Chapter 11 · 2010. 8. 4. · Chapter 11 Incorporating Dynamics Through the Use of Distributed Lags. Section 11.1 Introduction. ... RPC(a0) 0.003645 Object 1.916E-6 Trace(S) 0.000247

Chapter 11Chapter 11

Incorporating Dynamics

Through the Use of

Distributed Lags

Section 11.1Section 11.1

Introduction

Arise in theory when economic stimuli produce their effect only after some lag in time.

Effects are not felt all at once but instead are distributed over a period of time.

Types:

Distributed Lag Models

Types:

(1) Free-form

(2) Geometric Lag

(3) Almon (Polynomial Distributed) Lag

(4) Partial Adjustment Hypothesis

3

- SR Effects

- LR Effects

Information from Distributed Lag Models

- Mean Lag

4

tmtm

1t1t0

t2t10t

ADVw...

ADVwADVw

INCOMEPRICESALES

ε+++++

β+β+β=

−

−

0wffecte SR →

Example

effect

m

0ii

m

0ii

LR

iwagl MEAN

wfectef LR

∑

∑

=

=

→

→

5

1. Psychological

2. Technical

3. Institutional

Distributed Lags Attributed to Four Factors

3. Institutional

4. Imperfect Knowledge or Uncertainty

Griliches (1967)

Dhrymes (1971)

6


Approaches to Distributed

Lag Models

1. No assumptions as to the form of distribution of weights (free form)

2. Assume a distribution of weights: a general

form for the distribution of lag (geometric

Approaches

form for the distribution of lag (geometric

and Almon lags)

3. Nerlovian LagsExplicit dynamic model which implies a distributed lag only incidentally

8

(1) No assumption as to the form of the distributi on;weights associated with various lags are an empiric al issue .

....Xw...XwXwXwXwY

tmtm

3t32t21t1t00t

ε++++++++α=

−

−−−

Approaches to Estimating Distribution of a Lag

Incorporate all conceivable lags and estimate.

Problems

* Finite number of observations in most time serie s -reduces degrees of freedom

* Possible collinearity with X t, Xt-1, . . . , Xt-m.

9


SAMPLE PROBLEM:

Free- Form Lag

Orange Juice Problem: Free-Form Distributed Lag****************************************************************************************DEMAND ANALYSIS FOR ORANGE JUICE USING SCANNER DATA FROM RETAIL FOOD STORES

RETAIL FOOD STORES IN EXCESS OF $2 MILLION ANNUALLY

ALLOJGALPC--PER CAPITA ORANGE JUICE CONSUMPTION IN GALLONS

RALLOJPRICE--REAL PRICE OF ORANGE JUICE, DOLLARS PER GALLON

RALLGFJPRICE--REAL PRICE OF GRAPEFRUIT JUICE, DOLLARS PER GALLONRPCDPI--REAL PER CAPITA DISPOSABLE PERSONAL INCOME, DOLLARS

SC0996--DUMMY VARIABLE, BEGINNING IN SEPTEMBER 1996 WALMART PART OF THE DATA COLLECTION

ROJFDOCADEXP--REAL ADVERTISING EXPENDITURES ON ORANGE JUICE BY THE FLORIDA DEPT OF CITRUS,THOUSAND DOLLARS***********************************************************************************11

/* Read in the RAW DATA */ /* STATEMENT*/

data Orange_Juice_FDOC_Data;input MONTH YEAR ALLOJGALPC RALLOJPRICERALLGFJPRICE RPCDPI SC0996 ROJFDOCADEXP;

* logarithmic transformation of variables; * logarithmic transformation of variables; lallojgalpc=log(allojgalpc); lrallojprice=log(rallojprice);

lrallgfjprice=log(rallgfjprice); lrpcdpi=log(rpcdpi); lrojfdocadexp=log(rojfdocadexp);

* creation of lags of lrojfdocadexp variable;lag1rfdocad=lag1(lrojfdocadexp);lag2rfdocad=lag2(lrojfdocadexp);lag3rfdocad=lag3(lrojfdocadexp);lag4rfdocad=lag4(lrojfdocadexp);

12

* seasonality through the use of monthly dummy variables;m1=0; m2=0; m3=0; m4=0; m5=0; m6=0; m7=0; m8=0; m9=0;

m10=0; m11=0; m12=0;if month=1 then m1=1;if month=2 then m2=1;if month=3 then m3=1;if month=4 then m4=1;if month=5 then m5=1;if month=6 then m6=1; if month=6 then m6=1; if month=7 then m7=1;if month=8 then m8=1;if month=9 then m9=1;if month=10 then m10=1;if month=11 then m11=1;if month=12 then m12=1;

datalines;13

options nodate;run;

***********************************************************************GLS procedure to run a free-form distributed lag **********************************************************************;

* correction for serial correlation;

