 Springer Texts in Business and Economics

# Distributed Lags and Dynamic Models

6.1 a. Using the Linear Arithmetic lag given in Eq. (6.2), a 6 year lag

on income gives a regression of consumption on a constant and

6

Zt = J2 (7 — i) Xt_i where Xt denotes income. In this case,

i=0

Zt = 7Xt C 6Xt_i + .. + Xt_6,

The Stata regression output is given below:

. gen ^6=7*ly+6*l. ly+5*l2.ly+4*l3.ly+3*l4.ly+2*l5.ly+l6.ly (6 missing values generated)

. reg lc z_6

 Source SS df MS Number of obs F(1,41) Prob > F R-squared Adj R-squared Root MSE = 43 = 3543.62 = 0.0000 = 0.9886 = 0.9883 = .03037 Model Residual 3.26755259 .037805823 1 41 3.26755259 .000922093 Total 3.30535842 42 .07869901 lc Coef. Std. Err. t P>|t| [95% Conf. Interval] z_6 .cons .0373029 .0006266 -.4950913 .1721567 59.53 -2.88 0.000 0.006 .0360374 -.8427688 .0385684 -.1474138
 From Eq. (6.2)

Pi = [(s C 1) — iP for i = 0,.., 6 with P estimated as the coefficient of Zt (which is z_6 in the regression). This estimate is 0.037 and is statistically significant.

Now we generate the regressors for an Almon lag first-degree polynomial with a far end point constraint using Stata:

. gen Z0= ly+l. ly+l2.ly+l3.ly+l4.ly+l5.ly+l6.ly (6 missing values generated)

B. H. Baltagi, Solutions Manual for Econometrics, Springer Texts in Business and Economics, DOI 10.1007/978-3-642-54548-1—6, © Springer-Verlag Berlin Heidelberg 2015

. gen Z1=0*ly+l. ly+2*l2.ly+3*l3.ly+4*l4.ly+5*l5.ly+6*l6.ly (6 missing values generated)

.gen Z=Z1-7*Z0 (6 missing values generated)

. reg lc Z

 Source SS df MS Number of obs = 43 Model Residual 3.26755293 .037805483 1 41 3.26755293 .000922085 F(1,41) Prob > F R-squared = 3543.66 = 0.0000 = 0.9886 Total 3.30535842 42 .07869901 Adj R-squared Root MSE = 0.9883 = .03037

 lc Coef. Std. Err. t P>|t| [95% Conf. Interval] Z -.0373029 .0006266 -59.53 0.000 -.0385684 -.0360374 _cons -.4950919 .1721559 -2.88 0.006 -.8427679 -.147416

The EViews output for PDL(Y,6,1,2) which is a sixth order lag, first degree polynomial, with a far end point constraint is given by:

Dependent Variable: LNC Method: Least Squares

 Coefficient Std. Error t-Statistic Prob. C -0.495091 0.172156 -2.875817 0.0064 PDL01 0.149212 0.002507 59.52843 0.0000 R-squared 0.988562 Mean dependent var 9.749406 Adjusted R-squared 0.988283 S. D. dependent var 0.280533 S. E. of regression 0.030366 Akaike info criterion -4.105595 Sum squared resid 0.037806 Schwarz criterion -4.023678 Log likelihood 90.27028 Hannan-Quinn criter. -4.075386 F-statistic 3543.634 Durbin-Watson stat 0.221468 Prob(F-statistic) 0.000000

b. Using an Almon-lag second degree polynomial described in Eq. (6.4), a

6 year lag on income gives a regression of consumption on a constant,

6 6 6

Z0 = 2^ Xt_i, Zi = 2^ iXt_i and Z2 = i2Xt_i. This yields the Almon-lag

i=0 i=0 i=0

without near or far end-point constraints. A near end-point constraint imposes

0 in Eq. (6.1) which yields a0 — a1 + a2 = 0 in Eq. (6.4). Substituting

for a0 in Eq. (6.4) yields the regression in (6.5).

The following Stata code, generates the variables needed to estimate an Almon lag second-degree polynomial with a near end point constraint:

.gen Z2=0*ly+l. ly+2‘2*l2.ly+3‘2*l3.ly+4‘2*l4.ly+5‘2*l5.ly+6‘2*l6.ly (6 missing values generated)

. gen Z01= Z1+Z0 (6 missing values generated)

. gen Z02= Z2-Z0

(6 missing values generated) . reg lc Z01 Z02

43

1701.72

0.0000

0.9884

0.9878

.03098

 lc Coef. Std. Err. t P>|t| [95% Conf. Interval] Z01 .1708636 .0260634 6.56 0.000 .1181875 .2235397 Z02 -.0441775 .0085125 -5.19 0.000 -.0613819 -.026973 _cons -.8139917 .2306514 -3.53 0.001 -1.280156 -.3478277

The EViews output for PDL(Y,6,2,1) which is a sixth order lag, second degree polynomial, with a near end point constraint is given by:

Dependent Variable: LNC Method: Least Squares

Coefficient Std. Error t-Statistic Prob.

