**3 = (X’X + k R ’R) 1X ’y**
u
**(5.32)**
2
o

**here again k = — ^ ** **is assumed known.**

**If k = 0, then ** **ßg = ßgLg> however, if k ** **we can use a result**
**in Rao (1973, p.33) to get**

**which can be shown to be equivalent to the non-stochastic Almon **
**estimator (5.21).**

**In practice, the explanatory variables is an autocorrelated time **
**series so that columns of ** **X are expected to be highly correlated. **
**Although the straight ridge estimator is particularly useful in breaking **
**multicollinearity, the implied prior of that estimator is based on the **
**exchangeable assumption that all lag weights are equal which may **

**contradict our prior knowledge about lag distributions for most of the **
**economic variables. ** **Nevertheless, it can be regarded as a special case **
**of the Shiller estimator with R = I ** **.. ** **We would therefore expect **
**ßg^ i-n (5*24) or ** **ßg ** **in (5.32) to be a more flexible alternative to**
**the unreliable ** **ß ** **and to the over-restrictive Almon estimator. ** **The**
**remaining problem for using this ridge-type estimator is the selection **
**of a suitable value of k. ** **Of course, one can follow Hoerl and Kennard’s**

**(1970) suggestion in selecting the k where the ridge trace begins to**
**stabilise. ** **Alternatively, Maddala (1977) suggested that one can estimate **
**k ** **iteratively by setting**

**(t) = (X'X + k (t_ 1)R ’R) 1X ’y**

**and ** **(t) ** **is the iteration number.**

~ 2 ^2

**As ** **o ** **is fixed in each iteration, k ^ ^ ** **will be bigger as ** **o^(t)**

**„2** **-2** **"(t)**

**gets smaller, it is not hard to see that ** **o (t) < o (t-1) ** **because ** **R B V ***'*

**B ** **B**

**is being pulled closer and closer to the null vector. ** **This sequence is**

**monotonic decreasing and so the conventional tolerance limit is reached**
**very quickly. ** **This becomes clear when we consider the length of the**

**straight ridge estimator ** **3*** **(X'X + kl) 1X'XB,**

*** t**
**<Kk) = B ’B** **B ’X ’X ( X ’X+kI) 2X ’XB** **(**5**.**34**)**
**2-2**
**n+1 XTyf**
**2**
**i=l (X.+k)**
**d 4> (k)**
**dk**

**n + i **

**x**

2;2
**-2**

**2**

**—**

**^**

**<**

**0**

**i=l (X^+k)**

**(**5

**.**35

**)**

**where****A 's ** **are e-roots of ** **X ’X ** **and ** **y = GB.**
**l**

**Clearly, the length of ** **B* ** **is decreasing monotonically as a**
**function of ** **k. ** **The rate of decrease is faster at small values of ** **k**
**than at large values. ** **The same applies to the length of ** **R B ^ ** **as a**
**function of ** **k. ** **It is therefore not surprising to find that the**

**iterative ridge-type estimator converge quickly to the Almon Estimator**

**under the usual tolerance limit for convergence. ** **On the other hand,**
**if iteration were applied to a generalised ridge estimator**

**where**

**B* = ( X ' X + G K G ’) 1X ,y**

**K = diag{k1 . . . kn + 1 >**

**and iteration begins with ** **k^**

**-2**

**o**

**(**5**.**36**)**

**166.**

then it is possible that

**B * **

will converge to a value different from
zero. [See Hemmerle (1975)]. However, they are not very interesting
estimators in this context because they will give irregular lag distribu
tions. One way to avoid the convergence to the Almon estimator in the
case of iterated Shiller estimator is to truncate the iterative process
at a suitable point.
The Bayesian interpretation of the ridge type estimator has greatly enriched the class of alternative estimators suitable for the finite distributed lag model. We can now turn to some useful extensions of the technique in econometric analysis.

§5.3 Seasonal V a r i a b i l i t y

We consider the problem of estimating a finite distributed lag model with n lags corresponding to each season. Data are collected in s seasons and it is believed that the lag coefficients vary seasonally. The model can be written

y_{t} **2**
**j=l**
n

**2 B**

i=l
D. .X . + e
ij j,t-i t-i t -n, J1 »1 • * *. ) 1T (5.37)
where D . . . D g are seasonal dummy variables of the zero-one type. Under the restriction of a common distributed lag response for all s seasons the model reduces to

y_{t}
n

### 2

i=l 6**ix t-i +**e t (5.38)

*where*

**B**

il i2 is
Within each season, one can follow Shiller to impose a smoothness prior on the elements (3-, ...

**B **

}. Gersovitz and MacKinnon (1978)
**exploited the same idea by allowing for smoothness prior in the seasonal **
**dimension.1 ** **In this section, the problem of estimating (5.37) is **

**formulated in such a way that we can allow for 2-dimensional smoothness **
**priors on the ** **{ ß _ |i = 1 . . . n , j = 1 . . . s } ** **and thus extend previously **
**available results. ** **We also consider estimation under the assumption of **
**exchangeable priors on the ** **{ß^j} ** **either in the seasonal dimension or **
**in the lag dimension or simultaneously in both dimensions. ** **Finally, the **
**problem of estimation under a combination of exchangeable prior in the **
**seasonal dimension and a smoothness prior in the lag dimension is con**
**sidered. ** **The total result is that we obtain a family of distributed lag **
**estimators derived from a common Bayesian approach using the hierarchical **
**prior model developed in Lindley and Smith (1972). ** **The justification **
**for considering "smooth seasonality" is particularly strong when monthly **
**or even weekly data are being analysed.**

**Let us define the following matrices and vectors**

**Y**