14

* correction for serial correlation; Proc autoreg data = Orange_Juice_FDOC_Data;

model lallojgalpc = lrallojprice lrallgfjprice lrpcdpi

sc0996 lrojfdocadexp lag1rfdocad lag2rfdocad lag3rfdocad lag4rfdocad m1-m11 / nlag=2 method=ml;

run;

Graph of Per Capita Monthly Orange Juice

Consumption, January 1989 to September 2002

.28

.30

ALLOJGALPC

15

.20

.22

.24

.26

89 90 91 92 93 94 95 96 97 98 99 00 01 02

Graph of Real (Inflation-Adjusted) FDOC Advertising

Expenditures for Orange Juice Over the Period

January 1989 to September 2002

4,000

ROJFDOCADEXP

16

0

1,000

2,000

3,000

89 90 91 92 93 94 95 96 97 98 99 00 01 02

Free-Form Distributed Lag

Dependent Variable lallojgalpc

Ordinary Least Squares Estimates

SSE 0.10645428 DFE 140

17

SSE 0.10645428 DFE 140

MSE 0.0007604 Root MSE 0.02758

SBC -615.14479 AIC -679.85428

MAE 0.01899552 AICC -673.2068

MAPE 1.37380851 Regress R-Square 0.8537

Durbin-Watson 0.4622 Total R-Square 0.8537

continued...

Standard Approx

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 0.8317 0.9011 0.92 0.3577

lrallojprice 1 -0.3887 0.0314 -12.38 <.0001

lrallgfjprice 1 0.3157 0.0517 6.10 <.0001

lrpcdpi 1 -0.2314 0.0954 -2.43 0.0166

SC0996 1 0.0460 0.009401 4.89 <.0001

lrojfdocadexp 1 -0.000542 0.002529 -0.21 0.8305

lag1rfdocad 1 0.002803 0.002759 1.02 0.3113

lag2rfdocad 1 0.002247 0.002816 0.80 0.4262

lag3rfdocad 1 0.003565 0.002764 1.29 0.1992 lag3rfdocad 1 0.003565 0.002764 1.29 0.1992

lag4rfdocad 1 0.002968 0.002559 1.16 0.2480

m1 1 0.0420 0.0114 3.70 0.0003

m2 1 -0.1087 0.0115 -9.47 <.0001

m3 1 -0.0247 0.0110 -2.24 0.0270

m4 1 -0.1034 0.0114 -9.05 <.0001

m5 1 -0.0956 0.0112 -8.58 <.0001

m6 1 -0.1465 0.0110 -13.37 <.0001

m7 1 -0.1234 0.0112 -10.97 <.0001

m8 1 -0.0966 0.0109 -8.85 <.0001

m9 1 -0.0974 0.0111 -8.81 <.0001

m10 1 -0.0501 0.0110 -4.56 <.0001

m11 1 -0.0678 0.0113 -5.97 <.0001 18 continued...

Maximum Likelihood Estimates

SSE 0.0339825 DFE 138

MSE 0.0002463 Root MSE 0.01569

SBC -786.23376 AIC -857.10606

MAE 0.01091217 AICC -849.04767


19



continued...

Standard Approx


Intercept 1 1.5658 1.6965 0.92 0.3577

lrallojprice 1 -0.8069 0.0741 -10.89 <.0001

lrallgfjprice 1 0.3817 0.1041 3.67 0.0003

lrpcdpi 1 -0.2655 0.1783 -1.49 0.1386

SC0996 1 0.0368 0.0156 2.36 0.0198

lrojfdocadexp 1 0.001138 0.001364 0.83 0.4054

lag1rfdocad 1 0.002392 0.001359 1.76 0.0807

lag2rfdocad 1 0.001550 0.001335 1.16 0.2478

lag3rfdocad 1 0.002980 0.001333 2.23 0.0270

lag4rfdocad 1 0.001504 0.001362 1.10 0.2712

m1 1 0.0428 0.005399 7.92 <.0001

m2 1 -0.1006 0.005653 -17.79 <.0001

Sum of coefficients associated with advertising and promotion corresponds to the long-run elasticity 0.009564. The mean lag Is equal to 2.14 months.

20

m2 1 -0.1006 0.005653 -17.79 <.0001

m3 1 -0.0156 0.006137 -2.54 0.0121

m4 1 -0.0896 0.006814 -13.14 <.0001

m5 1 -0.0833 0.006984 -11.92 <.0001

m6 1 -0.1344 0.006879 -19.54 <.0001

m7 1 -0.1110 0.006906 -16.07 <.0001

m8 1 -0.0884 0.006520 -13.56 <.0001

m9 1 -0.0918 0.006132 -14.96 <.0001

m10 1 -0.0479 0.005454 -8.79 <.0001

m11 1 -0.0641 0.004931 -13.00 <.0001

AR1 1 -0.6502 0.0813 -8.00 <.0001

AR2 1 -0.3174 0.0815 -3.89 0.0002

(2) Assume a distribution to the lag, with its associated parameters.

