Harmonic alternatives to the Almon polynomial technique

Journal of Econometrics 7 (1978) 57-66. 0 North-Holland Publishing Company

HARMONIC ALTERNATIVES TO THE ALMON POLYNOMIAL TECHNIQUE*

Susan S. HAMLEN? and William A. HAMLEN, Jr.s

School of Management, State University of New York at Buffalo, NY 14214, USA

Received September 1976, final version received May 1977

The Almon technique for estimating distributed lag coefficients uses polynomial approximations to the true but unknown distributed lag function. The technique is used primarily to increase the efficiency of the estimators over OLS estimators. This paper proposes the use of harmonic approximations which is at least as reasonable a priori as approximations by polynomials. This method can be used as an alternative to the Almon method in certain cases and as a convenient check against misspecification problems.

1. Introduction

In this paper an alternative to the Almon distributed lag technique is proposed. The alternative is justified on basically the same grounds as the Almon polynomial technique and can be expected to yield more efficient estimates of the distributed lag coefficients in some cases. The technique can also be used as a

convenient check on the Almon method. Almon assumed that the true distributed lag function was a polynomial

function and then used Lagrangian interpolation polynomials to approximate the true function.’ The Weierstrass theorem is usually called upon to ensure that the approximating polynomial can be used to represent the true function. It states that a function continuous in a closed interval can be approximated by some polynomial of sufficient degree which differs from the function by less than any given positive quantity at every point of the interval. If the assumptions of the theorem are satisfied the extraneous information on the coefficients which

*Revised version of Working Paper No. 243, School of Management Working Paper Series, January 1976. The authors would like to thank John Boot for assistance in a particular aspect of the problem, and Negesh Revankar for supplying the capital appropriation-expenditures data. One of the authors was working under a Baldy Summer Fellowship, SUNYAB, when this paper was completed.

TAssistant Professor of Finance and Operations Analysis. lAssistant Professor of Environmental Analysis and Policy. ‘Johnston (1972, p. 297) recommends the simple power series for computational convenience.

For empirical comparisons between the Almon technique and the proposed alternative in the third section of this paper, this recommendation is followed.

58 S. Hamlen and W. Hamlen, Jr., Alternatives to Almon method

results by using the Almon technique represents a problem of restricted least squares [Goldberger (1964, p. 256)]. In this case the Almon technique will provide more efficient estimators than unconstrained ordinary least squares (OLS) on the lagged variables [Cargill-Meyer (1974, p. 1033)]. However, if the assumptions of the Weierstrass theorem are not met, use of the Almon technique may represent a misspecification of the problem and result in either biased estimators or estimators with higher generalized variance than OLS. The degree of the approximating polynomial is usually limited by the number of lags that are reasonable for a particular model and by the tendency for roundoff errors to occur in polynomials of a large degree. This, in turn, greatly reduces the value of the Weierstrass theorem in justifying the use of the Almon technique since a polynomial of ‘sufficient degree’ may not be feasible. A more practical approach is to examine the convergence properties of various polynomials in approximating normal functions and attempt to select those with the most rapid convergence properties and smallest approximation remainders.

The convergence properties of approximating functions are properly studied under the topic of ‘numerical analysis’. Approximation by rational functions and trigonometric functions is a standard component of this subject and represents a logical alternative to polynomial approximation. The use of any of the above three approximation functions requires that a criterion of approximation be chosen.’ The Almon technique uses the least squares criterion with a polynomial as an approximating function. The method proposed in this paper uses the least squares criterion with trigonometric approximations. Trigonometric sums, either Fourier series in the continuous case or harmonics in the present discrete case, have some of the desirable properties of polynomial functions such as being easy to compute and representing rapidly convergent series.3 Use of the trigonometric sums as approximating functions could be justified in a manner analogous to the polynomials through the Weierstrass theorem since any function which is given at every point in the interval can be represented by an infinite Fourier series [Thomas (1960, p. 821)]. However, the rapidly convergent property of trigonometric sums is a more realistic justification for their use. In addition, trigonometric functions possess orthogonalityand periodicity properties which are lacking in most polynomial functions.