 C -0.813961 0.230649 -3.529009 0.0011 PDL01 0.259216 0.043085 6.016333 0.0000 PDL02 -0.044177 0.008512 -5.189671 0.0000 R-squared 0.988384 Mean dependent var 9.749406 Adjusted R-squared 0.987803 S. D. dependent var 0.280533 S. E. of regression 0.030982 Akaike info criterion -4.043589 Sum squared resid 0.038396 Schwarz criterion -3.920714 Log likelihood 89.93716 Hannan-Quinn criter. -3.998276 F-statistic 1701.718 Durbin-Watson stat 0.377061 Prob(F-statistic) 0.000000 Lag Distribution of LNY i Coefficient Std. Error t-Statistic 0 0.21504 0.03457 6.21968 1 0.34172 0.05213 6.55566 2 0.38006 0.05266 7.21676 3 0.33003 0.03621 9.11501 4 0.19166 0.00402 47.7225 *. | 5 -0.03508 0.04810 0.72915 * . I 6 -0.35016 0.11562 3.02843 Sum of 1.07327 0.02249 47.7225

Lags

c. The far end-point constraint imposes "7 = 0. This translates into the fol­lowing restriction a0 + 7ai + 49a2 = 0. Substituting for a0 in (6.4) yields the regression in (6.6) with s = 6, i. e., the regression of consumption on a constant, (Z1 — 7Z0) and (Z2 — 49Z0).

The following Stata code, generates the variables needed to estimate an Almon lag second-degree polynomial with a far end point constraint:

. gen Z10_far=Z1-7*Z0 (6 missing values generated) . gen Z20_far=Z2-7‘2*Z0 (6 missing values generated) . reg lc Z10_farZ20_far

Source

Model

Residual

Total

 ------- l-c--- Coef. Std. Err. t P>|t| [95% Conf. Interval] Z10_far -.3833962 .0538147 -7.12 0.000 -.4921598 -.2746326 Z20_far .0381843 .0059371 6.43 0.000 .0261849 .0501837 _cons -1.237493 .1681062 -7.36 0.000 -1.577249 -.8977381

The EViews output for PDL(Y,6,2,2) which is a sixth order lag, second degree polynomial, with a far end point constraint is given by:

Dependent Variable: LNC Method: Least Squares

 Coefficient Std. Error t-Statistic Prob. C -1.237555 0.168112 -7.361483 0.0000 PDL01 0.006206 0.022306 0.278216 0.7823 PDL02 -0.154297 0.018196 -8.479577 0.0000

. gen Z_NF=-47* Z0-6* Z1 + Z2 (6 missing values generated)

. reg lc Z_NF

43

2663.90

0.0000

0.9848

0.9845

.03496

 ----- lc--- Coef. Std. Err. t P>|t| [95% Conf. Interval] Z_NF -.0028049 .0000543 -51.61 0.000 -.0029147 -.0026952 _cons -.2411805 .1936405 -1.25 0.220 -.6322455 .1498845

The EViews output for PDL(Y,6,2,3) which is a sixth order lag, sec­ond degree polynomial, with both near and far end point constraints is given by:

Dependent Variable: LNC Method: Least Squares

 Coefficient Std. Error t-Statistic Prob. C -0.233622 0.197176 -1.184841 0.2429 PDL01 0.097163 0.001918 50.64935 0.0000 R-squared 0.984269 Mean dependent var 9.749406 Adjusted R-squared 0.983886 S. D. dependent var 0.280533 S. E. of regression 0.035612 Akaike info criterion -3.786892 Sum squared resid 0.051996 Schwarz criterion -3.704975 Log likelihood 83.41817 Hannan-Quinn criter. -3.756683 F-statistic 2565.357 Durbin-Watson stat 0.214478 Prob(F-statistic) 0.000000

 Lag Distribution of LNY i Coefficient Std. Error t-Statistic 0 0.08502 0.00168 50.6494 1 0.14574 0.00288 50.6494 2 0.18218 0.00360 50.6494 3 0.19433 0.00384 50.6494 4 0.18218 0.00360 50.6494 5 0.14574 0.00288 50.6494 6 0.08502 0.00168 50.6494 Sum of Lags 1.02021 0.02014 50.6494

e. The RRSS for the Chow test for the arithmetic lag restrictions is given by the residual sum of squares of the regression in part (a), i. e., .037805823. The URSS is obtained from running consumption on a constant and six lags on income. The corresponding Stata regression is given below:

. reg lc ly l. ly l2.ly l3.ly l4.ly l5.ly l6.ly

 Source SS df MS Number of obs = ................... F(7 35) 43 1093.59 0.0000 0.9954 0.9945 .02073 Model Residual 3.29031477 .015043647 7 35 = .470044967 Prob > F = .000429818 R-squared = Total 3.30535842 42 .07869901 Root MSE =

 |c Coef. Std. Err. t P>|t| [95% Conf. Interval] ly 1.237818 .2192865 5.64 0.000 .7926427 1.682993 L1. .2504519 .310222 0.81 0.425 -.3793323 .8802361 L2. -.203472 .3005438 -0.68 0.503 -.8136084 .4066644 L3. -.0279364 .3041055 -0.09 0.927 -.6453034 .5894306 L4. .0312238 .3049614 0.10 0.919 -.5878808 .6503284 L5. -.0460776 .3048432 -0.15 0.881 -.6649422 .572787 L6. -.1270834 .2028801 -0.63 0.535 -.5389519 .2847851 _cons -1.262225 .1667564 -7.57 0.000 -1.600758 -.9236913

URSS = .015043647. The number of restrictions given in (6.2) is 6. Hence, the Chow F-statistic can be computed as follows:

. display (.037805483-.015043647)*35/(6*.015043647)

8.8261428

and this is distributed as F(6,35) under the null hypothesis. This rejects the arithmetic lag restrictions.

f. Similarly, the Chow test for the Almon lag second-degree polynomial with a near end point constraint can be computed using RRSS = .038395978 from part (b). URSS = .015043647 from the last regression in part (e), and the number of restrictions is 5:

. display (.038395978-.015043647)*35/(5*.015043647)

10.866136

and this is distributed as F(5,35) under the null hypothesis. This rejects the Almon lag second-degree polynomial with a near end point constraint.

g. The Chow test for the Almon lag second-degree polynomial with a far end point constraint can be computed using RRSS = .018586036 from part (c). URSS = .015043647 from the last regression in part (e), and the number of restrictions is 5:

. display (.018586036-.015043647)*35/(5*.015043647)

1.6483186

and this is distributed as F(5,35) under the null hypothesis. This does not reject the Almon lag second-degree polynomial with a far end point constraint.

