How to do time series regression without scikit and numpy in Python?

In case you want to perform a simple time-series regression without using any packages such as Numpy etc, you need to write and solve the model yourself. You can either use gradient descent or least squares to solve the model.

Let's look at a least squares solution. A simple model (omitting sibscripts) would look like:

$$ y = \beta_0 + \beta_1 y_{t-1} + u,$$

where $y$ is explained by a lag $y_{t-1}$ and $u$ is the error term (you can also use more lags if you want). You can "combine" this into a matrix with first column equal to one (the intercept) and the second column containing the lagged $y_{t-1}$. In matrix notation, the model looks like:

$$ y = \beta X + u .$$

Now you can solve by:

$$ \hat{\beta}=(X'X)^{-1} X'y.$$

Now it should be relatively easy (but still some work) to solve the problem without using packages such as numpy.

As an alternative to matrix notation and gradient descent, you can also solve a linear regression by other means, e.g. as demonstrated in this post.