Example 1 Geometric Lag

Geometric Lag

Weights declining geometrically with time

.10,WW 0i

i <λ<λ=

21

Coefficients decline geometrically; that is, the closest lags carry the heaviest weight. Expression for any coefficient

k λλ <<==

Geometric Lags: Koyck Distributed Lags

General lag structure

Nonlinear Model

.......

.10 ...2,1,0

0100

0

tmtm

ttt

kk

XWXWXWY

kWW

ελλα

λλ

++++++=

<<==

−−

22

Useful to describe the lag structure of a distributed lag model: (1) in terms of its mean or average lag, and (2) in terms of the long-run response of the dependent variable to a permanent change in one of the explanatory variables.

The long-run response in the geometric lag model is:

∑

=

m

ssW

0

The long-run response measures the change in Y associated with a one-unit change in X which stays in effect for all time.

Mean lag refers to the average length of time for a (unit) change in the explanatory variable X to be transferred to the dependent variable.

ss

m

s

m

sss WWWsWmean λ0

0 0

/lag ==∑ ∑= =

23


SAMPLE PROBLEM:

Geometric Lag

options nodate;run;***************************************************************************Conventional model with correction for serial correlation; proc model data = Orange_Juice_FDOC_Data;parms a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20;

lallojgalpc = a0 + a1*lrallojprice + a2*lrallgfjprice + a3*lrpcdpi + lallojgalpc = a0 + a1*lrallojprice + a2*lrallgfjprice + a3*lrpcdpi +

a4*sc0996 + a5*lrojfdocadexp + a6*lag1rfdocad + a7*lag2rfdocad

+ a8*lag3rfdocad + a9*lag4rfdocad + a10*m1 + a11*m2 + a12*m3

+ a13*m4 + a14*m5 + a15*m6 + a16*m7 + a17*m8 + a18*m9 +

a19*m10 + a20*m11;

%ar(lallojgalpc,2);fit lallojgalpc / dw;

25 continued...

run; GLS procedure to run a geometric distributed lag

* correction for serial correlation; proc model data=Orange_Juice_FDOC_Data;parms a0 a1 a2 a3 a4 a5 a6 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20;

lallojgalpc = a0 + a1*lrallojprice + a2*lrallgfjprice + a3*lrpcdpi +

a4*sc0996 + a5*lrojfdocadexp + a5*a6*lag1rfdocad +

a5*a6*a6*lag2rfdocad + a5*a6*a6*a6*lag3rfdocad + a5*a6*a6*lag2rfdocad + a5*a6*a6*a6*lag3rfdocad +

a5*a6*a6*a6*a6*lag4rfdocad + a10*m1 + a11*m2 + a12*m3 +

a13*m4 + a14*m5 + a15*m6 + a16*m7 + a17*m8 + a18*m9 +

a19*m10 + a20*m11;%ar(lallojgalpc,2);fit lallojgalpc start=(a0=1.56 a1=-.8 a2=.4 a3=-.3 a4=.04 a5=.001138 a6=.5) / dw; run;

26

The MODEL Procedure

OLS Estimation Summary

Data Set Options

DATA= ORANGE_JUICE_FDOC_DATA

Minimization Summary

Parameters Estimated 23

Method Gauss

Iterations 8

Final Convergence Criteria Final Convergence Criteria

R 0.000305

PPC(a0) 0.000624

RPC(a0) 0.003645

Object 1.916E-6

Trace(S) 0.000247

Objective Value 0.000212

Observations Processed

Read 165

Solved 165

Used 161

Missing 4 27

Nonlinear OLS Summary of Residual Errors

DF DF

Equation Model Error SSE MSE

lallojgalpc 23 138 0.0341 0.000247

28

lallojgalpc 23 138 0.0341 0.000247

Adj Durbin

Root MSE R-Square R-Sq Watson

0.0157 0.9531 0.9456 2.1052

Nonlinear OLS Parameter Estimates

Approx Approx

Parameter Estimate Std Err t Value Pr > |t| Label

a0 1.556093 1.7677 0.88 0.3802

a1 -0.82047 0.0735 -11.17 <.0001

a2 0.390448 0.1038 3.76 0.0002

a3 -0.25617 0.1869 -1.37 0.1728

a4 0.03731 0.0157 2.37 0.0191

a5 0.001134 0.00135 0.84 0.4037

a6 0.002355 0.00135 1.74 0.0834

a7 0.001524 0.00133 1.15 0.2527

a8 0.002975 0.00132 2.25 0.0261

a9 0.001466 0.00135 1.08 0.2800

a10 0.043003 0.00538 7.99 <.0001

29

a10 0.043003 0.00538 7.99 <.0001

a11 -0.1003 0.00560 -17.91 <.0001

a12 -0.01538 0.00608 -2.53 0.0125

a13 -0.08925 0.00674 -13.24 <.0001

a14 -0.083 0.00690 -12.03 <.0001

a15 -0.13409 0.00680 -19.73 <.0001

a16 -0.11061 0.00683 -16.20 <.0001

a17 -0.08814 0.00645 -13.66 <.0001

a18 -0.09156 0.00608 -15.06 <.0001

a19 -0.04773 0.00541 -8.83 <.0001

a20 -0.06393 0.00492 -13.00 <.0001

lallojgalpc_l1 0.662 0.0810 8.17 <.0001 AR(lallojgalpc)