Finally, The Chow test for the Almon lag second-degree polynomial with both near and far end point constraints can be computed using RRSS = .050101463 from part (d). URSS = .015043647 from the last regression in part (e), and the number of restrictions is 6:

. display (.050101463-.015043647)*35/(6*.015043647)

13.594039

and this is distributed as F(6,35) under the null hypothesis. This rejects the Almon lag second-degree polynomial with both near and far end point constraints.

6.2 a. For the Almon-lag third degree polynomial "i = a0 C a1i C a2i2 + a3i3 for i = 0,1,.., 5.

In this case, (6.1) reduces to

5

Yi = a C У (a0 + a1i + a2i2 + a3i3) Xt_i + ut

i=1

5 5 5 5

= a C a0 C У Xt_i C a1 У iXt_i C a2 У i2Xt_i C a3 У i3Xt_i C ut,

i= 0 i= 0 i= 0 i= 0

Now a, a0,a1,a2 and a3 can be estimated from the regression of Yt on a

5 5 5 5

constant, Z0 = Yl Xt_i, Z1 = Yl iXt_i, Z2 = Yl i2Xt_i and Z3 = Yl i3Xt_i.

i=0 i=0 i=0 i=0

The following Stata code generates the variables to run the OLS regression:

. gen Z5_0= ly+l. ly+l2.ly+l3.ly +l4.ly +l5.ly (5 missing values generated)

. gen Z5_1=l. ly+2*l2.ly +3*l3.ly +4*l4.ly +5*l5.ly (5 missing values generated)

. gen Z5_2=l. ly+2''2*l2.ly +3“2*l3.ly +4“2*l4.ly +5“2*l5.ly (5 missing values generated)

. gen Z5_3=l. ly+2‘3*l2.ly +3‘3*!3.ly +4‘3*!4.ly +5‘3*!5.ly

(5 missing values generated) . reg lc Z5_0 Z5_1 Z5_2 Z5.3

44

2063.28

0.0000

0.9953

0.9948

.02086

 ----- lc--- Coef. Std. Err. t P>|t| [95% Conf. Interval] Z5_0 1.27317 .1911863 6.66 0.000 .8864591 1.659881 Z5_1 -1.623359 .5714614 -2.84 0.007 -2.779249 -.4674696 Z5_2 .5693275 .2955923 1.93 0.061 -.0285644 1.167219 Z5_3 -.0599977 .0390623 -1.54 0.133 -.1390087 .0190133 _cons -1.128979 .1533499 -7.36 0.000 -1.439158 -.8187994

b. The estimate of "3 is "3 = a0 + 3a3 + 9a2 + 27a3 with var 3^ = var (a0) + 9 var(ai) + 81 var (a2) + 272var (a 3)

+ 2.3. cov (a0, a1) + 2.9. cov (a0, a2) + 2.27.cov (a0, a3) + 2.3.9. cov (ai, a2) + 2.3.27 cov (ai, a3) + 2.9.27 cov (a2, a3) .

The estimate of ^3 can be computed from this regression results as follows: . *beta3

. display 1.27317+3*(-1.623359)+9*.5693275+27*(-.0599977)

-.0928974

The var-cov matrix of the regression estimates are given by:

. matrix list e(V) symmetric e(V)[5,5]

 Z5_0 Z5_1 Z5_2 Z5_3 _cons Z5_0 .0365522 Z5_1 -.09755986 .32656818 Z5_2 .04577699 -.16633866 .08737482 Z5_3 -.00565526 .0214819 -.0114859 .00152587 _cons -.01320609 .02236579 -.00962469 .00120366 .02351619

The EViews output for PDL(Y,5,3) which is a fifth order lag, third degree polynomial, with no end point constraint is given by:

Dependent Variable: LNC Method: Least Squares

 Coefficient Std. Error t-Statistic Prob. C -1.129013 0.153388 -7.360517 0.0000 PDL01 -0.176078 0.122030 -1.442904 0.1570 PDL02 -0.066325 0.162878 -0.407206 0.6861 PDL03 0.209243 0.066943 3.125704 0.0033 PDL04 -0.059926 0.039088 -1.533091 0.1333 R-squared 0.995295 Mean dependent var 9.736786 Adjusted R-squared 0.994813 S. D. dependent var 0.289615 S. E. of regression 0.020859 Akaike info criterion -4.795464 Sum squared resid 0.016968 Schwarz criterion -4.592715 Log likelihood 110.5002 Hannan-Quinn criter. -4.720275 F-statistic 2062.698 Durbin-Watson stat 0.393581 Prob(F-statistic) 0.000000 Lag Distribution of LNY i Coefficient Std. Error t-Statistic ■ *| 0 1.27295 0.19130 6.65411 ■ * I 1 0.15942 0.15649 1.01868 * ■ I 2 -0.17608 0.12203 -1.44290 *■ I 3 -0.09309 0.11943 -0.77943 ■* | 4 0.04883 0.15599 0.31306 *■ I 5 -0.10987 0.17870 -0.61485 Sum of 1.10217 0.01492 73.8706 Lags

c. Imposing the near end-point constraint = 0 yields the following restriction on the third degree polynomial in a’s:

ao _ ai C a2 — a3 — 0.

solving for a0 and substituting above yields the following constrained regression:

Yt — a C ai. Zi C Zo) C a2(Z2 — Zo) C a3.Zi C Z3) C ut
The corresponding Stata regression is reported below.

