The post has two goals:

**(1)** Explain how to forecast volatility using a simple Heterogeneous Auto-Regressive (HAR) model. (Corsi, 2002)

**(2)** Check if higher moments like Skewness and Kurtosis add forecast value to this model.

The post has two goals:

**(1)** Explain how to forecast volatility using a simple Heterogeneous Auto-Regressive (HAR) model. (Corsi, 2002)

**(2)** Check if higher moments like Skewness and Kurtosis add forecast value to this model.

Five months ago I generated forecasts for the Eurozone Misery index. I used the built-in “FitAR” package in R. Using different models differing in their memory length (how many lags were considered for each model) 24 months ahead forecasts were generated. Might be interesting to see how accurate are the forecasts. The previous post is updated and few bugs corrected in the code. The updated data is public and can be found here. It is the sum of inflation rate and unemployment rate in the Euro-zone area.

Some knowledge about the bootstrapping procedure is assumed.

In time series analysis, Information Criteria can be found under every green tree. These are function to help you determine when to stop adding explanatory variables to your model.

Bootstrapping in its general form (“ordinary” bootstrap) relies on IID observations which staples the theory backing it. However, time series are a different animal and bootstrapping time series requires somewhat different procedure to preserve dependency structure.

Spurious Regression problem dates back to Yule (1926): “Why Do We Sometimes Get Nonsense Correlations between Time-series?”. Lets see what is the problem, and how can we fix it. I am using Morgan Stanley (MS) symbol for illustration, pre-crisis time span. Take a look at the following figure, generated from the regression of MS on the S&P, *actual prices* of the stock, *actual prices* of the S&P, when we use actual prices we term it regression in levels, as in price levels, as oppose to log transformed or returns.

A *beta* of a stock generally means its relation with the market, how many percent move we should expect from the stock when the market moves one percent.

Market, being a somewhat vague notion is approximated here, as usual, using the S&P 500. This aforementioned relation (henceforth, *beta*) is detrimental to many aspects of trading and risk management. It is already well established that volatility has different dynamics for rising markets and for declining market. Recently, I read few papers that suggest the same holds true for *beta*, specifically that the *beta* is not the same for rising markets and for declining markets. We anyway use regression for estimation of *beta*, so piecewise linear regression can fit right in for an investor/speculator who wishes to accommodate himself with this asymmetry.