# How “bad” is it to use an independent variable as part of a proxy dependent variable in forecasting?

I am fitting a logistic regression model in order to forecast the occurrence of an event.

Let the event I am interested in forecasting $Y$ be whether or not someone purchases stock in a given company in 2016. As we clearly cannot pull a sample of data from 2016 since 2016 has not yet happened, I want to create a proxy variable for purchasing stock in 2016.

Let's say that I believe you are likely to purchase stock in 2016 if you have purchased stock in 2015 or 2014. Thus, our proxy variable is $Y'$ where $Y'=1$ if they have purchased stock in 2015 or 2014, and $Y'=0$ if they have not purchased stock in either 2015 or 2014.

A likely important predictor of 2016 stock purchases is stock purchases in 2015 or 2014. Thus, my model would optimally include an independent variable for purchasing stock in 2015 and another independent variable for purchasing stock in 2014: $Y = logit[\beta_0 + \beta_1X_{buy in 2015} + \beta_2X_{buy in 2014} + \cdots]$, where $\cdots$ indicates other independent variables of interest.

Replacing $Y$ directly with the proxy $Y'$ would cause issues due to the correlation between our independent variables and the dependent variable. However, I don't think it's a great idea to remove $X_{buy in 2015}$ and $X_{buy in 2014}$ from the model entirely as they're arguably the best predictors of $Y$.

My questions are:

1) Let's assume that I proceed with the model $Y' = logit[\beta_0 + \beta_1X_{buy in 2015} + \beta_2X_{buy in 2014} + \cdots]$, where $Y'=1$ if they have purchased stock in 2015 or 2014, and $Y'=0$ if they have not purchased stock in either 2015 or 2014. How "bad" is it to disregard the correlation issues? What are the potential side effects?

2) What suggestions would you make to improve prediction of purchasing stock in 2016? Due to reasons that are too lengthy to describe here, I am limited to a couple of binary classification methods - namely logistic regression and random forests - but I'm open to suggestions about different proxy variables, different independent variables, transformations, etc.

Your situation is not unique, it generally covers the vast majority of situations in which predictive modeling is applied. In order for a model to be useful, it must, to some accuracy, predict the future, yet one only has information from the past available to actually construct the model.

How "bad" is it to disregard the correlation issues?

Catastrophic. The model is completely unidentifiable. Fitting such a model will return something like

$$\beta_0 = -5, \beta_1 = 10, \beta_2 = 10$$

where the particulars will depend on the details of the fitting algorithm. The implication

$$Y_{2004} \ \text{or} \ Y_{2005} \Rightarrow Y_{2004 \ \text{or} \ 2005}$$

cannot be expressed as a logit-linear function (at least not with finite coefficients), the fitting algorithm gets as close as it can and then terminates.

Extreme parameter estimates like this in a logistic regression are often a sign that something is wrong. I give extra scrutiny to any parameter estimate in a logistic regression greater in absolute value than $.5$.

However, I don't think it's a great idea to remove Xbuyin2015 and Xbuyin2014 from the model entirely as they're arguably the best predictors of Y.

Indeed they are, because they are $Y$. If the only concern in modeling was to pick the best predictor, you would always chose the response. My guess is this strikes your intuition as absurd.

What suggestions would you make to improve prediction of purchasing stock in 2016?

The general line of reasoning goes like this. If it is the case that purchasing stock in 2014 or 2015 makes one likely to purchase in 2016, then it should also be the case that purchasing in 2013 or 2014 made (in the past tense) one more likely to purchase in 2015. This is an assumption, and one should consider carefully whether it is valid, as shocks in the underlying process can disrupt its potential truth.

If true, the modeler is in the advantageous situation of being able to apply a model built on 2013, 2014, 2015 data to a 2014, 2015, 2016 situation.

A way to put this in your language would be: $Y_{2005}$ is used as a proxy for the quantity you want to know, $Y_{2006}$. The predictors $Y_{2004}$ and $Y_{2003}$ are used as proxies for $Y_{2005}$ and $Y_{2004}$, to be used in the production implementation.

Another way to say it: to predict whether someone purchased stock in a given year we use whether they purchased stock in the previous, and the previous, previous year.

• Thanks for the well-written post, Matthew! The problems are exceedingly clear and your thought process is easy to follow. To sum up, you're suggesting that I remove buying in 2015 and buying in 2014 as predictors, then rely on my remaining features for prediction? – Matt Brems Dec 29 '15 at 12:53
• A follow-up question: suppose we created a proxy variable that takes a weighted average of whether or not you bought stock in 2015, 2014, 2013, 2012, and so on. Would this create a proxy that is "diluted" enough such that we could include each of those years as a predictor or do you think this would exacerbate our problem? – Matt Brems Dec 29 '15 at 13:09

I would look more specifically at the individual characteristics of the companies you're trying to parametrize that aren't directly tied to making the decision to buy and start from there in your regression analysis.

Using past buying data as the main determinants of future demand is exactly what feeds bubbles. Observe what you must, but fall victim to them you should not.

If you're going to run a regression on the expected future price on past price increases (increase in demand, individual or otherwise) then you would only reasonably have variables that denoted when the stock was sold (the price went down).

By this point you're basing the likelihood that there will be a change in the future price of the stock (implicitly through demand for it) on it's history of ups/downs with some error parameter and a correlation problem that's bound to wash out any signal you'd want to use of the regression.