﻿ Time series regression

# Time series regression

This chapter looks at the various methods that are used in economics and wider statistics in order to perform regression analysis on data sets.
The main approaches used will be explained, with their underlying assumptions and limitations denoted.

Initially there is an explanation of what constitutes time series data, and basic time series regression models. Static models are presented, following the model Y = a + bX + e. This is then expanded on to include finite distributed lags, so that the model can take account of earlier explanatory variable values.

Following this the assumptions and issues with the typical ordinary least squares (OLS) regression method are considered. Worth noting is that this method requires there to be no collinearity between variables. That is, the explanatory variables cannot determine each other, they must be independent.
Considering homoskedasticity, the variance of errors is expected to be constant and not dependant on X.

Following consideration of the OLS method, this is progressed to consider ARIMA - Autocorrelated Integrated Moving Average.
ARIMA is used to identify where patterns may exist within error values following regression using methods such as OLS and/or classical decomposition. ARIMA uses lags of error values in order to identify autocorrelation, and a moving average component to provide smoothing based on errors.

The model is determined by first ensuring a stationary time series, we can then use the Augmented Dickey Fuller method to identify the length of lag that is used.
We can also view Autocorrelation Function (ACF) and Partial ACF (PACF) plots and observe the patterns present in order to determine the exact ARIMA model that best fits the data.
Through a trial and error approach and observation of resultant mean squared error we can identify the best fitting ARIMA function, with the aim to explain as much variance within the error term as possible and leave pure 'white noise' error.