How should I interpret the assumption of the regression?

Question

I read an econometrics book which states one of the basic assumptions of regression is that

$$E(u|x) = 0$$

In another book however I see it written that

$$E(u_i|x_i) = 0$$

Are these two saying the same thing? More particularly, is it saying

$$E(U|X=?)$$

Or does it mean for observation $i^{th}$, the expected values of error term $i^{th} = 0$ (which doesn't seem to really make sense to me)?. How should I interpret it? On the other hand for the assumption which two error terms have an expected value of $0$, I wonder how this is calculated?

Answer 1

The addition of subscripts on the variables is usually done to refer to particular data points in the regression. Reference to $u$ and $x$ are generic references to the error variable and explanatory variable, whereas reference to $u_i$ and $x_i$ are specific references to the error variable and explanatory variable for the $i$th data point.

Now, strictly speaking, all of the regression assumptions are conditional on the entire set of explanatory variables, so the actual conditioning should be on the entire design matrix. It is not sufficient to set the regression assumptions conditional on the explanatory variables for only one data point. Consequently, the actual "linearity" assumption in regression should look like this:

$$\mathbb{E}(u_i | \mathbf{x}) = 0 \quad \quad \quad \text{for all } i=1,...,n,$$

which means that when you condition on knowledge of all the explanatory variables in the regression, each of the error terms has zero expected value. For brevity, this assumption is sometimes written in generic form as in your first equation, but you should bear in mind that the relevant conditioning is always on all the explanatory variables.

Regarding your third equation, you should note that every conditional moment statement has a conditioning statement that relates to some random variable. This means that the first two equations must be referring to some explanatory variable/vector as the relevant random item for the conditioning statement. When the relevant random variable is not explicitly specified, as in your first two equations, this is because the author thinks that the correspondence is obvious from the notation (usually because the random variable at issue uses notation which is the upper-case version of the lower case conditioning value).

Answer 2

Revised Answer

I originally deleted this answer because I didn't feel it was helpful, but I will leave it here in case your discussion with whuber will make the answer more clear (deleting the answer unfortunately wipes out all the comments with it, so I have simply marked out the portion that isn't helpful). Since I seem to be misconstruing the language used in this setting, I will leave it to better minds to provide an answer here.

Old Answer

Both are generally saying the same thing, however they have slightly different meanings behind them which may or may not be important.

• $E(u|x)=0$ basically says that the error term, given $x$, should be expected to average to zero. This is the "general" case and doesn't have any specificity. The assumption fails if $x$ and $u$ are correlated.
• $E(u_i|x_i)=0$ essentially says the same thing, but includes $i$ as an index indicating each $i$ observation. This may matter if you get into specific regression contexts, such as if you include time $t$ or cluster $j$ into the mix of the linear equation. Then your values may switch, for example, to $y_{ij}$, a $y$ response belonging to $i$ individual in $j$ group.
• Usually $U$ and $X$ here mean the same thing (the vectors $u$ and $x$), they are just not indexed ($x_i$ signified an individual $x$ whereas $X$ just refers to the variable as a whole).

For your last question:

On the other hand for the assumption which two error terms have an expected value of $0$, I wonder how this is calculated?

Not sure what you mean, but I guess you are referring to error terms for both variables, which you would not have, as $x$ is fixed and known, whereas $y$ is random, hence the error term we usually stick onto a linear regression equation. Of course this is assuming that the $x$ here is measured with perfect precision, but that's a completely different topic.

Edit

Some of my answer may not have been very illustrative so I will just show you how this works in practice. Here I will use an example in R by fitting a regression and showing this in action.

#### Load Library ####
library(tidyverse)
library(broom)
Save Data as Tibble
cars <- mtcars %>% 
  as_tibble()
cars
Plot Points
cars %>% 
  ggplot(aes(x=wt,
             y=mpg))+
  geom_point()
Fit Model
fit <- lm(mpg ~ wt,cars)
resid(fit)

The residuals here, the $u$ here, is as we said not actually zero, but fluctuations around zero. This is why they are called expected errors, the $u_i$ individually and $u$ as a whole vector:

        1          2          3          4          5          6          7 
-2.2826106 -0.9197704 -2.0859521  1.2973499 -0.2001440 -0.6932545 -3.9053627 
         8          9         10         11         12         13         14 
 4.1637381  2.3499593  0.2998560 -1.1001440  0.8668731 -0.0502472 -1.8830236 
        15         16         17         18         19         20         21 
 1.1733496  2.1032876  5.9810744  6.8727113  1.7461954  6.4219792 -2.6110037 
        22         23         24         25         26         27         28 
-2.9725862 -3.7268663 -3.4623553  2.4643670  0.3564263  0.1520430  1.2010593 
        29         30         31         32 
-4.5431513 -2.7809399 -3.2053627 -1.0274952 

However, if you take their mean or sum, you get something very close to zero if the model is not horribly specified (note this is in scientific notation, hence the e here:

> sum(resid(fit))
[1] 2.303713e-15

You can plot the distance of the fitted terms from the regression line to visualize the errors and how they sum to zero:

#### Show Residuals ####
cars %>% 
  lm(mpg ~ wt, data = .) %>% 
  augment() %>% 
  ggplot(aes(wt, mpg)) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE, color = "purple") + 
  geom_segment(aes(xend = wt, yend = .fitted)) +
  labs(title = "Fitted Values and Distance from Line",
       x = "Weight of Car",
       y = "Miles per Gallon")

Shown here, where the black segments above the regression line are positive residuals and the segments below it are negative residuals: