# Can an omitted random variable cause "omitted variable bias"?

Suppose we have a linear regression:

Y = mx + b

where X is the independent variable of interest, in this case "scoops of ice cream per order" at an ice cream shop, b is the error term, and Y is the dependent variable of interest, "service time per order" at an ice cream shop. The goal is to find "m", the time it takes to scoop ice cream.

Incoming orders are treated as random, at least for the sake of this argument/question. An order could be for any number of scoops of ice cream. There are also "random" events that happen during the course of an order, for example, like when there is a miscommunication between a guest and a worker about what flavor ice cream they want. When the wrong flavor of ice cream is delivered, the worker will have to take the time to correct the mistake, and this is reflected in the data. There is no correlation between scoops ordered and mistakes like that. However, the mistakes only ever add time to order processing. They never reduce time of order processing.

My question is, it seems that not including the variable "mistakes" will not fit the definition of an omitted variable for the purposes of "omitted variable bias". There is no correlation between scoops and mistakes. However, since I am looking for the mistake-free time it takes to generate one scoop, will my "m" not be shifted upward relative to the theoretical "mistake-free" line? Is there not then some kind of bias here? How could that be corrected?

• It seems that your example demonstrates that it's too restrictive to define "omitted variable bias" in such a way as to require the omitted variable to be correlated with the independent variable of interest. Jul 20, 2017 at 19:01
• You also include an intercept, right? The intercept would capture the "starting costs" of an order, as well as the average time lost due to mistakes Jul 20, 2017 at 20:34

First, your way of writing the equation is unusual. That's not necessarily wrong but it can cause confusion. The usual way of writing a simple regression equation is $$Y = \beta_0 + \beta_1x + e$$ where $$\beta_0$$ is the intercept (and I agree with Ken that this should be in there), x is the IV and e is error.

Second, omitted variable bias does not seem to mean what you think it means. Here I (and Wikipedia, and some other sources) agree with Kodiologist. Wikipedia says

In statistics, omitted-variable bias (OVB) occurs when a statistical model leaves out one or more relevant variables. The bias results in the model attributing the effect of the missing variables to those that were included.

Third, whether omitting mistakes from the model causes bias is not clear.

Fourth, an order cannot be for "any number of scoops". First, it has to be a positive integer, second there are limits of scoops per order (at least, if an order is what one person will eat).

Fifth, I would guess that any kind of ice cream shop would have more than one server (even if not always at one time). So, "server" also needs to be added. Different people will scoop at different rates.

Sixth, there are other omitted variables too. Cone or cup? All one flavor or multiple flavors? Probably some other ones as well. (And mistakes may be related to both server and number of flavors).

Finally, and most important, if your goal is really:

However, since I am looking for the mistake-free time it takes to generate one scoop,

Then I think you only want to look at orders for a single scoop.