# Time Series equivalent of the Generalised Linear Model

I have a time series $$y_t$$ which is measured at regular intervals over a long period of time. The values of $$y$$ are between $$0$$ and $$1$$, it represents a proportion, and these values change slowly over time. I want to predict the value of $$y$$ at future time periods.

I wanted to start in the most simple way possible, so I creates a linear regression based on the previous time period:

$$y_t \sim y_{t-1}$$

This can be extended to include $$y_{t-2}$$ and some covariates.

To account for the fact that the dependent variable is between $$0$$ and $$1$$, I subsequently turned this into a generalised linear model by including a logit link function:

$$logit(y_t) \sim logit(y_{t-1}) + logit(y_{t-2}) + covariates$$

This already works quite well for predicting future values of $$y$$.

However, I realise linear regression and GLM are not appropriate for time series. Why not though? What assumptions are violated if one used such models on times series data with autocorrelation?

What would be an equivalent simple model to use for such time series? A model able to deal with a dependent variable between $$0$$ and $$1$$, but otherwise as simple as possible.

You are already using a popular and often very effective time series model: an autoregressive model of order two, typically referred to as AR(2). The only non-standard part of your procedure is that you have logit-transformed your data.

You can assess the validity of your AR(2) model by examining the residuals in the same way as you would for any other regression model. For example the residuals of your AR(2) model should not be heteroscedastic or serially correlated.

Autoregressive models belong to a wider class of linear time-series models called autoregressive-integrated-moving-average or ARIMA. These models are generally more complicated to fit than simple autoregressions, but statistics software packages such as R should be able to fit them out-of-the-box.

First of all, there's no assumption made when you fit a linear regression, i.e. optimize its coefficients to minimize some risk function. There are a few assumptions made when you want to apply tests over the coefficients.

However, I realise linear regression and GLM are not appropriate for time series. Why not though? What assumptions are violated if one used such models on times series data with autocorrelation?

You said it yourself. Independence of error terms is often violated when you perform a hypothesis test on the value of the coefficient. There are remedies to this though.

This already works quite well for predicting future values of $$y$$

If this was your objective all along, it's performing well on out of sample data, and you do not intend to perform hypothesis tests, then by all means keep with the linear regression.