# Strong vs Weak Assumptions

In my study of econometrics, I have encountered an assumption called the "Zero Conditional Mean" for time series regression models. The assumption states that the expected value of the error term, $$E[u_t|X]=0$$, $$t=1,2,\cdots,n$$. My professor said that this means that "the mean value of the unobserved factors is uncorrelated to the values of the explanatory variables in all periods." He defined this as strict exogeneity. He said that this (strict exogeneity), was a stronger assumption than "contemporaneous exogeneity," which said that $$E[u_t | x_t]=0$$. Contemporaneous exogeneity, he said, means that "the mean of the error term is uncorrelated to the explanatory variables of the same period. So what determines the strength of an assumption? And how can we compare the strengths of different assumptions?

As a sidenote, I was wondering if the phrase "the mean value of the unobserved factors is uncorrelated to the values of the explanatory variables in all periods" means all periods lumped together or for each and every individual period.

Let $$\mathbf u$$ be the $$T \times 1$$ column error vector and $$\mathbf X$$ be the $$T \times k$$ regressor matrix, where $$T$$ is the sample size.

Then strict exogeneity is defined as

$$E\left(\mathbf u \mid \mathbf X\right) = \mathbf 0$$

This can be decomposed and written perhaps more clearly as

$$E(u_t \mid \mathbf X) = 0,\;\;\; t=1,...,T$$

which shows that strict exogeneity requires that each error term is mean-independent from all regressors, "past present and future".

On the contrary contemporaneous exogeneity is defined as, denoting $$\mathbf x_t$$ a row of $$\mathbf X$$ (i.e. the regressors at one period in time),

$$E(u_t \mid \mathbf x_t) =0, \;\;\; t=1,...,T$$

This is weaker, because it incorporates only a subset of the assumptions implied by strict exogeneity.

Other important relations often encountered as assumptions or desiderata are

Contemporaneous uncorrelatedness (or orthgonality)

$$E(u_t \mathbf x_t) =0, \;\;\; t=1,...,T$$

This is sometimes also called "predetermined regressors" but this last term is also used for more stronger conditions in the literature.

This is weaker than mean-independence, because mean-independence implies non-correlation but not vice versa.

An "intermediate" in strength assumption is

$$E(u_t \mathbf x_s) = \mathbf 0 \;\;\; \forall (t,s)$$

Here we do require the relation to hold for each error and all periods, but we only require orthogonality and not mean-independence. Call it "strict orthogonality" perhaps?

Note: the interchangeable use of the terms "uncorrelatedness" and "orthogonality" depends critically on the assumption that the error term has zero-mean. Otherwise, the correct term is "orthogonality" and not "uncorrelatedness" (see here)

Think about a regression of sales of ice cream on advertising, where the error comes from the effect of weather, which is unobserved to you, but not to the ice cream man or his customers. You care about what advertising does to sales. Assume for the sake of simplicity that weather has no persistence across days, nor was it recorded to be included in our model as a control. The concern is that the ice cream man uses advertising to smooth the effect of weather, so days with high advertising are also cooler days and low advertising days are hotter, so the coefficient that comes from comparing days with more marketing to less marketing will be contaminated by that relationship. Advertising will look less effective than it really is since we don't know what the weather was. That is the most basic kind of endogeneity.

The weak version says advertising today cannot respond to today's weather. In other words, knowing the advertising level today tells me nothing about the weather today, on average. That might seem reasonable, since advertising takes time to roll out. Flyers need to be designed and printed and someone must be hired to hand them out. But advertising tomorrow could be correlated with today's cool weather. You can also have advertising today correlated with tomorrow's weather (or beliefs about it).

The strong version says that in addition to today, advertising yesterday and tomorrow does not tell you about weather today, so the ice cream parlor cannot select advertising tomorrow to compensate for the effect of cool weather today. That is a much more restrictive assumption, since it also rules out compensating behavior across time in addition to within each day.