Stochastic simulation study of martin' distributed lag model

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Stochastic simulation study of martin' distributed lagmodelBarbara L. Azalost aa Department of Statistics , Virginia Polytechnic institute and State University , Virginia,24061, U.S.A.Published online: 20 Mar 2007.

To cite this article: Barbara L. Azalost (1973) Stochastic simulation study of martin' distributed lag model, Journal ofStatistical Computation and Simulation, 2:1, 55-64, DOI: 10.1080/00949657308810034

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Department of Statistics, Virginia Polytechnic Institute and S b t e University+ Eiacksburg, Virg~nia 24067, u.S.A .

(Received January 25, 1972)

Iii l963, James E. hlai-tin presented the derivation of the most current of the distributed lag models. Martin's development was based first on a study made by Fuller and himself (1961) questioning the assumption of independent errors used in the conventional Nerlove or Koyck models m d second on the possibility that price and income are not identically lagged. The mode! developed by Martin contains two lag parameters and a first order auto- regressive error.

Mx?io's fo.mu!atim invo!ved ac extemi~n of the coiiveotioiid model wch that he considered the demand equation in which current consumption is a - . iuncllon sf current, as as past, prices for a set of n _ cGmmodlties, current. am! all past incomes, and an error.

t This is an extension of the Masters thesis in Statistics presented by Barbara L. Azalos at Virginia Polytechnic Institute and State University, August, 1970. Research was under the direction of Dr. Richard G. Krutchkoff and was supported by an NDEA feilowshp.

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- lhe above equation is noniinear in the originai parameters (ai, i = 0,. . .,

n, ,lp ,p). nerefore, t h e application of least sauares to Eq. I.?, does not inp2r.z ucique estimates.

Serious statistical problems are introduced if the errors a r t correlated in equa- tions containing lagged values of the dependent variable. Hurwicz (1950) has shown for small samples, and Griliches (1961) has shown for large samples, t h ~ t the estim~tion of the coefficient using least squares will be biased if the errors are correlated. Unfortunately, the usual test of hypothesis of inde-

- . . --- . ... , -. EeIi&nr rhe Durom-wais~n test, zpp&es lo re5Tesuion models in

which the independent variables can be regarded as fixed. Consequently, the Durbin-Watson test (Durbin and Watsont 1851) does not apply i u auto- regressive schemes nr schemes in which lagged values of the dependen t variable occur as independent variables.

It seems apparent from the form of U, that the error terms of the system of equations are correlated. Autoregressive least squares (A.L.S.) has been suggested as aprocedure to estimate the coefficients of an equation where the errors are assumed to follow the first order autoregressive scheme:

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where fl IS the autocorrelation coefident and e, has the srat~stica! propeztm generally associated with least squares. A.L.S. suppiies a valid asymptotic test ef the hyp=thes:s sf independmi: iEois arid can be used tu esiima~e the phrmeters of nonlinear equations wch as Eq. i.2. Combining Eq. 2.1 w ~ t h Eq i 2 resuiib in the equarion:

Fuiier and ~ a r z i n (1961) suggested the fojiowing :te:at~ve procedure of estimation whch is an application of the modified Gauss-Newton method of estimatio:: (see Hartky, !96 ! ) ,

1. Select a beginning vector of estimates of the u n k m w ~ parameters (@,,, . . ., Oi,j . i t was suggested that this set be seiected by inspecting the least squares solution to Eq. 2.2. Martin has suggested that this set be found by assuming the Nerlove model and using the estimates obtained from it using :he leas: sqa re s grocedure. 7Xs would give estimates of the ai and ;l and a Durbin-Watson statistic can be obtained. He proposed that p be set equal to the value obtained for i and that the beginning estimate for p be derived from the Durbin-Watson statistic- 4 such that ;f d =3+ 4 be set 2? -Q3 and if d = 1, /? b:: set at 0.5 with the intermit:ent va:ues of 8 to be 5ouiid by- inter- poiation. Throughout this procedure it is assumed that the variables are expressed as deviations from the mean, thus eliminating the need to directly estimate the constant term.