. gen ZZ01= Z5_0+Z5_1 (5 missing values generated) . gen ZZ03= Z5_0+Z5_3 (5 missing values generated) . gen ZZ02= Z5_2-Z5_0 (5 missing values generated) . reg lc ZZ01 ZZ03 ZZ02

44

2205.97

0.0000

0.9940

0.9935

.02327

 lc Coef. Std. Err. t P>|t| [95% Conf. Interval] ZZ01 .2551128 .0259506 9.83 0.000 .2026646 .307561 ZZ03 .0632448 .0123507 5.12 0.000 .0382831 .0882065 ZZ02 -.3852599 .0629406 -6.12 0.000 -.5124676 -.2580523 _cons -.9865419 .1641747 -6.01 0.000 -1.318351 -.6547324

d. Test the near end point constraint with a Chow test. TheURSS = .016963297 from part (a) and RRSS = .021668704 from part (c) and there is one restriction.

. display (.021668704-.016963297)*39/(1*.016963297)

10.818114

and this is distributed as F(1,39) under the null hypothesis. This rejects the near end point constraint.

e. The Chow test for the Almon 5 year lag third-degree polynomial with no end point constraints can be computed using RRSS = .016963297 from part (a). URSS = .016924337 from the unrestricted 5 year lag regression given below:

44

1308.00

0.0000

0.9953

0.9945

.02139

 lc Coef. Std. Err. t P>|t| [95% Conf. Interval] ly 1.303457 .2224286 5.86 0.000 .852774 1.75414 L1. .090694 .3099447 0.29 0.771 -.5373136 .7187016 L2. -.1401868 .3081789 -0.45 0.652 -.7646165 .484243 L3. -.0460399 .3104643 -0.15 0.883 -.6751003 .5830204 L4. -.0255622 .3129629 -0.08 0.935 -.6596853 .6085608 L5. -.0800021 .2087812 -0.38 0.704 -.5030331 .3430288 _cons -1.13104 .1574744 -7.18 0.000 -1.450114 -.8119665

and the number of restrictions is 3:

. display (.016963297- .016924337)*37/(3*.016924337)

.02839146

and this is distributed as F(3,37) under the null hypothesis. This does not reject the Almon 5 year lag third-degree polynomial restrictions.

6.3 a. From (6.18), Yt = "Yt-i C vt. Therefore, Yt_i

pYt-i = p"Yt_2 C pvt-1.

Subtracting this last equation from (6.18) and re-arranging terms, one gets

Yt = (" C p/ Yt-1 — p"Yt-2 C ©t.

Multiply both sides by Yt-1 and sum, we get

T T T T

I>tYt_i = (" C p)£ Y2-i — p"J^ Yt-iYt-2 C £ Yt-iet.

t=2 t=2 t=2 t=2

T

Divide by Yt2 i and take probability limits, we get

t=2

Hence,

T

t=2

Also, Vt_i = Yt_i — PolsYt_2. Multiply this equationby V and sum, we get

T T T

EVtVt_i = X Yt_i vt - Pol^~^Yt-2vt_i.

t=2 t=2 t=2

T

But, by the property of least squares Yt_iV = 0, hence

t= 2

X VtVt_i /T = - "ol^Yt_2Vt /T = - olsJ2 Yt_2Yt/T

t=2 t=2 t=2

T

C P 2l^Yt_2Yt_i/T

t2

and

From part (a), we know that

Yi = (P + p) Yt_i - p"Yt_2 + ©t multiply both sides by Yt_2 and sum and take plim after dividing by T, we get

"2 = (P C p) "i - pP"0 so that

"0"2 = (P C p)"i"0 C pP"2 and "2 - "0У2 = "2 - (P C p) "i"0 C pP"2.

But from part (a), plim"ols = yi/yo = (" + p)/(1 + p"). Substituting

(" + P)"o = (1 + Р")Уі above we get

- У0У2 = У? - (1 + Pp) У? + p""0 = P" ("2 - У?) ■

Hence,

plimp = — • p" = p" (" + p) / (1 + p")

У0

and

plim (p - p) = (p"2 + p2" - p - p2") / (1 + p") = p ("2 - 1) / (1 + p")

= - plim ("ols - .

The asymptotic bias of p is negative that of "ols.

T T T

d. Since d = ^2 (Vt - O-1) / ^2 and as T! 1, vt2 is almost identical

t=2 t=2 t=2

T

to ^2 v2_1, then plim d ^ 2(1 - plimp) where p was defined in (c). But

t=2

plimp = p - p(1-p2)/(1 + p") = ( p2" + pP2)/(1 + pP) = p"( p+")/(1 + p"). Hence, plim d = 2 1 - p"1(_C'"pp) -

e. Knowing the true disturbances, the Durbin-Watson statistic would be

T

d* = К (v‘- v‘-1)2 ^Zvt2

t=2 t=2

and its plim d* = 2(1 - p). This means that from part (d)

plim (d - d*) = 2 (1 - plim p) - 2 (1 - p) = 2 [ p - " p ( p + ") / (1 + p")]

= 2 [ p + p2" - " p2 - "2p ] / (1 + p") = 2 p (1 - "2) / (1 + p") = 2plim("ols - ")

from part (a). The asymptotic bias of the D. W. statistic is twice that of "ols. The plim d and plim d* and the asymptotic bias in d can be tabulated for various values of p and ".

6.4 a. From (6.18) with MA(1) disturbances, we get Yt = "Yt-1 + ©t + 0©t_1 with |"| < 1.