lallojgalpc lag1

parameter


lallojgalpc lag2

parameter

OLS Estimation Summary

Data Set Options

DATA= ORANGE_JUICE_FDOC_DATA

Minimization Summary

Parameters Estimated 20

Method Gauss

Iterations 10

Subiterations 2

Average Subiterations 0.2

Final Convergence Criteria

R 0.000968

PPC(a0) 0.001352

RPC(a2) 0.007989

Object 0.000022

Trace(S) 0.000244

Objective Value 0.000214

Observations Processed

Read 165

Solved 165

Used 161

Missing 4 30

Nonlinear OLS Summary of Residual Errors

DF DF

Equation Model Error SSE MSE

lallojgalpc 20 141 0.0344 0.000244

Adj Durbin


31


0.0156 0.9527 0.9463 2.1073

Nonlinear OLS Parameter Estimates

Approx Approx

Parameter Estimate Std Err t Value Pr > |t| Label

a0 1.579064 1.7490 0.90 0.3682

a1 -0.81986 0.0727 -11.28 <.0001

a2 0.389325 0.1027 3.79 0.0002

a3 -0.25878 0.1850 -1.40 0.1641

a4 0.037626 0.0155 2.42 0.0168

a5 0.001734 0.000987 1.76 0.0812

a6 1.058429 0.2088 5.07 <.0001

a10 0.043751 0.00521 8.41 <.0001

a11 -0.10001 0.00533 -18.78 <.0001

a12 -0.0148 0.00598 -2.47 0.0146

a5 = wo, and a6=λ, but since λ>1, the

geometric lag specification is

32

a12 -0.0148 0.00598 -2.47 0.0146

a13 -0.0881 0.00658 -13.39 <.0001

a14 -0.08289 0.00679 -12.21 <.0001

a15 -0.13322 0.00670 -19.89 <.0001

a16 -0.10975 0.00671 -16.37 <.0001

a17 -0.08795 0.00640 -13.74 <.0001

a18 -0.09079 0.00594 -15.30 <.0001

a19 -0.04783 0.00532 -9.00 <.0001

a20 -0.06275 0.00468 -13.42 <.0001


lallojgalpc lag1

parameter


lallojgalpc lag2

parameter

specification is not appropriate.

We assume that the lag weights can be specified by a continuous function and approximated by a polynomial function.

We must specify:

(A) the length of the lag

(B) degree of the polynomial

Polynomial Distributed Lags(development attributed to Shirley Almon)

desirable to have both of these fairly small.

We have flexibility; however:

• degree of the polynomial<the number of terms in dis tributed lag minus one, or else we get no reduction in the n umber of parameters to be estimated.

• possibility exists for endpoint restrictions.

Almon lag polynomials

.iC...iCiCCW nn

2210i ++++=

33

Y w X w X w X w X

Assume that

w c

w c c c

t t t t t t

o

= + + + + + + ∈

== + +

− − −α 0 1 1 2 2 6 6

0

1 0 1 2

...

Example:

w c c c

w c c c

w c c c

w c c c

w c c c

w c c c

= + += + += + += + += + += + +

1 0 1 2

2 0 1 2

3 0 1 2

4 0 1 2

5 0 1 2

6 0 1 2

2 4

3 9

4 16

5 25

6 3634

Endpoint Restrictions:

(1) Head Restriction w -1 = 0

(2) Tail Restriction w = 0

By Substitution,

(or by GLS if necessary to account for serial corre lationOnly 3 parameters to be estimated

[ ] [ ][ ] t6t2t1t2

6t2t1t16t1tt0t

X36...X4XCX6...X2XCX...XXCY

ε+++++++++++++α=

−−−

−−−−−

OLSby ,C ,C ,C Estimate 210

(2) Tail Restriction w 7 = 0

w c c c c c c

w c c c c c c− = − + = ⇒ = +

= + + = ⇒ = − −1 0 1 2 1 0 2

7 0 1 2 0 1 2

0

7 49 0 7 49

or both head & tail restrictions,

c c c c c c

c c

c c c c

1 1 2 2 1 2

1 2

1 2 0 2

7 49 7 48

8 48

6 7

= − − + = − −= −

= − = −

only need to estimate c 235

The polynomial lag model was used by Almon to estimate the relationship between current capital expenditures and current and past capital appropriations in the United States Manufacturing Industries. The degree of the polynomial in each case was 4, but the length of the lag was taken to be different for different industries. The data estimation was given by the quarterly observations for the years 1953-1961. The model was specified