Approximation of a function with the use of trigonometric sums is equivalent to decomposing the function into cyclical components which may or may not

‘Other criteria include collocations (functions agree at selected points), osculations (collocations for which derivatives also agree at selected points), minimax methods (minimizing the maximum difference of any two points) and the equal integral criteria. The Taylor theorem is a well-known example of an osculation using a simple power series as an approximating function. Fourier series is an example of an equal integral criterion with trigonometric sums as approximating functions.

‘Morganstem (1976, p. 814) believes that use of Fourier analysis to analyze economic data was unduly neglected until the acceptance of spectrum analysis techniques. The present authors believe this neglect in areas other than spectrum analysis is still prevalent.

S. Hamlen and W. Hamlen, Jr., Alternatives to Almon method 59

have causal explanations. This periodicity property may be used as a basis for selecting those cases where the harmonic approximations will yield more efficient distributed lag coefficients than either the Almon polynomials or OLS coefficient. Aggregate economic data frequently consists of data from various sectors which exhibit different lag structures. Aggregation of this data tends to yield a lag structure which hides those individual differences. The trigonometric approximation allows these differences to be embodied in the final approximating function. Of course, this approach faces the same potential problem of misspecification as the Almon polynomial method. Even in situations where there is no a priori reason to assume that there are cyclical components within the aggregate data the harmonic approximation method can be used as a convenient check on the results from the Almon method. If use of either or both methods represents a misspecification problem for a specific set of data there is no reason to believe that the resulting distributed lag coefficients would be equivalent for both methods. However, since both are capable of approximating any function with a reasonable degree of accuracy when the data truly requires a restricted least squares approach, the two methods would not be expected to yield vastly different distributed lag coefficients. Thus a convenient check on either method would be to compute the results of the other. Fortunately, both methods have similar, and fairly easy, computations.

While the orthogonality property of trigonometric sums does not aid the researcher in choosing between the polynomial and harmonic approximating functions, one subsequent attribute of the property will be noted. It can be shown [Hamlen-Hamlen (1976)] that the harmonic method eliminates any multicollinearity effect when OLS is applied to the transformed variables. On the other hand, this multicollinearity effect is found to be quite high in many problems using the Almon polynomial technique. It is not clear at this point what con- sequences, if any, the presence of multicollinearity in the transformed variables might have on the reliability of the final distributed lag coefficients. The answer to this question would best be sought through a comparative simulation approach similar to that used in the Cargill-Meyer (1974) study.

An attribute of the use of trigonometric sums for the approximating function is that a potential problem of large roundoff errors is mitigated relative to the use of polynomials. When a polynomial of large degree is used in the Almon method the new transformed variables have an extremely large range between the smallest and the largest variables. When OLS is applied to these transformed variables a well known potential for roundoff errors is present. Alter- natively, the use of trigonometric sums with a large number of harmonics does not exhibit this characteristic since the sine and cosine terms are fractions.

In the second section some of the analytical details of the above considerations are presented. In the third section the results of the Almon technique with a power series (polynomial) are compared to the proposed harmonic approach using two models taken from the literature.


2. Analytical considerations

The basic distributed lag model with lag m is given by

At> = PO + f B(Mt - 4 + u(t), t = 1, . . ..n. r=O

(1)

or in matrix notation:

y = Xfi+u, E(u) = 0, E(zfz4’) = &. (2)

The unconstrained ordinary least squares (OLS) estimators are given by

/I = (X’X)_‘X’y. (3)

As stated in the previous section the Almon distributed lag estimators are used to improve the efficiency of the estimators relative to those in (3). The Almon estimators can be obtained by forming the distributed lag power function of degree r ($ m),

B(z) ‘x k$Offkr*. Substituting (4) into (1) yields the transformed function

~(0 = PO +k$o akZkW+ f&>, t= l,...,n,

where in (5) each Zk(t) is defined by

z,(t) = F zkX(t-T), k=O,l,..., r. t=O

(4)

(5)

Alternatively, using harmonic functions, (4) would be replaced with

where B is a constant equivalent to the mean of the series p(O), b(l), . . . , j?(m), and m is the number of lags. S is the number of harmonics (sine and cosine terms in pairs) for S I m.

Substituting (7) into (1) yields a transformed function similar to (5),

J(t) = PO + BZO + f AkZk + 2 Bjzs+ j + U(t), t = 1, . . ..n. (8) k=l j=l

S. Hamlen and W. Hamlen, Jr., Alternatives to AImon method

In (8) the Zk’s are defined by

61

z, = t X(t-T), TZO

(94

Z, = $J X(t--z)sin r=O

(-$k+

Zs+j = f X(t-T)COS 7=0 ( > $+j*T .

