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Introduction Financial time series analysis is concerned with the theory and practice of asset valuation over time.
Key feature that distinguishes financial time series analysis from other time series analysis: it contains the element of uncertainty.
For example, there are various definitions of asset volatility, and for a stock return series, the volatility is not directly observable.
Recent developments in financial econometrics have led to the use of techniques that can model the attitude of investors not only towards returns but also towards risk.
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Introduction Most financial studies involve returns, instead of prices of assets.
Campbell, Lo, and MacKinlay (1997) give two main reasons for using returns.
First, return of an asset is a complete and scale-free summary of the investment opportunity.
Second, return series are easier to handle than price series because the former have more attractive statistical properties.
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Main Objective The objective of this lecture is to study some methods and econometric models available in the literature for modelling the volatility of an asset return. The models are referred to as conditional heteroscedastic models.
Volatility is an important factor in option trading. Here volatility means the conditional standard deviation of the underlying asset return.
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US stock market volatility
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Japan stock market volatility
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China stock market volatility
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Volatility clustering Financial market data often exhibit volatility clustering.
This stylized fact can be broadly summarized by the fact that large changes in prices tend to be followed by large changes (irrespective of the sign) and small changes tend to be followed by small changes.
This feature has been captured by ARCH and GARCH models, and this is one of the reasons why they are so popular in empirical finance.
In other words, in financial markets periods of high volatility tend to be followed by periods of high volatility and, conversely, periods of low volatility tend to be followed by periods of low volatility.
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Volatility clustering evidence
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Characteristics of financial series We will highlight some of the features shared by several time series of financial returns. There are three stylized facts which occur in most of the daily return series:
1 The distribution of returns is not normal. As the frequency of observation decreases (i.e. we consider weekly, or monthly returns), the distribution looks closer to the Normal (aggregational Gaussianity);
2 Returns are not autocorrelated; 3 Absolute values and squares of returns are positively autocorrelated and
nonzero autocorrelation persists even at very high lags.
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Normality Let’s start from the first empirical regularity. In general, distributions of returns display the following features:
They are symmetric; They have a high peak, generally at zero; They have fat tails.
The last property is the one that has captured a lot of attention by researchers and academics. It means that the probability of extreme events is higher than the one implied by a Gaussian law.
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Returns The second regularity is that returns are not autocorrelated. Table 4.8 reports some autocorrelations for di↵erent financial series.
Consider the following six categories: 1 r̂ < �0.1 2 �0.1 r̂ < �0.05 3 �0.05 r̂ < 0 4 0 r̂ < 0.05 5 0.05 r̂ 0.1 6 r̂ > 0.1
Notice that the third and the forth category indicate that there is no correlation, while the first and the second denote negative correlation and the fifth and the sixth display positive correlation.
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Absolute and squared returns I We consider two cases: absolute returns and square returns, i.e.
st =| rt |g, q = 1, 2
The results are reported respectively in Tables 4.10 and 4.11. It is clear that there is positive temporal dependence among absolute and square returns. It is more pronounced in the case of absolute returns, but it is also appreciable in the case of square returns. The results just shown confirm that financial returns are not generated by an i .i .d . sequence. Of course, this is not enough to reject the E�cient Market Hypothesis.
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Absolute and squared returns II We just know that large absolute returns tend to be followed by large absolute returns, but we are not able to predict the direction of price changes. One explanation for this positive dependence between non linear transformations of returns is the fact of volatility clustering. In periods of high volatility, which tend to cluster together, we observe a large variation in the returns and therefore this causes positive dependence among absolute returns.
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Distribution
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Autocorrelation of returns
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Autocorrelation of absolute returns
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Autocorrelation of squared returns
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ARCH(1) - Characteristics I The simplest version of the ARCH models was proposed by Engle (1982). Engle’s idea begins by allowing the variance of the residuals to depend on history or to have heteroscedasticity. The conditional distribution of asset returns is given by:
rt | rt�1, rt�2, . . . ⇠ N(µ, ht)
where
ht = w + a(rt�1 � µ)2,w > 0, a � 0 (1)
This implies that the volatility of the returns at time t depends on the square returns at the previous period.
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ARCH(1) - Characteristics II Rearranging (1), we get:
ht = w + ae 2 t�1
The model is stationary when | a |< 1. Weaknesses
The model assumes that positive and negative shocks have the same e↵ects on volatility because it depends on the square of the previous shocks.
The ARCH model does not provide any new insight for understanding the source of variations of financial time series.
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GARCH (1,1) - Characteristics I The GARCH (1,1) model is parametrized as follows:
rt | rt�1, rt�2, . . . ⇠ N(µ, ht)
where ht = w + ae
2 t�1 + bht�1,w > 0
and a, b > 0
We can notice that there exists one more parameter in the model above, i.e. b. The constraints are required to avoid the possibility of a negative variance (not meaningful).
The model is stationary whenever | a + b |< 1. Some of the properties enjoyed by the GARCH(1,1) models are:
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IGARCH(1,1) models (integrated) It is a special case when a + b = 1.
In this case, the process is clearly covariance non-stationary. In particular the conditional variance is given by:
ht = w + (1� b)e2t�1 + bht�1
where 1 > b > 0
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Asymmetric volatility I One of the most important empirical regularities which has been observed in equity data is the so-called leverage e↵ect. According to this e↵ect, future volatility reacts in a very di↵erent way to the sign of a shock in prices. In particular, negative shocks are associated with a much larger impact on future volatility than positive shocks of the same magnitude. This was discovered by Nelson (1991).
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Asymmetric volatility II Application of such a model (for instance):
The e↵ect of bad and good news as determinant of the volatility of a financial series.
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Other GARCH Models (among others) GARCH in mean
Exponential GARCH
Threshold GARCH
For more details on the models see Tsay (2010)
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References Asteriou, D. and Hall, S.G., 2015. Applied econometrics. Palgrave Macmillan.
Campbell, J.Y., Lo, A.W., and MacKinlay, A.C., 1997. The econometrics of financial markets. Princeton University Press, Princeton, NJ.
Engle, R.F., 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the Econometric Society, pp.987-1007.
Nelson, D.B., 1991. Conditional heteroskedasticity in asset returns: A new approach. Econometrica: Journal of the Econometric Society, pp.347-370.
Tsay, R.S., 2010. Analysis of financial time series. Third edition. John Wiley & Sons.
Vogelvang, B., 2005. Econometrics; theory and applications with eviews. Pearson Education Ltd.. UK.
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