How to interpret regression in a time series context? I am a newbie in stats and I have a hard time grasping regression in the context of time series, while reading "Time series analysis and its applications" by Robert H. Shumway; I came across:

When I think of regression I think of something like: 
$$housePrice=\beta_0 + \beta_1*houseSize$$
And in the example provided from the book it is as if the author is doing:
$$housePrice=\beta_0 + \beta_1*house1Size+\beta_2*house2Size...+\beta_n*houseNsize$$
It doesn't make sense to me. I know my logic is faulty, I do not know how to interpret the formula.
My question is: why in the formula there are several variables($z_{t1}, z_{t2}...z_{tq}$), if in time series we only have one variable.
 A: It's interesting to see how you conceptualize things when moving from cross-sectional to time series regression. 
In an attempt to help you make sense of this, I will stick with house prices and house sizes and give you three specific examples.
Cross Sectional Regression
In cross sectional regression (which is what you are familiar with), you would select $n = 10$ houses at random from some geographic region  of interest and then record for each house in your sample its price (in dollars) and its size (in square feet). If you believe that house price tends to increase with house size in the region of interest, then your (simple) linear regression model would be formulated like: 
$$housePrice_i =\beta_0 + \beta_1*houseSize_i + \epsilon_i$$
where $i$ is a house index referring to the $i$-th house in your sample, $\beta_0$ is the average price of a house having 0 square feet, $\beta_1$ is the change in average price of a house associated with an extra square foot of space and $\epsilon_i$ is an error term. (The intercept is not meaningful but you can make it meaningful by centering the variable $houseSize$ around a sensible reference value.)
The reason this type of regression is called cross-sectional is because the price and size of each house in your sample are measured at the same time (e.g., on January 1st, 2019). 
Time Series Regression
To move to a time series regression context, imagine now that you focus on a single house in that same area (it's a special house that can have rooms/annexes added to it each year for the purpose of this contrived example and can go on the market at the beginning of each year).  For 10 years, on January 1st of that year, you record the selling price of the special house as well as the current size and you formulate the following model: 
$$housePrice_t =\beta_0 + \beta_1*houseSize_t + \epsilon_t$$. 
This model postulates that the prices of this particular house tend to increase with the size of the house over the period of time spanned by the 10 years.
Comparison of Cross-Sectional and Time Series Regression
Notice the difference between the two regression models. 
Cross-sectional regression considers a random sample of $n$ different entities (in this case, houses) and uses response and predictor data collected on these entities at the same time point to determine the relationship between the response and predictor variables at that time point for all the entities represented by the ones included in the sample. 
Time series regression considers a single entity (in this case, a special house) and uses response and predictor data collected on this single entity at multiple time points to determine the relationship between the response and predictor variables for this entity across the entire time period spanned by the time points included in the study. 
With time series regression, sometimes the entity may be some sort of "aggregate entity". For example, if you measure the average annual price and average annual size of all houses in a region of interest over a period of 20 years and use time series regression to understand the relationship between these two variables, the entity would be the average house in that region. 
With time series regression, you would also need to worry about the possibility of spurious regression (e.g., the house price time series and the house size time series both show an increase over time, which would induce a strong positive correlation between house price and house size over the period of study; you would have to remove that increasing temporal trend from each time series and then correlate what is left over to get at the relationship of interest).  You would also need to worry about the possibility of temporal correlation among the model errors. 
In time series regression, the observations included in the model are indexed by the time index $t$.  For example, if time points are the years 2000, 2001, 2002, etc., the time index could be set as: 


*

*0, 1, 2, etc. 

*1, 2, 3, etc.

*2000, 2001, 2002, etc. 


Extension to Time Series Regression
If you select $n = 10$ houses at random and you measure their price and size annualy for 20 years (assuming their size could change over time through add-on construction), then you would be in a panel time series setting (also known as longitudinal regression) - an extension to time series regression whereby multiple entities are measured repeatedly over time. 
