604293881970D+02
V3 = .10051158752168130D+03
Calculator: Computed 3 scalar results
*/==================================================================
/*==================================================================
Example 4.23. Two Step Estimation.
No computations, as no data used. The following shows how to do it.
*/==================================================================
? 1. Set up the data after reading them in.
Namelist ; X = the list of variables $
Namelist ; Z = the list of variables $
Create ; Y1 = the Poisson variable $
Create ; Y2 = the binary variable $
? 2 Fit the Poisson model. LIMDEP has a simple command
? for this: Poisson ; Lhs = Y1 ; Rhs = Z $
? Note a trick, Gma(y+1)=y!. Lgm(.)=Log(Gma)
Calc ; KZ = Col(Z) $
Matrix ; Delta0 = Init(KZ,1,0.0) $
Maximize ; Labels = KZ_Delta ; Start = Delta0
; Fcn = ey = Exp(Delta1'Z) | -ey+y1*log(ey)-lgm(y1+1)$
? (Note, B is the full coefficient vector estimated)
Create ; ey1 = Exp(b'Z) ; uisqrd = (y1 - ey1)^2$
Matrix ; V1 = <Z'[uisqrd]Z>$
?
? 3. Fit the logit model. Once again, there is a simple
? command, Logit ; Lhs=Y2 ; Rhs = X2,Ey1 $ But, we'll
? do it a little more elaborately.
?
Namelist ; X21 = X,Ey $
Calc ; K1 = Col(X) $
Matrix ; Beta0 = Init(K1,1,0.0)$
Maximize ; Labels = K1_Beta,Gamma ; Start = Beta0, 0
; Fcn = P = 1/(1+exp(-Beta0'X1-Gamma*Ey)) |
log( y2*P + (1-y2)*(1-P)) $
Create ; Prob = 1/(1+exp(-B'X21)) ; visqrd = (y2 - Prob)^2 $
Matrix ; V2 = <X21'[Visqrd]X21> $
?
? 4. Compute R and C to do the correction to V2
?
Create ; ci = visqrd*Gamma*Ey1
; ri = Sqr(visqrd)*Sqr(visqrd) $
Matrix ; C = X21'[ci]Z ; R = X21'[ri]Z
; V2S = C*V1*C' - R*V1*C1 - C*V1*R'
; V2S = V2 + V2 * V2S * V2
; Stat(b,V2S) $
/*==================================================================
Example 4.24. Method of Moments Estimator for N[mu,sigma^2]
No computations.
*/==================================================================
/*==================================================================
Example 4.25. Mixture of Normal Distributions.
No computations.
*/==================================================================
/*==================================================================
Example 4.26. Gamma distribution.
Method of moments and MLE.
*/==================================================================
Read ; Nobs=20 ; Nvar=3 ; Names=
I, Y, X $
1 20.5 12
2 31.5 16
3 47.7 18
4 26.2 16
5 44.0 12
6 8.28 12
7 30.8 16
8 17.2 12
9 19.9 10
10 9.96 12
11 55.8 16
12 25.2 20
13 29.0 12
14 85.5 16
15 15.1 10
16 28.5 18
17 21.4 16
18 17.7 20
19 6.42 12
20 84.9 16
?
?-------------------------------------------------------------
? First compute moments. With 'i' = variable, then means.
?-------------------------------------------------------------
Create ; m1i=y ; m2i=y*y ; mstari=log(y) ; m_1i=1/y$
Calc ; list ; m1=xbr(m1i) ; m2=xbr(m2i) ; mstar=xbr(mstari)
; m_1=xbr(m_1i) $
?-------------------------------------------------------------
? Starting value for solutions to moment equations. If
? P=1, Lambda = 1/y-bar. Use these as initial guesses.
?-------------------------------------------------------------
Calc ; l0 = 1/m1$
?-------------------------------------------------------------
? Maximum likelihood estimation. Results are shown.
?-------------------------------------------------------------
Maximize ; fcn=p*log(l)-lgm(p)-l*y+(p-1)*log(y)
; labels=p,l ; start = 1,l0$
/*
Normal exit from iterations. Exit status=0.