Example:

observations for the years 1953-1961. The model was specified as:

where Y represents capital expenditures, the S’s represent seasonal dummy variables, and the X’s represent capital appropriations. The weights were restricted by the conditions that w-1=0 and w m+1=0, but they were not required to add up to unity.

tmtmtttttt XWXwXwSSSSY ∈++++++++= −− ...110443322111 αααα

36

The result for “All Manufacturing Industries” was as follows:

ttttt

ttttttttt

XXXX

XXXXSSSSY

∈+++++

+++++−+−=

−−−−

−−−

7654

)023.0(3

)013.0(2

)016.0(1

)023.0(4321

053.0105.0146.0167.0

165.0141.0099.0048.03205013283

As can be seen, the chosen length of the lag in th is case was 7 periods. The weights add up to 0.922.

Mean lag = 3.54

ttttt XXXX ∈+++++ −−−−)024.0(

7)016.0(

6)013.0(

5)023.0(

4 053.0105.0146.0167.0

37

38

39

PDLREG ProcedurePROC PDLREG can include any number of explanatory

variables with distribution lags and any number of explanatory variables without lag distributions (called covariates).

The PDLREG procedure supports endpoint restrictions. This procedure also allows the imposition of linear restrictions on the parameter estimates for the covariates.

40

the parameter estimates for the covariates.The PDLREG procedure allows the specification of a minimum

degree and a maximum degree for the lag distribution polynomial, and the procedure fits polynomials for all degrees in the specified range.

The PDLREG procedure also conducts tests for serial correlation, and allows the specification of any order autoregressive error model. In addition, the procedure allows for the specification of several different estimation methods for the autoregressive model.

The Syntax for the Specification of a Polynomial

Distributed Lag (PDL) for PROC PDLREG is as

follows:

Variable (length, degree, minimum-degree, constraint)

� Length – specifies the number of lags of the variable to include in the lag distribution.

41

the lag distribution.� Degree – specifies the maximum degree of the distribution

polynomial. If not specified, the degree defaults to the lag length.� Minimum-degree – specifies the minimum degree of the

polynomial. By default, minimum degree is the same as degree.� Constraint – specifies endpoint restrictions on the polynomial.

The value of constraint can be FIRST, LAST, or BOTH. If a value is not specified, there are no endpoint restrictions.


SAMPLE PROBLEM:

Polynomial Distributed Lag

options nodate;run;

*********************************************************************************GLS procedure to run a polynomial distributed lag *********************************************************************************;

* correction for serial correlation; Proc pdlreg data=Orange_Juice_FDOC_Data;*3 lags and 2nd degree polynomial with both end point restrictions;

model lallojgalpc = lrallojprice lrallgfjprice lrpcdpi sc0996

43

model lallojgalpc = lrallojprice lrallgfjprice lrpcdpi sc0996 lrojfdocadexp(3,2,,both) m1-m11 / nlag=2 method=ml; *4 lags and 2nd degree polynomial with both end point restrictions;



model lallojgalpc = lrallojprice lrallgfjprice lrpcdpi sc0996 lrojfdocadexp(6,2,,both) m1-m11 / nlag=2 method=ml;

*7 lags and 2nd degree polynomial with both end point restrictions;


lrojfdocadexp(7,2,,both)

m1-m11 / nlag=2 method=ml;



lrojfdocadexp(8,2,,both) m1-m11 / nlag=2 method=ml; lrojfdocadexp(8,2,,both) m1-m11 / nlag=2 method=ml;



lrojfdocadexp(9,2,,both) m1-m11 / nlag=2 method=ml;



lrojfdocadexp(10,2,,both) m1-m11 / nlag=2 method=ml;

44

Maximum Likelihood Estimation ProcedureAR(1), AR(2) Correction

Lag Length

RMSE SBC AIC MAE MAPE R 2

3 0.0156 -809.34 -868.00 0.0111 0.8077 0.9522

4 0.0155 -805.49 -864.04 0.0109 0.7977 0.9530

5 0.0157 -796.23 -854.66 0.0112 0.8125 0.9521

45

5 0.0157 -796.23 -854.66 0.0112 0.8125 0.9521

6 0.0158 -788.72 -847.03 0.0113 0.8211 0.9517

7 0.0158 -783.09 -841.28 0.0114 0.8293 0.9518

8 0.0160 -775.07 -833.14 0.0115 0.8376 0.9513

9 0.0161 -768.07 -826.02 0.0116 0.8451 0.9511

10 0.0161 -761.66 -819.49 0.0117 0.8513 0.9510

Lag Lengths of three or four are optimal.