Pb)

(9c)

In addition, the orthogonality property of harmonics is given by the following relationships :

fsin($*j-r)sin(s*k*r) = 0 if j # k, j,k s S, r=O

= (m+ 1)/2 if j = k # 0,

j, k <= S, (104

5 cos(--$*k*r)cos(--$*j*r)=O if j#k, j5S, k>S, I=0

= (m+ 1)/Z if j = k # 0

= mfl

The use of OLS to estimate the parameters to written as

d = (Z’Z)_‘Z’y.

The vector of new independent Almon variables, original vector of independent variables, X, by

z= XA,

where for the Almon transformation,

m m2 m3 . . . m’ 1

k > S, (lObI

if j = k = 0, k > S.

either (5) or (7) can be

(11)

Z, can be related to the

(12)

(13)

62 S. Hamlen and W. Hamlen, Jr., Alternatives to Ah-non method

and for the harmonic transformation

-1 I

0 0 . . . 0

/ 1 sin(s.l.1) sin(--$*l*2) . .

A= 1 sin($i.2*l) sin(--$-i.2.2)

1: i : 1 sin (--j$.rn*l) sin (&*m2)

c-(ik ) . . .

-*l*l . . .

Substituting either (13) or (14) into (11) yields

& = (14’X’XA)-1A’X’y.

sin 2n ( > -*l*s

m+l

.

0

( )I 2n: cos -*l*s

m+l

(15)

The distributed lag coefficients for the Almon and harmonic transformations are related to d by

/!? = Ad. (16)

The estimated variance-covariance matrix of the OLS coefficients, &, of (11) is

2, = a2,(A’X’XA)?

Therefore the variance-covariance coefficients is

2~ = A.f A’.

(17)

matrix of the Almon and harmonic

(18)

In order to gain some insight as to the nature of the variance and covariance matrices in (17) and (18) consider the matrix A’X’XA. A typical term in the jth row and kth column of A’X’XA (not k = j = 1) can be written using (13) and (14), respectively,


(--&*r*_j) sin (s*r’*k). (19b)

To the extent that x(t- r) and x($-r’) are highly correlated for any t and independent of z and z’, the terms in (19b) will tend toward zero when z = z’ and k # j. This result makes use of the orthogonality condition (lOa). These cases occur in the off-diagonal elements of A’X’XA. However, by (lOa) the diagonal terms tend to (m+ 1)/2. An equivalent statement can be made con- cerning (19b) with cosine terms and using (lob). This property results in the off-diagonal terms becoming small relative to the diagonal terms in the matrix which, in turn, prevents the small determinant which is associated with the multicollinearity problem. Thus, there is no multicollinearity effect in the OLS problem associated with (15) when (14) is used. Such a result does not occur when (13) is used and (19a) considered. As mentioned above, it is presently unclear what effect if any a high degree of multicollinearity in OLS problems associated with (15) might have the final distributed lag coefficients of (16) on variance*ovariance matrix of (IS).

3. Empirical comparisons

The results of using the Almon polynomial technique and the proposed harmonic alternative were examined for two models from the existing literature. The first model consists of sixty observations on quarterly amounts of capital appropriations and expenditures for all manufacturing industries, 1953-1967. This hypothesis that current expenditures are a distributed lag function of current and past appropriations was the well-known relationship examined by Almon in her original paper. The Almon technique was used here principally to increase the efficiency of the distributed lag coefficients by imposing linear restrictions which yield the expected shape of the distributed lag function. The degree of the polynomial used was set equal to the total number of sine and cosine terms to make the degrees of freedom equal. Both methods gave statistically significant relationships (at the 5 % level) for the OLS problem on the transformed variables using lags of six, eight, twelve and fifteen. The relevant sums of squared residuals were calculated as the difference between the observed values of expenditures y(t) and the estimated expenditures J(t) using the estimated distributed lag coefficients of (16). These were smaller for the polynomial approximation relative to the harmonic approximation for lags six and eight. They were larger for lags of twelve and fifteen4 Fig. 1 shows the

4This is analogous to the common practice of using Taylor (polynomial) series to approximate functions in local regions and Fourier series to approximate functions over wider intervals [Thomas (1960, p. 822)].


