Mean model
1. No pain, no gain
No pain, no gain. This applies to finance as well. If you want a higher return, you need to take more risk.2. GARCH-in-mean model
This risk-reward trade-off can be quantified using GARCH-in-mean models. They use the GARCH variance sigma square t to measure risk, and the expected return mu_t to measure the reward. The predicted mean mu_t is then equal to a constant mu plus lambda times the GARCH variance. The parameter lambda expresses the increase in return you can expect per unit of variance risk taken. This parameter needs to be positive such that higher returns can only be achieved by taking more risk.3. How?
To estimate the GARCH-in-mean model you need to set the mean-dot-model argument archm equal to TRUE and archpow equal to 2.4. Application to daily S&P 500 returns
Let's estimate this model for the S&P 500 returns using the ugarchfit function. We then print the first two coefficients, which are the constant mu and the risk reward parameter lambda. We find that the estimated lambda equals 1-point-995 and thus that for each additional unit of variance risk, the expected return increased with 1-point-995 units.5. Time series plot of predicted returns
By using the method fitted on the estimation output you can then plot the series of predicted returns. Note the spike around the 2008 financial crisis. Investors then require a higher return in compensation for the high risk.6. Today's return predicts tomorrow's return
The GARCH in mean model is based on the financial theory of a risk-reward trade-off. A different model is obtained using the statistical approach that exploits the correlation between today's return and tomorrow's return. The most popular approach to do so is the AR(1) model, where AR(1) stands for autoregressive model of order 1. It predicts the next return as the long term mean mu plus the autoregressive coefficient rho times the deviation of the current return from that long term mean mu.7. A positive autoregressive coefficient
If the autoregressive coefficient is positive, we predict that when the return is above its mean value, the next one will also be above the mean value, and vice versa. This positive relationship between today and tomorrow's returns occurs when markets underreact to news. Because they underreact, the effect continues the next day, leading to momentum in returns.8. A negative autoregressive coefficient
The reverse interpretation is in case the autoregressive coefficient is negative. Then a higher than average return is followed by a lower than average return. This happens if markets overreact to news and thus on the day following the overreaction there is a reversal in the market.9. Application to daily S&P 500 returns
Whether the autoregressive coefficient is positive or negative depends on the application. Here we analyze this for the daily S&P 500 returns by setting the argument armaOrder in mean.model to (1, 0). From the estimation follows that the coefficient is negative which hints towards overreaction and thus a reversal on the next day.10. MA(1) and ARMA(1,1) model
Other popular models for the conditional mean are the Ma(1) and ARMA(1,1) models. MA(1) stands for moving average model of order 1. It predicts the next return using the deviation of the past return from its conditional mean. The Arma(1,1) predicts the return using combination of the AR(1) and MA(1) model as you can see in the formula.11. How?
Just like the AR(1) model, the MA(1) and ARMA(1,1) model are easy to estimation using the rugarch package. It suffices to set the armaOrder argument to 0 1 for the MA(1) and to 1 1 for the ARMA(1,1) model.12. Your turn to change the mean.model argument
Please get your hands dirty and change the value of the mean model in ugarchspec!Create Your Free Account
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