2. Expand Eq. 2.2 about the beginning set of estimates of the parameters in a Taylor series expansion retaining only the first order terms.

3. Compute the regression of the resulting dependent variable on the first derivatives of Eq. 2.2 with respect to each unknown parameter to obtain estimates of the A@,,, the deviations from the true parameters.

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4, EXACT MODELS

. ' 7 .

r w v yarlailon~ of i&e mooel were c~nsideied. Model I considered the prices of a given commodity, two substitute goods for the commodity, and income. Theory indicates that the parameter, a,, for the commodity would be negative, while those for the two substitute goods (a, and a,) and income (a,) would be posithe. Model II considered the prices of a given commodity, one substitute good, one complementary good, and income. For this model, theory indicates that a, and a, (the parameter for the complementary good) are negative while a, and a& are positive. It was decided ts keep these four parameters constant at:

-0.9, 0.1, 0.2, 0.3 for Model I

-0.9, 0.4, -0.5, 0.3 for Model I1

throughout and concentrate our efforts on the effects of variations of the values of 1, p, and ,B, the two lag parameters and the parameter for the autoregressive error.

Throughout the tests, t-type confidence intervals were used, i.e. a check was made to see if the true values of the parameters, Qi , were contained in:

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-tZ,: x (the srandard error of P,? A +t, x (the standard error of 8Jj. - . For :he generated data for both models, the stasdard errors far the seven par;irnzters %ere most ;;f;sr x i the i=!!~w:ng ranges:

Paramerer Range

The standard errors were slightly larger only when the errors were generated frool bimodal distributions. - . . prom the above ranges, ;I ;s that the estimates to be qiiite ciose to the parameters in order to lie within L5e desigii&ed confidence interval. In all cases, the number of iterations was iirnited to 25, i.e. if the desired srnal!r,ess for AO, was not reached after 25 iterations, the estimates -C+O- +La + ~ r r a - + . r G A L ; ~ m r g f ; n r t X U P ~ P IICPC; a i r b l ulr Lr.rur,-Allnl ..,,,,,,, .,,,, -,-... m general, as the tme value of .A grew closer to the try value of p, fewer iterations were needed.

The first situation to be tested was the case where the assumptions were held to be true, i.e. the data was generated so that the errors, U,, were first order autoregressive and the e , were normal, independent and identically distributed with a mean of zero. For both models, the a, were held constant at the values stated previously. In model I, A was varied over the ten values, -0.99, -0.7, -0.5, -0.2, -0.1, 0.O5 0:059 0.2, 0.5, and 0.95; while ,u and P were fixed at 0.5. Then, ,u was varied over the same values, excluding 0.5, ...h llG :;.. : cllu -.-2 /j - . - were - iixcu z..-; at ".,. - = Finajjy, the same p;oze&re Te"'.7":: ?:A,? y-a..w... A"'

j?. Each of the resulting 28 sets of parameters was checked for confidences of 997; and 95 %. That is, a total of 39"Lcordidence intervals were iested. For mode! !I, the s2me procedure was used. While each. of R j pi and 0 was varied over the five values, -0.99, -0.3, 0.0, 0.5 and 0.95, the other two were fixed at 0.5. Therefore, 182 intervals were tested.

After reviewing the results of the above simulation procedure, it was de- cided that the procedure was almost always accurate at the 99 % level and we therefore felt justified in making tests of robustness on the mode!. Two general deviations from the assumption were introduced: (1) adding a constant term, u, to the error equation and (2) generating the errors from a

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distribution other than normal. For each model, the fo!lowing five sets of parameters were estiiiiated :

As 2r~t:ousl.$ sra:ed. the values fc3r (a,, i = i j 2; 3, 4) heid soneract fsr model t h~oughc ;~ i the study.

