Prof. George Karras
Consider the following regression model:
stdgdpi = 1 + 2*stdsolowi + 3*stdm1i + 4*meangovi + i, i=1,2,...,N, (1)
where stdgdp is the standard deviation of detrended GDP (a measure of business-cycle volatility), stdsolow is the standard deviation of the Solow residual, stdm1 is the standard deviation of the detrended M1 money supply, meangov is average government consumption as a fraction of GDP, is an error term, and 1, 2, 3, and 4 are parameters. The following is a RATS output which estimates (1) using the cross-sectional OECD data from G.Karras and F.Song (Fall 1996), "Sources of Business-Cycle Volatility: An Exploratory Study on a Sample of OECD Countries" Journal of Macroeconomics:
allocate 24
*
* cross section data from G.Karras and F.Song, Journal of Macroeconomics,
* Fall 1996 (Appendix)
*
* Note: N=24, K=4.
*
data(unit=input,org=obs) / number stdgdp stdsolow stdm1 meangov
1 0.0175898335346 2.3588138191884 0.0543157559544 0.1569046197688 (Australia)
2 0.0169331673977 2.0860331691907 0.0453915686851 0.1625823916291 (Austria)
3 0.0167926798370 1.8673502679234 0.0431795903515 0.1509336430254 (Belgium)
4 0.0194867550979 1.6762360647194 0.0895004826866 0.1807037033556 (Canada)
5 0.0179411133736 2.0093689346665 0.0447975353847 0.2189690791726 (Denmark)
6 0.0248948646821 2.2531112291235 0.1226715868950 0.1670768824372 (Finland)
7 0.0125789655760 1.6345309753864 0.0512042108332 0.1651290702207 (France)
8 0.0183784650456 1.8362044275719 0.0417975409105 0.1789682462017 (Germany)
9 0.0219341162869 4.0656503371884 0.0402864692620 0.1519544022388 (Greece)
10 0.0419316041961 3.6793784978823 0.1038025616835 0.1478924694045 (Iceland)
11 0.0223727069730 2.0875114394318 0.0586297877174 0.1605421280493 (Ireland)
12 0.0183036147471 2.5356831042386 0.0728270014533 0.1457770768282 (Italy)
13 0.0256984255909 3.3821794920687 0.0510256704137 0.0884022813210 (Japan)
14 0.0281542447889 3.0984816295201 0.0449064361729 0.1363891290764 (Luxembourg)
15 0.0177404506142 2.1480772183521 0.0381423313581 0.1532562086739 (Netherlands)
16 0.0306835312134 3.2139640709227 0.1768667034676 0.1453466284482 (N.Zealand)
17 0.0165243320437 1.6500173929878 0.0682760106228 0.1789848701432 (Norway)
18 0.0273089906354 3.9663215803002 0.0606549403413 0.1391622409978 (Portugal)
19 0.0244528411518 2.7065317524957 0.0637879072683 0.1134299233788 (Spain)
20 0.0144892333545 1.9019994121437 0.0621086944958 0.2365157205679 (Sweden)
21 0.0274475844095 1.9450768451571 0.0539401662015 0.1178606711949 (Switzerland)
22 0.0302366336809 3.0070092375755 0.1236136549118 0.1313023449675 (Turkey)
23 0.0204076379533 1.7481745014348 0.1321026401195 0.1913590390690 (U.K.)