In this case,

T T T T

P ols =J2 YtYt-1^; Y_1 = P +T, Yt-i©t /J2 Y?-1

t=2 t=2

TT

t=2

so that plim ols - ") = plim ^ E Yt_1©t /Tj /plim ^E Y2_1 /Tj

+ 0plim IE Yt_1©t_1 /^ /plim ^ X Y2_1 /^ . Now the above model can be written as (1 - "L) Yt = (1 C 0L)©

or

i=0

From the last expression, it is clear that E(Yt) = 0 and

T

var(Yt) = oe2[1 C (0 C ")2/(1 - P2)] = plim£ Y2_1/T.

t=2

Also,

TT

t= 2 t= 2

Since the ©t’s are not serially correlated, each term on the right hand side has

T

zero plim. Hence, plim Yt_1©t/T = 0 and the first term on the right hand

t= 2

side of plim (pols — is zero. Similarly,

T T

£Yt_i8t_i/T = E ©2-1/T c (0 c P)

=2

TT

t=2

T

which yields plim£ Yt-i©t-i/T = a©2 since the second term on the right

t= 2

hand side has zero plim. Therefore,

plim (pols — P) = 0a©2 /a©2 [l + (0 + P)2 / (1 — p2)"

= 0 (1 — P2) / (1 — P2 C 02 C P2 C 20p)

= 0 (1 — P2) / (1 C 02 C 20P) = 8 (1 — P2) / (1 C 2P8)

where 8 = 0/(1 C 02).

The asymptotic bias of Pols derived in part (a) can be tabulated for various values of P and 0 < 0 < 1.

Let vt = Yt — PolsYt-1 = PYt-1 — PolsYt-1 + vt = vt — (pols — P) Yt-1. But

TT

P ols — P = E Yt-1V^]Y2-1.

t=2 t=2

Therefore,

T T / T 2 T T

E v 2 = E v2 + E Yt-1 v^EY2-1 — 2 (E Yt-1 vt)2 / E Y2-1

t=2 t=2 t=2 t=2 t=2

T / T 2 T

= Ev2 — EYt-1 vt I /E! Y2-1.

t=2 t=2 t=2

T

pHmEv2/T = a2 +02 aE2 = ae2(1 +02).

t= 2

Also,

T T T

plim^^Yt-iVt/T = plim£Yt_iEt/T + 9plim EY‘-1©‘/T

t=2 t=2 t=2

from part (a). Therefore,

T

1 C (0 C ")2/(1 - "2)

t=2

= a©2 [1 C 02 - 02 (1 - "2) / (1 - 02 C 20")]

= a©2 [1 C 02 - 08 (1 - "2) / (1 C 28")] = a©2 [1 C 02 - 00*]

where 8 = (1_|0Q2) and 0* = 8(1 - "2)/(1 C 2"8).

6.5 a. The stata commands to generate the Durbin h statistic from scratch are the

following:

. gen lc_lag=l. lc (1 missing value generated) . reg lc ly lcJag

Source

Model

Residual

Total

 ----- lc--- Coef. Std. Err. t P>|t| [95% Conf. Interval] ly .3091105 .0816785 3.78 0.000 .1446015 .4736194 lcJag .7045042 .0765288 9.21 0.000 .5503673 .8586411 _cons -.1475238 .0870474 -1.69 0.097 -.3228462 .0277986

. display.0765288*2 .00585666 . predict miu, resid (1 missing value generated) . reg miu l. miu, noconstant

. * Durbin’s h

. display.5065964*(48*(1-.00585666))“0.5 3.4995099

This is asymptotically distributed as N(0,1) under the null hypothesis of p = 0. This rejects Ho indicating the presence of serial correlation. b. The EViews output for the Breusch and Godfrey test for first-order serial correlation is given below. The back up regression is given below the test statistics. This regresses the OLS residuals on their lagged value and the regressors in the model including the lagged dependent variable. This yields an R2 = 0.278576. The number of observations is 48. Therefore, the LM statistic = TR2 = 48(0.278576) = 13.372. This is asymptotically dis­tributed as Chi-Square(1) under the null hypothesis of p = 0. The p-value is 0.0003, and we reject the null hypothesis.

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 16.99048 Prob. F(1,44) 0.0002

Obs*R-squared 13.37165 Prob. Chi-Square(1) 0.0003

Test Equation:

Dependent Variable: RESID Method: Least Squares

Sample: 1960 2007 Included observations: 48

Presample missing value lagged residuals set to zero.

 Coefficient Std. Error t-Statistic Prob. C -0.061130 0.076227 -0.801941 0.4269 LNC(-1) -0.081685 0.068658 -1.189747 0.2405 LNY 0.086867 0.073256 1.185810 0.2421 RESID(-1) 0.552693 0.134085 4.121951 0.0002 R-squared 0.278576 Mean dependent var -5.00E-16 Adjusted R-squared 0.229388 S. D. dependent var 0.013797 S. E. of regression 0.012112 Akaike info criterion -5.909614 Sum squared resid 0.006455 Schwarz criterion -5.753680 Log likelihood 145.8307 Hannan-Quinn criter. -5.850686 F-statistic 5.663494 Durbin-Watson stat 1.773229 Prob(F-statistic) 0.002272

c. The EViews output for the Breusch and Godfrey test for second-order serial correlation is given below. The back up regression is given below the test statistics. This regresses the OLS residuals et on et_i and et_2 and the regres­sors in the model including the lagged dependent variable. This yields an R2 = 0.284238. The number of observations is 48. Therefore, the LM statistic = TR2 = 48(0.284238) = 13.643.

This is asymptotically distributed as Chi-Square(2) under the null hypothesis of no second-order serial correlation. The p-value is 0.0011, and we reject the null hypothesis.

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 8.537930 Prob. F(2,43) 0.0008

Obs*R-squared 13.64344 Prob. Chi-Square(2) 0.0011

Test Equation:

Dependent Variable: RESID Method: Least Squares

Sample: 1960 2007 Included observations: 48

Presample missing value lagged residuals set to zero.

