Forecasting Exchange Rates and Conditional Variance with GARCH

Summary

The document outlines an ARMA(1,0)-GARCH(1,1) model for forecasting exchange-rate movements and volatility. The conditional mean equation uses the previous observation to forecast the next exchange rate, while the variance equation combines a constant with the latest squared innovation and prior conditional variance. This structure captures volatility persistence, where large or small changes tend to cluster over time.

After estimating the model, the conditional mean gives a one-step-ahead rate forecast and the variance recursion gives a conditional volatility forecast. The response also presents the long-run variance as the constant divided by one minus the sum of the ARCH and GARCH coefficients, under the model’s stationarity condition. The discussion is a compact formula explanation rather than an empirical demonstration: it gives no fitted parameters, data diagnostics, forecast evaluation, or caveats about distributional assumptions and model specification.

Key ideas

• GARCH models represent persistence in conditional volatility.
• The conditional mean equation forecasts the next exchange-rate observation from the preceding value.
• The conditional variance depends on the previous squared innovation and previous conditional variance.
• The long-run variance is defined when the relevant persistence coefficients sum to less than one.
• Model calibration and forecast evaluation are not demonstrated in the document.

Tags

# What is the formula for variance in estimating exchange rate?


# What is the formula for variance in estimating exchange rate?












I was watching this Youtube Video. He used a exchange rates of Euro to Dollar for a few days and apply GARCH(1,1) to get the predicted price. However, I didnt understand variance that he calculates from the table. WHat is the formula?

## Answer by Anonymous (score 2)

https://quant.stackexchange.com/a/23160

GARCH model is used to model persistence in volatility. If you square demean exchange rate and calculate autocorrelation you will find significant autocorrelation upto many lags that indicates the clustering of volatility in data.

A simple ARMA(1,0)-GARCH(1,1) model can be written as : $$y_t=\mu + \phi y_{t-1}+e_t $$ $$e_t \sim N(0, \sigma^2_t)$$ $$\sigma_t^2=\gamma + \beta e_{t-1}^2+ \eta\sigma_{t-1}^2$$

After calibrating the above model, you can use the first equation to forecast one day ahead exchange rate(expected exchange rate). For example: $$\mathbb{E}(y_t|\mathscr{F_{t-1}})=\mu_t+\phi y_{t-1}$$

Similarly, using last equation you can get conditional standard deviation. To get unconditional variance take expectation one more time:$$\mathbb{E[\sigma_t^2|\mathscr{F}_{t-1}}]=\gamma + \beta \, \mathbb{E}[e^2_{t-1}]+\eta \,\mathbb{E}[\sigma^2_{t-1}]$$ solving above equation, you will get $$\sigma_t^2=\frac{\gamma}{1-\beta -\eta}$$

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