# EC316 TOPICS IN MACROECONOMICS & TIME SERIES

**EC316 TOPICS IN MACROECONOMICS & TIME SERIES **

**ECONOMETRICS EXAM**

**Question 5**

The analysis of volatility clustering in time series data is an important area of research, particularly in financial economics, where volatility clustering can impact risk management and portfolio allocation decisions. In the case of crude oil spot prices, ARCH and GARCH models can be used to examine whether there is evidence of volatility clustering over time. ARCH models are used to model time-varying variances in the data as a function of the past squared residuals. At the same time, GARCH models extend this idea also to include past variances. These models can capture the tendency of high volatility to cluster in time, where periods of high volatility tend to be followed by more periods of high volatility. Various statistical criteria and tests can be used to select the best model. The AIC, BIC, and HQIC are model selection criteria that balance the model’s goodness of fit with its complexity. Lower values of these criteria indicate a better trade-off between fit and complexity. The likelihood ratio test compares the fit of a more complex model, such as GARCH, to that of a simpler model, such as ARCH. If the test indicates that the more complex model significantly improves the fit of the data, then it may be preferred. Once a model has been selected, the estimated conditional variance series can be examined to determine if there is evidence of volatility clustering in the data. Suppose the estimated conditional variance series displays periods of high and low volatility, with clusters of high volatility followed by more periods of high volatility. In that case, this suggests the presence of volatility clustering.

**Question 9**

**Part (a)**

Autocorrelation refers to the degree of similarity between a time series and its past values at different lags. In finance, autocorrelation is important when modeling volatility because it indicates whether past volatility shocks still affect the current volatility. If the autocorrelation of volatility is high, it suggests that the shocks from the past continue to impact the current volatility, and hence the volatility is more persistent. This means that volatility forecasts must account for past volatility levels to predict future volatility accurately. One event during the sample period when the volatility of stock returns was likely to be high and autocorrelated was the Black Monday crash of 1987. On October 19, 1987, the S&P 500 experienced a sudden and severe price drop, losing over 20% of its value in a single day. The crash was triggered by a combination of factors, including rising interest rates, an overvalued market, and automated trading strategies.

The volatility of stock returns during this event was high because the sudden price drop created uncertainty and panic in the market, leading to a rapid increase in the VIX (a measure of market volatility). Furthermore, the autocorrelation of volatility was also high, as the shock from the crash had a persistent impact on market volatility in the following months. This autocorrelation can be seen in the persistence of high volatility levels in the weeks following the crash as investors continued to process the information and adjust their expectations. The Black Monday crash of 1987 was a significant event that led to high and autocorrelated volatility in the S&P 500, highlighting the importance of accounting for past volatility levels when modeling volatility in financial markets.

**Part (b)**

The output provided corresponds to the estimation of an Autoregressive Conditional Heteroskedasticity (ARCH) model for the S&P 500 stock returns. The dependent variable is the stock returns; an intercept is included in the regression equation. The volatility equation for the ARCH model is given by:

- σ_t^2 = ω + α_1ε^2,t-1 + α_2ε^2,t-2 + … + α_pε^2,t-p

Where σ_t^2 is the conditional variance at time t, ω is the constant term, α_1 to α_p are the coefficients corresponding to the lagged squared residuals ε^2, and p is the lag order. The output provided shows the optimal parameters estimated for the ARCH model. The constant term (mu) is estimated at 0.008003, indicating that the average daily return for the S&P 500 over the sample period is 0.8 basis points. The estimates for the lagged squared residuals (alpha1 to alpha5) coefficients are all positive, indicating that past squared residuals positively impact current volatility. However, the estimate for alpha4 is not statistically significant at the 5% level, indicating that it may not be necessary to include it in the model.

The estimate for omega is 0.000557, which represents the unconditional variance or long-run volatility of the S&P 500. The sum of the coefficients of the lagged squared residuals (alpha1 to alpha5) is 0.66508, indicating that about 66.5% of the variance in the stock returns can be attributed to past volatility shocks. To determine whether to increase or reduce the number of lagged terms included in the volatility equation, we should evaluate the coefficients’ significance and the model’s information criteria. A coefficient may be removed from the model if it is not statistically significant. Additionally, we can use information criteria such as the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC) to select the optimal number of lags to include in the model. These criteria penalize the inclusion of additional parameters to prevent overfitting of the model. Thus, we should compare the AIC or BIC values of different models with varying lags and select the model with the lowest information criterion value.

**Part (c)**

The GARCH model is an extension of the ARCH model that allows for the persistence of volatility shocks. The volatility equation for the GARCH model is given by:

- σ_t^2 = ω + α_1ε^2,t-1 + β_1σ^2,t-1

Where σ_t^2 is the conditional variance at time t, ω is the constant term, α_1 and β_1 are the coefficients corresponding to the lagged squared residuals ε^2 and the lagged conditional variance σ^2, respectively. The output provided corresponds to the estimation of a GARCH model for the S&P 500 stock returns. The constant term (mu) is estimated at 0.007834, indicating that the average daily return for the S&P 500 over the sample period is 0.78 basis points. The estimates for the coefficients of the lagged squared residuals (alpha1) and the lagged conditional variance (beta1) are statistically significant at the 5% level, indicating that past shocks to the volatility and past conditional variance have a positive impact on current volatility.

The estimate for omega is 0.000103, which represents the unconditional variance or long-run volatility of the S&P 500. The sum of the coefficients of the lagged squared residuals (alpha1) and the lagged conditional variance (beta1) is 0.938114, indicating that about 93.8% of the variance in the stock returns can be attributed to the past shocks to the volatility and past conditional variance. To determine whether to increase or reduce the number of lagged terms included in the volatility equation, we should evaluate the coefficients’ significance and the model’s information criteria. A coefficient may be removed from the model if it is not statistically significant. Additionally, we can use information criteria such as the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC) to select the optimal number of lags to include in the model. These criteria penalize the inclusion of additional parameters to prevent overfitting of the model. Thus, we should compare the AIC or BIC values of different models with varying lags and select the model with the lowest information criterion value.