Estimating an AR(1) Model from Log Returns

Summary

The document explains how to apply an AR(1) model to share-price log returns. First calculate each return as the logarithm of the current adjusted price divided by the previous adjusted price; these returns form the time series to model. Then fit a linear relationship between each return and the preceding return to estimate the intercept and autoregressive coefficient, with a residual term representing variation the model does not explain.

The response also points out that the magnitude of the autoregressive coefficient bears on whether the series is stationary. No dataset, fitted parameter values, diagnostic checks, or detailed estimation procedure are provided, so the guidance is conceptual. A linear fit alone does not establish that the AR(1) model describes the data well; further assessment would be needed before relying on it.

Key ideas

• Model the sequence of log returns rather than the share-price levels.
• Calculate each log return from consecutive adjusted prices.
• Estimate the intercept and lag coefficient with a linear fit of returns on their lagged values.
• The magnitude of the lag coefficient provides information about stationarity.
• The document gives no fitted estimates or model diagnostics.

Tags

# Estimate an AR(1) model from returns


# Estimate an AR(1) model from returns












I am studying share price log returns and AR(1) model. I downloaded data from $FTSE100$ and I used the Adj.close column to find the Ln returns:

Now I am trying to understand how can I estimate an AR(1) model using this information.

I understand the AR(1) model. I did a couple of example in excel, but I do not understand how the ln returns are related to that.

AR(1) is given by:

$X_t=\phi+\alpha*X_{t-1}+\epsilon$

I assume that I need to find values for $\phi$ and $\alpha$ to try to fit the AR(1) model but I am confused.

Can anyone help me on this?

Thanks.

## Answer by caverac (score 3)

https://quant.stackexchange.com/a/36925

You're on the right track. The time series you're trying to fit is the one formed by the returns $X_t = \ln (P_{t}/P_{t-1})$, where $P_t$ the tabulated price.

Once you calculate the returns just use a linear fit to estimate $\alpha$, $\phi$ and you're set. Probably you want to check $|\alpha|$ as well, it tells you something about the stationarity of the series

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.