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.
Time Series – Almon Model for Distributed Time Series Analysis
referencestime series
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I would recommed the following books:
- Time Series Analysis and Its Applications: With R Examples, Third Edition, by Robert H. Shumway and David S. Stoffer, Springer Verlag.
- Time Series Analysis and Forecasting by Example, 1st Edition, by Søren Bisgaard and Murat Kulahci, John Wiley & Sons.
I hope it helps you. Best of luck!
As far as I understand, by irregular time series you mean unevenly spaced time series, also referred to as irregularly sampled time series. Since I am curious about time series in general, I have performed a brief research on the topic of your (and now mine) interest. The results follow.
Despite high popularity of dynamic time warping (DTW) approach in time series analysis, clustering and classification, irregular time series present some challenges to direct application of DTW to such data type (for example, see this paper and this paper). Based on my relatively brief research efforts, it is not totally clear to me, whether it is impossible to apply DTW directly, as some research suggests otherwise (also see this paper/chapter). For more comprehensiveness, I also would like to mention an IMHO excellent and relevant to the topic dissertation on irregular time series.
Nevertheless, it seems that this topic is mostly covered by the following two research streams:
- proposing and evaluating approaches, alternative to DTW, such as model-based ones (see this paper and this paper);
- proposing and evaluating modified DTW approaches, such as cDTW, EDR, ERP, TWED, envelope transforms, CDTW (continuous DTW - do not confuse with cDTW - constrained DTW!) and others variants (for example, see this paper). An overview of the above-mentioned approaches and results of some empirical comparisons can be found in this paper.
Finally, I would like to touch on the subject of open source software, available for research or system implementation, focused on DTW and supporting some of the above-mentioned algorithms for irregular time series. Such software include Python/NumPy-based cDTW module project as well as GPU-focused CUDA-based CUDA-DTW project. For R enthusiasts, a comprehensive Dynamic Time Warp project also should be mentioned (corresponding package dtw is available on CRAN). Even though it might not support many DTW algorithms for irregular time series at the moment (though I think it supports cDTW), I think it is just a matter of time until this project will offer more comprehensive support for DTW algorithms, focused on such type of data. I hope that you have enjoyed reading my answer as much as I have enjoyed researching the topic and writing this post.
Best Answer
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:
$Y_t = \alpha + \beta c_0 X^*_{0,t} + \beta c_1 X^*_{1,t} + \beta c_2 X^*_{2,t} + \beta c_3 X^*_{3,t} + e_t$
where
$X^*_{0,t} = X_t + X_{t-1} + X_{t-2} + X_{t-3} + X_{t-4}$
$X^*_{1,t} = X_{t-1} + 2 X_{t-2} + 3 X_{t-3} + 4 X_{t-4}$
$X^*_{2,t} = X_{t-1} + 4X_{t-2} + 9X_{t-3} + 16X_{t-4}$
$X^*_{3,t} = X_{t-1} + 8 X_{t-2} + 27X_{t-3} + 64X_{t-4}$
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.