FDOC Orange Juice Problem

The PDLREG Procedure


Lag Length of Three Months for Advertising and Promotion

Ordinary Least Squares Estimates Ordinary Least Squares Estimates

SSE 0.11591805 DFE 145

MSE 0.0007994 Root MSE 0.02827

SBC -627.05461 AIC -679.54375

MAE 0.0197688 AICC -675.29375



46

Standard Approx


Intercept 1 1.1278 0.9049 1.25 0.2147

lrallojprice 1 -0.3823 0.0320 -11.93 <.0001

lrallgfjprice 1 0.2992 0.0527 5.67 <.0001

lrpcdpi 1 -0.2599 0.0961 -2.70 0.0077

SC0996 1 0.0462 0.009587 4.82 <.0001

lrojfdocadexp**0 1 0.004549 0.001372 3.31 0.0012

lrojfdocadexp**1 1 -1E-18 0 -Infty <.0001

lrojfdocadexp**2 1 -0.000910 0.000274 -3.31 0.0012

47

lrojfdocadexp**2 1 -0.000910 0.000274 -3.31 0.0012

m1 1 0.0413 0.0112 3.71 0.0003

m2 1 -0.1072 0.0112 -9.61 <.0001

m3 1 -0.0246 0.0112 -2.20 0.0297

m4 1 -0.0937 0.0110 -8.51 <.0001

m5 1 -0.0907 0.0110 -8.26 <.0001

m6 1 -0.1427 0.0110 -13.01 <.0001

m7 1 -0.1173 0.0110 -10.70 <.0001

m8 1 -0.0927 0.0110 -8.43 <.0001

m9 1 -0.0945 0.0110 -8.62 <.0001

m10 1 -0.0499 0.0112 -4.48 <.0001

m11 1 -0.0670 0.0111 -6.02 <.0001

o 1 2co, c1, and c2

Tests of Endpoint Restrictions

Standard Approx

Restriction DF L Value Error t Value Pr > |t|

lrojfdocadexp(-1) -1 -0.2506 0.1662 -1.51 0.1320

lrojfdocadexp(4) -1 0.2606 0.1655 1.57 0.1158

Estimate of Lag Distribution Estimate of Lag Distribution

Standard Approx

Variable Estimate Error t Value Pr > |t|

lrojfdocadexp(0) 0.001819 0.000549 3.31 0.0012

lrojfdocadexp(1) 0.002729 0.000823 3.31 0.0012

lrojfdocadexp(2) 0.002729 0.000823 3.31 0.0012

lrojfdocadexp(3) 0.001819 0.000549 3.31 0.0012

48

wo=0.001819; w1= 0.002729, w2= 0.002729; and w3= 0.001819



SSE 0.03476417 DFE 143

MSE 0.0002431 Root MSE 0.01559

SBC -809.33613 AIC -868.00046

49

SBC -809.33613 AIC -868.00046

MAE 0.01107909 AICC -862.64835



Standard Approx


Intercept 1 1.6456 1.6799 0.98 0.3289

lrallojprice 1 -0.8153 0.0730 -11.16 <.0001

lrallgfjprice 1 0.3720 0.1025 3.63 0.0004

lrpcdpi 1 -0.2699 0.1766 -1.53 0.1286

SC0996 1 0.0380 0.0154 2.47 0.0146

lrojfdocadexp**0 1 0.003564 0.001386 2.57 0.0112

lrojfdocadexp**1 1 -2.8E-18 0 -Infty <.0001

lrojfdocadexp**2 1 -0.000713 0.000277 -2.57 0.0112

m1 1 0.0436 0.005156 8.46 <.0001

50

m1 1 0.0436 0.005156 8.46 <.0001

m2 1 -0.0996 0.005357 -18.60 <.0001

m3 1 -0.0149 0.006070 -2.45 0.0156

m4 1 -0.0860 0.006454 -13.33 <.0001

m5 1 -0.0806 0.006613 -12.19 <.0001

m6 1 -0.1321 0.006652 -19.86 <.0001

m7 1 -0.1081 0.006559 -16.48 <.0001

m8 1 -0.0868 0.006370 -13.62 <.0001

m9 1 -0.0905 0.005973 -15.16 <.0001

m10 1 -0.0480 0.005363 -8.94 <.0001

m11 1 -0.0639 0.004671 -13.67 <.0001

AR1 1 -0.6509 0.0797 -8.16 <.0001

AR2 1 -0.3182 0.0799 -3.98 0.0001


Standard Approx


lrojfdocadexp(-1) -1 -0.0571 0.0928 -0.62 0.5402

lrojfdocadexp(4) -1 0.0991 0.0917 1.08 0.2814

Estimate of Lag Distribution

51


Standard Approx


lrojfdocadexp(0) 0.001426 0.000554 2.57 0.0112

lrojfdocadexp(1) 0.002138 0.000832 2.57 0.0112

lrojfdocadexp(2) 0.002138 0.000832 2.57 0.0112

lrojfdocadexp(3) 0.001426 0.000554 2.57 0.0112

sum 0.007128 (Long-Run)