distributed lag functions for the case of fifteen lags. It might be noted that the multicollinearity effect [Theil (1971, p. 179)J associated with (15) was compared for the two methods. The ratio of the ‘multicollinearity effect’ [Murphy (1973, p. 379)] to the coefficient of determination, R’, was examined. The ratio for the Almon method was 0.991, indicating a serious degree of multicollinearity. The ratio for the harmonic technique was not significantly different from zero, indicating an absence of multicollinearity. Since the distributed lag functions shown in fig. 1 are nearly equivalent, it might suggest that the multicollinearity problem does not affect the final coefficients of (16).

.2

Harmonic transformation

Polynomial transformation

.5

I

T 1 2 3 4 5 6 7 8 9 10 11 12 -;;- 14 15

Fig. 1. Lag 15 capital expenditures.

The second model comes from the study of Yohe and Karnosky (1969) which tests the well-known Fisher hypothesis that the nominal interest rate is a distributed lag function of the past rates of price changes. The sample consists of monthly data for the period 1952 to 1969. The nominal interest rates are represented by four-to-six month commercial paper and by the yield to maturity on Aaa-rated corporate bonds. The rate of change in the overall consumer price index is used to measure price changes. Lags of 24, 36 and 48 months were tested for both long- and short-term interest rates.

In fig. 2 the two resulting distributed lag functions are given for a lag of 48 and long-term interest rates. Except for the end points, it can be seen that they are roughly equivalent. The computed sums of squared residuals

were slightly less for the harmonic method than for the Almon method but in no case was the difference significant. Again the fact that there was a high


degree of multicollinearity in the problem associated with (15) and using the Almon technique but none in the harmonic problem did not seem to effect the final distributed lag coefficients.

As noted in the previous section, there is no reason to believe that if the use of either or both of these techniques represented a misspecification problem the resulting distributed lag functions would be so nearly equivalent.

Polynomid transformation

I T

8 16 24 32 40 48

Fig. 2. Lag 48 long-termiates.

4. Conclusion

The purpose of this paper has been to introduce an alternative to the Almon distributed lag technique. The alternative is obtained by replacing the polynomial approximation of the unknown distributed lag function by a harmonic approximation. The authors offer no definite theoretical basis upon which to decide, a priori, when the polynomial or the harmonics of the same degree should be used to maximize the efficiency of the estimators and minimize the risk of a misspecification error. If the relationship to be studied can be expected to have periodicity properties or distributed lag functions which cover a large interval with relatively few observations, then the choice of harmonics over polynomials would be consistent with methods in numerical analysis. 5

‘The authors have applied the technique to the problem of predicting incoming solar radiation, which has definite periodicity properties [Hamlen-Hamlen (forthcoming)].

References

Almon, S., 1965, The distributed lag between capital appropriations and expenditures, Econometrica 33, 178-196.

Cargill, T.F. and R.A. Meyer, 1974, Some time and frequency domain distributed lag estimators: A comparative Monte Carlo Study, Econometrica 42, 1031-1047.


Dhrymes, P.J., 1971, Distributed lags (Holden-Day, San Francisco, CA). Goldberger, A.S., 1964, Economic theory (Wiley, New York). Hamlen, S. and W. Hamlen, forthcoming, A distributed lag model of solar radiation, Solar

Energy. Hamlen, S. and W. Hamlen, 1976, Harmonic substitution in the Almon polynomial technique,

School of Management Working Pauer no. 243 (State University of New York at Buffalo, NY). -

- _

Johnston, J., 1972, Econometric methods, 2nd ed. (McGraw-Hill, New York). Morganstern, Oscar, 1976, The collaboration between Oscar Morganstem and John Von

Neumann on the theory of games, Journal of Economic Literature XIV, no. 3,805-816. Murphy, J.L., 1973, Introductory econometrics (Irwin, Homewood, IL). Theil, H., 1971, Principles of econometrics (Wiley, New York). Thomas, G.B., 1960, Calculus and analytic geometry (Addison-Wesley, Reading, MA). Yohe, W.P. and D.S. Karnosky, 1969, Interest rates and price level changes, 1952-69, Review

of Federal Reserve Bank of St. Louis 51. 18-38.

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Harmonic alternatives to the Almon polynomial technique