Analysis of financial data Anders Lundquist Spring 2010

ARCH-modelling Recall that EWMA and MA are not model based. They are merely the methods invented to track the changes in volatility observed in returns. Another way to tackle the problem of estimating and forecasting volatilities is to build models for the conditional variance Var (Y t | Y t-1, Y t-2,...) in the same way that we did for the conditional expectation E(Y t | Y t-1, Y t-2,...) (linear autoregressions).

The sample variance was defined as (assuming the returns have zero mean) This is an estimator of  2 = V(Y t ) = E( ). It is clear from the definition of  2 that (the squared returns) contain information about the variance. Exploring and finding models for is then equivalent to studying the volatility of the returns, in the same way that exploring and finding models for Y 1, Y 2,..., Y n is equivalent to studying the mean of the returns.

We have already earlier explored the squared returns. We have, for instance, computed s 2 and sometimes assumed that it converged (stationarity) to  2. Assuming stationarity of the variance is equivalent to assume the stationarity of the mean of the squared returns! What we have not done yet is to explore the dependence structure of In the same way as we did for returns, we can look at the autocorrelations of squared returns.

ACF / PACF of log-returns

ACF / PACF of squared log-returns

According to the partial autocorrelation plots, we conclude that the log-returns of Nordea seem to be uncorrelated while the squared log-returns look correlated. This is a fairly common behavior of financial asset log-returns!

Now, we are ready to define a model for conditional variance by using the model for linear dependence of the squared log-returns. The squared log-returns are linearly dependent. A model for this linear dependence can be specified as we have done for the plain log-returns. Var (Y t | Y t-1, Y t-1,...) = Alternatively, we may write

It is more convenient to write down a model for the log-return so that, in particular, both the conditional mean and variance may be modeled together. Thus, we write a “total” model Y t = μ t +  t  t, with The parameters  0,  1,...,  p must be positive.

This model for the returns specify both the mean and the variance of the log-returns. In general, one may have μ t = f (Y t−1,...,Y t−p ). The conditional mean may vary depending on the history of log-returns.

Also, the model considers a variance. If, then the variance is constant through time. Otherwise, the conditional variance varies depending on the history of the log-returns. If the mean of the log-returns (μ t ) can be assumed to be zero, we have an ARCH model (AutoRegressive Conditional Heteroskedasticity model):

We have seen that the log-returns of Nordea seemed uncorrelated and that their sample mean is very close to zero. The squared log-returns, on the other hand, seemed correlated which implies that an ARCH model is suitable. (model the linear dependence in the squared log-returns). To decide: How many days p should we include in the model for the variance

This question is solved with the partial autocorrelations. Partial autocorrelations of the squared log-returns are used to determinate p in the same way that partial autocorrelations of log- returns were used to determine the order of a linear autoregression. Justification: we want to find a linear autoregression model for the squared log- returns!

PACF

The parameters of an ARCH model can be estimated as the parameters of a linear autoregression, again because an ARCH model is a linear model for the squared log-returns. Least squares can be used. This can be done with SPSS. When a negative value for  i is obtained, it should be set to zero. (since ”anything” squared must be above, or possibly equal to, zero) However, for ARCH models, a better fit is obtained by assuming a distribution for  t in To use this you need specialized software (SPSS cannot do it).

From the partial autocorrelations of the squared log-returns we decide that we should use an ARCH model of order 11. We estimate the parameters using SPSS and obtain:

SPSS output

A few estimates negative (but the variables look non-significant) Alternatively, we can use only those coefficients that show significance for respective parameters. In this case, we would use lag 1, lag 4, lag 6, lag 7, lag 9, lag 10 and lag 11 plus a constant. In order to do this, we have to save all these lagged variables and perform linear regression.

SPSS output

Note: the estimated coefficients from this reduced model are somewhat different from the corresponding coefficients from the full model with 11 lags…

Forecast, squared log-returns

A usual model for the error term  t is that it is normally distributed. An ARCH model with normal error term  t will typically imply that Y t (the log-returns) are not normally distributed but instead have large kurtosis. So eventually we have a model which explains the observed behaviour of the log-returns. Consider a series of simulated log-returns, using the significant estimated parameters from the ARCH(11) as input.

Simulated observations, ARCH(11)

Comparison, observed in the left column, simulated ARCH in the right

Comparison, numerical summaries Simulated observations, ARCH(11) MeanStDevSkewnessKurtosis 0, Observed values for the log-returns

Now, the volatility of returns may be estimated using different estimators, moving averages, exponentially weighted moving averages and also by using a model for conditional volatility (ARCH). A natural question arise: How can we decide which estimator to choose? There are several ways to proceed to compare the performance of ARCH and EWMA (MA) estimators but this topic is not included in this course (see, e.g., Market Models by Carol Alexander, 2001) We just make simple comparisons by plotting different estimators.

Order of graphs on next page Observed squared log- returns ARCH MA(30) EWMA(0.94)

Summary We have seen that ARCH models give an account of excess kurtosis often present in financial data. These models succeed in capturing departures from normality in asset log-returns. Consequently, they may be useful alternatives to MA-methods