Can anybody give me a good reference on Almon model for time-series analysis. Also, what's the basic difference between Koyck model & Almon model.

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    $\begingroup$ Almon, Shirley (1965), "The distributed lag between capital appropriations and net expenditures," Econometrica 33: 178-196. $\endgroup$ – onestop Mar 3 '12 at 16:24
  • $\begingroup$ Chapter 17 of Gujarati's Basic Econometrics (5th ed) covers both the Koyck model and the Almon model. Examples are provided and the data can be downloaded at the book's website. I recall it being quite easy to replicate the results shown in the book. Definitely check it out. $\endgroup$ – Graeme Walsh May 19 '13 at 1:22

The basic difference between Koyck and Almon is that the former is a geometric lag model and the latter is a polynomial lag model. I dug out an old copy of Pindyck and Rubenfeld (1976) for this, but I'm sure they are outlined in more recent texts.

Koyck transform: Consider the infinite-lag model:

$Y_t = \alpha + \beta_0X_t + \beta_1X_{t-1} + \beta_2X_{t-2} + \dots + e_t$

rewritten as:

$Y_t = \alpha + \beta(X_t + wX_{t-1} + w^2 X_{t-2} + \dots) + e_t = \alpha + \beta\sum_{i=0}^{\infty}w^iX_{t-i} + e_t$

Rearranging terms gives us the Koyck transformed model:

$Y_t = \alpha(1-w) + wY_{t-1} + \beta X_t + e_t - we_{t-1}$

Almon distributed lag: With a polynomial lag, we might assume, for example, that the weights follow a cubic polynomial going back some specified number of periods, then become equal to zero:

$w_i = c_0 + c_1i + c_2i^2 + c_3i^3$

with $w_i = 0$ for $i>5$, let us say for the sake of an example. Lots of rearranging of terms, which I'm sure you've seen in the Almon paper, allows us to construct new variables and estimate a linear regression:

$Y_t = \alpha + \beta c_0 X^*_{0,t} + \beta c_1 X^*_{1,t} + \beta c_2 X^*_{2,t} + \beta c_3 X^*_{3,t} + e_t$


$X^*_{0,t} = X_t + X_{t-1} + X_{t-2} + X_{t-3} + X_{t-4}$

$X^*_{1,t} = X_{t-1} + 2 X_{t-2} + 3 X_{t-3} + 4 X_{t-4}$

$X^*_{2,t} = X_{t-1} + 4X_{t-2} + 9X_{t-3} + 16X_{t-4}$

$X^*_{3,t} = X_{t-1} + 8 X_{t-2} + 27X_{t-3} + 64X_{t-4}$

More advanced formulations can force the coefficients to decline to zero ("endpoint restrictions") at some point; with the above formulation, it's possible (although not likely if you're sort-of close to correct with your specification) that you would have your coefficients increase sharply, then suddenly go to zero after the maximum lag.

The Almon transform is clearly far more flexible, but some models, e.g., adaptive expectations models, can give rise to the geometric lag model in a direct and clearcut manner.

When you've got a lot of data, you can probably do better than the Almon model, as IrishStat suggests. However, if you don't have a lot of data, say you're stuck with 50 or 100 data points, the Almon model can be quite helpful. (Simulation of plausible models may help you decide on a specific approach). There's a bias-variance tradeoff at work here; more flexible models may reduce bias due to model misspecification, but increase variability by having to estimate more terms (and how many more terms), and make your model, overall, worse.

  • $\begingroup$ The problem of having too many parameters can often be abated by stepdown modelling and/or efficient anomaly detection. In my experience 50-100 is ordinarily quite sufficient to support model identification. $\endgroup$ – IrishStat Mar 3 '12 at 18:29
  • $\begingroup$ Thanks a lot jbowman! Your answer was really helpful. I took time to reply back as it took me sometime to understand your argument. $\endgroup$ – Beta Mar 11 '12 at 8:45

@Ari These models are anachronistic as they are presumptive of model form thus leading to probable Model Specification Bias . You might consider allowing your data to help you identify the appropriate Polynomial Distributed Lag structure via robust Transfer Function Identification.

  • $\begingroup$ Thanks IrishStat.I'll keep your argument in mind while working with this kind of analysis. $\endgroup$ – Beta Mar 11 '12 at 8:46

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