+---------------------------------------------+
| User Defined Optimization |
| Number of observations 20 |
| Iterations completed 6 |
| Log likelihood function -85.37567 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
P 2.410601626 .87682554 2.749 .0060
L .7707018675E-01 .27077098E-01 2.846 .0044
*/
?-------------------------------------------------------------
? Alternative estimators using the method of moments.
? Just finding solution to two equations, so set sample to 1.
?-------------------------------------------------------------
?
Sample ; 1 $
?
?-------------------------------------------------------------
? Based on m1 and m2. Note, equations are m1=P/l = 0 and
? m2=P(P+1)/l^2 = 0. But, the solution is the same for
? l*m1 - P = 0 and l*l*m2 - P(P+1) = 0. Also, the solutions
? are forced by minimizing the sum of squares. For two
? equations and two unknowns, this zeros exactly.
?-------------------------------------------------------------
?
Minimize ; fcn = (l*m1 - p)^2 + (l*l*m2 - p*(p+1))^2
; labels = p,l ; Start = 1,l0 $
/*
+---------------------------------------------+
| Iterations completed 81 |
| Log likelihood function -.2122069E-14 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
P 2.056818141 1.0000000 2.057 .0397
L .6575925992E-01 1.0000000 .066 .9476
*/
?-------------------------------------------------------------
? Based on m_1 and m2
?-------------------------------------------------------------
?
Minimize ; fcn = ((p-1)*m_1 - l)^2 + (l*l*m2 - p*(p+1))^2
; labels = p,l ; Start = 1,l0 $
/*
+---------------------------------------------+
| Number of observations 1 |
| Iterations completed 11 |
| Log likelihood function -.9040069E-09 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
P 2.607765493 1.0000000 2.608 .0091
L .8044102328E-01 1.0000000 .080 .9359
*/
?-------------------------------------------------------------
? Based on m1 and m*
?-------------------------------------------------------------
?
Minimize ; fcn = (l*m1 - P)^2 + (mstar - (psi(p)-log(l)))^2
; labels = p,l ; Start = 1,l0 $
/*
+---------------------------------------------+
| Number of observations 1 |
| Iterations completed 9 |
| Log likelihood function -.4243523E-12 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
P 2.410597740 1.0000000 2.411 .0159
L .7707008485E-01 1.0000000 .077 .9386
*/
?-------------------------------------------------------------
? Based on m2 and m*
?-------------------------------------------------------------
?
Minimize ; fcn = (l*l*m2 - P*(P+1))^2 + (mstar - (psi(p)-log(l)))^2
; labels = p,l ; Start = 1,l0 $
/*
+---------------------------------------------+
| Number of observations 1 |
| Iterations completed 10 |
| Log likelihood function -.1754566E-10 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
P 2.264470105 1.0000000 2.264 .0235
L .7130398476E-01 1.0000000 .071 .9432
*/
?-------------------------------------------------------------
? Based on m_1 and m*
?-------------------------------------------------------------
?
Mini ; fcn = ((p-1)*m_1 - l)^2 + (mstar - (psi(p)-log(l)))^2
; labels = p,l ; Start = 1,l0 $
/*
+---------------------------------------------+
| Number of observations 1 |
| Iterations completed 11 |
| Log likelihood function -.2031249E-12 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
P 3.035841208 1.0000000 3.036 .0024
L .1018203332 1.0000000 .102 .9189
*/
?-------------------------------------------------------------
? Next set of computations is based on the MLEs, which use
? m1 and m*. So, we redo the computations of l and P with
? these two moments.
?-------------------------------------------------------------
?
Sample ; 1 $
Minimize ; fcn = (l*m1 - P)^2 + (mstar - (psi(p)-log(l)))^2
; labels = p,l ; Start = 1,l0 $
?
?-------------------------------------------------------------
? Get estimated covariance matrix for moments
?-------------------------------------------------------------
Sample ; 1 - 20 $
Matrix ; list; V = 1/n*Xvcm(m1i,mstari)$
/*
Matrix V has 2 rows and 2 columns.
1 2
+----------------------------
1| .2503390D+02 .7155037D+00
2| .7155037D+00 .2387296D-01
*/
?-------------------------------------------------------------
? Elements of derivative matrix.