 Coefficient Std. Error t-Statistic Prob. C -0.049646 0.079289 -0.626136 0.5345 LNC(-1) -0.067455 0.073355 -0.919582 0.3629 LNY 0.071648 0.078288 0.915192 0.3652 RESID(-1) 0.591123 0.150313 3.932602 0.0003 RESID(-2) -0.092963 0.159390 -0.583241 0.5628 R-squared 0.284238 Mean dependent var -5.00E-16 Adjusted R-squared 0.217656 S. D. dependent var 0.013797 S. E. of regression 0.012204 Akaike info criterion -5.875827 Sum squared resid 0.006404 Schwarz criterion -5.680910 Log likelihood 146.0198 Hannan-Quinn criter. -5.802167 F-statistic 4.268965 Durbin-Watson stat 1.867925 Prob(F-statistic) 0.005364

Ordinary Least Squares Estimates

 SSE 0.014269 DFE 32 MSE 0.000446 Root MSE 0.021117 SBC -167.784 AIC -175.839 Reg Rsq 0.9701 Total Rsq 0.9701 Durbin h 2.14769 PROB>h 0.0159

Godfrey's Serial Correlation Test Alternative LM Prob>LM

 AR(+ 1) 4.6989 0.0302 AR(+ 2) 5.1570 0.0759 Variable DF B Value Std Error t Ratio Approx Prob Intercept 1 0.523006 1.1594 0.451 0.6550 RGNP. POP 1 0.050519 0.1127 0.448 0.6571 CAR. POP 1 -0.106323 0.1005 -1.058 0.2981 PMG. PGNP 1 -0.072884 0.0267 -2.733 0.0101 LAG. DEP 1 0.907674 0.0593 15.315 0.0001

c. LM Test for AR(1) by BREUSCH & GODFREY

Dependent Variable: RESID

 Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 5 0.00178 0.00036 0.863 0.5172 Error 30 0.01237 0.00041 C Total 35 0.01415 Root MSE 0.02031 R-square 0.1257 Dep Mean -0.00030 Adj R-sq -0.0200 C. V. -6774.59933

Parameter Estimates

 Variable DF Parameter Estimate Standard Error T for HO: Parameter=0 Prob > |T| INTERCEP 1 -0.377403 1.15207945 -0.328 0.7455 RESID.1 1 0.380254 0.18431559 2.063 0.0479 RGNP. POP 1 -0.045964 0.11158093 -0.412 0.6833 CAR. POP 1 0.031417 0.10061545 0.312 0.7570 PMG. PGNP 1 0.007723 0.02612020 0.296 0.7695 LAG. DEP 1 -0.034217 0.06069319 -0.564 0.5771

SAS PROGRAM Data RAWDATA;

Input Year CAR QMG PMG POP RGNP PGNP; Cards;

Data USGAS; set RAWDATA; LNQMG=LOG(QMG);

LNCAR=LOG(CAR);

LNPOP=LOG(POP);

LNRGNP=LOG(RGNP);

LNPGNP=LOG(PGNP);

LNPMG=LOG(PMG);

QMG_CAR=LOG(QMG/CAR);

RGNP_POP=LOG(RGNP/POP);

CAR_POP=LOG(CAR/POP);

PMG_PGNP=LOG(PMG/PGNP);

LAG_DEP=LAG(QMG_CAR);

Proc reg data=USGAS;

Model QMG_CAR=RGNP_POP CAR. POP PMGPGNP; TITLE ‘ STATIC MODEL’;

Proc autoreg data=USGAS;

Model QMG_CAR=RGNP_POP CAR_POP PMG_PGNP

LAG_DEP/LAGDEP=LAG_DEP godfrey=2; OUTPUT OUT=MODEL2 R=RESID;

TITLE ‘ DYNAMIC MODEL’;

RUN;

DATA DW_DATA; SET MODEL2;

RESID_1=LAG(RESID);

PROC REG DATA=DW_DATA;

MODEL RESID=RESID_1 RGNP. POP CAR. POP PMG. PGNP LAG. DEP; TITLE ‘LM Test for AR(1) by BREUSCH & GODFREY’;

RUN;

6.7 a. Unrestricted Model Autoreg Procedure Dependent Variable = QMG. CAR

Ordinary Least Squares Estimates

 SSE 0.031403 DFE 22 MSE 0.001427 Root MSE 0.037781 SBC -96.1812 AIC -110.839 Reg Rsq 0.9216 Total Rsq 0.9216 Durbin-Watson 0.5683 Std t Approx Variable DF B Value Error Ratio Prob Intercept 1 -7.46541713 3.0889 -2.417 0.0244 RGNP. POP 1 -0.58684334 0.2831 -2.073 0.0501 CAR. POP 1 0.24215182 0.2850 0.850 0.4046 PMG. PGNP 1 -0.02611161 0.0896 -0.291 0.7734 PMPG.1 1 -0.15248735 0.1429 -1.067 0.2975 PMPG.2 1 -0.13752842 0.1882 -0.731 0.4726 PMPG.3 1 0.05906629 0.2164 0.273 0.7875 PMPG.4 1 -0.21264747 0.2184 -0.974 0.3408 PMPG.5 1 0.22649780 0.1963 1.154 0.2609 PMPG.6 1 -0.41142284 0.1181 -3.483 0.0021

 Almon Lag (S = 6, P = : 2) PDLREG Procedure Dependent Variable = QMG. CAR Ordinary Least Squares Estimates SSE 0.04017 DFE 26 MSE 0.001545 Root MSE 0.039306 SBC -102.165 AIC -110.96 Reg Rsq Durbin-Watson 0.8998 0.5094 Total Rsq 0.8998 Variable DF B Value Std Error t Ratio Approx Prob Intercept 1 -5.06184299 2.9928 -1.691 0.1027 RGNP. POP 1 -0.35769028 0.2724 -1.313 0.2006 CAR. POP 1 0.02394559 0.2756 0.087 0.9314 PMG_PGNP**0 1 -0.24718333 0.0340 -7.278 0.0001 PMG_PGNP**1 1 -0.05979404 0.0439 -1.363 0.1847 PMG_PGNP**2 1 -0.10450923 0.0674 -1.551 0.1331