24 0.0202967057965 1.6960134063870 0.0309017914202 0.1785955083676 (U.S.A.)
linreg stdgdp
# constant stdsolow stdm1 meangov
Dependent Variable STDGDP - Estimation by Least Squares
Usable Observations 24 Degrees of Freedom 20
Centered R**2 0.646052 R Bar **2 0.592960
Uncentered R**2 0.972792 T x R**2 23.347
Mean of Dependent Variable 0.0221907707
Std Error of Dependent Variable 0.0065412907
Standard Error of Estimate 0.0041733238
Sum of Squared Residuals 0.0003483326
Regression F(3,20) 12.1685
Significance Level of F 0.00009402
Durbin-Watson Statistic 2.217786
Variable Coeff Std Error T-Stat Signif
*******************************************************************************
1. Constant 0.015138492 0.007544442 2.00658 0.05850603
2. STDSOLOW 0.004136110 0.001381421 2.99410 0.00717051
3. STDM1 0.069799137 0.024255859 2.87762 0.00930712
4. MEANGOV -0.049979664 0.032630638 -1.53168 0.14126723
C1. Input the vector y and the matrix X. Display X.
declare vector y(24)
declare rectangular x(24,4)
do i=1,24
(01.0023) compute y(i) = stdgdp(i)
(01.0045) compute x(i,1) = 1
(01.0064) compute x(i,2) = stdsolow(i)
(01.0088) compute x(i,3) = stdm1(i)
(01.0112) compute x(i,4) = meangov(i)
(01.0136) end do i
write x
1.0000 2.3588 0.0543 0.1569
1.0000 2.0860 0.0454 0.1626
1.0000 1.8674 0.0432 0.1509
1.0000 1.6762 0.0895 0.1807
1.0000 2.0094 0.0448 0.2190
1.0000 2.2531 0.1227 0.1671
1.0000 1.6345 0.0512 0.1651
1.0000 1.8362 0.0418 0.1790
1.0000 4.0657 0.0403 0.1520
1.0000 3.6794 0.1038 0.1479
1.0000 2.0875 0.0586 0.1605
1.0000 2.5357 0.0728 0.1458
1.0000 3.3822 0.0510 0.0884
1.0000 3.0985 0.0449 0.1364
1.0000 2.1481 0.0381 0.1533
1.0000 3.2140 0.1769 0.1453
1.0000 1.6500 0.0683 0.1790
1.0000 3.9663 0.0607 0.1392
1.0000 2.7065 0.0638 0.1134
1.0000 1.9020 0.0621 0.2365
1.0000 1.9451 0.0539 0.1179
1.0000 3.0070 0.1236 0.1313
1.0000 1.7482 0.1321 0.1914
1.0000 1.6960 0.0309 0.1786
C2. Compute X'X and invert to get (X'X)-1.
compute xprimex = tr(x)*x
write xprimex
24.0000 58.5537 1.6747 3.7980
58.5537 156.3632 4.1992 8.9593
1.6747 4.1992 0.1477 0.2645
3.7980 8.9593 0.2645 0.6245
compute xprimexinv = inv(tr(x)*x)
write xprimexinv
3.2681 -0.4663 -0.7730 -12.8578
-0.4663 0.1096 -0.3782 1.4242
-0.7730 -0.3782 33.7807 -4.1795
-12.8578 1.4242 -4.1795 61.1346
C3. Compute ^=(X'X)-1X'y, y^=X^, and ^=y-y^.
declare vector betahat(4)
compute betahat = inv(tr(x)*x)*tr(x)*y
write betahat
0.0151 4.1361e-003 0.0698 -0.0500
declare vector yhat(24)
declare vector epsilonhat(24)
compute yhat = x*betahat
compute epsilonhat = y - yhat
do i=1,24
(01.0023) display y(i) yhat(i) epsilonhat(i)
(01.0066) end do i
0.01759 0.02084 -0.00325
0.01693 0.01881 -0.00188
0.01679 0.01833 -0.00154
0.01949 0.01929 1.99621e-004
0.01794 0.01563 0.00231
0.02489 0.02467 2.25334e-004
0.01258 0.01722 -0.00464
0.01838 0.01671 0.00167
0.02193 0.02717 -0.00524
0.04193 0.03021 0.01172
0.02237 0.01984 0.00253
0.01830 0.02342 -0.00512
0.02570 0.02827 -0.00257
0.02815 0.02427 0.00388
0.01774 0.01903 -0.00129
0.03068 0.03351 -0.00283
0.01652 0.01778 -0.00126
0.02731 0.02882 -0.00151
0.02445 0.02512 -6.63314e-004
0.01449 0.01552 -0.00103
0.02745 0.02106 0.00639
0.03024 0.02964 5.95143e-004
0.02041 0.02203 -0.00162
0.02030 0.01538 0.00491
C4. Demonstrate that X'^=0, i^i=0, and ^2s2=i^i2/(N-K).