?-------------------------------------------------------------
Calc;list ; g11=-1/l;g12=p/l^2;g21=-psp(P);g22=1/l$
/*
G11 = -.12975203049693140D+02
G12 = .40583833808939320D+03
G21 = -.51240866266339470D+00
G22 = .12975203049693140D+02
*/
?-------------------------------------------------------------
? G matrix, and covariance matrix.
?-------------------------------------------------------------
Matrix ;G=[g11,g12/g21,g22];Gt=G';list;Var=<G>*V*<Gt>$
/*
Matrix VAR has 2 rows and 2 columns.
1 2
+----------------------------
1| .3897375D+00 .1460351D-01
2| .1460351D-01 .6874036D-03
*/
?
?-------------------------------------------------------------
? Covariance matrix based on Hessian of the log likelihood.
?-------------------------------------------------------------
Sample ; 1-20$
Calc ; h11=psp(p);h12=-1/l;h21=h12;h22=p/l^2$
Matrix ; Hessian=[h11,h12/h21,h22];list;1/n*<Hessian>$
/*
Matrix Result has 2 rows and 2 columns.
1 2
+----------------------------
1| .5124327D+00 .1638317D-01
2| .1638317D-01 .6469939D-03
*/
/*==================================================================
Example 4.27. Characterizing normality of a distribution
No computations, but here is how they could be applied. We use the data from table 4.1
*/==================================================================
Read ; Nobs=20 ; Nvar=3 ; Names=
I, Y, E $
1 20.5 12
2 31.5 16
3 47.7 18
4 26.2 16
5 44.0 12
6 8.28 12
7 30.8 16
8 17.2 12
9 19.9 10
10 9.96 12
11 55.8 16
12 25.2 20
13 29.0 12
14 85.5 16
15 15.1 10
16 28.5 18
17 21.4 16
18 17.7 20
19 6.42 12
20 84.9 16
Create ; d = y - Xbr(y)
; m2 = d^2 ; m3 = d^3 ; m4 = d^4 ; m5 = d^5
; m6 = d^6 ; m7 = d^7 ; m8 = d^8$
Calc ; mu2 = Xbr(m2) ; mu3 = Xbr(m3) ; mu4 = Xbr(m4)
; mu5 = Xbr(m5) ; mu6 = Xbr(m6) ; mu7 = Xbr(m7)
; mu8 = Xbr(m8) $
Calc ; Theta1 = Mu3 / Mu2^(1.5)
; Theta2 = Mu4 / Mu2^2 -3$
Names ; M = m2,m3,m4 $
Matrix ; V = 1/n* XVCM(M) $
Calc ; j11 = -1.5 / mu2^2.5 * mu3
; j12 = mu2^(-1.5)
; j21 = -2/mu2^3 * mu4
; j23 = mu2^(-2) $
Matrix ; J = [j11,j12,0 / j21,0,j23]
; Vart = J * V * J'
; Theta12 = [Theta1 / Theta2]
; Stat(Theta12,Vart)$
/*
Matrix statistical results: Coefficients=TH2 Variance=VART
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
TH2 _ 1 1.368929905 .36055351 3.797 .0001
TH2 _ 2 1.071718598 .90811548 1.180 .2379
*/
/*==================================================================
Example 4.28. Confidence Intervals for the Normal Mean.
For N[mu,sigma^2]
*/==================================================================
?
? Data and sample set up in variable named X.
?
Proc = CI(x,sig) $
Calc ; xbar = Xbr(x)
; sdev = Sdv(x) $
Calc ; list ; z = ttb(sig,(n-1))
; Lower = xbar - z*sdev/sqr(n)
; Upper = xbar + z*sdev/sqr(n) $
Calc ; list ; c1 = ctb(sig,(n-1))
; c2 = ctb((1-sig),(n-1))
; Lower = sqr(n-1)*sdev/c1
; Upper = sqr(n-1)*sdev/c2 $
EndProc
?
? We now execute our procedure for the income data of the
? previous examples.
?
