Lecture 6: Modelling Volatility and Economic Forecasting

This is lecture 6 in my Econometrics course at Swansea University. Watch the lecture Live on The Economic Society Facebook page Every Monday 2:00 pm (UK time) between October 2nd and December 2017.

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In this lecture, I covered two topics: Modelling volatility and Economic Forecasting

Topic 1: Modelling Volatility
Financial time series, such as stock prices, interest rates, foreign
exchange rates, often exhibit volatility clustering (periods of turbulence & periods of tranquillity).
Various sources of news and other economic events may have an
impact on the time series pattern of asset prices; news can lead to various interpretations, and economic events like an oil crisis can last for some time. So we often observe the large positive and large negative observations in financial time series to appear in clusters.

Such swings in oil prices and credit crises have serious effects. Investors are concerned about the rate of return on their investment, and the risk of investment and the variability or volatility of risk. Therefore, it is important to measure asset price and asset returns volatility.

A simple measure of asset return volatility is its variance over time. The variance by itself does not capture volatility clustering. It does not take into account the past history (time-varying volatility).

The ARCH Model:
A measure that takes into account the past history (time-varying
volatility). In time series data involving asset returns, such as returns on stocks or foreign exchange, we observe autocorrelated heteroscedasticity.

Autocorrelated Heteroscedasticity:
Heteroscedasticity, or unequal variance, in cross section data because of the heterogeneity among individual cross-section units. In time series data, we usually observe autocorrelation. In financial data, we observe autocorrelated heteroscedasticity (i.e., heteroscedasticity observed over different periods is autocorrelated). In the literature, this phenomenon called ARCH effect.

Drawbacks of ARCH Model:
It requires estimation of the coefficients of p autoregressive terms, which consumes several degrees of freedom. It may be difficult to interpret all the coefficients, especially if some of them are
negative. The OLS estimating procedure does not lend itself to estimate the mean and variance function simultaneously. The literature suggests that any model higher than ARCH(3) is better
estimated by GARCH.

GARCH Model:
Generalised autoregressive conditional heteroscedasticity. We modify the variance equation to get GARCH(1,1) by expressing the conditional variance at time t in terms of the lagged squared error term at time (t − 1), and the lagged variance term at time (t − 1).

It can be shown that ARCH(p) model is equivalent to GARCH(1,1) as
p increases. In ARCH(p) we have to estimate (p + 1) coefficients, whereas in GARCH(1,1) model we estimate only 3 coefficients. GARCH(1,1) can be extended to GARCH(p,q) model (p lagged squared error terms, q lagged conditional variance terms). In practice, GARCH(1,1) has proved useful to model returns on financial assets.

The GARCH-M Model:
Modify the mean equation by explicitly introducing the risk factor, the
conditional variance, to take into account the risk.

Topic 2: Economic Forecasting

Based on past and current information, the objective of forecasting is to provide quantitative estimate(s) of the likelihood of the future
course of the object of interest (e.g. personal consumption expenditure). We develop econometric models and use one or more methods of forecasting its future course.

Methods of Forecasting:
There are several methods of forecasting. We will consider three prominent methods of forecasting: 1. regression models, 2. autoregressive integrated moving average (ARIMA) models [Box–Jenkins (BJ) methodology], 3. vector autoregression (VAR) models (Sims).

Point & Interval Forecasts:
In point forecasts we provide a single value for each forecast period. In interval forecasts we obtain a range, or an interval, that will include
the realized value with some probability. The interval forecast provides a margin of uncertainty about the point forecast.

ex post & ex ante
Estimation period: we have data on all the variables in the model. Ex post forecast period: we also know the values of the regressand
and regressors (the holdover period - used to get some idea about the performance of the fitted model) Ex ante forecast we estimate the values of the depend variable beyond the estimation period but we may not know the values of the regressors with certainty.

Conditional & Unconditional Forecasts
Conditional forecasts: we forecast the variable of interest conditional on the assumed values of the regressors. Recall that all along we have conducted our regression analysis, conditional on the given values of the regressors.