Estimate of Lag Distribution

Variable -0.184 0

c. Almon Lag (S=4,P=2) PDLREG Procedure Dependent Variable = QMG_CAR

Ordinary Least Squares Estimates

 SSE 0.065767 DFE 28 MSE 0.002349 Root MSE 0.048464 SBC -94.7861 AIC -103.944 Reg Rsq 0.8490 Total Rsq 0.8490 Durbin-Watson 0.5046 Std t Approx Variable DF B Value Error Ratio Prob Intercept 1 -6.19647990 3.6920 -1.678 0.1044 RGNP_POP 1 -0.57281368 0.3422 -1.674 0.1053 CAR_POP 1 0.21338192 0.3397 0.628 0.5351 PMG_PGNP**0 1 -0.19423745 0.0414 -4.687 0.0001 PMG_PGNP**1 1 -0.06534647 0.0637 -1.026 0.3138 PMG_PGNP**2 1 0.03085234 0.1188 0.260 0.7970

Estimate of Lag Distribution

Variable -0.116 0

ALMON LAG(S=8,P=2)

PDLREG Procedure Dependent Variable = QMG_CAR

Ordinary Least Squares Estimates

 SSE 0.020741 DFE 24 MSE 0.000864 Root MSE 0.029398 SBC -112.761 AIC -121.168 Reg Rsq 0.9438 Total Rsq 0.9438 Durbin-Watson 0.9531

 Variable DF B Value Std Error t Ratio Approx Prob Intercept 1 -7.71363805 2.3053 -3.346 0.0027 RGNP. POP 1 -0.53016065 0.2041 -2.597 0.0158 CAR_POP 1 0.17117375 0.2099 0.815 0.4229 PMG_PGNP**0 1 -0.28572518 0.0267 -10.698 0.0001 PMG_PGNP**1 1 -0.09282151 0.0417 -2.225 0.0358 PMG_PGNP**2 1 -0.12948786 0.0512 -2.527 0.0185
 Parameter Std t Approx Variable Value Error Ratio Prob PMG_PGNP(0) -0.11617 0.028 -4.09 0.0004 PMG_PGNP(1) -0.07651 0.016 -4.73 0.0001 PMG_PGNP(2) -0.05160 0.015 -3.34 0.0027 PMG_PGNP(3) -0.04145 0.018 -2.30 0.0301 PMG_PGNP(4) -0.04605 0.017 -2.63 0.0146 PMG_PGNP(5) -0.06541 0.013 -4.85 0.0001 PMG_PGNP(6) -0.09953 0.012 -8.10 0.0001 PMG_PGNP(7) -0.14841 0.025 -5.97 0.0001 PMG_PGNP(8) -0.21204 0.047 -4.53 0.0001 Estimate of Lag Distribution Variable -0.212 0 PMG_PGNP(0) 1 **********************1 PMG_PGNP(1) 1 ***************j PMG_PGNP(2) 1 **********j PMG_PGNP(3) 1 PMG_PGNP(4) 1 *********j PMG_PGNP(5) 1 *************j PMG_PGNP(6) 1 *******************j PMG_PGNP(7) 1 *****************************j PMG_PGNP(8) *****************************************

d. Third Degree Polynomial Almon Lag(S = 6,P = 3/ PDLREG Procedure

Dependent Variable = QMG_CAR

Ordinary Least Squares Estimates

 SSE 0.034308 DFE 25 MSE 0.001372 Root MSE 0.037045 SBC -103.747 AIC -114.007 Reg Rsq 0.9144 Total Rsq 0.9144 Durbin-Watson 0.6763 Std t Approx Variable DF B Value Error Ratio Prob Intercept 1 -7.31542415 3.0240 -2.419 0.0232 RGNP_POP 1 -0.57343614 0.2771 -2.069 0.0490 CAR_POP 1 0.23462358 0.2790 0.841 0.4084 PMG_PGNP**0 1 -0.24397597 0.0320 -7.613 0.0001 PMG_PGNP**1 1 -0.07041380 0.0417 -1.690 0.1035 PMG_PGNP**2 1 -0.11318734 0.0637 -1.778 0.0876 PMG_PGNP**3 1 -0.19730731 0.0955 -2.067 0.0493

 Parameter Std t Approx Variable Value Error Ratio Prob PMG_PGNP(0) -0.03349 0.059 -0.57 0.5725 PMG_PGNP(1) -0.14615 0.041 -3.54 0.0016 PMG_PGNP(2) -0.12241 0.043 -2.87 0.0082 PMG_PGNP(3) -0.04282 0.026 -1.64 0.1130 PMG_PGNP(4) -0.01208 0.045 0.27 0.7890 PMG_PGNP(5) -0.03828 0.043 -0.90 0.3788 PMG_PGNP(6) -0.27443 0.067 -4.10 0.0004

Estimate of Lag Distribution

Variable -0.274 0.0121

PMG_PGNP(0)

PMG_PGNP(1)

PMG_PGNP(2)

PMG_PGNP(3)

PMG_PGNP(4)

PMG_PGNP(5)