compute xprimeepsilonhat = tr(x)*epsilonhat
write xprimeepsilonhat
-1.2681e-015
-2.6221e-015
-8.2206e-017
-1.9284e-016
compute mean = 0.
compute sigmahatsq = 0.
do i=1,24
(01.0023) compute mean = mean + epsilonhat(i)/24.
(01.0049) compute sigmahatsq = sigmahatsq + (epsilonhat(i)**2)/(24.-4.)
(01.0084) end do i
compute sigmahat = sqrt(sigmahatsq)
display mean sigmahat
-5.28820e-017 0.00417
C5. Derive Var(^)=s2(X'X)-1.
declare rectangular varbetahat(4,4)
compute varbetahat = sigmahatsq*inv(tr(x)*x)
write varbetahat
5.6919e-005 -8.1216e-006 -1.3464e-005 -2.2394e-004
-8.1216e-006 1.9083e-006 -6.5876e-006 2.4806e-005
-1.3464e-005 -6.5876e-006 5.8835e-004 -7.2793e-005
-2.2394e-004 2.4806e-005 -7.2793e-005 1.0648e-003
C6. Derive R2 and R_2.
compute Rsquare = (%corr(y,yhat))**2
compute Rsquarebar = 1. - (24.-1.)*(1.-Rsquare)/(24.-4.)
display Rsquare Rsquarebar
0.64605 0.59296
*
end
Normal Completion
A SPEAKEASY Alternative:
:_x
X (A 24 by 4 Matrix)
1 2.3588 .054316 .1569
1 2.086 .045392 .16258
... etc. ...
1 1.7482 .1321 .19136
1 1.696 .030902 .1786
:_xprimex = transpose(x)*x
:_xprimex
XPRIMEX (A 4 by 4 Matrix)
24 58.554 1.6747 3.798
58.554 156.36 4.1992 8.9593
1.6747 4.1992 .14767 .2645
3.798 8.9593 .2645 .62452
:_1/xprimex
1/XPRIMEX (A 4 by 4 Matrix)
3.2681 -.46632 -.77303 -12.858
-.46632 .10957 -.37824 1.4242
-.77303 -.37824 33.781 -4.1795
-12.858 1.4242 -4.1795 61.135
:_(1/xprimex)*transpose(x)*vfam(yr_sd_hp)
(1/XPRIMEX)*TRANSPOSE(X)*VFAM(YR_SD_HP) (A Vector with 4 Components)
.015138 .0041361 .069799 -.04998
:_olsq yr_sd_hp c solow_sd m1_sd_hp govmean
ORDINARY LEAST SQUARES
VARIABLES...
YR_SD_HP #C SOLOW_SD M1_SD_HP GOVMEAN
INDEPENDENT COL ESTIMATED STANDARD T- CONTRIBUTION
VARIABLE COEFFICIENT ERROR STATISTIC TO R**2
#C 1 .015138 .0075444 2.0066
SOLOW_SD 1 .0041361 .0013814 2.9941 .15865
M1_SD_HP 1 .069799 .024256 2.8776 .14655
GOVMEAN 1 -.04998 .032631 -1.5317 .041519
R-SQUARED = .64605
R-SQUARED(CORRECTED) = .59296
MULTICOLLINEARITY EFFECT = .29934
DURBIN-WATSON STATISTIC = 2.2178
NUMBER OF OBSERVATIONS = 24
SUM OF SQUARED RESIDUALS = 3.4833E-4
STANDARD ERROR OF THE REGRESSION = .0041733
:_quit