# Least stupid way to forecast a short multivariate time series

I need to forecast the following 4 variables for the 29th unit of time. I have roughly 2 years worth of historical data, where 1 and 14 and 27 are all the same period (or time of year). In the end, I am doing a Oaxaca-Blinder style decomposition on $W$, $wd$, $wc$, and $p$.

time    W               wd              wc               p
1       4.920725        4.684342        4.065288        .5962985
2       4.956172        4.73998         4.092179        .6151785
3       4.85532         4.725982        4.002519        .6028712
4       4.754887        4.674568        3.988028        .5943888
5       4.862039        4.758899        4.045568        .5925704
6       5.039032        4.791101        4.071131        .590314
7       4.612594        4.656253        4.136271        .529247
8       4.722339        4.631588        3.994956        .5801989
9       4.679251        4.647347        3.954906        .5832723
10      4.736177        4.679152        3.974465        .5843731
11      4.738954        4.759482        4.037036        .5868722
12      4.571325        4.707446        4.110281        .556147
13      4.883891        4.750031        4.168203        .602057
14      4.652408        4.703114        4.042872        .6059471
15      4.677363        4.744875        4.232081        .5672519
16      4.695732        4.614248        3.998735        .5838578
17      4.633575        4.6025          3.943488        .5914644
18      4.61025         4.67733         4.066427        .548952
19      4.678374        4.741046        4.060458        .5416393
20      4.48309         4.609238        4.000201        .5372143
21      4.477549        4.583907        3.94821         .5515663
22      4.555191        4.627404        3.93675         .5542806
23      4.508585        4.595927        3.881685        .5572687
24      4.467037        4.619762        3.909551        .5645944
25      4.326283        4.544351        3.877583        .5738906
26      4.672741        4.599463        3.953772        .5769604
27      4.53551         4.506167        3.808779        .5831352
28      4.528004        4.622972        3.90481         .5968299


I believe that $W$ can be approximated by $p\cdot wd + (1 - p)\cdot wc$ plus measurement error, but you can see that $W$ always considerably exceeds that quantity because of waste, approximation error, or theft.

Here are my 2 questions.

1. My first thought was to try vector autoregression on these variables with 1 lag and an exogenous time and period variable, but that seems like a bad idea given how little data I have. Are there any time-series methods that (1) perform better in the face of "micro-numerosity" and (2) would be able to exploit the link between the variables?

2. On the other hand, the moduli of the eigenvalues for the VAR are all less than 1, so I don't think I need to worry about non-stationarity (though the Dickey-Fuller test suggest otherwise). The predictions seem mostly in line with projections from a flexible univariate model with a time trend, except for $W$ and $p$, which are lower. The coefficients on the lags seem mostly reasonable, though they are insignificant for the most part. The linear trend coefficient is significant, as are some of the period dummies. Still, are there any theoretical reasons to prefer this simpler approach over the VAR model?

Full disclosure: I asked a similar question on Statalist with no response.

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Hi there, could you give some more context around the decomposition you wish to do, as I have not seen it applied to time series data? – Michelle Mar 15 '12 at 6:38
I am breaking up the change into components in the following way: $W^{′}-W=p^{′}∗(w^{′}_{D}-w_{D})+(1-p^{′})∗(w^{′}_{C}-w_{C})+(w_{D}-w_{C})∗(p^{′‌​}-p)+(\epsilon^{′}-\epsilon)$, where primes denote current value of the variables. – Dimitriy V. Masterov Mar 15 '12 at 13:09
hmmm, how about exclude the outliers first, before regression? – athos Aug 31 '13 at 14:37
What level of precision do you require? I'm asking because as you know, you can use ARIMA models and get a very low MSE. However, since those models are usually fit using maximum likelihood, it is almost certain that you will overfit. Bayesian models are robust when dealing with little data, but I think you will get a MSE an order of magnitude higher than in ARIMA models. – Robert Smith Sep 29 '14 at 4:56