Exec;proc=ci(y,.95)$
/*
Z = .17291328084071510D+01
LOWER = .22626476765584580D+02
UPPER = .39929523234415430D+02
C1 = .30143527215110020D+02
C2 = .10117013148930000D+02
LOWER = .32356531161353670D+01
UPPER = .96405921717317770D+01
*/
/*==================================================================
Example 4.29. Estimated Confidence Intervals for A Normal Mean.
No computation required.
*/==================================================================
/*==================================================================
Example 4.30. Confidence Intervals for A Normal Variance
No computation required.
*/==================================================================
/*==================================================================
Example 4.31. Power Function for a Test about a Mean
We compute the power function for a test based on the
results in Example 4.19. Sampling 25 observations from
the normal distribution, we obtain xbar=1.63 and s=1.51.
The test is H0:mu = 1.5. The hypothesis will be rejected
if the t-ratio, sqr(n)*(xbar-1.5)/s is > 2.064.
*/==================================================================
?
? The t statistic is shown first.
?
Calc list ; tstat=sqr(25)*(1.63-1.5)/.51$
?
? Now, compute and plot the power function.
?
? A grid of values for mu. We use .5 to 2.5, steps of .02
?
Sample ; 1-100$
Create ; mu=trn(.5,.02)$
Create ; pwr=0$
?
? We'll use the calc function for the t-table, in a loop
?
Proc $
Sample ; i$
? In the loop, Prob|t| > 2.064 if mu = mui equals
? Prob|t*| > 2.064 - sqr(n)(mu-1.5)/s
?
Calc ; upper = 2.064-sqr(25)*(mui-1.5)/.51
; lower = -2.064-sqr(25)*(mui-1.5)/.51
; mui=mui+.02
; power = 1-tds(upper,24) + tds(lower,24)$
? Computed for this one observation, then put in the data.
Create ; pwr=power$
endproc
? Start the loop, then produce 100 values.
Calc ; mui=.5$
Exec ; i=1,100$
Sample ; 1-100$
Plot ; lhs=mu;rhs=pwr;fill;Title=Power Function for t test$
/*==================================================================
Example 4.32. Consistent Test About a Mean.
No computations done.
*/==================================================================
/*==================================================================
Example 4.33. Testing a Hypothesis about a Mean with a Confidence
Interval.
No computations done.
*/==================================================================
/*==================================================================
Example 4.34. One Sided Test About a Mean
No computations done.
*/==================================================================
/*==================================================================
Example 4.35. Wald Test for a Restriction
No computations done.
*/==================================================================
/*==================================================================
Example 4.36. Testing a Hypothesis About a Mean
No computations done.
*/==================================================================
Chapter 5. Computation and Optimization
/*==================================================================
Example 5.1. Random Number Generator
No computations done. For the interested reader, here is a Fortran
subroutine that does the computation. The 'COMMON' is used to make
the routine remeber the seed between calls. Many programs use
different 'library' programs, such as the IMSL routines.
C
C RANDOM NUMBER GENERATOR
C
SUBROUTINE RANDOM (X)
IMPLICIT DOUBLE PRECISION ( A-H, O-Z )
COMMON/RNSEED/SEED
RPM = 2147483647.D0
RPM2 = 2147483655.D0
SEED = MOD( 16807.D0*SEED, RPM )
X = SEED / RPM2
RETURN
END
*/==================================================================
/*==================================================================
Example 5.2. Gibbs Sampler for a Bivariate Distribution
No computations done.
*/==================================================================
/*==================================================================
Example 5.3. Monte Carlo Study of the Mean versus
the Median. We do this with a procedure with variable
parameters, and some useful tools from matrix algebra.
*/==================================================================
?
? The entire operation is contained in the procedure
?
Proc = MeanMed(r,nr,d)
?
? Initialize targets and set ultimate sample size.
?
Calc ; Vmean=0;Vmedian=0$
Sample ; 1 - nr $
?
? Matrix command generates a sample of nr observations
? from t distribution with d degrees of freedom.
? Repeat operation R times.
?
Do For ; Rep ; ir = 1,r $
Matrix ; Zc = Rndm(nr,d) ; Zt = Rndm(nr) ? Std. Normal
; Zc = Dirp(Zc,Zc) ? Squares of standard normals
; i =Init(d,1,1.0) ? Sqrt of sum of squares.