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

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

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

Sample (adjusted): 1965 2007 Included observations: 43 after adjustments

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

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

Sample (adjusted): 1965 2007 Included observations: 43 after adjustments

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

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

Sample (adjusted): 1965 2007 Included observations: 43 after adjustments

. 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

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

Sample (adjusted): 1965 2007 Included observations: 43 after adjustments

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

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

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]

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

Sample (adjusted): 1964 2007 Included observations: 44 after adjustments

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

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

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

plim X Х /T = "0 - 2"2/уо - "і/"о = "0 - ("2/"0) = ("0 - Yi) /"о.

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

C 0J2 Yt_1©t_1/£ Y2_1

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

Yt = (1 C 0L)^]piLi©t

i=0

Yt = (1 C 0L) (©t C P©t_1 + "2©t_2 + ..)

Yt = ©t C (0 C ") [©t_1 c "©t_2 c "2©t_3 c ..]

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

Yt_1©t/T = ©t [©t_1 C (0 C P) (©t_2 C "©t_3 C ..)] /T

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-1 ©t-2/T C P ©t-1 ©t-3/T C..

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

But, vt = ©t C 0©t-1 withvar(vt) = a©2(1 C 02). Hence,

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

= 0 C 0a©2 = 0a©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

. 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.

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.

Godfrey's Serial Correlation Test Alternative LM Prob>LM

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

Dependent Variable: RESID

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

Estimate of Lag Distribution

Variable -0.184 0

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

Estimate of Lag Distribution

Variable -0.116 0

ALMON LAG(S=8,P=2)

PDLREG Procedure Dependent Variable = QMG_CAR

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

Dependent Variable = QMG_CAR

Ordinary Least Squares Estimates

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

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

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