--- 1 he &g tegt of r~bus:i.ess was a d & ~ ~ a cons-;afir ierj-jl, z, LO the err.37 T h e c 3 -- ~ ----

quantity N ~ a s varied we; tbs va!ues - 8, - 6, -4, - 2 , 2, 4, 6 and 8, T'ine second of robus'_ness generating eE9i-s from Sis;nbutions other than a normal distbbution. Four distributions were used: (1) a U-shaped distribiition, (2) Bimodal Oistribution I, (3j Bimodal Distribution 11, and (4) Bimodal Distribution 111. The three bimodal distributions were obtained by combining, with varying weights, two normal distributions, the first having a =em of 2.0 and a ~ariance of i .O, and the second having a mean of -2.0 and a variance of i.6 For Bimodai Distribution I, the weights were both 0.5, for Bimodai Distribution IIj the weights were 0.6 and 0.4, respec- tively, and f ~ r Bimodal Distribution III &e weights were 0.4 and 0.6, re- spectively. These tests of robustness added to each model 840 confidence intervals to be tested. Cunsequentiy, for modei I, we tested 1232 intervals and for model 11, we tested 1022 intervals.

5. RESULTS .AND CONCLUSIONS

The results of the Stochastic Simulation tests indicated that the procedure of e s t i~a t i on propcsed by Martin is indeed almost aiways accurate at the 99 % level. A situation did occur, however, which caused certain sets of parameters

; 3.; "-' 'je& Ti-i;s S;f--i - .uacion arose in model I for the fourth set\of pararr?eters when a = 2, 4, 6 and 8 and for the U-shaped and Bimodai III distributions and in rnociel II for 1 = 0.95 N h e s we varied 2.) and f r ~ r the fourth set of parax te rs wher, c: = 2,4, 6 aiid 8 and for the U-shaped distribution. For the above sets s f parameters, the prxedure did nst 70ik diie to the fact that it produced a negative variance for some of the parameters. Consequently, 84 intervals for each model were invalid. Therefore, only 1148 intervals for model I and 938 for model I1 were actually checked.

The check used was to compare the claimed confidence for each parameter with that which was obtained by simulation. If the actual confidence plus one standard error vas geater than or equal to the claimed confidence, it was

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V ~ N 1 3 10 0 1 (These figures include the situation where I, p, and j? = 005, while :hose for Vary p and Vary ,9 do not.)

Vary u 1 4 1 2 vary jl' 0 3 0 1 g = -9 5 5, 4 8

a = -6 0 4 1 8 a = -4 1 3 0 3

= -2 0 6 1 3 a : = 2 0 1 0 2 2 = 4 0 I 0 2 s( = 6 0 1 0 2 a = 8 0 1 9 2 U-shaped 0 0 0 3 ED I 1 2 0 d < BD I1 I 2 3 2 - - -.. - --- u s ! ;;: .? -- A*- d - 9

Totals 1 < 1 J 50 A ! - 7 52

Obviously, not all situations that may arise in the use of this model were examined. The results, however, do indicate that the 99 % confidence claimed for the procedure of estimation is justified for the majority of data. For the situation which arose with the negative variances, Dr. Martin has suggested that instead of using the Nerlove estimate for the first set of estimates in the iteration process, we vary the first set until the process does work. No attempt

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Acknowledgement

A-n expressicr, of appreciatisn is due io Dr. Xicnard G, Krutchkofi'for his guidance, and to Dr. James E. Marrin for his cooperation and encouragement during this research.

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Fuller, Wayne A. and Martin, James E. (1961), "The Effects of Autocorrelated Errors on the Statistical Zs;iinatio~~ of Distributed Lag Modeis". J. Farm Econ. 43, 71-82.

Fuller, Wayne, A. and Martin, James E. (!962), "A Notc on the EiTects of Autocorrelated Errors on the Statistical Estimation of Iiisrributed Lag ilIodelsn. J. Farm Econ. 44, 407410.

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