Lag Length of Four Months for Advertising and Promotion


SSE 0.1078914 DFE 144

MSE 0.0007492 Root MSE 0.02737

52

MSE 0.0007492 Root MSE 0.02737

SBC -633.31147 AIC -685.69534

MAE 0.01904292 AICC -681.41562



Standard Approx


Intercept 1 0.9254 0.8824 1.05 0.2960

lrallojprice 1 -0.3903 0.0311 -12.54 <.0001

lrallgfjprice 1 0.3157 0.0513 6.15 <.0001

lrpcdpi 1 -0.2413 0.0936 -2.58 0.0109

SC0996 1 0.0469 0.009293 5.04 <.0001

lrojfdocadexp**0 1 0.004967 0.001269 3.92 0.0001

lrojfdocadexp**1 1 7.35E-19 0 Infty <.0001

lrojfdocadexp**2 1 -0.001187 0.000303 -3.92 0.0001

53

lrojfdocadexp**2 1 -0.001187 0.000303 -3.92 0.0001

m1 1 0.0418 0.0108 3.87 0.0002

m2 1 -0.1076 0.0108 -9.96 <.0001

m3 1 -0.0248 0.0108 -2.29 0.0235

m4 1 -0.1010 0.0109 -9.27 <.0001

m5 1 -0.0927 0.0107 -8.68 <.0001

m6 1 -0.1439 0.0106 -13.53 <.0001

m7 1 -0.1191 0.0106 -11.22 <.0001

m8 1 -0.0938 0.0106 -8.83 <.0001

m9 1 -0.0949 0.0106 -8.96 <.0001

m10 1 -0.0496 0.0108 -4.59 <.0001

m11 1 -0.0663 0.0108 -6.15 <.0001


Standard Approx


lrojfdocadexp(-1) -1 -0.2915 0.2237 -1.30 0.1935

lrojfdocadexp(5) -1 0.2714 0.2223 1.22 0.2235


54


Standard Approx


lrojfdocadexp(0) 0.001587 0.000405 3.92 0.0001

lrojfdocadexp(1) 0.002538 0.000648 3.92 0.0001

lrojfdocadexp(2) 0.002856 0.000729 3.92 0.0001

lrojfdocadexp(3) 0.002538 0.000648 3.92 0.0001

lrojfdocadexp(4) 0.001587 0.000405 3.92 0.0001



SSE 0.0342103 DFE 142

MSE 0.0002409 Root MSE 0.01552

SBC -805.48903 AIC -864.03572

MAE 0.01094611 AICC -858.64565

MAPE 0.79770193 Regress R-Square 0.9381 MAPE 0.79770193 Regress R-Square 0.9381


55

Standard Approx


Intercept 1 1.6988 1.6567 1.03 0.3069

lrallojprice 1 -0.8064 0.0729 -11.06 <.0001

lrallgfjprice 1 0.3772 0.1020 3.70 0.0003

lrpcdpi 1 -0.2790 0.1743 -1.60 0.1116

SC0996 1 0.0374 0.0153 2.45 0.0155

lrojfdocadexp**0 1 0.004243 0.001419 2.99 0.0033

lrojfdocadexp**1 1 -3.87E-19 0 -Infty <.0001

lrojfdocadexp**2 1 -0.001014 0.000339 -2.99 0.0033

m1 1 0.0436 0.005120 8.51 <.0001 m1 1 0.0436 0.005120 8.51 <.0001

m2 1 -0.1005 0.005357 -18.75 <.0001

m3 1 -0.0156 0.006048 -2.59 0.0107

m4 1 -0.0883 0.006595 -13.39 <.0001

m5 1 -0.0828 0.006732 -12.30 <.0001

m6 1 -0.1336 0.006695 -19.95 <.0001

m7 1 -0.1098 0.006568 -16.72 <.0001

m8 1 -0.0878 0.006324 -13.88 <.0001

m9 1 -0.0908 0.005933 -15.30 <.0001

m10 1 -0.0478 0.005339 -8.96 <.0001

m11 1 -0.0635 0.004660 -13.62 <.0001

AR1 1 -0.6472 0.0799 -8.10 <.0001

AR2 1 -0.3202 0.0802 -3.99 0.0001

56


Standard Approx


lrojfdocadexp(-1) -1 -0.0282 0.1143 -0.25 0.8059

lrojfdocadexp(5) -1 0.0405 0.1140 0.36 0.7236

Estimate of Lag Distribution Estimate of Lag Distribution

Standard Approx


lrojfdocadexp(0) 0.001355 0.000453 2.99 0.0033

lrojfdocadexp(1) 0.002169 0.000725 2.99 0.0033

lrojfdocadexp(2) 0.002440 0.000816 2.99 0.0033

lrojfdocadexp(3) 0.002169 0.000725 2.99 0.0033

lrojfdocadexp(4) 0.001355 0.000453 2.99 0.0033

57

Sum = 0.009488 LR advertising elasticity


The Partial Adjustment

Model

uQYPQ

Q)1(YaPaaQ

YaPaaQ

QQ)QQ)(/1(

10)QQ(QQQ

t1t3t2t10t

t1tt2t10t

