I believe the confusion arises when distinguishing between population and sample levels. There is some inconsistencies with notation (when to use capital/cursive/bold letters), but once you determine on which level the notation is, everything should be clear. Usually, it is clear when we are on the sample levels since the equations would have to iterate through all observations $\{1 \dots n\}$ or $\{1 \dots T\}$, whereas on population levels only a single $i$ or $t$ as index is shown for models.

On population level every observation is a random variable/vector, so the model $y_t=\beta^Tx_t+\varepsilon_t$ holds for every $t$. On population level, the distinction between indexed and non-indexed is very important when describing properties of the model. It is best to understand this when dealing with indexing by time $t$:

1. Indexed values show a relation that holds only between variables for the same time point. For example: $E[\varepsilon_t|x_t]=0$ is the innovation property of the noise $\varepsilon_t$ (or predetermined property of the regressors $x_t$) and it states that $\varepsilon_t$ has no influence (in the mean) on any of the current and previous responses $\{ y_{t-1}, y_{t-2}, ... \}$. It does not rule out no influence on future values.

2. Non-indexed values show a relation that holds between variables for all time points. For example: $E[\varepsilon|x]=0$ is the exogeneity assumption of the regressors $x$, meaning that $\varepsilon$ has no influence (in the mean) on any of the responses $y$, both past, present and future. Here, $\varepsilon$ and $x$ are vectors that include all $\varepsilon_t$ respectively $x_t$.

Obviously, 2 is stronger than 1. Note that both $x_t$,$\varepsilon_t$ and $x$,$\varepsilon$ can be vectors.

I believe the confusion arises when distinguishing between population and sample levels. There is some inconsistencies with notation (when to use capital/cursive/bold letters), but once you determine on which level the notation is, everything should be clear. Usually, it is clear when we are on the sample levels since the equations would have to iterate through all observations $\{1 \dots n\}$ or $\{1 \dots T\}$, whereas on population levels only a single $i$ or $t$ as index is shown for models.

On population level, the distinction between indexed and non-indexed is very important when describing properties of the model. It is best to understand this when dealing with indexing by time $t$. Consider $y_t=\beta^Tx_t+\varepsilon_t$. On population level every observation is a random variable/vector, so the model holds for every $t$. Then:

1. Indexed values show a relation that holds only between variables for the same time point. For example: $E[\varepsilon_t|x_t]=0$ is the innovation property of the noise $\varepsilon_t$ (or predetermined property of the regressors $x_t$) and it states that $\varepsilon_t$ has no influence (in the mean) on any of the current response $y_{t}$. It does not rule out no influence on past/future values.

2. Non-indexed values show a relation that holds between variables for all time points. For example: $E[\varepsilon|x]=0$ is the exogeneity assumption of the regressors $x$, meaning that $\varepsilon$ has no influence (in the mean) on any of the responses $y$, both past, present and future. Here, $\varepsilon$ and $x$ are vectors that include all $\varepsilon_t$ respectively $x_t$.

Obviously, 2 is stronger than 1. Note that both $x_t$,$\varepsilon_t$ and $x$,$\varepsilon$ can be vectors.

I believe the confusion arises when distinguishing between population and sample levels. There is some inconsistencies with notation (when to use capital/cursive/bold letters), but once you determine on which level the notation is, everything should be clear. Usually, it is clear when we are on the sample levels since the equations would have to iterate through all observations $\{1 \dots n\}$ or $\{1 \dots T\}$, whereas on population levels only a single $i$ or $t$ as index is shown for models.

On population level every observation is a random variable/vector, so the model $y_t=\beta^Tx_t+\varepsilon_t$ holds for every $t$. On population level, the distinction between indexed and non-indexed is very important when describing properties of the model. It is best to understand this when dealing with indexing by time $t$:

1. Indexed values show a relation that holds only between variables for the same time point. For example: $E[\varepsilon_t|x_t]=0$ is the innovation property of the noise $\varepsilon_t$ (or predetermined property of the regressors $x_t$) and it states that $\varepsilon_t$ has no influence (in the mean) on all previous responses $\{ y_{t-1}, y_{t-2}, ... \}$. It does not rule out no influence on future values.

2. Non-indexed values show a relation that holds between variables for all time points. For example: $E[\varepsilon|x]=0$ is the exogeneity assumption of the regressors $x$, meaning that $\varepsilon$ has no influence (in the mean) on any of the responses $y$, both past, present and future. Here, $\varepsilon$ and $x$ are vectors that include all $\varepsilon_t$ respectively $x_t$.

Obviously, 2 is stronger than 1. Note that both $x_t$,$\varepsilon_t$ and $x$,$\varepsilon$ can be vectors.

I believe the confusion arises when distinguishing between population and sample levels. There is some inconsistencies with notation (when to use capital/cursive/bold letters), but once you determine on which level the notation is, everything should be clear. Usually, it is clear when we are on the sample levels since the equations would have to iterate through all observations $\{1 \dots n\}$ or $\{1 \dots T\}$, whereas on population levels only a single $i$ or $t$ as index is shown for models.

On population level every observation is a random variable/vector, so the model $y_t=\beta^Tx_t+\varepsilon_t$ holds for every $t$. On population level, the distinction between indexed and non-indexed is very important when describing properties of the model. It is best to understand this when dealing with indexing by time $t$:

1. Indexed values show a relation that holds only between variables for the same time point. For example: $E[\varepsilon_t|x_t]=0$ is the innovation property of the noise $\varepsilon_t$ (or predetermined property of the regressors $x_t$) and it states that $\varepsilon_t$ has no influence (in the mean) on any of the current and previous responses $\{ y_{t-1}, y_{t-2}, ... \}$. It does not rule out no influence on future values.

2. Non-indexed values show a relation that holds between variables for all time points. For example: $E[\varepsilon|x]=0$ is the exogeneity assumption of the regressors $x$, meaning that $\varepsilon$ has no influence (in the mean) on any of the responses $y$, both past, present and future. Here, $\varepsilon$ and $x$ are vectors that include all $\varepsilon_t$ respectively $x_t$.

Obviously, 2 is stronger than 1. Note that both $x_t$,$\varepsilon_t$ and $x$,$\varepsilon$ can be vectors.