PMG_PGNP(6)

e. Almon Lag(S = 6,P = 2/ with Near End-Point Restriction

PDLREG Procedure

Dependent Variable = QMGCAR

Ordinary Least Squares Estimates

 SSE 0.046362 DFE 27 MSE 0.001717 Root MSE 0.041438 SBC -101.043 AIC -108.372 Reg Rsq 0.8843 Total Rsq 0.8843 Durbin-Watson 0.5360 Variable DF B Value Std Error t Ratio Approx Prob Intercept 1 -3.81408793 3.0859 -1.236 0.2271 RGNP_POP 1 -0.28069982 0.2843 -0.988 0.3322 CAR_POP 1 -0.05768233 0.2873 -0.201 0.8424 PMG_PGNP**0 1 -0.21562744 0.0317 -6.799 0.0001 PMG_PGNP**1 1 -0.07238330 0.0458 -1.581 0.1255 PMG_PGNP**2 1 0.02045576 0.0268 0.763 0.4519 Restriction DF L Value Std Error t Ratio Approx Prob PMG. PGNP(-l) -1 0.03346081 0.0176 1.899 0.0683

 Variable Parameter Value Std Error t Ratio Approx Prob PMG_PGNP(0) -0.02930 0.013 -2.31 0.0286 PMG_PGNP(1) -0.05414 0.020 -2.75 0.0105 PMG_PGNP(2) -0.07452 0.021 -3.49 0.0017 PMG_PGNP(3) -0.09043 0.018 -4.91 0.0001 PMG_PGNP(4) -0.10187 0.015 -6.80 0.0001 PMG_PGNP(5) -0.10886 0.022 -4.88 0.0001 PMG_PGNP(6) -0.11138 0.042 -2.66 0.0130

Estimate of Lag Distribution

Variable -0.111 0

PMG_PGNP(0)

PMG_PGNP(1)

PMG_PGNP(2)

PMG_PGNP(3)

PMG_PGNP(4)

PMG_PGNP(5)

PMG_PGNP(6)

ALMON LAG(S=6,P=2) with FAR END-POINT RESTRICTION

PDLREG Procedure Dependent Variable = QMG_CAR

Ordinary Least Squares Estimates

 SSE 0.050648 DFE 27 MSE 0.001876 Root MSE 0.043311 SBC -98.2144 AIC -105.543 Reg Rsq 0.8736 Total Rsq 0.8736 Durbin-Watson 0.5690 Std t Approx Variable DF B Value Error Ratio Prob Intercept 1 -6.02848892 3.2722 -1.842 0.0764 RGNP_POP 1 -0.49021381 0.2948 -1.663 0.1079 CAR_POP 1 0.15878127 0.2982 0.532 0.5988 PMG_PGNP**0 1 -0.20851840 0.0337 -6.195 0.0001 PMG_PGNP**1 1 -0.00242944 0.0418 -0.058 0.9541 PMG_PGNP**2 1 0.06159671 0.0240 2.568 0.0161 Restriction DF L Value Std Error t Ratio Approx Prob PMG_PGNP(7) -1 0.03803694 0.0161 2.363 0.0256

Estimate of Lag Distribution Variable -0.106

PMG_PGNP(0)

PMG_PGNP(1)

PMG_PGNP(2)

PMG_PGNP(3)

PMG_PGNP(4)

PMG_PGNP(5)

PMG_PGNP(6)

SAS PROGRAM Data RAWDATA;

Input Year CAR QMG PMG POP RGNP PGNP; Cards;

Data USGAS; set RAWDATA; LNQMG=LOG(QMG);

LNCAR=LOG(CAR);

LNPOP=LOG(POP);

LNRGNP=LOG(RGNP);

LNPGNP=LOG(PGNP);

LNPMG=LOG(PMG);

QMG_CAR=LOG(QMG/CAR);

RGNP_POP=LOG(RGNP/POP);

CAR_POP=LOG(CAR/POP);

PMG_PGNP=LOG(PMG/PGNP);

PMPG_1=LAG1(PMG_PGNP);

PMPG_2=LAG2(PMG_PGNP);

PMPG_3=LAG3(PMG_PGNP);

PMPG_4=LAG4(PMG_PGNP);

PMPG_5=LAG5(PMG_PGNP);

PMPG_6=LAG6(PMG_PGNP);

Proc autoreg data=USGAS;

Model QMG_CAR=RGNP_POP CAR_POP PMG_PGNP PMPG_1 PMPG_2 PMPG_3 PMPG_4 PMPG.5 PMPG_6;

TITLE‘UNRESTRICTED MODEL’;

PROC PDLREG DATA=USGAS;

MODEL QMG_CAR=RGNP_POP CAR_POP PMG_PGNP(6,2);

TITLE ‘ALMON LAG(S=6,P=2)’;

PROC PDLREG DATA=USGAS;

MODEL QMG_CAR=RGNP_POP CAR_POP PMG_PGNP(4,2);

TITLE ‘ALMON LAG(S=4,P=2)’;

PROC PDLREG DATA=USGAS;

MODEL QMG_CAR=RGNP_POP CAR_POP PMG_PGNP(8,2);

TITLE ‘ALMON LAG(S=8,P=2)’;

PROC PDLREG DATA=USGAS;

MODEL QMG_CAR=RGNP_POP CAR_POP PMG_PGNP(6,3);

TITLE ‘Third Degree Polynomial ALMON LAG(S=6,P=3)’;

PROC PDLREG DATA=USGAS;

MODEL QMG_CAR=RGNP_POP CAR_POP PMG_PGNP(6,2,,FIRST); TITLE ‘ALMON LAG(S=6,P=2) with NEAR END-POINT RESTRICTION’;

PROC PDLREG DATA=USGAS;

MODEL QMG_CAR=RGNP_POP CAR_POP PMG_PGNP(6,2,,LAST); TITLE ‘ALMON LAG(S=6,P=2) with FAR END-POINT RESTRICTION’;

RUN;

CHAPTER 7

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