tt2t10*t

*t1t1tt

1t*tt1tt

+β+β+β+β=γε+γ−+γ+γ+γ=

ε+++=

→γ

=+−γ

<γ<−γ=∆=−

−

−

−−

−−

Coefficient or Elasticity of Adjustment

Partial Adjustment

Model – Nerlove (1958)

Q

Q

a

1 Sinceˆ1ˆ

t

*t

33

⇒

⇒

β

γ−=ββ−=γ

Coefficients LR CoefficientsCoefficients SR Coefficients

Quantity demanded in Long-Run Equilibrium

(Desired Level)Current Quantity Demanded

59

γ Represents the Proportion of Adjustment Made Toward Long-Run Equilibrium in Various Time

Periods

)1(2

1

Adjustment of ProportionPeriod Time

γ−γγ

1s

3

2

)1(S

::

)1(4

)1(3

)1(2

−γ−γ

γ−γγ−γγ−γ

60


SAMPLE PROBLEM:

Partial Adjustment Model

*************************************************************************GLS procedure to run a partial adjustment model **************************************************************************;

* no correction for serial correlation is necessary from the Durbin h-test; Durbin h-test; proc autoreg data=Orange_Juice_FDOC_Data;

model lallojgalpc=lrallojprice lrallgfjprice lrpcdpi sc0996 lrojfdocadexp m1-m11

laglojgalpc / lagdep=laglojgalpc;

62

The AUTOREG Procedure



SSE 0.05803079 DFE 146

MSE 0.0003975 Root MSE 0.01994

SBC -746.04083 AIC -801.83843

MAE 0.01326703 AICC -797.12119

63

SBC -746.04083 AIC -801.83843

MAE 0.01326703 AICC -797.12119


Total R-Square 0.9219

Miscellaneous Statistics

Statistic Value Prob Label

Durbin h 1.4128 0.0789 Pr > h

NOTE: No serial correlation

Standard Approx


Intercept 1 1.1001 0.6328 1.74 0.0842

lrallojprice 1 -0.1350 0.0274 -4.93 <.0001

lrallgfjprice 1 0.1357 0.0380 3.57 0.0005

lrpcdpi 1 -0.1547 0.0680 -2.28 0.0243

SC0996 1 0.0214 0.006909 3.10 0.0023

lrojfdocadexp 1 0.002706 0.001483 1.82 0.0701

m1 1 -0.0111 0.008701 -1.28 0.2034

m2 1 -0.1793 0.009578 -18.72 <.0001

m3 1 0.007710 0.007833 0.98 0.3266

64

m3 1 0.007710 0.007833 0.98 0.3266

m4 1 -0.1312 0.008225 -15.95 <.0001

m5 1 -0.0736 0.007787 -9.45 <.0001

m6 1 -0.1262 0.007804 -16.17 <.0001

m7 1 -0.0637 0.008670 -7.35 <.0001

m8 1 -0.0569 0.008215 -6.92 <.0001

m9 1 -0.0730 0.007929 -9.21 <.0001

m10 1 -0.0309 0.007994 -3.86 0.0002

m11 1 -0.0777 0.007947 -9.77 <.0001

laglojgalpc 1 0.7159 0.0497 14.40 <.0001

2841.07159.01ˆ =−=γ

Information Gleaned from the Estimation of the Partial Adjustment Model

SR Elasticity LR ElasticityOrange Juice -0.1350 -0.4752

Grapefruit Juice 0.1357 0.4776

65

2841.07159.01ˆ =−=γ

Grapefruit Juice 0.1357 0.4776

Advertising 0.002706 0.0095248

Proportion of Adjustment to LR Equilibrium in

Various Time PeriodsTime Period Proportion of Adjustment

1 0.2841

2 0.2034

3 0.1456

4 0.1042

5 0.0746

The length of time to adjust to LR

66

5 0.0746

6 0.0534

7 0.0382

8 0.0274

9 0.0196

10 0.0140

11 0.0100

12 0.0072

sum 0.9819

adjust to LR equilibrium is roughly 1 year (12 months). Roughly 98% of the adjustment to LR equilibrium takes place after one year.

Documents

Chapter 11 · 2010. 8. 4. · Chapter 11 Incorporating Dynamics Through the Use of Distributed Lags. Section 11.1 Introduction. ... RPC(a0) 0.003645 Object 1.916E-6 Trace(S) 0.000247