where

= 2 (« + a,/ + a2/)x, , + ,

1 = 0

= aoZo, + a,z„ + aiZi, + ,

Zo, = 2 X,-, Zl, = Yj iX,i Z21 = Yj X, ,

(10.27)

Thus we regress y, on the constructed variables Zo„ z,„ and Zi,- After we get the estimates of the as we use equation (10.26) to get estimates of the p,.

Following the suggestion of Almon, often some "endpoint constraints" are used. For instance, if we use the constraints p , = 0 and p+i = 0 in (10.26), we have the following two linear relationships between the as [substituting i = - 1 and / = A: + I in (10.26)].

ao - a, + aj = 0 and ao + ,( + 1) + a2(A: + 1) = 0 (10.28)

These give the conditions

«0 = -«2( + 1) and a, -aik (10.29)

S. Almon, "The Distributed Lag Between Capital Appropriations and Net Expenditures," Econometrica. January 1965, pp. 178-196.

In addition to the geometric lag distribution, there are some alternative forms of lag distributions that have been suggested in the literature. We will discuss these with reference to:

1. The finite lag distribution (10.7).

2. The infinite lag distribution (10.8).

Finite Lags: The Polynomial Lag

Consider the estimation of the equation

y, = PoX, + P,x, , + • • + , + , (10.25)

The problem with the estimation of this equation is that because of the high correlations between x, and its lagged values (multicollinearity), we do not get reliable estimates of the parameters p,. As we discussed in Section 10.3, Irving Fisher assumed the p, to be decHning arithmetically. Almon generaUzed this to the case where the p, follow a polynomial of degree r in /. This is known as the Almon lag or the polynomial lag. We denote this by PDF (k, r), where PDF denotes polynomial distributed lag, is the length of the lag, and r the degree of the polynomial. For instance, if r = 2, we write

P, = ao + a,/ + ajJ (10.26)

Substituting this in (10.25), we get

Figure 10.2. Polynomial distributed lag.

Thus we can simplify (10.27) to

y, = ttjZ, + u,

where

= 2 (P - ki-k - l)x,„,.

We get an estimate of a, by regressing y, on z, and we get estimates of ao and a, from (10.29). Using these we get estimates of p, from (10.26).

Figure 10.2 shows a polynomial distributed lag. The curve shown is that where p, is a quadratic function of /. A quadratic function can take many shapes and it has been argued that the imposition of the endpoint restrictions (10.29) is often responsible for the "plausible" shapes of the lag distribution fitted by the Almon method." Instead of imposing endpoint constraints a priori, one can actually test them because once equation (10.27) has been estimated, tests of the hypotheses like (10.28) are standard tests of linear hypotheses (discussed in Section 4.8).

Many of the problems related to polynomial lags can be analyzed in terms of the number of restrictions that it imposes on p, in (10.25). For instance, with a quadratic polynomial we estimate three as, whereas we have {k + \) ps in (10.27). Thus there areA:+l-3 = A:-2 restrictions on the Ps. With an rth-degree polynomial, we have (k - r) restrictions.

Suppose that we fit a quadratic polynomial for lag length and lag length {k + 1). The residual sum of squares may increase or decrease."* The reason is

"P. Schmidt and R. N. Waud, "The Almon Lag Technique and the Monetary Versus Fiscal Policy Debate," Journal of the American Statistical Association, March 1973, pp. 11-19. "This is demonstrated with an empirical example in J. J. Thomas, "Some Problems in the Use of Almons Technique in the Estimation of Distributed Lags," Empirical Economics, Vol. 2, 1977, pp. 175-193.

Figure 10.3. Long-tailed lag distribution.

that linear restrictions are being imposed on two different parameter sets: (p,, p,, . . . , p,) and (p„ , . . . , , +,).

Apart from the problem of endpoint restrictions, there are three other problems with polynomial lags:

1. Problems of long-tailed distributions. It is difficult to capture long-tailed lag distributions like the one shown in Figure 10.3. This problem can be solved by using a piecewise polynomial. Another procedure is to have a polynomial for the initial p, and a Koyck or geometric lag for the latter part.

2. Problem of choosing the lag length k. Schmidt and Waud suggest choosing on the basis of maximum R: Frost" did a simulation experiment using this criterion and found that a substantial upward bias in the lag length occurs. As we discussed in Chapter 4, the R criterion implies that a regressor is retained if the F-ratio is > 1. The bias that Frost suggests can be corrected by using F-ratios greater than 1, says F = 2.

3. Problem of choosing r, the degree of the polynomial. If the lag length is correctly specified, then choosing the degree of the polynomial r is straightforward. What we do is start with a sufficiently high-degree polynomial as in (10.27). Construct Zf,„ Z\„ Zi,, z,, ... as defined in (10.27) and start dropping the higher terms sequentially. Note that the proper way of testing is to start with the highest-degree polynomial possible and then go backward, until one of the hypotheses is rejected. This sequential test was suggested by Anderson,* who showed that the resulting se-

"P. A. Frost, "Some Properties of the Almon Lag Technique When One Searches for Degree of Polynomial and Lag," Journal of ttie American Statistical Association, Vol. 70, 1975, pp. 606-612.

"T. W. Anderson, "The Choice of the Degree of a Polynomial Regression as a Multiple Decision Problem," Annals of Matliematical Statistics. Vol. 33, No. I, 1966, pp. 255-265. This test is also discussed in T. W. Anderson, Statistical Analysis of Time Series (New York: Wiley, 1971), pp. 34-t3.