What model should one use for this short time series?

Below I have quarterly total sales on the left (dependent variable), and a sample of the sales on the right. The two variables share a correlation of 98.7%. What model should I use to predict X? For that model, should I include a constant? Seasonal adjustments? Remove outliers? The most important criteria is minimizing out of sample prediction error.

Q3'10   40.19   0.2386
Q4'10   39.36   0.2000
Q1'11   51.25   0.2173
Q2'11   54.99   0.2630
Q3'11   50.38   0.2242
Q4'11   50.77   0.2623
Q1'12   67.39   0.3548
Q2'12   77.14   0.3716
Q3'12   72.54   0.3451
Q4'12   80.21   0.3816
Q1'13   94.57   0.4661
Q2'13   102.13  0.4919
Q3'13E  X       0.4424

• Is this all you have? If you know total sales, what is the interest in a sample? Is the problem knowing how good the sample might be for future use? (By the way, I can't reproduce "a correlation of 98.7%". It's pretty much universal practice in mainstream statistics, to report correlations on a scale from -1 to 1. I get 0.979 for the correlation: no question that it's quite good.) – Nick Cox Oct 29 '13 at 1:32
• I do not know total sales for Q3'13E, but we have our sample of sales beforehand of .4424. I am trying to predict Q3'13E sales. – Hype Oct 29 '13 at 1:41
• Why not use the average multiplier for total sales/sample sales? It's not absolutely clear that you need time series forecasting here. If you do, the sample really is very small to fit a model with seasonality too. – Nick Cox Oct 29 '13 at 7:38
• My comment is directed at the specific question here, where (unusually in my experience) the OP already has some information for the time period for which a forecast is desired, which cannot be used by your method. FWIW, the average multiplier total/sample is 207.74 which itself leads to a forecast that is 204.7404 * 0.4424 = 90.58. No question that there is some information in the time series. My larger point is that there is no harm, and much good, in using quite different methods, to produce forecasts that can be checked against each other. I imply nothing about OLS. – Nick Cox Oct 29 '13 at 14:48
• @IrishStat Thanks for the information. Naturally you are the absolute authority on what AUTOBOX can do and I most willingly retract that assertion. I take it that a forecast can be based on both "total" and "sample" in OP's terminology. That's useful. I remain a little queasy about overfitting. Using the variable's own history and a covariate sounds like about 5 parameters (or more) for 12 data points. – Nick Cox Oct 29 '13 at 17:34

You can just use the history of Y or also your suggested causal. I have not seen "sample of sales" before as a causal so I am hesitant to want to use that variable, but I am sure you know what you are doing.

Yes, you should consider the adjustment of outliers. Yes, you should allow for a constant. Yes, you should consider seasonal impacts.

The ACF/PACF doesn't show that the lag of 4 is important so autoregressive seasonality is weak. The data are short so this can be expected. Q4 is flat and then the last year Q4 is high which might due to the short data or a change in the behavior of Q4. Tough to tell.

A possible model (automatically developed using AUTOBOX (http:/www.autobox.com) , a piece of software I have helped develop is providing . There are two seasonal dummies detected consistent 1st and 2nd qtr positive effects.

If one did not use the predictor then a very similar forecast is developed using this equation enter image description here . It is interesting (at least to me !) that the two quarterly negative seasonal pulses ( qtrs 3 and 4 ) are the "reflection" of the two quarterly seasonal pulses developed using the predictor series.

EDITED to respond to Nick's OLS MODEL:

If you take Y and divide it by X to get a new variable called Z and THEN run an OLS model restricting the intercept to be 0., you in fact will obtain . The residuals from this assumed model (as you have wisely said in previous posts it is always a good thing to bring the residuals to your "doctor" for a checkup) have a serious violation/malady at period 1 and clearly evident non-randomness. The whole idea is to avoid entertaining insufficient models and adequately capturing the signal. Clearly, the simple OLS model for Z ignores the very clear need for seasonal/quarterly dummies which are lost in translation when converting Y and X to Z.

• Idle curiosity: How many estimated parameters in this fit? – Nick Cox Oct 29 '13 at 13:24
• @Nick the univariate solution has three paremeters while the causal had 4 . Admittedly this might be a bit much BUT the models seems to reflect the data and the forecasts seem reasonable. – IrishStat Oct 29 '13 at 14:16

Why not use a simple linear regression, in the form

y = ax + b


or

total_sales = a*sample + b


since your correlation is so high?

I would consider using Gaussian Process regression. Carl Rasmussen's excellent book and associated Matlab software are freely available here: http://www.gaussianprocess.org/gpml/. You would probably want to use a mixture of a periodic kernel for the seasonal effect plus a linear kernel since there looks to be a roughly linear growth. If you want to use the "sample of sales" variable as a covariate that is possible too.

• Any idea what such a model would predict for X above? – Hype Oct 29 '13 at 1:45